Latest revision as of 16:13, 13 February 2020

"Non quia difficilia sunt non audemus, sed quia non audemus difficilia sunt."
Seneca (CIV,26)

Acknowledgement

It has been a long and challenging way, but, fortunately, during these years I have met many people, who encouraged me to go ahead and taught me not to give up. I am very grateful for this.

I would like to thank my advisor, Prof. Eugenio Oñate, for giving me the opportunity to study in CIMNE. During the years spent at this research center, I could find many sources of knowledge and inspiration, that I will take with me wherever I will be in the future. A special thanks goes to my co-advisor, Prof. Antonia Larese, for introducing me to the field of computational mechanics and for supporting me in the most important moments of this experience. I would like to thank those colleagues of CIMNE, who with great patience, helped me dedicating part of their time, Riccardo, Stefano, Charlie, Pablo, Lorenzo, Jordi and Alessandro.

I would like to thank Prof. Massimiliano Cremonesi for his supervision and kind welcome during my visit at the Department of Civil and Environmental Engineering, at Politecnico di Milano. I want to express my gratitude to Dr. Fausto Di Muzio for his kindness and warming welcome during my brief stay at Nestlé. This experience allowed me to look at the research world from a different perspective and to recognize how important is to conjugate fundamental research with more practical applied aspects. In this respect, I would like to thank Dr. Julien Dupas for his collaboration and support in providing some experimental data.

I have been enough fortunate to be part of the European project T-MAPPP, which greatly contributed not only to the development of my technical knowledge, but also of my soft skills. I would like to thank all the researchers involved in the project. In particular, Prof. Stefan Luding and Prof. Vanessa Magnanimo, for the valuable discussions and the interest shown in my work; Yousef, Kostas, Kianoosh, Behzad, Somik, Sasha and Niki, because without them it would have not been the same.

A special thank you goes to all the people who made my day with a smile and let me see the positive side in any situation. Thank you Josie, Manu, Edu, Vicente, Nanno, Giulia, Eugenia and Alessandra.

Last but not least, all my gratitude goes to my family: Marta, Gianni and Claudia, for their wholehearted support and love.

The research was supported by the Research Executive Agency through the T-MAPPP project (FP7 PEOPLE 2013 ITN-G.A.n607453).

Resumen

El manejo, el transporte y el procesamiento de materiales granulares y en polvo son operaciones fundamentales en una amplia gama de procesos industriales y de fenómenos geofísicos. Los materiales particulados, que pueden encontrarse en la naturaleza, generalmente están caracterizados por un tamaño de grano, que puede variar entre varios órdenes de magnitud: desde el nanómetro hasta el orden de los metros. En función de las condiciones de fracción volumétrica y de deformación de cortante, los materiales granulares pueden tener un comportamiento diferente y a menudo pueden convertirse en un nuevo estado de materia con propiedades de sólidos, de líquidos y de gases. Como consecuencia, tanto el análisis experimental como la simulación numérica de medios granulares es aún una tarea compleja y la predicción de su comportamiento dinámico representa aun hoy día un desafío muy importante. El principal objetivo de esta monografía es el desarrollo de una estrategia numérica con la finalidad de estudiar el comportamiento macroscópico de los flujos de medios granulares secos en régimen cuasiestático y en régimen dinámico. El problema está definido en el contexto de la mecánica de medios continuos y las leyes de gobierno están resueltas mediante un formalismo Lagrangiano. El Metodo de los Puntos Materiales (MPM), método basado en el concepto de discretización del cuerpo en partículas, se ha elegido por sus características que lo convierten en una técnica apropiada para resolver problemas en grandes deformaciones donde se tienen que utilizar complejas leyes constitutivas. En el marco del MPM se ha implementado una formulación irreducible que usa una ley constitutiva de Mohr-Coulomb y que tiene en cuenta no-linealidades geométricas. La estrategia numérica está verificada y validada con respecto a tests de referencia a resultados experimentales disponibles en la literatura. También, se ha implementado una formulación mixta para resolver los casos de flujo granular en condiciones no drenadas. Por último, la estrategia MPM desarrollada se ha utilizado y evaluado con respecto a un estudio experimental realizado para caracterizar la fluidez de diferentes tipologías de azúcar. Finalmente se presentan unas observaciones y una discusión sobre las capacidades y las limitaciones de esta herramienta numérica y se describen las bases para una investigación futura.

Abstract

Bulk handling, transport and processing of granular materials and powders are fundamental operations in a wide range of industrial processes and geophysical phenomena. Particulate materials, which can be found in nature, are usually characterized by a grain size which can range across several scales: from nanometre to the order of metre. Depending on the volume fraction and on the shear strain conditions, granular materials can have different behaviours and often can be expressed as a new state of matter with properties of solids, liquids and gases. For the above reasons, both the experimental and the numerical analysis of granular media is still a difficult task and the prediction of their dynamic behaviour still represents, nowadays, an important challenge. The main goal of the current monograph is the development of a numerical strategy with the objective of studying the macroscopic behaviour of dry granular flows in quasi-static and dense flow regime. The problem is defined in a continuum mechanics framework and the balance laws, which govern the behaviour of a solid body, are solved by using a Lagrangian formalism. The Material Point Method (MPM), a particle-based method, is chosen due to its features which make it very suitable for the solution of large deformation problems involving complex history-dependent constitutive laws. An irreducible formulation using a Mohr-Coulomb constitutive law, which takes into account geometric non-linearities, is implemented within the MPM framework. The numerical strategy is verified and validated against several benchmark tests and experimental results, available in the literature. Further, a mixed formulation is implemented for the solution of granular flows that undergo undrained conditions. Finally, the developed MPM strategy is used and tested against the experimental study performed for the characterization of the flowability of several types of sucrose. The capabilities and limitations of this numerical strategy are observed and discussed and the bases for future research are outlined.

List of Symbols

${\textstyle t}$ :	time;
${\textstyle \phi }$ :	internal friction angle;
${\textstyle d}$ :	particle diameter;
${\textstyle \rho _{p}}$ :	particle density;
${\textstyle {\dot {\boldsymbol {\gamma }}}}$ :	shear rate;
${\textstyle \Omega ^{0}}$ :	undeformed configuration;
${\textstyle \varphi \left(\Omega \right)^{n}}$ or ${\textstyle \varphi _{n}}$ :	deformed configuration at time ${\textstyle t^{n}}$ ;
${\textstyle \varphi \left(\Omega \right)^{n+1}}$ or ${\textstyle \varphi _{n+1}}$ :	deformed configuration at time ${\textstyle t^{n+1}}$ ;
${\textstyle \chi _{p}}$ :	characteristic function in the Generalized Interpolation Material Point Method;
${\textstyle V_{p}}$ :	particle volume;
${\textstyle \Omega _{p}}$ :	current particle domain;
${\textstyle \Omega }$ :	current domain occupied by the continuum;
${\textstyle N_{I}({\boldsymbol {x}})}$ :	shape function relative to node ${\textstyle I}$ evaluated at position at position ${\textstyle {\boldsymbol {x}}}$ ;
${\textstyle \nabla N_{I}({\boldsymbol {x}})}$ :	gradient shape function relative to node ${\textstyle I}$ evaluated at position at position ${\textstyle {\boldsymbol {x}}}$ ;
${\textstyle {\widehat {\nabla N_{I}}}({\boldsymbol {x}})}$ :	gradient shape function from node-based calculation;
${\textstyle {\overline {\nabla N_{I}}}({\boldsymbol {x}})}$ :	gradient shape function evaluated in Dual Domain Material Point Method;
${\textstyle \left(\cdot \right)_{p}}$ :	${\textstyle \left(\cdot \right)}$ relative to the material point ${\textstyle p}$ ;
${\textstyle \left(\cdot \right)_{I}}$ :	${\textstyle \left(\cdot \right)}$ relative to the node ${\textstyle I}$ ;
${\textstyle \left(\cdot \right)_{I}^{n}}$ :	${\textstyle \left(\cdot \right)}$ relative to the node ${\textstyle I}$ at time ${\textstyle t^{n}}$ ;
${\textstyle ^{it}\left(\cdot \right)_{I}^{n}}$ :	${\textstyle \left(\cdot \right)}$ relative to the node ${\textstyle I}$ at time ${\textstyle t^{n}}$ and iteration ${\textstyle it}$ ;
${\textstyle {\boldsymbol {u}}}$ :	displacement vector;
${\textstyle \Delta {\boldsymbol {u}}}$ :	increment displacement vector;
${\textstyle {\boldsymbol {v}}}$ :	velocity vector;
${\textstyle {\boldsymbol {a}}}$ :	acceleration vector;
${\textstyle p}$ :	pressure;
${\textstyle {\boldsymbol {q}}}$ :	momentum vector;
${\textstyle {\boldsymbol {f}}}$ :	inertia vector;
${\textstyle m}$ :	mass;
${\textstyle \left(\lambda ,\zeta \right)}$ :	stability parameters of Bossak method;
${\textstyle \delta {\boldsymbol {u}}}$ :	vector of unknown incremental displacement;
${\textstyle \mathbf {K} ^{tan}}$ :	tangent matrix of the linearised system of equation;
${\textstyle \mathbf {R} }$ :	residual vector of the linearised system of equation;
${\textstyle \left(\xi _{p},\eta _{p}\right)}$ :	local coordinates relative to a material point;
${\textstyle \beta }$ :	on-negative locality coefficient in Local Maximum Entropy technique;
${\textstyle h}$ :	measure of nodal spacing in Local Maximum Entropy technique;
${\textstyle \gamma }$ :	dimensionless parameter in Local Maximum Entropy technique;
${\textstyle {\boldsymbol {F}}}$ :	total deformation gradient;
${\textstyle J}$ :	determinant of the total deformation gradient;
${\textstyle {\boldsymbol {f}}}$ :	incremental deformation gradient;
${\textstyle {\boldsymbol {P}}}$ :	First Piola-Kirchhoff stress tensor;
${\textstyle {\boldsymbol {S}}}$ :	Second Piola-Kirchhoff stress tensor;
${\textstyle {\boldsymbol {\tau }}}$ :	Kirchhoff stress tensor;
${\textstyle {\boldsymbol {\sigma }}}$ :	Cauchy stress tensor;
${\textstyle \Psi }$ :	specific strain energy function;
${\textstyle {\boldsymbol {Q}}}$ :	rotation tensor;
${\textstyle {\boldsymbol {R}}}$ :	rotation tensor from polar decomposition of ${\textstyle {\boldsymbol {F}}}$ ;
${\textstyle {\boldsymbol {U}}}$ :	right stretch tensor;
${\textstyle {\boldsymbol {V}}}$ :	left stretch tensor;
${\textstyle {\boldsymbol {C}}}$ :	right Cauchy-Green tensor;
${\textstyle {\boldsymbol {I_{C}}}}$ :	first principal invariant of ${\textstyle {\boldsymbol {C}}}$ ;
${\textstyle {\boldsymbol {I_{C}^{*}}}}$ :	first invariant of ${\textstyle {\boldsymbol {C}}}$ ;
${\textstyle {\boldsymbol {b}}}$ :	left Cauchy-Green tensor;
${\textstyle \left(\lambda ,\mu \right)}$ :	Lame constant;
${\textstyle K}$ :	bulk modulus;
${\textstyle G}$ :	shear modulus;
${\textstyle \mathrm {C} ^{CE}}$ :	material incremental constitutive tensor;
${\textstyle \mathrm {C} ^{\tau }}$ :	spatial incremental constitutive tensor;
${\textstyle {\boldsymbol {I}}}$ :	symmetric second order unit;
${\textstyle \mathrm {I} }$ :	fourth order identity tensor;
${\textstyle \mathrm {I} _{s}}$ :	fourth order symmetric identity tensor;
${\textstyle \mathrm {I} _{d}}$ :	fourth order deviatoric projector tensor;
${\textstyle {\boldsymbol {d}}}$ :	symmetrical spatial velocity gradient;
${\textstyle {\boldsymbol {D}}}$ :	rate of deformation tensor;
${\textstyle {\boldsymbol {W}}}$ :	spin tensor;
${\textstyle {\boldsymbol {L}}}$ :	velocity gradient tensor;
${\textstyle \left(\cdot \right)_{vol}}$ :	volumetric part of ${\textstyle \left(\cdot \right)}$ ;
${\textstyle \left(\cdot \right)_{dev}}$ :	deviatoric part of ${\textstyle \left(\cdot \right)}$ ;
${\textstyle dev\left(\cdot \right)}$ :	deviatoric part of ${\textstyle \left(\cdot \right)}$ ;
${\textstyle \left(\cdot \right)^{e}}$ :	elastic part of ${\textstyle \left(\cdot \right)}$ ;
${\textstyle \left(\cdot \right)^{p}}$ :	plastic part of ${\textstyle \left(\cdot \right)}$ ;
${\textstyle \left({\overline {\cdot }}\right)}$ :	volume preserving part of ${\textstyle \left(\cdot \right)}$ ;
${\textstyle \sigma _{Y}}$ :	flow stress;
${\textstyle H}$ :	isotropic hardening;
${\textstyle \alpha }$ :	hardening parameter;
${\textstyle \gamma }$ :	plastic multiplier;
${\textstyle {\boldsymbol {n}}}$ :	unit vector of ${\textstyle {\boldsymbol {\tau }}_{dev}}$ ;
${\textstyle \left(\cdot \right)^{trial}}$ :	${\textstyle \left(\cdot \right)}$ in trial state;
${\textstyle tr\left(\cdot \right)}$ :	trace of ${\textstyle \left(\cdot \right)}$ ;
${\textstyle \mathrm {g} }$ :	metric tensor in current configuration;
${\textstyle \mathrm {C} ^{ep}}$ :	spatial algorithmic elastoplastic moduli;
${\textstyle c}$ :	cohesion;
${\textstyle \psi }$ :	dilation angle;
${\textstyle {\boldsymbol {\sigma }}_{n}}$ :	normal stress;
${\textstyle {\boldsymbol {\epsilon }}^{e}}$ :	principal Hencky strain;
${\textstyle \mathrm {a} }$ :	Hencky elastic constitutive tensor;
${\textstyle \mathrm {a} ^{ep}}$ :	elasto-plastic fourth order constitutive tensor;
${\textstyle \mathrm {C} ^{cep}}$ :	consistent elasto-plastic tangent;
${\textstyle {\boldsymbol {\lambda }}}$ :	eigenvalues vector of ${\textstyle {\boldsymbol {b}}}$ ;
${\textstyle {\boldsymbol {n}}}$ :	eigenvector vector of ${\textstyle {\boldsymbol {b}}}$ ;
${\textstyle {\boldsymbol {m}}}$ :	eigenbases vector of ${\textstyle {\boldsymbol {b}}}$ ;
${\textstyle {\boldsymbol {N}}}$ :	eigenvector vector of ${\textstyle {\boldsymbol {C}}}$ ;
${\textstyle {\boldsymbol {M}}}$ :	eigenbases vector of ${\textstyle {\boldsymbol {C}}}$ ;
${\textstyle {\mathcal {B}}}$ :	body;
${\textstyle {\mathcal {E}}}$ :	3D Euclidean space;
${\textstyle {\mathcal {R}}^{3}}$ :	real coordinate space in 3D;
${\textstyle \rho }$ :	mass density;
${\textstyle \rho _{0}}$ :	mass density in underformed configuration;
${\textstyle {\boldsymbol {b}}}$ :	body force;
${\textstyle g}$ :	gravity;
${\textstyle \varphi (\partial \Omega _{N})}$ :	Neumann boundary in deformed configuration;
${\textstyle \varphi (\partial \Omega _{D})}$ :	Dirichlet boundary in deformed configuration;
${\textstyle {\boldsymbol {\overline {t}}}}$ :	prescribed normal tension on Neumann boundary;
${\textstyle {\boldsymbol {\overline {u}}}}$ :	prescribed displacement on Dirichlet boundary;
${\textstyle {\boldsymbol {w}}}$ :	displacement weight function;
${\textstyle q}$ :	pressure weight function;
${\textstyle {\mathcal {V}}}$ :	displacement space;
${\textstyle {\mathcal {V}}_{h}}$ :	displacement finite element space;
${\textstyle \left(\cdot \right)_{h}}$ :	${\textstyle \left(\cdot \right)}$ in the finite element space;
${\textstyle {\mathcal {B}}_{h}}$ :	geometrical representation of ${\textstyle {\mathcal {B}}}$ ;
${\textstyle \Omega _{p}}$ :	finite volume assigned to a material point;
${\textstyle \mathbf {\mathbf {H} } ^{1}({\mathcal {B}})}$ :	Hilbert space;
${\textstyle dv}$ :	differential volume in deformed configuration;
${\textstyle da}$ :	differential boundary surface in deformed configuration;
${\textstyle dV}$ :	differential volume in undeformed configuration;
${\textstyle \nabla _{X}\left(\cdot \right)}$ :	material gradient operator;
${\textstyle \nabla _{x}\left(\cdot \right)}$ :	spatial gradient operator;
${\textstyle \nabla \left(\cdot \right)^{s}}$ :	symmatric part of ${\textstyle \left(\cdot \right)}$ gradient;
${\textstyle \mathbb {C} }$ :	fourth order incremental constitutive tensor relative to ${\textstyle {\boldsymbol {S}}}$ ;
${\textstyle {\widehat {\mathbb {C} }}}$ :	fourth order incremental constitutive tensor relative to ${\textstyle {\boldsymbol {\tau }}}$ ;
${\textstyle {\overline {\widehat {\mathbb {C} }}}}$ :	fourth order incremental constitutive tensor relative to ${\textstyle {\boldsymbol {\sigma }}}$ ;
${\textstyle \mathbf {D} }$ :	matrix form of ${\textstyle {\overline {\widehat {\mathbb {C} }}}}$ ;
${\textstyle \mathbf {B} }$ :	deformation matrix;
${\textstyle \mathbf {K} ^{G}}$ :	geometric stiffness matrix;
${\textstyle \mathbf {K} ^{M}}$ :	material stiffness matrix;
${\textstyle \mathbf {K} ^{static}}$ :	static part of ${\textstyle \mathbf {K} ^{tan}}$ ;
${\textstyle \mathbf {K} ^{dynamic}}$ :	dynamic part of ${\textstyle \mathbf {K} ^{tan}}$ ;
${\textstyle {\mathcal {Q}}}$ :	pressure space;
${\textstyle \delta p}$ :	vector of unknown pressure;
${\textstyle ^{m}\mathbf {K} ^{G}}$ :	geometric stiffness matrix in mixed formulation;
${\textstyle ^{m}\mathbf {K} ^{M}}$ :	material stiffness matrix in mixed formulation;
${\textstyle {\boldsymbol {R}}_{\boldsymbol {u}}}$ :	residual vector relative to the momentum balance equation;
${\textstyle {\boldsymbol {R}}_{p}}$ :	residual vector relative to the pressure continuity equation;
${\textstyle {\boldsymbol {R}}_{p}^{\mathrm {stab} }}$ :	residual vector relative to the stabilization term;
${\textstyle \mathbf {M} }$ :	mass matrix;
${\textstyle \mathbf {M} ^{\mathrm {stab} }}$ :	mass matrix relative to the stabilization term;
${\textstyle R_{p}^{\mathrm {stab} }}$ :	residual vector relative to the stabilization term;
${\textstyle \alpha }$ :	stabilization parameter;
${\textstyle \left(\mathbf {B} ,\mathbf {B} ^{*}\right)}$ :	mixed terms in the mixed formulation
${\textstyle d_{10}}$ :	diameter at which 10% of the sample's mass is comprised of particles with a diameter less than this value;
${\textstyle d_{50}}$ :	diameter at which 50% of the sample's mass is comprised of particles with a diameter less than this value;
${\textstyle d_{90}}$ :	diameter at which 90% of the sample's mass is comprised of particles with a diameter less than this value;
${\textstyle \mathbf {D} ^{A}}$ :	matrix form of ${\textstyle {\overline {\widehat {\mathbb {C} }}}}$ in axisymmetric case;
${\textstyle \mathbf {K} ^{A,tan}}$ :	tangent stiffness matrix in axisymmetric case;
${\textstyle \mathbf {K} ^{A,G}}$ :	geometric stiffness matrix in axisymmetric case;
${\textstyle \mathbf {K} ^{A,M}}$ :	material stiffness matrix in axisymmetric case;

1 Introduction

Bulk handling, transport and processing of particulate materials, such as, granular materials and powders, are fundamental operations in a wide range of industrial processes [1] or geophysical phenomena and hazards, such as, landslides, debris flows, etc. [2]. Particulate systems are difficult to handle and they can show an unpredictable behaviour, representing a great challenge in the industrial production, concerning both design and functionality of unit operations in plants, but also in the research community of Powders and Grains [3,4]. Granular materials and powders consist of discrete particles such as, e.g., separate sand-grains, agglomerates (made of several primary particles), natural solid materials like sandstone, ceramics, metals or polymers sintered during additive manufacturing. The primary particles can be as small as nano-metres, micro-metres, or millimetres [5] covering multiple scales in size and a variety of mechanical and other interaction mechanisms, such as, friction and cohesion [6], which become more and more important the smaller the particles are. All those particle systems have a particulate, usually disordered, possibly inhomogeneous and often anisotropic micro-structure; nowadays, the research community is working actively in order to have a deeper understanding and aware knowledge of bulk behaviour affected by micro-scale parameters. Indeed, particle systems as bulk show a completely different behaviour as one would expect from the individual particles. Collectively, particles either flow like a fluid or rest static like a solid. In the former case, for rapid flows, granular materials are collisional, inertia dominated and compressible similar to a gas. In the latter case, particle aggregates are solid-like and, thus, can form, e.g., sand piles or slopes that do not move for long time. Between these two extremes, there is a third flow regime, dense and slow, characterized by the transitions (i) from static to flowing (failure, yield) or vice-versa (ii) from fluid to solid (jamming). At the particle and contact scale, the most important property of particle systems is their dissipative, frictional, and possibly cohesive nature. In this context, dissipation shall be understood as kinetic energy, at the particle scale, which converts into heat, for instance, due to plastic deformations. The transition from fluid to solid can be caused by dissipation alone, which tends to slow down motion. The transition from solid to fluid (start of flow) is due to failure and instability when dissipation is not strong enough and the solid yields and transits to a flowing regime.

In this Chapter, the granular flow theory is presented more in detail and the main attempts, available in the literature, for the modelling of granular matter behaviour in the different regimes are discussed. Afterwards, objectives and layout of the current monograph are presented.

1.1 The granular flows

The heavy involvement of particle materials in many different industrial processes makes the granular matter, nowadays, a remarkable object of study. Particulate materials exist in large quantity in nature and it is established that most of the industrial processes, such as, pharmaceutical, agricultural, chemical, just to cite a few, deal with materials that are particulate in structure. In the industrial field, dealing with processes at large scales and huge quantities of raw material, any issue, encountered in the production line, may cause losses in terms of productivity, and, thus, of money. During the last decades, it has been documented that also in processes of granular matters, a lack of knowledge implies non-optimal production quality. In older industrial surveys, Merrow [7] found that the main factor causing long start-up delays in chemical plants is represented by the processing granular materials, especially due to the lack of reliable predictive models and simulations, while Ennis et al. [8] reported that 40 ${\textstyle \%}$ of the capacity of industrial plants is wasted because of granular solid phenomena. More recently, Feise [9] analysed the changes in chemical industry and predicted an increase in particulate solids usage along with new challenges due to new concepts like versatile multi-purpose plants, and fields like nano and bio-technology. For these reasons, it is clear that it is fundamental to have a better understanding of particulate materials behaviour under different conditions and to be able to improve the production quality through experimental campaigns and numerical modelling works. However, due to the wide variety of intrinsic properties of particulate materials a unified constitutive description, under any condition, has not been established yet.

With granular flow, we refer to motions where the particle-particle interactions play an important role in determining the flow properties and the flow patterns which are quite different from those of conventional fluids. The most evident differences between granular systems and simple fluids affecting the macroscopic properties of the flow, as pointed out in [10], are:

The size of grains is typically approximately ${\textstyle 10^{18}}$ more massive and voluminous than a water molecule. As both fluid and particle motions can be studied according to the laws of classical mechanics, this is not a fundamental difference, but could represent an important factor while evaluating the applicability of continuum hypothesis, as explained below;
In granular systems, when granules collide, a loss of kinetic energy converted in true heat is observed. This difference determines the main feature which deeply distinguishes the granular flows modelling from the fluid flows one;
In nature, grains are not identical: particle shape, particle roughness and solid density are only some of the particle properties which characterize a grain from the other. As real particles are not exactly spherical and typically the surface is rough, in grain-grain interactions frictional forces and torque are created and grains rotate during the collision.

The above comparison is useful to emphasise that the main assumptions on the basis of fluid flow modelling can not coincide with those of granular flow modelling. Further, this comparison can also be useful to set the bounds of the continuum assumptions enforceability. Three length scales have to be considered for the definition of these hypotheses. The first one is related to the particle size. Typically the value of density in grain systems is much smaller than in molecular fluids; this means that, for instance, in a cubic mm of fluids the number of molecules is much higher than the number of grains. If a macroscopic quantity changes significantly over a 1 mm of length, the variation over the molecules is small, but in the granular materials, if the number of particles in a 1 mm is low, a bigger variation is registered falling out of the continuum assumptions. The second length is related to the container confining the system and the third one to the inelasticity in grain-grain collisions. The latter can be defined as the radius of the pulse, related to a degradation of factor ${\textstyle e^{-}}$ of the total kinetic energy in a system of grains after a localized input of energy. If the inelasticity is not small this length covers just a few particles. This implies substantial changes in macroscopic quantities over distances measured over a small number of grain diameters, which do not allow to respect the continuum assumptions.

When one wants to study the flow of granular materials has to bear in mind that the bulk in motion is represented by an assembly of discrete solid particles interacting with each other. Depending on the intrinsic properties of the grains and the macroscopic characteristic of the system (i.e. geometry, density, velocity gradient), the internal forces can be transmitted in different ways within the granular material. Depending on this, three main flow patterns can be observed experimentally and numerically [11]. At large solids concentration and low shear rate, the stresses are not evenly distributed, but are concentrated along networks of particles, called force chains. The force chains are dynamic structures, which rotate, become unstable and, finally, collapse as a result of the shear motion. When granular material fails it is observed that the failure occurs along narrow planes, within the material, which have not infinitesimal thickness, but are zones of the order of ten particles across called shear bands. Within the shear bands, the stresses ${\textstyle {\boldsymbol {\tau }}}$ are still distributed along the force chains and the shear ${\textstyle \tau _{xy}}$ and normal stresses ${\textstyle \tau _{yy}}$ are related in non-cohesive material as

(1.1)

where ${\textstyle \phi }$ is the internal friction angle. Observing Equation 1.1, it is noted that ${\textstyle \mathrm {tan} \phi }$ depends on the geometry of the force chains; thus, the friction-like response of the bulk is a result of the internal structure of force chains, as well.

This flow pattern takes the name of Quasi-static regime because the rate of formation of the force chain divided by their lifetime is independent on the shear rate ${\textstyle {\dot {\boldsymbol {\gamma }}}}$ and with this also the generated stresses. If on one hand, the shear rate does not affect the response of the system, on the other hand, it is worth highlighting that the inter-particle stiffness k plays an important role, as the stresses within the force chains show a linear dependence with k. The inter-particle stiffness, in turn, can be expressed as a linear function of the Young modulus E of the material and also depends on the local radius of curvature, thus, on the geometry of the contact. Bathurst and Rothenburg [12] have derived an expression for the bulk elastic modulus K, which linearly depends on the stiffness and can be used in the definition of the sound speed in a static granular material. As shown in the data of Goddard [13] and Duffy ${\textstyle \&}$ Mindlin [14], the wave velocity is dependent on the pressure applied to the bulk assembly. Thus, increasing the confining pressure, the elastic bulk modulus increases along with the inter-particle stiffness k and the force chains lifetime.

Until the shear rates ${\textstyle {\dot {\boldsymbol {\gamma }}}}$ are kept low, the inertia effects are small and force chains are the only mechanism available to balance the applied load. Increasing the shear rate, the particles are still locked in force chains, but the forces generated have to take into account the inertia introduced in the system. Further increasing ${\textstyle {\dot {\boldsymbol {\gamma }}}}$ , the inertial component of the internal forces linearly increases with the shear rate and when these are comparable with the static forces the flow transition to the Inertial regime takes place. The Inertial regime encompasses flows where force chains cannot form and the momentum is transported largely by particle inertia. In this regime, the shear stresses are independent of the stiffness, but dependent on the second power of the shear rate, as expressed in the Bagnold scaling ${\textstyle \tau _{xy}/\rho d^{2}{\dot {\boldsymbol {\gamma }}}^{2}}$ [15], where ${\textstyle d}$ and ${\textstyle \rho }$ are the particle diameter and particle density, respectively. Even if force chains are not present, multiple simultaneous contacts between the particles still coexist allowing longer contact period ${\textstyle t_{c}}$ . By defining with ${\textstyle T_{bc}}$ the binary contact time, i.e., the duration of a contact between two freely colliding particles, the ratio ${\textstyle t_{c}/T_{bc}>1}$ . If this ratio has the value of 1, the dominant particle collisions are binary and instantaneous and the flow is defined with the name of Rapid Granular Flow [16], which can be considered as an asymptotic case of the Inertial Regime. This flow path is controlled by the property of granular temperature, which represents a measure of the unsteady components of velocity. The granular temperature is generated by the shear work and it drives the transport rate in two principal modes of internal (momentum) transport: a collisional and a streaming mode. In the first case, the granular temperature provides the relative velocity that drives particle to collide; while, in the second case, it generates a random velocity that makes the particles move relatively to the velocity gradient. In the Rapid Granular Flow the coexistence of contact and streaming stresses can be observed; obviously, the collisional mode dominates at high concentrations, while the streaming mode at low concentrations. It is generally assumed that, at small shear rates, a flow behaves quasi-statically, and that by increasing the shear rate, one will eventually end up in the Rapid Flow regime. As pointed out by Campbell [11], the transition through the regimes is regulated by the volume fraction and the shear rate. However, by fixing the first or the second field, the transition may take place in a different way.

1.2 Granular flow modelling

Despite the prevalence of granular materials in most of the industrial applications, there is still a large discrepancy between results predicted by analytical or numerical solutions and their real behaviour [17]. Thus, structures and facilities for dealing with particulate material handling are not functioning efficiently and there is always a probability of structural failure and an unexpected arrest of the production line. Due to the intrinsic nature of granular materials, the prediction of their dynamic behaviour represents nowadays an important challenge for two main reasons. Firstly, the characteristic grain size has an excessively wide span: from nanoscale powders (such as colloids with a typical size of nanometre) to large blocks of coal extracted from mines. This feature gives rise to some difficulties in defining a unique model able to properly work across many scales. Secondly, although these materials are solid in nature, they behave differently in various circumstances and often changes in a new state of matter with properties of solids, liquids and gases [17]. Indeed, as with solids, they can withstand deformation and form heap; as with liquids, they can flow; as with gases, they can exhibit compressibility. This second aspect makes the modelling of granular matter even more difficult to define, as the macroscopic behaviour is affected by a set of microscopic parameters which often are not directly measurable from laboratory tests. Nevertheless, these challenges encouraged the research community to work actively in the particle technology field, developing and improving several numerical and experimental techniques for the characterization of granular materials.

As explained in Section 1.1, a granular flow can undergo three main regimes in different domains of volume fraction and shear rate. When the grains have very little kinetic energy, the assembly of the particle is dense, and if the structure is dominated by the force chains the response of the bulk is independent on the shear rate. In this case, the flow pattern is known as Quasi-static regime and the behaviour is well described by classical models used in soil mechanics [18]. On the other hand, if a lot of energy is brought to the grains, the system is dilute, granular materials are collisional, inertia-dominated and compressible similar to a gas. In this case, the stresses vary as the square of the shear rate ${\textstyle {\dot {\boldsymbol {\gamma }}}}$ [15] and the flow falls under the Rapid granular flow theory. The principal approach, provided in the literature, for modelling granular flow under these conditions is represented by the Kinetic Theory of granular gases [16]. In the definition of such a model, the formalism of gas kinetic theory is used with the constraint to consider the particles perfectly rigid and the kinetic theory formalism leads to a set of Navier-Stokes equations. Between these two regimes, we can find the dense and slow flow regime, characterized by the presence of multiple particles contact, but also by the absence of force chains. For the modelling of granular flows under this regime, in [19] the constitutive relation of a viscoplastic fluid is proposed, commonly known with the name of ${\textstyle \mu (I)}$ rheology. The idea comes from the analogy observed with Bingham fluids, characterized by a yield criterion and a complex dependence on the shear rate. By assuming the particles perfectly rigid and a homogeneous and steady flow, a set of Navier-Stokes equations is provided. Despite it has been demonstrated that the model can successfully reproduce the results of some experimental tests [19], the model can only qualitatively predict the basic features of granular flows. In fact, some phenomena, such as, the formation of shear bands, flow intermittence and hysteresis in the transition solid to fluid and vice-versa, cannot be modelled through the ${\textstyle \mu (I)}$ law.

Even if there are models able to predict the flow behaviour in single regimes, a comprehensive rheology, able to gather together all three regimes, is still missing in the literature. Many attempts have been done during the last years. To cite a few of them, it is worth mentioning the contribution of Vescovi and Luding [20] where a homogeneous steady shear flow of soft frictionless particles is investigated; both fluid and solid regimes are considered and merged into a continuous and differentiable phenomenological constitutive relation, with a focus on the volume fraction close to the jamming value. Also Chialvo and coworkers [21] turn the attention to the interface between the quasi-static and the inertial regime in the context of a jamming transition, still neglecting any time dependency (under the assumption of steady-state flow), but considering soft friction particles. Other proposals are based on the relaxation of some hypotheses at the base of the ${\textstyle \mu (I)}$ rheology, such as, for instance, the constitutive laws provided by Kamrin et al. [22] where the non-local effects are considered and by Singh et al. [23] where the particle stiffness influence is included in the model.

In order to perform a numerical investigation of the granular flow problem, one has to keep in mind that not only a constitutive model is needed for its accomplishment, but also a numerical technique used to solve the system of algebraic equations which govern the problem. Numerical methods can be distinguished according to the kinematic description adopted and to the spatial and time scales that balance laws are based on. As previously observed, particulate materials can be studied at different scales and depending on this the selection of the numerical technique may change.

Granular materials have a discrete nature and it is of paramount importance to have a clear description at the microscale for a deep understanding of the physics behind the bulk behaviour. In recent decades, the most common and used numerical tool for such investigation is represented by the Discrete Element Method (DEM) [24], which considers a finite number of discrete interacting particles, whose displacement is described by the Newton's equations related to translational and rotational motions.

In the research community, granular materials are studied by using a continuum approach, as well. For instance, all the aforementioned constitutive models, proposed under the condition which range between the Quasi-static regime and Rapid granular flow, are based on a continuous description of the granular matter behaviour. On one hand these approaches, obviously, do not allow to predict the material response on the point, where two particles collide, i.e., at the microscale, but are able to provide a mesoscale response, constant on a representative elemental volume (REV), where the continuum assumptions are still valid [10].

Other methods, available in the literature, are used to scale-up from the micro-level. In this regard, it is worth mentioning the Population Balance Method (PBM) [25], able to describe the evolution of a population of particles from the analysis of single particle in local conditions. For instance, PBM is widely used to track the change of particle size distribution during processes where agglomeration or breakage of particles are involved, by using information, such as, impact velocity distribution, provided by DEM analysis [26,27].

1.3 Objectives

In the present work, we focus on the macroscale analysis of granular flows. More specifically, the current monograph aims at providing a verified and validated numerical model able to predict the behaviour of highly deforming bulk of granular materials in their real scale systems. Some examples, objects of this study, are represented by hopper flows, the collapse of granular columns and measurement of bearing capacity of a soil undergoing the movement of a rigid strip footing. These tests are characterized by some common features which are essential for the definition of the numerical tool to be developed. All the examples of granular flow mentioned above are densely packed with solid concentrations well above 50% of the volume and they can be considered dense granular flow where forces are largely generated by inter-particle contacts. This implies that a collisional state of the matter, where the principal mechanism of momentum transport is based on binary particle contacts, will never be reached. Moreover, in all cases an elastic/quasi-static regime usually coexists with a plastic/flowing regime. In order to be able to model the quasi-static and the flowing behaviours simultaneously in different parts of the material domain, a constitutive law which accounts for both the elastic and plastic regime is needed. The use of viscous fluid materials, as those described by the ${\textstyle \mu (I)}$ rheology and its different versions, can predict the granular flow behaviour under the inertial regime. However, the good reliability of these models is still limited to the steady case and to volume fractions whose values never exceed the jamming point. Further, if a viscous material is chosen, some difficulties might be encountered in the evaluation of internal forces where a zero strain rate is present. With the picture described above, the numerical model is conceived in a continuum mechanics framework in order to optimize the high, not to say prohibitive, computational cost which might be induced by the high value of density in the grain system, if a discrete technique is, then, selected. Moreover, the transition between the solid-like and fluid-like behaviour induces large displacement and huge deformation of the continuum which, from the numerical viewpoint, is well established to be tough to handle with standard techniques, such as the well-known Finite Element Method (FEM). Last but not least, not only geometric, but also material non-linearities should be considered. To address to this concern, the choice falls on those elasto-plastic laws, defined within the solid mechanics framework, whose stress response depends on the total strain history and historical parameters characteristic of the material model. The numerical model to adopt has to be based not only on a continuum mechanics framework, but also has to be able to track with high accuracy the huge deformation of the medium and the spatial and time evolution of its own material properties. After a search focused on the numerical model which closest fits with the features outlined above, it is found that the Material Point Method (MPM) [28], a continuum-based particle method, might be a good candidate in solving granular flows problems under multi-regime conditions and an optimal platform for the numerical implementation of new constitutive laws which attempt to include a bridge between different scales (from the particle-particle contact (micro) to the bulk (macro) scale). In the current work, an implicit MPM is developed by the author in the multi-disciplinary Finite Element codes framework Kratos Multiphysics [29,30,31]. Unlike most MPM codes, which make use of explicit time integration, in this monograph it is decided to adopt an implicit integration scheme. The choice is made with the aim of analysing cases characterized by a low-frequency motion and providing results with a higher stability and better convergence properties. Two formulations are implemented within the MPM framework by taking into account the geometric non-linearity, which allows to treat problems of finite deformation, usually not considered in many MPM codes that one can find in the literature. Firstly, an irreducible formulation and a Mohr-Coulomb constitutive law are developed. Further, a mixed formulation is proposed for the analysis of granular flows under undrained conditions, which represents, to the knowledge of the author, an original solution in the context of the MPM technique. The MPM strategy, with both the formulations, is validated by using experimental results or solutions of other studies, available in the literature. Last but not least, as final objective of this monograph, the developed MPM numerical tool is successfully tested in an industrial framework, in the context of a collaboration with Nestlé. A comparison is performed against an unpublished experimental study conducted for the characterization of flowability of several types of sucrose. Advantages and limitations of the numerical strategy provided are observed and discussed.

1.4 T-MAPPP project

The current work has been funded by the T-MAPPP (Training in Multiscale Analysis of MultiPhase Particulate Processes and Systems, FP7 PEOPLE 2013 ITN-G.A. n60) project. This project has been conceived in order to bring together European organizations leading in their respective fields of production, handling and use of particulate systems. T-MAPPP is an Initial Training Network funded by FP7 Marie Curie Actions with 10 full partners and 6 associate partners. The role of the network is to train the next generation of researchers who can support and develop the emerging inter- and supra-disciplinary community of Multiscale Analysis (MA) of multi Phase Particulate Processes. The goal is to develop skills to progress the field in both academia and industry, by devising new multiscale technologies, improving existing designs and optimising dry, wet, or multiphase operating conditions. One aim of the project is to train researchers who can transform multiscale analysis and modelling from an exciting scientific tool into a widely adopted industrial method; in other words, the establishment of an avenue able to increasingly link academic to real world challenges.

1.5 Layout of the monograph

The layout of the document is as follows: in Chapter 2, after a brief review of the state of the art in particle methods, the focus is put on those methods which are more consistently used for the prediction of granular flows behaviour, such as, the Discrete Element Method (DEM), the Particle Finite Element Method (PFEM), the Galerkin Meshless Methods (GMM) and the Material Point Method (MPM). The latter is the chosen approach, used and developed in this monograph. The choice is discussed and the details of the proposed formulations are provided. In Chapter 3 the theory of constitutive laws used in the current work is presented with their implementations under the assumption of finite strains. In Chapter 4 and Chapter 5 an irreducible and a mixed stabilized formulation, respectively, are presented and verified with solid mechanics benchmark examples. Then, in Chapter 6 the numerical model of MPM, presented in the previous chapters, is applied and validated (with experimental and numerical results available in the literature) against granular flow examples, such as, the granular column collapse and the rigid strip footing test. In Chapter 7 the MPM strategy is applied in an industrial framework. The numerical results are compared against experimental measurements performed for the assessment of the flowability performance of different types of sucrose. Finally, in Chapter 8 some conclusions are drawn, where observations and limitations of the numerical strategy are provided, and the bases for future research are outlined.

2 Particle Methods

Computer modelling and simulation are now an indispensable tool for resolving a multitude of scientific and challenging problems in science and engineering. During the last decades the importance of computer-based science has exponentially grown in the engineering field and, nowadays, it is widely adopted in the study of different processes because of its advantages of low cost, safety and efficiency over the experimental modelling. The numerical simulation of solid mechanics problems involving history-dependent materials and large deformations has historically represented one of the most important topics in computational mechanics. Depending on the way deformation and motion are described, existing spatial discretisation methods can be classified into Lagrangian, Eulerian and hybrid ones. Both Lagrangian and Eulerian methods have been widely used to tackle different examples characterized by extreme deformations. In this chapter, firstly, the most common numerical techniques used in the modelling of granular flows are presented. Then, the focus is put on the Material Point Method, which is the object of the present study.

2.1 Lagrangian and Eulerian approaches

In continuum mechanics two fundamental descriptions of the kinematic and the material properties of the body, under analysis, are possible. The first one is represented by the Lagrangian approach. In this case the description is made as the observer were attached to a material point forming part of the continuum. Lagrangian algorithms, traditionally employed in structural mechanics, make use of a moving deforming mesh dependent on the motion of the body and are distinguished by the ease with which the material interfaces can be tracked and the boundary conditions can be imposed. According to [32] three Lagrangian formulations can be defined:

the Total Lagrangian formulation, where all the variables are written with reference to the undeformed configuration ${\textstyle \Omega ^{0}}$ at the initial time ${\textstyle t_{0}}$
the Updated Lagrangian formulation, where all the variables are written with reference to the deformed configuration ${\textstyle \varphi \left(\Omega \right)^{n}}$ at the previous time ${\textstyle t_{n}}$
the Updated Lagrangian formulation, where all the variables are written with reference to the deformed configuration ${\textstyle \varphi \left(\Omega \right)^{n+1}}$ at the current time ${\textstyle t_{n+1}}$

Moreover, history-dependent constitutive laws can be readily implemented and, since there is not advection between the grid and the material, no advection term appears in the governing equations. In this regard, Lagrangian methods are more simple and more efficient than Eulerian methods. The greatest drawback of this approach is represented by the high distortion of the mesh and element entanglement when the material undergoes really large deformation, which makes more difficult to obtain a stable solution with an explicit integration scheme. The second approach lies on an Eulerian description, i.e., the observer is located at a fixed spatial point. Thus, Eulerian techniques, mostly employed in fluid mechanics, are characterized by the use of a fixed grid and no mesh distortion or element entanglement are observed neither in the case of very large deformation. On the other hand, due to its intrinsic nature, it is difficult to identify the material interfaces and the definition of history-dependent behaviour is computationally intensive if compared with Lagrangian methods. As can be seen, each of the two approaches has advantages and drawbacks; thus, depending on the problem to solve, one technique is preferable over the second one.

In the framework of granular flow modelling the Lagrangian viewpoint presents, in this context, a rather obliged choice, since the adoption of such a framework greatly simplifies the constitutive modelling and the tracking of the entire deformation process. In the case of mesh-based methods, the natural limitation of the Lagrangian approach is related to the deformation of the underlying discretisation, which tends to get tangled as the deformation increases. Massive remeshing procedures have proved to be capable of further extending the realm of applicability of Lagrangian approaches, effectively extending the limits of the approach well beyond its original boundaries. Nevertheless, while, on one hand, it is possible to alleviate the distortion of the mesh, on the other hand, additional numerical errors arise from the remeshing and the mapping of state variables from the old to the new mesh. In this regard, the Arbitrary Lagrangian–Eulerian method (ALE) [32], a generalization of the two approaches described earlier, has been developed in the attempt to overcome the limitation of the Total Lagrangian (TL) and Updated Lagrangian (UL) techniques when severe mesh distortion occurs by making the mesh independent of the material, so that the mesh distortion can be minimized. However, for very large deformation severe computational errors are introduced by the distorted mesh. Furthermore, the convective transport effects can lead to spurious oscillations that need to be stabilized by artificial diffusion or by other stabilization techniques. Such disadvantages make the ALE methods less suitable than other techniques which can be found in the literature.

In the current work, the Lagrangian framework is considered, but the focus is on the so-called particle methods, a series of techniques which represent a natural choice for the solution of granular flow problems involving large displacement, large deformation and history-dependent materials. The next section introduces a brief state of the art of the most common particle methods with their distinguished features and fields of application.

2.2 Particle methods. A review of the state of the art

Particle methods are techniques which have in common the discretisation of the continuum by only a set of nodal points or particles. According to [33], they can be classified based on two different criteria: physical principles or computational formulations. For those methods classified according to physical principles a further distinction is made if the model is deterministic or probabilistic; while according to the computational formulations, the particle methods can be distinguished in two subcategories, those serving as approximations of the strong forms of the governing partial differential equations (PDEs), and those serving as approximations of their weak forms. In Tables 2.1 and 2.2 the classification is graphically shown with a list of the main approaches which fall under each category.

Table. 2.1 Physical principles based particle methods.
Physical principles
Deterministic models	Probabilistic models
Discrete Element Method (DEM)	Molecular Dynamics
	Monte Carlo methods
	Lattice Boltzmann Equation method

Table. 2.2 Computational formulations based particle methods.
Computational formulations
Approximations of the strong form	Approximations of the weak form
Smooth Particle Hydrodynamics	Meshfree Galerkin Method:
Vortex Method	- Diffusive Element Method
Generalized finite Difference Method	- Element Free Galerkin Method
Finite Volume PIC	- Reproducing Kernel Method
	- h-p Cloud Method
	- Partition of Unity Method
	- Meshless Local Petrov-Galerkin Method
	- Free Mesh Method
	Mesh-based Galerkin Method
	- Material Point Method
	- Particle Finite Element Method

In the following sections, a bibliographic review of the most common and widely used particle methods in granular flow modelling is presented. The first method to be presented is the Discrete Element Method. Then, the Smooth Particle Hydrodynamics, the Meshfree Galerkin Method, the Particle Finite Element Method and the Particle Finite Element Method 2 are briefly introduced. For each of those advantages and disadvantages are discussed. Finally, the Material Point Method and a meshless variation of it and their algorithms are extensively described.

2.2.1 The Discrete Element Method (DEM)

The numerical approach which considers the problem domain as a conglomeration of independent units is known as Discrete Element Method (DEM), developed by Cundall and Strack in 1979 [24]. DEM was initially used for studying of rock mechanics problems using deformable polygonal-shaped blocks. Later, it has been widely utilized to study geomechanics, powder technology and fluid mechanics problems. Each particle is identified separately having its own mass, velocity and contact properties and, during the calculation, it is possible to track the displacement of particles and evaluate the magnitude and direction of forces acting on them. The main distinction between DEM and continuum approaches is the assumption on material representation; in DEM every particles represents a physical entity, e.g., the single grains in the granular system, while in a continuum method particles take the place of material points, which have instead just a numerical purpose in the computation of the solution.

According to [24] the time step must be chosen in a way that disturbances from an individual particle cannot propagate further than their neighbours. Usually, in order to avoid significant instability in the granular system the time step should be smaller than a critical time step, called the Rayleigh time step.

DEM is a good example of numerical technique that treats the bulk solid as a system of distinct interacting bodies. Thus, with DEM it is possible to simulate interaction at the particle level (at a spatial scale which ranges from ${\textstyle 10^{-6}m}$ to ${\textstyle 10^{-1}m}$ , depending on the size of the grains) and, at the same time, to obtain an insight into overall response, bulk properties such as stresses and mean velocities [34]. Therefore, it can provide a clear explanation on particle-scale behaviour of granular solids to characterize bulk mechanical responses, as it is done in several contributions [35,5]. Moreover, this technique is really useful and interesting in the research field of granular matter since DEM can be seen as a tool for performing numerical experiments that allow contact-less measurements of microscopic quantities that are usually impossible to quantify using physical experiments. Discrete element modelling has been also used extensively to analyse various handling and processing systems that deals with multiple bulk solids [36,37,38]. However, the extremely high computational cost, proportional to the number of particles, leads to the limitation of considering relatively small system sizes and idealized geometries.

The schematic flowchart, which has to be followed in order to execute a DEM calculation [39] at each time step, is displayed in Figure 2.1a. Even if the algorithm looks to be straightforward to run and easy to implement, the computational cost is proportional to the number of discrete elements and to their shapes. Thus, simplified assumptions have been made in the mathematical models in order to reduce the computational efforts. The primary idealized factor in DEM simulations is the shape of particles which is considered as spheres to simplify the contact detection process, which is the most time consuming step in DEM simulations. Among diverse physical properties of individual particles in particulate materials, the shape and morphology play important roles in shear strength and flowability of the bulk. In order to improve this aspect several approaches have been utilized in DEM, such as, clumped spheres [40], polyhedral shapes [41], super-quadric function [42]. However, in order to obtain accurate results the computational cost may arise significantly. Another important step in DEM is to realistically simulate the physical impact between particles. This is usually approximated by defining spring and dashpots between contacting surfaces, as it is done in the linear-spring dash-pot models [43] or the Hertzian visco-elastic models. In the literature other contact models can be found, such as meso-scale models [35,44,45] or realistic contact models [46,47,48], which can provide a high accuracy both at the particle and bulk level, but valid only for the limited class of materials they are particularly designed for.

Moreover, for a proper understanding of a process and to study realistic behaviour through DEM simulations, the input parameters, listed in Figure 2.1b, play a vital role. The input parameters are often assumed without careful assessment or calibration which often leads to unrealistic behaviours and erroneous results. Designing of equipment or of a process route with an un-calibrated DEM model may lead to serious handling and processing operations such as segregations, unexpected wear, irregular density of products, flow blockages and etc. Thus, a correct definition of the input parameters by experimental characterization and/or calibration, using particle-level tests, directly affect the reliability of the final response at the bulk level. However, this might result in an extremely time and cost consuming procedure, that not always it is possible to perform for a lack of time and/or money.

In conclusion, DEM is a valid and useful numerical tool for research in the granular matter field and development of new contact models. However, the drawbacks aforementioned in the current section make the DEM still not an easy and limited tool to use in the engineering and industrial framework, mainly when real scale systems are under study.


(a)

(b)
Figure 2.1: Dem algorithm (a), with $t$ the time instant and $\Delta t$ the time interval, and input parameters in DEM model (b).

2.2.2 The Smoothed Particle Hydrodynamics

Among the methods which serve as approximations of the strong of PDEs, the Smoothed Particle Hydrodynamics (SPH) [49,50] is one of the earliest particle methods in computational mechanics. It was initially designed for solving hydrodynamics problems [51,52], such as astrophysical applications [53,54,55]; later, SPH has been also applied to solid mechanics problems involving impact, penetration and large deformation of geomaterials [56,57,58,59] or compressible and incompressible flow problems [60,61]. The PDE is usually discretized by a specific collocation technique: the essence of the method is to choose a smooth kernel which not only smoothly discretized a PDE, but also furnishes an interpolant scheme on a set of moving particles. Even if the method is widely used by the computational mechanics community, it is well established in the literature that the SPH suffers of some pathologies such as tensile instability [62,63], lack of interpolation consistency [64,65], zero-energy mode [66] and difficulty in enforcing essential boundary condition [67,68]. To solve the aforementioned fundamental issues several improvements have been provided through the years. To mitigate the tensile instability and the zero-mode issues the stress point approach [63,69] has been proposed. To correct completeness, or consistency, closely related to convergence, the use of corrective kernels are considered; for instance Liu et al. [64] proposed new interpolants named the Reproducing Kernel Particle Method and many other approaches can be found in the literature [70,71,72] addressing this shortcoming. To enforce essential boundary conditions it is worth mentioning the contribute of Randles and Libersky [67], where the so-called ghost particle approach is proposed.

2.2.3 The Meshfree Galerkin Methods

The Meshfree Galerkin Methods, unlike the SPH, were mainly developed only in the early of 1990s. The first meshless methods appeared in the literature are represented by the Diffusive Element Method [73], where moving least square (MLS) interpolants [74] are employed, and the Reproducing Kernel Particle Method (RKPM) [64,75], defined in the attempt to provide a corrective SPH. Later, other techniques were proposed, such as the Element Free Galerkin Method (EFGM) [76], in which the MLS interpolants are for the first time used in a Galerkin procedure, or the Partition of Unity Method [77], where a partition of unity is taken and multiplied by any independent basis. Usually most meshfree interpolants do not satisfy the Kronecker delta property ¹ and the impossibility of a correct imposition of the essential boundary conditions represents one of the principal bottleneck of these approaches. Some remedies for the enforcement of the EBCs are given by the Lagrange Multipliers and Penalty method [78], the Transformation method [79], the Boundary singular kernel method [74] and the Coupled finite element and particle approach [80]. Most Meshfree Galerkin Methods make use of background grid to locate the quadrature points to integrate the weak form. From this aspect some problems in terms of accuracy as well as invertibility of the stiffness matrix may arise, due to the arbitrariness in locating the Gauss quadrature points. If these points are not enough in a compact support or are not evenly distributed spurious modes may also occur. In order to completely eliminate quadrature points some approaches have been proposed in the literature, e.g., the one proposed by Chen et al. [81] based on a stabilized nodal integration method. Despite the typical drawbacks of Meshfree Galerkin Methods, e.g., the aforementioned issue of quadrature integration and the higher computation cost in comparison with standard FEM, during the last decades meshless methods have been increasingly used to solve applied mechanics problem due to some key advantages which distinguish them from other techniques. For instance to mention some of them, in these methods the connectivity changes with time as they do not have a fixed topological data structure, the accuracy can be controlled easily given a h-adaptivity procedure and the meshfree discretisation can provide accurate representation of geometric object. Initially, meshfree methods have been used to address the challenging field of computational fracture mechanics. In this regards, the EFGM and the Partition of Unity Methods have been applied to crack growth and propagation problems [82,83,84]. The great advantage of not using a remeshing procedure has been also exploited in the application of large deformation problems; in particular, it is worth mentioning the use of the RPKM to metal forming, extrusion [85] and soil mechanics problems [86,87]. Meshfree methods have been also extensively applied to simulation of strain localization problems [88] since meshfree interpolants can successfully reduce the mesh alignment sensitivity in the formation of the shear bands.

(¹) Let us define the Lagrange polynomials of degree $n-1$ , $L_{k}(x)\,(k=1,....,n)$ . $L_{k}(x)$ satisfy the Kronecker delta property if

(2.1)

i.e., $L_{k}(x_{i})=\delta _{ik}$ . With regards to the shape functions, these ones lack of the Kronecker delta property when the weight function $w_{j}({\boldsymbol {x}})$ associated with the nodal points ${\boldsymbol {x}}_{j}\,{\mbox{for }}\left(j\neq i\right)$ is not zero at the location of nodal point of interest ${\boldsymbol {x}}_{i}$ .

2.2.4 The Particle Finite Element Method

The Particle Finite Element Method (PFEM) is a particle method which falls under the category of mesh-based Galerkin approaches. In PFEM the domain is modelled using an Updated Lagrangian formulation and the continuum equations are solved by means of a FEM approach on the mesh built up from the underlying node, also called ${\textstyle particles}$ . The main feature of the method is based on the employment of a fast remeshing procedure to relieve the typical issue of high distortion of the mesh and a boundary recognition method, i.e., the alpha-shape technique [89], needed to define the free surfaces and the boundaries of the material domain. Given an initial mesh, the remeshing procedure can be used arbitrarily at every time step [90] or when the mesh starts affecting the accuracy of the numerical solution, as in the case of explicit formulations [91]. This Lagrangian technique was first developed for the simulation of free surface flows and breaking waves [92,93], and then successfully adapted for structural mechanics problems involving large deformations [94,95], for the simulation of viscoplastic materials [96,91,97,98,99,100], in geomechanics [101,102] and Fluid-Structure Interaction (FSI) applications [103,104,105]. Although the method has broad capabilities, some disadvantages come from the use of remeshing procedures [106,107]. In the practice, although possible, the application of PFEM in problems with elastic or elasto-plastic behaviour faces difficulties related to the storage of historical variables, since information on the integration points is not preserved and needs to be remapped at the moment of remeshing. Moreover the alpha-shape technique, and the remeshing itself, lead to intrinsic conservation problems related to the arbitrariness of the reconnection patterns [108]. Last but not least, the characteristics of the remeshing approach at the base of the PFEM make it very hard to parallelize, thus, limiting the possibility of the method in terms of computational efficiency.

2.2.5 The Particle Finite Element Method 2

A second generation of PFEM (PFEM2) has been recently introduced [109,110,111,112], which tries to repair to the shortcomings observed in the previous version. PFEM2 is a hybrid particle method, which exploits the combination of both the Eulerian and Lagrangian approaches, as in the Material Point Method. The method has been tested in the analysis of interaction between different materials, such as, incompressible multifluids and fluid-structure interaction [113], and simulation of landslides and granular flows problems [114]. This technique is based on the use of a set of Lagrangian particles, in order to track properties of the continuum, and by a fixed finite element mesh, employed for the solution of the governing equations. Moreover, a projection of the data between the two spaces is performed during the calculation of a time step in order to keep updated the kinematic information between the particles and the nodes of the Eulerian mesh. One of the main differences which distinguishes PFEM2 from the Material Point Method lies on the definition of Lagrangian particle itself. In the case of PFEM2, the particles represent material points without a fixed amount of mass; in order to guarantee a good particle distribution in the computational domain, the number of particle might change during the simulation time. These Lagrangian entities do not represent integration points, but are just used with the purpose of convecting all the historical material properties and kinematic information through the simulation. This feature makes PFEM2 particularly adapted for the modelling of incompressible fluids, with Newtonian and non-Newtonian rheology and FSI problems.

2.3 The Materal Point Method

An alternative particle method proposed in the literature is represented by the Material Point Method (MPM) [115,28], which is object of study of this monograph. The Material Point Method (MPM) is a particle-based method, whose origin goes back to the paper of [116], where the particle-in-cell method (PIC), a technique for the solution of fluid flow problems, was proposed for the first time. Some decades after, in the works of [115,28], the PIC method is redefined within the solid mechanics framework, and after that, it is known to the computational community with the name of Material Point Method. MPM combines a Lagrangian description of the body under analysis, which is represented by a set of particles, the so-called material points, with the use of a computational mesh, named background grid, as can be observed in Figure 2.2.

Figure 2.2: MPM. The shape functions on the material point

are evaluated using FE shape function of element I-J-K.

This distinctive feature allows one to track the deformation of the body and retrieve the history-dependent material information at each time instant of the simulation, without committing mapping information errors, typical of methods which make use of remeshing techniques. This makes the method particularly attractive for the solution of problems, characterized by very large deformations and by the use of complex constitutive laws [117,118]. For instance, the method has been extensively used for geotechnical problems [119,120,118] for its capabilities in tracking extremely large deformation while preserving material properties of the material points.

In the key works of Sulsky and co-workers [115,28], the MPM has been applied for the first time in the solid mechanics framework. Even if through the original MPM it was possible to solve complex problems involving, for instance, contact [121], interaction between different materials [122,123] and the use of history-dependent material laws [115], it was observed that the first version of MPM suffers from some intrinsic shortcomings. Indeed, due to the use of piece-wise linear shape functions, the latter are only locally defined and their gradients are discontinuous. This implies that a material point on the cell boundary would not be covered by the local shape functions defined within the respective cells around the particle. This would produce a noise in the numerical solution, which is called cell-crossing error. Recently, many improvements to the original MPM have been provided to alleviate the cell-crossing noise and to have a more efficient and algorithmically straightforward evaluation of grid node integrals in the weak formulation. The Generalized Interpolation Material Point method (GIMP) [124] represents the first attempt to provide an improved version of the original MPM. The essence of this method is based on the definition of a characteristic function ${\textstyle \chi _{p}(x)}$ which has to satisfy the partition of unity criterion, i.e.,

(2.2)

The particle characteristic function defines the spatial volume occupied by the particle ${\textstyle V_{p}}$ as

(2.3)

where ${\textstyle \Omega _{p}}$ and ${\textstyle \Omega }$ are the current particle domain and the current domain occupied by the continuum, respectively. Moreover, since a material property ${\textstyle f({\boldsymbol {x}})}$ can be approximated by its particle value ${\textstyle f_{p}}$ as

(2.4)

${\textstyle \chi _{p}(x)}$ acts as a smoothing of the particle properties and it determines the smoothness of the spatial variation. The full version of GIMP requires integration over the current support of ${\textstyle \chi _{p}(x)}$ , which deforms and rotates according to the deformation of the background grid. To do that, a tracking of the particle shape is mandatory, but in a multi-dimensional problem this could be very difficult to accomplish. Thus, an alternative version of the GIMP is represented by the uniform GIMP (uGIMP), where shear deformation and rotation of the particles are neglected. The uGIMP assumes that the sizes of particles are fixed during the material deformation. In this way, the particle characteristic function, whose support may overlap or leave gaps for very large deformation, is no longer able to satisfy the partition of unity criterion, and, thus, the ability of computing rigid body motion is lost. Therefore, the uGIMP is unable to completely eliminate the cell-crossing error.

In the attempt to improve the issues left by the GIMP, the Convected Particle Domain Interpolation technique (CPDI) [125] is proposed. In the CPDI the particle has an initial parallelogram shape and a constant deformation gradient is assumed over the particle domain. This technique is a first-order accurate approximation of the particle domain ${\textstyle \Omega _{p}}$ . Even if in the CPDI a more accurate approximation of ${\textstyle \Omega _{p}}$ is obtained, the issues of overlaps and gaps are not overcome. Only with the second-order Convected Particle Domain Interpolation (CPDI2) [126], an enhanced CPDI, which provides a second-order approximation of the particle domain, these issues are totally corrected. It is also worth mentioning the Dual Domain Material Point Method (DDMPM) [127], an alternative technique which is able to definitely eliminate the cell-crossing error. Unlike the GIMP or CPDI, the DDMPM does not make use of particle characteristic functions and the issue of tracking the particle domain through the whole simulation is avoided. The essence of this technique relies on the use of modified gradient of the shape function, defined as follows

(2.5)

where ${\textstyle \nabla N_{I}({\boldsymbol {x}})}$ is the gradient of the shape function evaluated as in the original MPM, ${\textstyle {\widehat {\nabla N_{I}}}({\boldsymbol {x}})}$ is the gradient from the node-based calculation as used in FLIP ( FLuid-Implicit Particle)[128].

Most MPM codes make use of explicit time integration, due to the ease of the formulation and implementation [115,129,130]. Explicit methods are preferable to solve transient problems, such as impact or blast, where high frequencies are excited in the system [131,132,121]. However, when only the low-frequency motion is of interest, the adoption of an implicit time scheme may reduce the computational cost in comparison to the employment of an explicit time scheme [133]. Some implicit versions of MPM can be found in the literature. For instance, Guilkey [134] exploits the similarities between MPM and FEM in an implicit solution strategy. Beuth [135] proposes an implicit MPM formulation for quasi-static problems using high order elements and a special integration procedure for partially filled boundary elements. Sanchez [136] presented an implicit MPM for quasi-static problems using a Jacobian free algorithm and in [137] a GIMP method is used together with an implicit formulation.

In order to assess the features of MPM, as reference for a comparison, a standard Lagrangian FEM is chosen. In Table 2.3, the two methods are compared and a list of differences is made, according to the basic formulation, computational efficiency and computational accuracy. It is observed that in the small deformation range the MPM has a lower accuracy and efficiency than a Lagrangian FEM. Nevertheless, the FEM procedure shows its advantageous use only in a narrow range of strain magnitude, established by a critical deformed configuration for which the element quality is seriously compromised, which may cause a drastic deterioration of accuracy or even the end of the computation. In this regard, it is evident that MPM finds its natural field of application in large deformation problems. However, it is important to highlight that an extra computational cost is expected in MPM compared to FEM. This is due to additional steps in the MPM algorithmic procedure, in order to be able to track the kinematic and historical variables through the deformation process, and to a number of material points higher than the number of Gauss points normally employed in a FEM simulation.

Table. 2.3 Comparison between the Finite Element Method and the Material Point Method
STANDARD LAGRANGIAN FEM	MPM
BASIC FORMULATION
$\bullet$ Gauss quadrature	$\bullet$ Particle quadrature
$\bullet$ The Lagrangian computational mesh is attached to the continuum during the whole solution process	$\bullet$ No fixed mesh connectivity is required
$\bullet$ Higher accuracy and efficiency for small deformations	$\bullet$ Lower accuracy and efficiency of the MPM for small deformations
$\bullet$ For large deformations accuracy rapidly deteriorates and computational cost increases dramatically due to mesh distortion and the need for remeshing	$\bullet$ It naturally deals with large deformation problems
$\bullet$ Contact between different bodies can be modelled only by applying a contact technique	$\bullet$ Unphysical material interpenetration and non-slip contact constraint are inherent in the MPM
COMPUTATIONAL EFFICIENCY
$\bullet$ Mass and momentum are carried by the mesh nodes and they are calculated only at the beginning of the analysis. Further Gauss points move according to the mesh such that it is not necessary to update their positions and velocities	Additional steps have to be performed such as mapping of particle info (mass and momentum) on the grid and the update of particle info at the end of the Lagrangian step
$\bullet$ Less Gauss points per element	$\bullet$ Minimum number of particles per element higher than number of Gauss points per element
$\bullet$ In explicit time scheme, the critical time step decreases with the element deformation	$\bullet$ In MPM the characteristic element length does not change, thus, the critical time step size in MPM is constant
COMPUTATIONAL ACCURACY
$\bullet$ The Gauss quadrature can integrate accurately the weak form	$\bullet$ The particle quadrature is less accurate than the Gauss one for integrating the weak form
$\bullet$ For large deformations, in order to avoid element entanglement, a remeshing technique has to be adopted. However, this can lead to conservation issues of mass, momentum and energy. Further, the remapping of material properties of history-dependent material will result in significative errors	$\bullet$ The original MPM suffers from a cell crossing instability

As earlier discussed, the MPM is a particle method, whose advantages are evident in applications at large strain and displacement regime. Moreover, the MPM is characterized by some features which make this technique able to overcome all the typical disadvantages of other particle methods, listed in the previous sections. MPM does not employ any kind of remeshing procedure, the calculation is performed always at a local level, allowing an easy adaptation of the code to parallel computation and a good conservation of the properties. A conservation of the mass is also guaranteed during the whole simulation time, as the total mass is distributed between the material points representing the volume of the entire continuum under study. A remapping of the state variables is avoided and the employment of complex time dependent constitutive laws can be used without committing any mapping error. In addition, since this technique is a grid-based method all the issues, related to the meshless methods, such as, lack of interpolation consistency and difficulties in enforcing the Essential Boundary conditions are avoided. Last but not least, MPM is a technique defined in the continuum mechanics framework, thus, it can be easily applied to real scale problems at a not prohibitive computational cost. Given the fulfillment of the aforementioned features, MPM represents a suitable choice for the solution of real scale large deformation problems and particularly attractive for the modelling of granular flow problems.

2.4 The Material Point Method in Kratos Multiphysics

In the current work, an implicit MPM is developed in the multi-disciplinary Finite Element codes framework Kratos Multiphysics [29,30,31]. Two formulations are investigated: a displacement-based [138] and a mixed displacement-pressure (u-p) [139,140] formulations, presented in Chapters 4 and 5, respectively. In both the numerical strategies, the original version of the MPM is implemented, i.e., a particle integration is adopted, where the particle mass is assumed to be concentrated only on one spatial point, the particle position. As earlier discussed, this may lead to the so-called cell-crossing error; however, it is demonstrated that GIMP method is not able to definitely fix this issue and only other more computationally expensive techniques, such as, the CPDI2 and the DDMPM can remarkably mitigate this inherent error. In this work, it is made the choice to focus on the capabilities of the method in the modelling of granular flows under large deformation and large displacement regime and to leave, as future work, the exploration and investigation of other versions of MPM, which can improve the accuracy of the numerical results. The MPM in Kratos Multiphysics is developed in an Updated Lagrangian finite deformation framework and the matrix system to be solved is built-up from taking into account the contribution of each material point, to be considered as integration point, as well. In the initialization of the solving process, the initial position of the material points is chosen to coincide with the Gauss points of a FE grid and the mass, which remains constant during the simulation, is equally distributed between the material points, falling, initially, in the same element. At each time step, the governing equations are solved on the computational nodes, while history dependent variables and material information are saved on the particles during the entire deformation process. Thus, in the MPM the material points shall be understood as the integration points of the calculation, each carrying information about the material and kinematic response. Each material point represents a computational element with one single integration point (the material point itself), whose connectivity is defined by the nodes of the elements in which it falls. In Figure 2.2, for instance, the case of the i-th material point, which falls in a triangular element with a connectivity of nodes ${\textstyle I,J}$ and ${\textstyle K}$ , is depicted. In the evaluation of the FEM integrals, the shape functions are evaluated at the material point location on the basis of the grid element the material point falls into (Figure 2.2). At the end of every time step, in order to prevent mesh distortion, the undeformed background grid is recovered, i.e., the nodal solution is deleted.

In what follows, the algorithm of MPM for an implicit time scheme discretization is presented in detail.

2.4.1 MPM Algorithm

Traditionally, the MPM algorithm is composed of three different phases [28], as graphically represented in Figure 2.3 and below described:

Initialization phase (Fig.2.3a): at the beginning of the time step the connectivity is defined for each material points and the initial conditions on the FE grid nodes are created by means of a projection of material points information obtained at the previous time step ${\textstyle t_{n}}$ ;
UL-FEM calculation phase (Fig.2.3b): the local elemental matrix, represented by the left-hand-side ( ${\textstyle lhs}$ ) and the local elemental force vector, constituted by the right-hand-side ( ${\textstyle rhs}$ ) are evaluated in the current configuration according to the formulation presented in Chapters 4 and 5; the global matrix ${\textstyle LHS}$ and the global vector ${\textstyle RHS}$ are obtained by assembling the local contributions of each material point, as in a classical FEM approach, and, finally, the solution system is iteratively solved. During the iterative procedure, the nodes are allowed to move, accordingly to the nodal solution, and the material points do not change their local position within the geometrical element until the solution has reached convergence;
Convective phase (Fig.2.3c): during the third and last phase the nodal information at time ${\textstyle t_{n+1}}$ are interpolated back to the material points. The position of the material points is updated and, in order to prevent mesh distortion, the undeformed FE grid is recovered.


(a) Initialization phase	(b) Updated Lagrangian FEM phase

(c) Convective phase
Figure 2.3: MPM phases.

Many features of the MPM are connected to the Finite Element Method [115]. Indeed, phase b coincides with the calculation step of a standard non-linear FE code, while phases a and c define the MPM features. At the beginning of each time step ( ${\textstyle t_{n}}$ ), during phase a, the degrees of freedom and the variables on the nodes of the fixed mesh are defined gathering the information from the material points (Figure 2.3a).

For the sake of clarity, hereinafter, ${\textstyle p}$ and ${\textstyle I}$ subscripts are used to refer to variables attributed to material points and computational nodes, respectively, while ${\textstyle n}$ superscript refers to the time instance in which the variable is defined. The momentum ${\textstyle {\boldsymbol {q}}_{p}}$ and inertia ${\textstyle {\boldsymbol {f}}_{p}}$ on the material points, which are expressed as functions of mass ${\textstyle m_{p}}$ , velocity ${\textstyle {\boldsymbol {v}}_{p}}$ and acceleration ${\textstyle {\boldsymbol {a}}_{p}}$

(2.6)

(2.7)

are projected on the background grid by evaluating in a first step, the global values of mass ${\textstyle m_{I}}$ , momentum ${\textstyle {\boldsymbol {q}}_{I}}$ and inertia ${\textstyle {\boldsymbol {f}}_{I}}$ on node ${\textstyle I}$ as described in Algorithm 2.1.

Once ${\textstyle m_{I}}$ , ${\textstyle {\boldsymbol {q}}_{I}}$ and ${\textstyle {\boldsymbol {f}}_{I}}$ are obtained, it is possible to compute the values of nodal velocity ${\textstyle {\widetilde {\boldsymbol {v}}}_{I}^{n}}$ and nodal acceleration ${\textstyle {\widetilde {\boldsymbol {a}}}_{I}^{n}}$ of the previous time step as

(2.8)

(2.9)

It is worth mentioning that, the initial nodal conditions are evaluated at each time step using material point information in order to have initial values even on grid elements empty at the previous time step ( ${\textstyle t_{n-1}-t_{n}}$ ).

The MPM makes use of a predictor/corrector procedure, based on the Newmark integration scheme. The prediction of the nodal displacement, velocity and acceleration reads

(2.10)

(2.11)

(2.12)

where the upper-left side index ${\textstyle it}$ indicates the iteration counter, while the upper-right index ${\textstyle n}$ the time step. ${\textstyle \lambda }$ and ${\textstyle \zeta }$ are the Newmark's coefficients equal to 0.5 and 0.25, respectively.

Once the nodal velocity and acceleration are predicted (Equations 2.10-2.12), the system of linearised governing equations is formulated, as in classic FEM, and the local matrix ${\textstyle \mathbf {K} ^{tan}}$ and the residual ${\textstyle \mathbf {R} _{I}}$ are evaluated and assembled, respectively (phase b, Figure 2.3b).

The solution in terms of increment of nodal displacement is found iteratively solving the residual-based system. Once the solution ${\textstyle ^{it+1}\delta {\boldsymbol {u}}_{I}^{n+1}}$ is obtained, a correction of the nodal increment of displacement is performed

(2.13)

Velocity and acceleration are corrected according to Equations 2.11 and 2.12, respectively. This procedure has to be repeated until convergence is reached.

Unlike a FEM code, the nodal information is available only during the calculation of a time step: at the beginning of each time step a reset of all the nodal information is performed and the accumulated displacement information is deleted. The computational mesh is allowed to deform only during the iterative procedure of a time step, avoiding the typical element tangling of a standard FEM. When convergence is achieved, the position of the nodes is restored to the original one (phase c, Figure 2.3c). Before restoring the undeformed configuration of the FE grid, the solution in terms of nodal displacement, velocity and acceleration is interpolated on the material points, as

(2.14)

(2.15)

(2.16)

where ${\textstyle n_{n}}$ is the total number of nodes per geometrical element, ${\textstyle (\xi _{p},\eta _{p})}$ are the local coordinates of material point ${\textstyle p}$ and ${\textstyle N_{I}\left(\xi _{p},\eta _{p}\right)}$ is the shape function evaluated at the position of the material point ${\textstyle p}$ , relative to node ${\textstyle I}$ .

Finally the current position of the material points is updated as

(2.17)

The details of the MPM algorithm are presented in Algorithm 2.1.

(we will use $(\bullet )^{n}=(\bullet )(t_{n})$ ), Material DATA: E, $\nu$ , $\rho$

Initial data on material points:

,

and

Initial data on nodes: NONE - everything is discarded in the initialization phase

OUTPUT of calculations:

INITIALIZATION PHASE

Clear nodal info and recover undeformed grid configuration

Calculation of initial nodal conditions.

(a) for p = 1:

Calculation of nodal data

${\textstyle \mathbf {q} _{I}^{n}=\sum _{p}N_{I}\,m_{p}\mathbf {v} _{p}^{n}}$

${\textstyle \mathbf {f} _{I}^{n}=\sum _{p}N_{I}m_{p}\mathbf {a} _{p}^{n}}$

${\textstyle m_{I}^{n}=\sum _{p}N_{I}m_{p}}$

(b) for I = 1:

${\textstyle {\widetilde {\mathbf {v} }}_{I}^{n}={\dfrac {\mathbf {q} _{I}^{n}}{m_{I}^{n}}}}$

${\textstyle {\widetilde {\mathbf {a} }}_{I}^{n}={\dfrac {\mathbf {f} _{I}^{n}}{m_{I}^{n}}}}$

Newmark method: PREDICTOR. Evaluation of ${\textstyle ^{it+1}\Delta \mathbf {u} _{I}^{n+1},^{it+1}\mathbf {v} _{I}^{n+1}}$ and ${\textstyle ^{it+1}\mathbf {a} _{I}^{n+1}}$ using Equations (MPM disp predictor)–(MPM acc predictor)

UL-FEM PHASE

for p = 1: ${\textstyle N_{p}}$

(a) Evaluation of local residual (

)

(b) Evaluation of local Jacobian matrix of residual (

)

(c) Assemble

and

to the global vector

and global matrix

Solving system ${\textstyle (\Delta \mathbf {u} _{I}^{n+1})}$

Newmark method: CORRECTOR (Equations (MPM vel predictor)–(MPM disp corrector))

Check convergence

(a) NOT converged: go to Step 2

(b) Converged: go to Step 3

CONVECTIVE PHASE

Update the kinematics on the material points by means of an interpolation of nodal information (Equations (displacement mapping)–(material point position update))

Save the stress ${\textstyle {\boldsymbol {\sigma }}_{p}^{n+1}}$ , strain ${\textstyle {\boldsymbol {\epsilon }}_{p}^{n+1}}$ and total deformation gradient ${\textstyle \mathbf {F} _{p}^{n+1}}$ on material points (the latter by ${\textstyle \mathbf {F} _{p}^{n+1}=\Delta \mathbf {F} _{p}\cdot \mathbf {F} _{p}^{n}}$ )

Algorithm. 2.1 MPM algorithm.

2.5 The Galerkin Meshless Method

In Section 2.3 the Material Point Method and its state of the art is presented. Different versions of the method, proposed with the attempt to overcome the cell-crossing error issue, are discussed. An alternative to the approaches aforementioned, such as GIMP, CPDI and DDMPM, is represented by the Galerkin Meshless Method (GMM) [141]. The GMM can be seen as the MPM, where the Eulerian background grid is replaced by a Lagrangian one, defined by a cloud of nodes. In GMM, the material points move together with the computational nodes and the shape functions are evaluated once the surrounding cloud of nodes is defined (Figure 2.4a). In this case, unlike MPM, the nodes preserve their history through the whole simulation, as in FEM. GMM is a continuum method and it does not make use of a remeshing technique which gives all the advantages of MPM, previously discussed, such as, local level calculation, good conservation properties and an easy adaptation to parallel computing. The GMM is a truly meshless method, based on a Galerkin formulation. Unlike other methods, such as, the Element-Free Galerkin Method [76] or the Reproducing Kernel Particle Method [64], this technique does not need element connectivity for integration or interpolation purposes. However, as it belongs to the class of techniques described in Section 2.2.3, it may suffer from the drawbacks which typically affects all the meshless methods (e.g. tension instability, difficulty in enforcing the Essential Boundary Conditions (EBCs), lack of interpolation consistency, etc.). Apart from these shortcomings, GMM with MPM might represent a suitable choice for the solution of real scale large deformation problems and a comparison within a unified framework would be beneficial for an objective evaluation of the capabilities of each method.

Figure 2.4: GMM. The shape functions on the material point

are evaluated using the information on the nodes sufficiently closed to the material point itself.

2.6 The Galerkin Meshless Method in Kratos Multiphysics

In this section the Galerkin Meshless Method, implemented in Kratos Multiphysics, is described. The algorithm presented in [141] to simulate fluid-structure interaction problems is taken as a starting point and adapted to the simulation of deformable solids [138].

As explained in Section 2.5, the GMM is a truly meshless method and it can be seen as the application of the MPM idea extended to the case in which both the nodes and the material points behave as purely Lagrangian through the whole analysis. Thus, it is relatively easy to enforce conservation properties at the integration points, while also maintaining the history of nodal results during all the simulation time, provided that a reliable technique is chosen for the computation of the meshless shape functions. The difficulty is, hence, moved to the construction of such an effective meshless base, which is addressed in Section 2.6.2. In what follows, the algorithm used for the implementation of the GMM in Kratos Multiphysics is described. Differences and analogies with the MPM algorithm procedure are highlighted. Moreover, the construction of effective meshless intepolants is discussed in Section 2.6.2, where two techniques are presented: the Moving Least Square (MLS) technique and the Local Max-Ent (LME) method. Further, in Chapter 4 a comparison between the MPM and the GMM is performed through some benchmark tests and an assessment in terms of computational cost, accuracy and robustness is provided.

2.6.1 GMM Algorithm

The GMM algorithm is based on three principal steps (see Figure 2.5).

Initialization phase (Fig.2.5a): this is the step which mostly distinguishes GMM from MPM. During this phase the connectivity of each integration point (i.e., each material point) is computed as the "cloud of nodes", centred on the material point, and obtained by a search-in-radius. Such a cloud is then employed for the calculation of the shape functions. Unlike MPM, the Newmark prediction is performed by using the nodal information of the previous time step, as in FEM. Once ${\textstyle N}$ and ${\textstyle \nabla N}$ are suitably defined, MPM and GMM essentially coincide in the following steps;
UL-FEM calculation phase (Fig.2.5b): the local matrix, represented by the left-hand-side ( ${\textstyle lhs}$ ) and the local vector, constituted by the right-hand-side ( ${\textstyle rhs}$ ) are evaluated in the current configuration according to the formulation presented in Chapter 4; the global matrix ${\textstyle LHS}$ and the global vector ${\textstyle RHS}$ are obtained by assembling the local contributions of each material point and, finally, the system is iteratively solved. During the iterative procedure, the nodes are allowed to move, accordingly to the nodal solution, and the material points do not change their local position within the geometrical element until the solution has reached convergence;
Convective phase (Fig.2.5c): during the third and last phase the nodal information at time ${\textstyle t_{n+1}}$ are interpolated back to the material points. The position of the material points is updated. Unlike MPM, the nodal information are not deleted, but used as initial conditions in the next time step.

The details of the GMM algorithm are presented in Algorithm 2.2.


(a) Initialization phase	(b) Updated Lagrangian FEM phase

(c) Convective phase
Figure 2.5: GMM phases.

Material DATA: E, $\nu$ , $\rho$

Initial data on material points:

,

and

Initial data on nodes:

OUTPUT of calculations:

INITIALIZATION PHASE

for every material point with position ${\textstyle \mathbf {x} _{p}^{n}}$ gather the cloud of nodes with position ${\textstyle \mathbf {x} _{I}^{n}}$ such that ${\textstyle \left|\mathbf {x} _{p}-\mathbf {x} _{I}\right|<R}$
compute the shape functions ${\textstyle N_{I}\left(\mathbf {x} _{p}^{n}\right)}$ for all nodes ${\textstyle I}$ in the cloud
Newmark method: PREDICTOR

for the prediction of displacement, unlike Equation (2.10),

,

while for the prediction of nodal velocity and nodal acceleration, see Equations (2.11) and (2.12)

UL-FEM PHASE (identical to MPM, see Algorithm 2.1 )

CONVECTIVE PHASE

Update position of the material points by means of an interpolation of nodal solution

(a) for p = 1:

Save state of stress ${\textstyle {\boldsymbol {\sigma }}_{p}^{n+1}}$ , state of strain ${\textstyle {\boldsymbol {\epsilon }}_{p}^{n+1}}$ and total deformation gradient ${\textstyle \mathbf {F} _{p}^{n+1}}$ on material points

(the latter by

)

Algorithm. 2.2 GMM algorithm.

2.6.2 Calculation of GMM shape functions

While the computation of the shape functions is trivial for the standard MPM (as in FEM), thanks to the presence of a background grid (Figure 2.2), the evaluation of the shape functions in GMM is more complex. From a technical point of view, GMM is based on a conceptually simple operation: given an arbitrary position ${\textstyle x_{p}}$ in space (which will, in the practice, coincide with the position of the material point) and a search radius ${\textstyle R}$ , one may find all of the Nodes ${\textstyle I}$ such that ${\textstyle ||x_{I}-x_{p}||<R}$ . Given such a cloud of nodes, one may then compute, at the position ${\textstyle x_{p}}$ , the shape functions ${\textstyle N_{I}(x_{p})}$ (together with their gradients), such that, a given function ${\textstyle u}$ , whose nodal value is ${\textstyle u_{I}}$ , can be interpolated at the position ${\textstyle x_{p}}$ as ${\textstyle u(x_{p})=\sum _{I}N_{I}(x_{p})\,u_{I}}$ (Figure 2.4a).

However, in order to construct a convergent solution, some guarantees must be provided by the shape functions. In particular, they shall comply with the Partition of Unity (PU) property, as a very minimum at all of the positions ${\textstyle x_{p}}$ at which the shape functions are evaluated. A number of shape functions exist complying with such property [142,143,33]. Among the available options, two appealing class of meshless functions are considered in this work: the first choice is constituted by the so called Moving Least Square (MLS) method and the second one represented by the Local Maximum Entropy (LME) technique.

The first technique is based on the MLS approach, first introduced by Lancaster [74] and Belytschko [76,143]. The MLS-approximation fulfils the reproducing conditions by construction, so no corrections are needed. The fundamental principle of MLS approximants is based on a weighted least square fitting of a target solution, sampled at a given, possibly randomly distributed, set of points, via a function of the type

(2.18)

where the coordinates ${\textstyle x,y}$ are to be understood as relative to the sampling position.

The reconstruction of a continuous function ${\textstyle h(\mathbf {x} )}$ can be obtained considering the data ${\textstyle h_{I}}$ be located at points ${\textstyle \mathbf {x} _{I}}$ and an arbitrary, smooth and compactly supported, weight function ${\textstyle W_{I}(\mathbf {x} )}$ , such that the ${\textstyle \mathbf {x} _{I}}$ fall within the support of ${\textstyle W}$ . Assuming now that the reconstructed function ( ${\textstyle h_{x}^{h}}$ ) is computed as

(2.19)

the fitting to ${\textstyle h(\mathbf {x} )}$ is done by minimizing the error function ${\textstyle J}$ , defined as

(2.20)

where ${\textstyle \mathbf {P} _{I}=\mathbf {P} (\mathbf {x} _{I})}$ .

This allows defining a set of approximating shape functions ${\textstyle N}$ such that

(2.21)

where

(2.22)

with ${\textstyle \mathbf {M} }$ defined as

(2.23)

It can be readily verified that the shape functions are able to reproduce exactly a polynomial up to the order used in the construction. This fact can also be used to prove compliance to the partition of unity property. Namely, if one assumes ${\textstyle h_{I}=\mathbf {P} _{I}(\mathbf {x_{I}} )}$ and substitutes into Equation 2.21 then

(2.24)

Hence, considering the special case of a constant polynomial ${\textstyle \mathbf {P} (\mathbf {x} )=1}$ , or of a linear variation in ${\textstyle \mathbf {x} }$ , ${\textstyle \mathbf {P} (\mathbf {x} )=\mathbf {x} }$ we obtain respectively

(2.25)

A similar reasoning also gives

(2.26)

thus proving the compliance with the PU property.

However, MLS shape functions are not able to guarantee the Kronecker delta property at the nodes. This implies that two nodal shape functions may be simultaneously non zero at a given nodal position. This has practical implications at the moment of imposing Dirichlet Boundary conditions, namely, in order to impose ${\textstyle u(x_{d})=0}$ at a given point on the Dirichlet boundary ${\textstyle x_{d}}$ , one must impose that ${\textstyle \sum (N_{I}(x_{d})u_{I})=0}$ , which constitutes a classical multipoint constraint [144].

Interestingly, the choice of different shape functions could ease this particular problem. An appealing choice could be the use of LME approximants, which guarantees complying with a weak Kronecker delta property until the cloud of nodes is represented by a convex hull.

The LME technique is based on the evaluation of the local max-ent approximants [145], which represents the solution that exhibits a (Pareto) compromise between competing objectives: the principle of max-ent subject to the constraints:

(2.27)

and the objective function interpreted as a measure of locality of the shape functions of the Delaunay triangulation

(2.28)

The solution to the problem can be found minimizing ${\textstyle \beta U({\boldsymbol {x}},N_{1},N_{2},...,N_{m})-H(N_{1},N_{2},...,N_{m})}$ subjected to the usual constrains. The optimization problem takes the form

(2.29)

with ${\textstyle \beta =\gamma /h^{2}}$ representing a non-negative locality coefficient, where ${\textstyle \gamma }$ is a dimensionless parameter and ${\textstyle h}$ is a measure of nodal spacing. The value of ${\textstyle \gamma }$ is always chosen in a range between ${\textstyle 0.6}$ , relative to spread-out meshfree shape functions, and ${\textstyle 4}$ , relative to linear finite elements basis functions. Unlike MLS approximants, the LME basis functions possess the weak Kronecker delta property at the boundary of the convex hull of the nodes and they are ${\textstyle C^{\infty }}$ function of ${\textstyle \beta }$ in ${\textstyle (0,+\infty )}$ . However, the computation of the LME approximation scheme is more onerous than MLS basis functions, as the problem described by Equation 2.29 is a convex problem.

3 Constitutive Models

As explained in the previous Chapters, granular material can be modelled by continuum approach on a macroscopic scale. In the continuum approach, the macroscopic behaviour of granular flow is described by the balance equations (introduced in Chapter 4) facilitated with boundary conditions and constitutive laws. These latter ones characterize the relation between the stress and strain, thus, defining the behaviour of the material. In this Chapter, the constitutive models employed in this monograph for verification, validation and application of the MPM strategy are presented and their numerical implementation is explained.

The irreducible and mixed formulations, presented and verified in Chapters 4 and 5, respectively, are written in an Updated Lagrangian framework, e.g, the last known configuration is considered to be the reference one, and are valid under the hypothesis of large deformations, since the strain information, used for the evaluation of the material response, is represented by the total deformation gradient ${\textstyle {\boldsymbol {F}}:={\frac {\partial \phi \left({\boldsymbol {X}},t\right)}{\partial {\boldsymbol {X}}}}}$ and not by the symmetric gradient of displacement ${\textstyle \left[\Delta {\boldsymbol {u}}\right]^{sym}={\frac {1}{2}}\left(\Delta {\boldsymbol {u}}+\Delta {\boldsymbol {u}}^{T}\right)}$ . In this monograph, homogeneous isotropic elasic and elasto-plastic materials are considered. More specifically, a hyperelastic Neo-Hookean, a hyperelastic-plastic J2 and Mohr-Coulomb plastic laws have been implemented in the framework of MPM and they are presented in Sections 3.1, 3.2 and 3.3.

3.1 Hyperelastic law

The first constitutive law to be presented is an elastic law under finite strain regime. In this regard, in the current work a hyperelastic material is considered. Typically, these models are suitable to describe the behaviour of engineering materials which can undergo deformation of several times their initial configuration, such as, rubber-like solids, elastomers, sponges and other soft flexible materials. Hyperelastic materials are non-dissipative and their state of stress solely depends on the current deformation, as the Cauchy elastic models, and do not depend on the path of deformation. For this reason, in this case it is possible to derive the stress-strain relation from a specific strain energy function

(3.1)

which relates the specific strain energy to the total deformation gradient as unique state variable, since dissipative related state variables can be neglected and the assumption of isothermal processes is made [146]. As stated in [146], any constitutive law must satisfy the axioms of thermodynamic determinism, material objectivity and material symmetry. The compliance of the first axiom implies the stress constitutive equation in terms of the first Piola-Kirchhoff stress tensor ${\textstyle {\boldsymbol {P}}}$

(3.2)

Accordingly, the stress constitutive relation can be expressed also by the Kirchhoff stress tensor

(3.3)

which for a hyperelastic material is given by

(3.4)

and the Cauchy stress tensor, ${\textstyle {\boldsymbol {\sigma }}={\boldsymbol {\tau }}/J}$

(3.5)

where ${\textstyle J\equiv det{\boldsymbol {F}}}$ .

In order to comply with the second axiom, the specific strain energy function has to be invariant under changes in the observer. This can be expressed by the following equation

(3.6)

which has to be valid for any rotation tensor ${\textstyle {\boldsymbol {Q}}}$ . If we consider ${\textstyle {\boldsymbol {Q}}={\boldsymbol {R}}^{T}}$ with ${\textstyle {\boldsymbol {R}}}$ the rotation obtained from the polar decomposition ${\textstyle {\boldsymbol {F}}={\boldsymbol {RU}}}$ , the following identity is obtained

(3.7)

Thus, the material objectivity or frame invariance implies that ${\textstyle \Psi }$ depends on ${\textstyle {\boldsymbol {F}}}$ solely through the right stretch tensor ${\textstyle {\boldsymbol {U}}}$ and in an equivalent way, ${\textstyle \Psi }$ can be expressed as a function of the right Cauchy-Green strain tensor ${\textstyle {\boldsymbol {C}}\equiv {\boldsymbol {F}}^{T}{\boldsymbol {F}}={\boldsymbol {U}}^{2}}$ :

(3.8)

The stress constitutive equations in terms of ${\textstyle {\boldsymbol {P}}}$ , ${\textstyle {\boldsymbol {\tau }}}$ and ${\textstyle {\boldsymbol {\sigma }}}$ can be expressed as

(3.9)

(3.10)

and,

(3.11)

where the definition of ${\textstyle {\frac {\partial C_{ij}}{\partial F_{kl}}}=\delta _{il}F_{kj}+\delta _{jl}F_{ki}}$ is used. The specific strain energy function has to be further constrained by considering the third and last axiom of material symmetry. With a focus on material isotropy, ${\textstyle \Psi }$ must satisfy

(3.12)

for all rotations ${\textstyle {\boldsymbol {Q}}}$ . By choosing ${\textstyle {\boldsymbol {Q}}={\boldsymbol {R}}^{T}}$ , it is established that

(3.13)

which states that the specific strain energy function of an isotropic hyperelastic material must depend only on ${\textstyle {\boldsymbol {F}}}$ through the left stretch tensor ${\textstyle {\boldsymbol {V}}}$ . In an equivalent way, ${\textstyle \Psi }$ can also be a function of the left Cauchy-Green strain tensor ${\textstyle {\boldsymbol {b}}\equiv {\boldsymbol {F}}{\boldsymbol {F}}^{T}={\boldsymbol {V}}^{2}}$ as follows

(3.14)

By considering both the axiom of frame invariance and material symmetry it implies that

(3.15)

The hyperelastic law, considered in the current work, is a Neo-Hookean model, of the form

(3.16)

with dependence only on the first invariant of ${\textstyle {\boldsymbol {C}}}$ , ${\textstyle I_{C}=\mathrm {tr} {\boldsymbol {C}}}$ , which exhibits the following specific strain energy function [147]

(3.17)

where ${\textstyle \lambda }$ and ${\textstyle \mu }$ are the Lamé constants.

According to Equation 3.9, it is known the stress constitutive relation in terms of the first Piola-Kirchhoff stress tensor and, thus, also in terms of the second Piola-Kirchhoff stress tensor

(3.18)

and the Kirchhoff stress tensor

(3.19)

In order to derive the strain energy function of Equation 3.17 with respect to ${\textstyle {\boldsymbol {C}}}$ , the following expression needs to be solved

(3.20)

By using these identities

(3.21)

(3.22)

and substituting them into the Equation 3.20, the final expression is obtained

(3.23)

By considering the result of Equation 3.23, and substituting it in the expression of Equation 3.19, the Kirchhoff stress tensor reads

(3.24)

which, after reminding the definition of the left Cauchy-Green strain tensor ${\textstyle {\boldsymbol {b}}={\boldsymbol {F}}{\boldsymbol {F}}^{T}}$ and the identity ${\textstyle {\boldsymbol {F}}{\boldsymbol {C}}^{-1}{\boldsymbol {F}}^{T}={\boldsymbol {F}}({\boldsymbol {F}}^{-1}{\boldsymbol {F}}^{-T}){\boldsymbol {F}}^{T}={\boldsymbol {I}}}$ , the final expression reads

(3.25)

where ${\textstyle {\boldsymbol {I}}}$ denotes the symmetric second order unit tensor.

As shown in Chapter 4, where the linearisation of the weak form is derived, a rate form of the constitutive equation is needed, and it can be obtained by taking the material derivative of the stress tensor ${\textstyle {\boldsymbol {S}}}$ [148]

(3.26)

where, with ${\textstyle {\boldsymbol {\mathrm {C} }}^{CE}}$ , it is denoted the tangent modulus. The push-forward operation of Equation 3.26, ${\textstyle {\boldsymbol {F}}\cdot {\dot {\boldsymbol {S}}}\cdot {\boldsymbol {F}}^{T}}$ , yields to the Lie derivative of ${\textstyle \tau }$ in compact form

{\mathcal {L}}_{v}{\boldsymbol {\tau }}:={\boldsymbol {F}}\cdot {\dot {\boldsymbol {S}}}\cdot {\boldsymbol {F}}^{T}={\boldsymbol {F}}\left(4{\frac {\partial ^{2}\Psi \left({\boldsymbol {C}}\right)}{\partial {\boldsymbol {C}}\partial {\boldsymbol {C}}}}:{\frac {\dot {\boldsymbol {C}}}{2}}\right){\boldsymbol {F}}^{T}=4{\boldsymbol {F}}{\frac {\partial ^{2}\Psi \left({\boldsymbol {C}}\right)}{\partial {\boldsymbol {C}}\partial {\boldsymbol {C}}}}{\boldsymbol {F}}^{T}:{\frac {\dot {\boldsymbol {C}}}{2}}

(3.27)

After expressing the rate of ${\textstyle {\boldsymbol {C}}}$ as

(3.28)

with ${\textstyle {\boldsymbol {d}}}$ the symmetrical spatial velocity gradient, defined as

(3.29)

and ${\textstyle {\boldsymbol {l}}={\dot {\boldsymbol {F}}}{\boldsymbol {F}}^{-1}}$ , the Lie derivative can be expressed as

{\mathcal {L}}_{v}{\boldsymbol {\tau }}=4{\boldsymbol {F}}{\frac {\partial ^{2}W\left({\boldsymbol {C}}\right)}{\partial {\boldsymbol {C}}\partial {\boldsymbol {C}}}}{\boldsymbol {F}}^{T}:\left({\boldsymbol {F}}^{T}{\boldsymbol {d}}{\boldsymbol {F}}\right)=4\left[\left({\boldsymbol {F}}\otimes {\boldsymbol {F}}^{T}\right):{\frac {\partial ^{2}W\left({\boldsymbol {C}}\right)}{\partial {\boldsymbol {C}}\partial {\boldsymbol {C}}}}:\left({\boldsymbol {F}}^{T}\otimes {\boldsymbol {F}}\right)\right]:{\boldsymbol {d}}

(3.30)

where ${\textstyle \mathrm {C} ^{\tau }}$ denotes the spatial incremental constitutive tensor. In order to define ${\textstyle \mathrm {C} ^{\tau }}$ , the second derivative of ${\textstyle \Psi }$ with respect to ${\textstyle {\boldsymbol {C}}}$ , ${\textstyle \displaystyle {\frac {\partial ^{2}\Psi \left({\boldsymbol {C}}\right)}{\partial {\boldsymbol {C}}\partial {\boldsymbol {C}}}}}$ , has to be computed. By recovering Equation 3.23, it can be written as

(3.31)

The derivative of ${\textstyle {\boldsymbol {C}}^{-1}}$ follows from the rule which is valid for any second order tensor ${\textstyle {\boldsymbol {A}}}$ , which, in Cartesian components, can be expressed as

(3.32)

Since ${\textstyle {\boldsymbol {C}}}$ is a symmetric tensor, only the symmetrical part is needed. This latter is given by the fourth order tensor ${\textstyle \mathrm {I} _{{\boldsymbol {C}}^{-1}}}$ defined in component form as

(3.33)

According to Equations 3.21 and 3.33, it is possible to write again the expression of ${\textstyle \displaystyle {\frac {\partial ^{2}\Psi \left({\boldsymbol {C}}\right)}{\partial {\boldsymbol {C}}\partial {\boldsymbol {C}}}}}$ as:

(3.34)

By performing a push-forward operation of this last expression, which defines the material tangent modulus ${\textstyle {\boldsymbol {\mathrm {C} }}^{CE}}$ (see Equation 3.26), the spatial incremental constitutive tensor is obtained

(3.35)

with ${\textstyle \mathrm {I} _{s}}$ the fourth order symmetric identity tensor.

The spatial counterparts ${\textstyle {\boldsymbol {I}}\otimes {\boldsymbol {I}}}$ and ${\textstyle \mathrm {I} _{s}}$ of the fourth-order tensors ${\textstyle {\boldsymbol {C}}^{-1}\otimes {\boldsymbol {C}}^{-1}}$ and ${\textstyle \mathrm {I} _{{\boldsymbol {C}}^{-1}}}$ , have been evaluated through the push-forward operation:

(3.36)

and

(3.37)

In the case of nearly-incompressible materials, the numerical treatment is not trivial. With regard to the finite element analysis, as it is addressed in Chapter 5, the use of mixed finite element formulations is required where the mean stress is considered as an additional primary variable, together with the displacements. In the context of the specific strain energy function and the stress constitutive relation, the isochoric/volumetric split permits to treat in a different way the incompressible part. As follows, the split of the total deformation gradient is exploited

(3.38)

and the expression of the volumetric ${\textstyle {\boldsymbol {F}}_{vol}}$ and deviatoric ${\textstyle {\boldsymbol {F}}_{dev}}$ terms respectively read

(3.39)

(3.40)

Accordingly, it is possible to define the deviatoric left Cauchy-Green strain tensor as

(3.41)

and its first principal invariant

(3.42)

With this last expression at hand, the specific strain energy function of a Neo-Hookean model is written as

(3.43)

with ${\textstyle K}$ representing the bulk modulus and ${\textstyle U(J)}$ the volumetric part of ${\textstyle {\tilde {\Psi }}}$ dependent on ${\textstyle J}$ .

It is possible to find the expression of the Kirchhoff stress tensor ${\textstyle {\boldsymbol {\tau }}}$ by using Equation 3.19, which in the case of ${\textstyle {\tilde {\Psi }}}$ is derived:

(3.44)

where ${\textstyle \mathrm {I} _{d}=\mathrm {I} _{s}-{\frac {1}{3}}{\boldsymbol {I}}\otimes {\boldsymbol {I}}}$ is the fourth-order deviatoric projector tensor, ${\textstyle {\boldsymbol {b}}_{dev}=J^{-{\frac {2}{3}}}{\boldsymbol {b}}}$ the volume-preserving part of ${\textstyle {\boldsymbol {b}}}$ and the quantity ${\textstyle p=KU'(J)}$ , the only contribution to the volumetric part of ${\textstyle {\boldsymbol {\tau }}}$ , which in the case of a nearly-incompressible material is represented by the mean stress primary variable, as addressed in Chapter 5.

As it was previously done, in order to find the expression of the spatial incremental constitutive tensor ${\textstyle \mathrm {C} ^{\tau }}$ , given by the sum of the volumetric and deviatoric contribution

(3.45)

the second derivative of ${\textstyle {\tilde {\Psi }}}$ , ${\textstyle \displaystyle {\frac {\partial ^{2}{\tilde {\Psi }}({\boldsymbol {C}}_{dev})}{\partial {\boldsymbol {C}}\partial {\boldsymbol {C}}}}}$

{\frac {\partial ^{2}{\tilde {\Psi }}({\boldsymbol {C}}_{dev})}{\partial {\boldsymbol {C}}\partial {\boldsymbol {C}}}}={\frac {\partial \left({\frac {1}{2}}J\,p{\boldsymbol {C}}^{-1}\right)}{\partial {\boldsymbol {C}}}}+{\frac {\partial \left({\frac {\mu }{2}}J^{-{\frac {2}{3}}}{\boldsymbol {I}}:\mathrm {I} _{d}\right)}{\partial {\boldsymbol {C}}}}

(3.46)

needs to be computed. The first addend of Equation 3.46 refers to the volumetric component and the second addend to the deviatoric component of the second derivative of ${\textstyle {\tilde {\Psi }}}$ . With regards to the first term, by performing the derivative with respect to ${\textstyle {\boldsymbol {C}}}$ leads to

{\frac {\partial \left({\frac {1}{2}}J\,p{\boldsymbol {C}}^{-1}\right)}{\partial {\boldsymbol {C}}}}={\frac {1}{2}}p\left({\frac {\partial J}{\partial {\boldsymbol {C}}}}{\boldsymbol {C}}^{-1}+J{\frac {\partial {\boldsymbol {C}}^{-1}}{\partial {\boldsymbol {C}}}}\right)={\frac {1}{2}}J\,p\left({\frac {1}{2}}{\boldsymbol {C}}^{-1}\otimes {\boldsymbol {C}}^{-1}+{\frac {\partial {\boldsymbol {C}}^{-1}}{\partial {\boldsymbol {C}}}}\right)

(3.47)

With respect to the deviatoric term of Equation 3.46, it is possible to derive it, by firstly rewriting it as

{\frac {\partial \left({\frac {\mu }{2}}J^{-{\frac {2}{3}}}{\boldsymbol {I}}:\mathrm {I} _{d}\right)}{\partial {\boldsymbol {C}}}}={\frac {\partial \left({\frac {\mu }{2}}J^{-{\frac {2}{3}}}{\boldsymbol {I}}:\left(-{\frac {1}{3}}{\boldsymbol {C}}^{-1}\otimes {\boldsymbol {C}}+\mathrm {I} _{s}\right)\right)}{\partial {\boldsymbol {C}}}}

(3.48)

and, then, by deriving with respect to ${\textstyle {\boldsymbol {C}}}$

(3.49)

By substituting the results of Equations 3.47 and 3.49 in the definition of ${\textstyle \mathrm {C} ^{\tau }}$

(3.50)

the final expressions of the volumetric and deviatoric spatial incremental constitutive tensors are obtained

(3.51)

(3.52)

For the sake of clarity, Equations 3.36 and 3.37 are used in the push forward operation.

3.2 Hyperelastic - J₂ plastic law

In this section a hyperelastic - ${\textstyle J_{2}}$ metal plastic law in finite strains regime is presented. In this context, the main hypothesis, on which the elastoplastic constitutive framework is based, is represented by the notion of the multiplicative decomposition of the total deformation gradient ${\textstyle {\boldsymbol {F}}}$ in an elastic and plastic component of the form

(3.53)

This theory, introduced for the first time in [149,150], lies on the concept of a local intermediate stress-free configuration defined by the plastic total deformation gradient ${\textstyle {\boldsymbol {F}}^{P}}$ , which can be recovered by performing a purely elastic loading from the fully deformed configuration. As pointed out in [151], by formulating the ${\textstyle J_{2}}$ plastic flow theory based on the concept of elastic-plastic multiplicative decomposition of ${\textstyle {\boldsymbol {F}}}$ , some features can be observed. The stress-strain relation derives from the specific strain energy function, decoupled into its volumetric and deviatoric parts and the integration algorithm reduces to the radial return mapping in which the elastic predictor is computed through the elastic stress-strain relation. In addition to that, as in the infinitesimal theory, it is possible to linearise the algorithm which allows to define the algorithmic tangent elastoplastic moduli in a closed-form. In the case that the plastic flow does not take place, the solution of finite elasticity is recovered and the procedure presented in Section 3.1 is valid. In this section, the equations which are used to derive the algorithmic procedure are presented under the hypothesis of isotropic stress response and isochoric plastic flow, i.e., ${\textstyle \mathrm {det} {\boldsymbol {F}}^{P}=\mathrm {det} {\boldsymbol {C}}^{P}=1}$ .

The first equation to be introduced is the specific strain energy function, from which it is possible to derive the expression of the stress tensor. As previously discussed in Section 3.1, according to the axioms of thermodynamic determinism, material objectivity, material symmetry and the aforementioned notion of local stress-free configuration, the following specific strain energy function ${\textstyle \Psi }$ with uncoupled volumetric ( ${\textstyle U(J^{e}))}$ and deviatoric ( ${\textstyle {\tilde {\Psi }}({\bar {{\boldsymbol {C}}^{e}}})}$ ) part is considered

(3.54)

with ${\textstyle {\bar {{\boldsymbol {C}}^{e}}}\equiv J^{-{\frac {2}{3}}}{\boldsymbol {C}}^{e}=J^{-{\frac {2}{3}}}{{\boldsymbol {F}}^{e}}^{T}{\boldsymbol {F}}^{e}}$ being the volume preserving part of the elastic right Cauchy-Green strain tensor. According to Equation 3.44, the elastic Kirchhoff stress tensor is derived as

(3.55)

where ${\textstyle {\bar {{\boldsymbol {b}}^{e}}}\equiv J^{-{\frac {2}{3}}}{\boldsymbol {F}}^{e}{{\boldsymbol {F}}^{e}}^{T}}$ is the volume preserving part of ${\textstyle {\boldsymbol {b}}^{e}}$ .

Once defined the stored energy function of Equation 3.54, it is necessary to introduce the yield condition, which is characteristic of the ${\textstyle J_{2}}$ plastic theory; the classical Mises-Huber yield condition in terms of ${\textstyle {\boldsymbol {\tau }}}$ , graphically represented in Figure 3.1, can be expressed as

(3.56)

with ${\textstyle \sigma _{Y}}$ being the flow stress, ${\textstyle H>0}$ the isotropic hardening and ${\textstyle \alpha }$ the hardening parameter. The last governing equation, which makes the plastic problem determined, is given by the fundamental form of the corresponding associative flow rule, which can be derived uniquely by satisfying the principle of the maximum dissipation [152]. As shown in [153,154], the associative flow rule in strain space reads

\left\{{\begin{array}{rcll}{\dfrac {\partial }{\partial t}}\left({\overline {\boldsymbol {C}}}^{p}\right)^{-1}&=&-{\frac {2}{3}}\gamma \,{\mathrm {tr} }({\boldsymbol {b}}^{e}){\boldsymbol {F}}^{-1}{\boldsymbol {n}}{\boldsymbol {F}}^{-T}\\{\dot {\alpha }}&=&{\sqrt {\frac {2}{3}}}\gamma \end{array}}\right.

(3.57)


(a)	(b)
Figure 3.1: $J_{2}$ model in the principal stress space (a) and in $\pi$ plane (b)

where ${\textstyle {\overline {\boldsymbol {C}}}^{p}}$ is the volume preserving part of the plastic right Cauchy-Green deformation tensor ${\textstyle {\overline {\boldsymbol {C}}}^{p}={{\overline {\boldsymbol {F}}}^{p}}^{T}{\overline {\boldsymbol {F}}}^{p}}$ , ${\textstyle \gamma }$ is the plastic multiplier, ${\textstyle {\boldsymbol {n}}:={\dfrac {\mathrm {dev} {\boldsymbol {\tau }}}{\Vert \mathrm {dev} {\boldsymbol {\tau }}\Vert }}}$ is the unit vector of ${\textstyle {\mathrm {dev} }({\boldsymbol {\tau }})}$ . In order to complete the formulation of the model, it is assumed that the parameter ${\textstyle \alpha }$ , as in the linear theory, is governed by a rate equation [151], where ${\textstyle \gamma }$ is subjected to the standard Kuhn-Tucker loading/unloading condition:

(3.58)

and consistency condition

(3.59)

The algorithmic procedure to be established in order to solve the plastic problem has to respect the material frame indifference principle. This can be accomplished by defining the discrete form of the evolution equation (see Equation 3.57) in material description. Accordingly, a time stepping algorithm is conducted by applying a backward Euler difference scheme on Equation 3.57:

\left\{{\begin{array}{rcll}({\overline {\boldsymbol {C}}}_{n+1}^{p})^{-1}-({\overline {\boldsymbol {C}}}_{n}^{p})^{-1}&=&-{\frac {2}{3}}\Delta \gamma \left(\left({\boldsymbol {C}}_{n+1}^{p}\right)^{-1}:{\boldsymbol {C}}_{n+1}\right){\boldsymbol {F}}_{n+1}^{-1}{\boldsymbol {n}}_{n+1}{\boldsymbol {F}}_{n+1}^{-T},\\\alpha _{n+1}-\alpha _{n}&=&{\sqrt {\frac {2}{3}}}\Delta \gamma \end{array}}\right.

(3.60)

and by operating a push-forward transformation it is possible to recover the spatial form of the discrete spatial evolution equation through some useful and fundamental steps, such as

(3.61)

{\overline {\boldsymbol {F}}}_{n+1}\left({\overline {\boldsymbol {C}}}_{n}^{p}\right)^{-1}{\overline {\boldsymbol {F}}}_{n+1}^{T}={\overline {\boldsymbol {f}}}_{n+1}\left({\overline {\boldsymbol {F}}}_{n}({\overline {\boldsymbol {C}}}_{n}^{p})^{-1}{\overline {\boldsymbol {F}}}_{n}^{T}\right){\overline {\boldsymbol {f}}}_{n+1}^{T}={\overline {\boldsymbol {f}}}_{n+1}{\overline {\boldsymbol {b}}}_{n}^{e}{\overline {\boldsymbol {f}}}_{n+1}^{T}

(3.62)

(3.63)

where in Equation 3.62, the relation ${\textstyle {\overline {\boldsymbol {F}}}_{n+1}={\overline {\boldsymbol {f}}}_{n+1}{\overline {\boldsymbol {F}}}_{n}}$ and the results of Equation 3.61 are used. By the use of the expressions earlier derived, the spatial form of Equation 3.60 is

\left\{{\begin{array}{rcll}{\overline {\boldsymbol {b}}}_{n+1}^{e}&=&{\overline {\boldsymbol {f}}}_{n+1}{\overline {\boldsymbol {b}}}_{n}^{e}{\overline {\boldsymbol {f}}}_{n+1}^{T}-{\frac {2}{3}}\Delta \gamma \mathrm {tr} {\overline {\boldsymbol {b}}}_{n+1}^{e}{\boldsymbol {n}}_{n+1},\\\alpha _{n+1}&=&\alpha _{n}+{\sqrt {\frac {2}{3}}}\Delta \gamma \end{array}}\right.

(3.64)

With the evolution equation written in spatial configuration, the yield condition and the definition of the Kirchhoff stress tensor it is possible to define the return mapping algorithm, through which the plastic problem is solved in the time interval ${\textstyle \left[t_{n},t_{n+1}\right]}$ . Let ${\textstyle {\boldsymbol {F}}_{n}}$ , ${\textstyle \alpha _{n}}$ , ${\textstyle {\boldsymbol {b}}_{n}^{e}}$ and the configuration ${\textstyle \phi _{n}}$ be known data at time ${\textstyle t_{n}}$ and the incremental displacement of the configuration ${\textstyle \phi _{n}}$ , ${\textstyle {\boldsymbol {u}}_{n}\circ \phi _{n}}$ , at time ${\textstyle t_{n+1}}$ used to defined the incremental deformation gradient ${\textstyle {\boldsymbol {f}}_{n+1}}$ which can be evaluated as ${\textstyle {\boldsymbol {f}}_{n+1}={\boldsymbol {1}}+\nabla _{x_{n}}{\boldsymbol {u}}_{n}}$ . In order to compute the elasto-plastic response, a trial elastic state is defined where no plastic flow takes place. Through this assumption, the intermediate configuration remains unchanged, i.e.:

(3.65)

which, by a push-forward transformation, corresponds to the spatial form:

\left({\overline {\boldsymbol {b}}}_{n+1}^{e}\right)^{trial}:={\overline {\boldsymbol {F}}}_{n+1}\left({\overline {\boldsymbol {C}}}_{n+1}^{p^{-1}}\right)^{trial}{\overline {\boldsymbol {F}}}_{n+1}^{T}={\overline {\boldsymbol {f}}}_{n+1}\left[{\overline {\boldsymbol {F}}}_{n}{\overline {\boldsymbol {C}}}_{n}^{p^{-1}}{\overline {\boldsymbol {F}}}_{n}^{T}\right]{\overline {\boldsymbol {f}}}_{n+1}^{T}

(3.66)

According to Equation 3.66, the trial state of stress is defined as

(3.67)

and the discrete governing equations are written as

\left\{{\begin{array}{rcll}{\overline {\boldsymbol {b}}}_{n+1}^{e}&=&{\overline {\boldsymbol {b}}}_{n+1}^{e^{trial}}-{\frac {2}{3}}\Delta \gamma \mathrm {tr} {\overline {\boldsymbol {b}}}_{n+1}^{e}{\boldsymbol {n}}_{n+1},\\\alpha _{n+1}&=&\alpha _{n}+{\sqrt {\frac {2}{3}}}\Delta \gamma \end{array}}\right.

(3.68)

\left\{{\begin{array}{rcll}{\boldsymbol {\tau }}_{n+1}&=&U'(J_{n+1})J_{n+1}{\boldsymbol {I}}+\mu \mathrm {dev} \left({\overline {\boldsymbol {b}}}_{n+1}^{e}\right)=U'(J_{n+1})J_{n+1}{\boldsymbol {I}}+\mathrm {dev} \left({\boldsymbol {\tau }}_{n+1}\right),\\{\boldsymbol {n}}_{n+1}&=&{\dfrac {\mathrm {dev} \left({\boldsymbol {\tau }}_{n+1}\right)}{\Vert \mathrm {dev} \left({\boldsymbol {\tau }}_{n+1}\right)\Vert }}\end{array}}\right.

(3.69)

(3.70)

By exploiting the Kuhn-Tucker condition in discrete form of Equation 3.70, two cases might be encountered. The first one, it is identified with the case the yield condition ${\textstyle f_{n+1}^{trial}\leq 0}$ :

(3.71)

then, the condition ${\textstyle \Delta \gamma =0}$ is directly satisfied and no plastic flow takes place, leading to a completely elastic response. In the second alternative case, where the yield condition ${\textstyle f_{n+1}^{trial}>0}$ , it is clear that ${\textstyle {\boldsymbol {\tau }}_{n+1}^{trial}}$ can not be admitted and, according to Equation 3.68, ${\textstyle {\overline {\boldsymbol {b}}}_{n+1}^{e}\neq {\overline {\boldsymbol {b}}}_{n+1}^{e^{trial}}}$ leading to the condition of ${\textstyle \Delta \gamma {>0}}$ . Thus, in this case the radial return mapping has to be performed. By considering Equation 3.68 and applying the trace operator to it, it is immediately demonstrated that

(3.72)

since ${\textstyle \mathrm {tr} \,{\boldsymbol {n}}_{n+1}=0}$ . By substituting the expression of Equation 3.72 in Equation 3.68 it is found that

(3.73)

which used for the definition of the deviatoric part of ${\textstyle {\boldsymbol {\tau }}}$ leads to

(3.74)

By considering the expression ${\textstyle {\overline {\mu }}={\dfrac {1}{3}}\mu \mathrm {tr} \,{\overline {\boldsymbol {b}}}_{n+1}^{e^{trial}}}$ , Equation 3.74 can be rewritten as

(3.75)

with ${\textstyle {\boldsymbol {n}}_{n+1}^{trial}={\dfrac {\mathrm {dev} {\boldsymbol {\tau }}_{n+1}^{trial}}{\Vert \mathrm {dev} {\boldsymbol {\tau }}_{n+1}^{trial}\Vert }}}$ . From this last equation it is deduced that

(3.76)

along with

(3.77)

Thus, it is possible to rewrite the yield condition as

(3.78)

and, finally, the unknown ${\textstyle \Delta \gamma }$ is determined

(3.79)

which is used for the determination of ${\textstyle {\overline {\boldsymbol {b}}}_{n+1}^{e}}$ and ${\textstyle \mathrm {dev} {\boldsymbol {\tau }}_{n+1}}$ (see Equations 3.73 and 3.74, respectively). The procedure defined above is able to guarantee the preservation of the material frame indifference framework and it results in an extension to finite strains of the classical radial return mapping method of infinitesimal plasticity. The governing equations have been derived and the algorithmic procedure to be followed step-by-step is described in Algorithm 3.1.

Initial data on material points: ${\boldsymbol {F}}_{n}$ , ${\boldsymbol {b}}_{n}^{e}$

OUTPUT of calculations:

,

UPDATE THE CURRENT CONFIGURATION

Compute the the current configuration: ${\textstyle {\boldsymbol {\varphi }}_{n+1}={\boldsymbol {\varphi }}_{n}+{\boldsymbol {u}}_{n}\circ {\boldsymbol {\varphi }}_{n}}$
Compute the relative deformation gradient: ${\textstyle {\boldsymbol {f}}_{n+1}={\boldsymbol {1}}+\nabla _{{\boldsymbol {x}}_{n}}{\boldsymbol {u}}_{n}}$
Compute the total deformation gradient in updated configuration: ${\textstyle {\boldsymbol {F}}_{n+1}={\boldsymbol {f}}_{n+1}{\boldsymbol {F}}_{n}}$

COMPUTE THE ELASTIC PREDICTOR

Compute the volume preserving part of ${\textstyle {\boldsymbol {f}}_{n+1}}$ : ${\textstyle {\overline {\boldsymbol {f}}}_{n+1}=\left[\mathrm {det} {\boldsymbol {f}}_{n+1}\right]^{-{\frac {1}{3}}}{\boldsymbol {f}}_{n+1}}$
Compute the volume preserving part of ${\textstyle {{\boldsymbol {b}}^{e}}^{\mathrm {trial} }}$ : ${\textstyle {{\overline {\boldsymbol {b}}}^{e}}_{n+1}^{\mathrm {trial} }={\overline {\boldsymbol {f}}}_{n+1}{{\overline {\boldsymbol {b}}}^{e}}_{n}{{\overline {\boldsymbol {f}}}_{n+1}}^{T}}$
Compute the deviatoric part of the ${\textstyle {\boldsymbol {\tau }}}$ : ${\textstyle \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }=\mu \mathrm {dev} \left[{{\overline {\boldsymbol {b}}}^{e}}_{n+1}^{\mathrm {trial} }\right]}$

CHECK FOR PLASTIC LOADING

Evaluate the Mises-Huber yield condition

a If

, no plastic loading is observed. Thus,

and

b If

, plastic loading is observed.

Setting

Compute the plastic multiplier ${\textstyle \Delta \gamma ={\dfrac {f_{n+1}^{\mathrm {trial} }/2{\overline {\mu }}}{1+H/3{\overline {\mu }}}}}$
Compute the unit tensor ${\textstyle {\boldsymbol {n}}={\dfrac {\mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }}{\lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert }}}$

Computation of the return mapping

Correct

:

Correct

:

EVALUATE THE STRESS IN CURRENT CONFIGURATION
${\textstyle {\boldsymbol {\tau }}_{n+1}=J_{n+1}U'(J_{n+1})+\mathrm {dev} {\boldsymbol {\tau }}_{n+1}}$
UPDATE THE INTERMEDIATE CONFIGURATION
${\textstyle {{\overline {\boldsymbol {b}}}^{e}}_{n+1}={\dfrac {\mathrm {dev} {\boldsymbol {\tau }}_{n+1}}{\mu }}+{\dfrac {1}{3}}\mathrm {tr} \left[{{\overline {\boldsymbol {b}}}^{e}}_{n+1}^{\mathrm {trial} }\right]{\boldsymbol {1}}}$

Algorithm. 3.1 Return Mapping algorithm.

Finally, the closed form expression for the algorithmic elasto-plastic moduli ${\textstyle {\boldsymbol {\mathrm {C} ^{ep}}}}$ is derived through the linearisation of the Kirchhoff stress tensor

(3.80)

According to the definition of the tangent modulus in current configuration

(3.81)

where ${\textstyle {\boldsymbol {g}}}$ denotes the metric tensor in the current configuration, the expression of ${\textstyle {\boldsymbol {\mathrm {C} }}^{\tau }}$ in the context of the Hyperelastic- ${\textstyle J_{2}}$ plastic law reads

{\boldsymbol {\mathrm {C} }}^{\tau ,ep}=2{\frac {\partial {\boldsymbol {\tau }}_{n+1}^{trial}}{\partial {\boldsymbol {g}}_{n+1}}}-{\boldsymbol {n}}\otimes 2{\overline {\mu }}2{\frac {\partial \Delta \gamma }{\partial {\boldsymbol {g}}_{n+1}}}-2{\overline {\mu }}\Delta \gamma 2{\frac {\partial {\boldsymbol {n}}}{\partial {\boldsymbol {g}}_{n+1}}}-2\Delta \gamma {\boldsymbol {n}}\otimes 2{\frac {\partial {\overline {\mu }}}{\partial {\boldsymbol {g}}_{n+1}}}

(3.82)

It is deduced that the first term of Equation 3.82 is given by the sum of Equations 3.51 and 3.52 and it reads

(3.83)

In what follows, the derivation of the three last terms is presented. By rewriting Equation 3.79 as

(3.84)

and deriving it with respect to ${\textstyle {\boldsymbol {g}}_{n+1}}$ , the derivative ${\textstyle {\dfrac {\partial \Delta \gamma }{\partial {\boldsymbol {g}}_{n+1}}}}$ is obtained

{\frac {\partial \Delta \gamma }{\partial {\boldsymbol {g}}_{n+1}}}=\left({\frac {1}{1+{\dfrac {H}{3{\overline {\mu }}}}}}\right)\left({\frac {1}{2}}{\frac {\partial \lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert }{\partial {\boldsymbol {g}}_{n+1}}}-\Delta \gamma {\frac {\partial {\overline {\mu }}}{\partial {\boldsymbol {g}}_{n+1}}}\right)

(3.85)

The derivative ${\textstyle {\dfrac {\partial {\overline {\mu }}}{\partial {\boldsymbol {g}}_{n+1}}}}$ is solved by firstly rewriting it in reference configuration with the use of Equation 3.63 as

{\frac {\partial {\overline {\mu }}}{\partial {\boldsymbol {C}}_{n+1}}}={\frac {\partial \left({\frac {1}{3}}\mu \left({\boldsymbol {C}}_{n}^{p^{-1}}:{\boldsymbol {C}}_{n+1}\right)J_{n+1}^{-{\frac {2}{3}}}\right)}{\partial {\boldsymbol {C}}_{n+1}}}={\frac {1}{3}}\mu \left({\frac {\partial J_{n+1}^{-{\frac {2}{3}}}}{\partial {\boldsymbol {C}}_{n+1}}}+J_{n+1}^{-{\frac {2}{3}}}{\frac {\partial \left({\boldsymbol {C}}_{n}^{p^{-1}}:{\boldsymbol {C}}_{n+1}\right)}{\partial {\boldsymbol {C}}_{n+1}}}\right)

(3.86)

and, then, by performing a push-forward transformation with ${\textstyle {\boldsymbol {F}}_{n+1}}$ , the derivative in current configuration is obtained

(3.87)

The derivative ${\textstyle {\dfrac {\partial \lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert }{\partial {\boldsymbol {g}}_{n+1}}}}$ is solved by introducing the following equation

(3.88)

where ${\textstyle {\overline {\boldsymbol {g}}}:=J^{-{\frac {2}{3}}}{\boldsymbol {g}}}$ and the derivative ${\textstyle {\dfrac {\partial \lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert }{\partial {\overline {\boldsymbol {g}}}_{n+1}}}}$ , according to [153], is defined as

{\frac {\partial \lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert }{\partial {\overline {\boldsymbol {g}}}_{n+1}}}={\frac {1}{2\lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert }}(2\mu \mathrm {tr} {\boldsymbol {b}}^{e}\lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert {\boldsymbol {n}}+2J^{\frac {2}{3}}\left(\lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert ^{2}{\boldsymbol {n}}^{2}\right)

(3.89)

By substituting this last equation in 3.88, the final expression is obtained

(3.90)

Having defined Equations 3.87 and 3.90, it is possible to write again Equation 3.85 as

{\frac {\partial \Delta \gamma }{\partial {\boldsymbol {g}}_{n+1}}}=\left({\frac {1}{1+{\dfrac {H}{3{\overline {\mu }}}}}}\right)\left({\frac {1}{2}}{\overline {\mu }}{\boldsymbol {n}}+{\frac {\lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert }{2{\overline {\mu }}}}\mathrm {dev} {\boldsymbol {n}}^{2}-{\frac {\Delta \gamma }{3}}\lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert {\boldsymbol {n}}\right)

(3.91)

The last derivative to be solved is

2{\frac {\partial {\boldsymbol {n}}}{\partial {\boldsymbol {g}}_{n+1}}}=2{\frac {\partial \left({\dfrac {\mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }}{\lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert }}\right)}{\partial {\boldsymbol {g}}_{n+1}}}={\frac {1}{\lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert }}\left({\frac {\partial \left(\mathrm {dev} {\boldsymbol {\tau }}_{n+1}^{trial}\right)}{\partial {\boldsymbol {g}}_{n+1}}}-{\boldsymbol {n}}\otimes 2{\frac {\partial \lVert \mathrm {dev} {\boldsymbol {\tau }}^{\mathrm {trial} }\rVert }{\partial {\overline {\boldsymbol {g}}}_{n+1}}}\right)

(3.92)

By substituting the expressions of Equations 3.87, 3.91 and 3.92 in Equation 3.82, the spatial constitutive tensor is finally obtained

(3.93)

where the coefficients ${\textstyle \beta _{1},\beta _{3}}$ and ${\textstyle \beta _{4}}$ are expressed as

(3.94)

(3.95)

(3.96)

(3.97)

${\textstyle {\boldsymbol {\mathrm {C} }}^{\mathrm {trial} }}$ and ${\textstyle {\boldsymbol {\mathrm {C} }}_{dev}^{\mathrm {trial} }}$ as in Equations 3.45 and 3.52, respectively.

When the use of a mixed formulation, with mean stress ( ${\textstyle p}$ ) and displacements ( ${\textstyle {\boldsymbol {u}}}$ ) as primary variables, might be required, the reader has to refer to Equation 3.44 for the definition of the Kirchhoff stress tensor and to Equation 3.93 for the definition of the spatial constitutive tensor, keeping in mind that its volumetric part, ${\textstyle {\boldsymbol {\mathrm {C} }}{\mathrm {trial} }_{vol}}$ , is obtained as prescribed by Equation 3.51.

3.3 Hyperelastic - Mohr-Coulomb plastic law

The ${\textstyle J_{2}}$ plastic law, presented in the previous section, is based on the theory which works fine for the modelling of plastic behaviour of metals. As it can be noted, this kind of theory is pressure insensitive since the yield limit does not depend on the mean stress, also graphically represented by Figure 3.1. For materials, such as soils and other granular materials, whose behaviour is pressure-dependent, the employment of the plastic law, presented in Section 3.2, might be inappropriate. In this regard, one of the most used plastic model in geotechnical engineering is the Mohr-Coulomb strength theory. This constitutive law is a phenomenological model, based on the fundamental assumption that the macroscopic plastic behaviour is the result of the microscopic mechanism of friction sliding between the single grains which compose the bulk. This concept is expressed by one of the most important failure criteria proposed by Coulomb in 1776

(3.98)

which states that the plastic flow starts when the state of stress on a specific plane exceeds the shear strength ( ${\textstyle \tau }$ ) which is a function of the normal stress ${\textstyle \sigma _{n}}$ , the material constants of cohesion ${\textstyle c}$ and internal friction angle ${\textstyle \phi }$ . By using the Mohr plane representation, where it is possible to visualize the shear stresses as function of the normal stresses, the Coulomb's failure criterion is shown in Figure 3.2.

Figure 3.2: Coulomb's failure criterion in Mohr's plane representation

According to this stress representation, Equation 3.98 can be seen as a failure envelope, which can be experimentally determined: the failure occurs when the Mohr's circle is just tangent to the failure envelope.

In this Section, a Mohr-Coulomb plastic law for finite strains is presented. For the derivation of the formulas which are the expressions of the stress return and elasto-plastic moduli, the theory presented by Simo in [155,156] is exploited. In these works, the main idea lies on modelling the elastic response by the use of principal stresses, which allows to extend the use of a small strain return mapping in stress space to the finite deformation regime. In Appendix A, the plastic flow rule within the multiplicative plastic framework is presented and the form in terms of Hencky strains is derived. According to this simplification, the following uncoupled form of the specific strain energy function is assumed

(3.99)

quadratic in the principal Hencky strains ${\textstyle {\boldsymbol {\epsilon }}^{e}}$ , defined as

(3.100)

with ${\textstyle {\boldsymbol {\lambda }}}$ the eigenvalues of the left Cauchy–Green deformation tensor ${\textstyle {\boldsymbol {b}}^{e}}$ , which can be calculated according to its spectral decomposition

(3.101)

where ${\textstyle {\boldsymbol {m}}^{A}}$ are the eigenbases associated with ${\textstyle {\boldsymbol {\lambda }}_{A}}$ . If an isotropic elastic response is assumed, by defining the spectral decomposition of the Kirchhoff stress tensor ${\textstyle {\boldsymbol {\tau }}}$ as

(3.102)

it can be observed that the eigenbasis ${\textstyle {\boldsymbol {m}}^{A}}$ of Equation 3.101 are the same of those of Equation 3.102.

With the specific strain energy function of Equation 3.99 and the aforementioned assumption of isotropy, the stress-strain relation in principal axes takes the form

(3.103)

with ${\textstyle \mathrm {a} =K{\boldsymbol {I}}\otimes {\boldsymbol {I}}+2\mu \left[\mathrm {I} -{\frac {1}{3}}{\boldsymbol {I}}\otimes {\boldsymbol {I}}\right]}$ being the Hencky elastic tensor.

As depicted in Figure 3.3, the Mohr-Coulomb criterion comprises six planes in principal stress space, forming six corners and a common vertex on the tension side of the hydrostatic axis.


(a)	(b)
Figure 3.3: Mohr Coulomb model in the principal stress space (a) and in $\pi$ plane (b)

If the principal stresses are rearranged as

(3.104)

the stresses are returned to only one of the six faces, the primary yield plane.

Following the reordering of principal stresses as described by Equation 3.104, the yield function and the plastic potential, in the case of Mohr-Coulomb plasticity, are respectively written as

(3.105)

(3.106)

where ${\textstyle \phi }$ is the angle of internal friction, ${\textstyle c}$ the cohesion and ${\textstyle \psi }$ the dilation angle. The expressions of Equations 3.105 and 3.106 in the principal stress space are represented by planes and, accordingly, they can be rewritten as

(3.107)

(3.108)

where ${\textstyle {\boldsymbol {a}}_{1}=\left[k\quad 0\quad -1\right]^{T}}$ and ${\textstyle {\boldsymbol {b}}_{1}=\left[m\quad 0\quad -1\right]^{T}}$ are the gradients, and the constants ${\textstyle k}$ and ${\textstyle m}$ are respectively defined as

(3.109)

(3.110)

3.3.1 The stress regions

In the case of the Mohr-Coulomb plastic law, four types of stress returns and constitutive matrices are possible.In Figure 3.4a the stress regions are shown: return to a yield plane (with condition of ${\textstyle f({\boldsymbol {\tau }}=0)}$ ), to the line which corresponds to triaxial compression ( ${\textstyle l_{1}}$ ), to the line which corresponds to triaxial tension ( ${\textstyle l_{2}}$ ) and to the apex point (point A). In order to be able to know to which region to apply the return, boundary planes between these regions need to be defined. In the case of a linear yield criterion, boundary planes in the principal stress space are planes (Figure 3.4b) and the solution of the problem is found by simply applying geometric arguments.


(a)	(b)
Figure 3.4: Stress regions (a) and boundary planes (b) in the principal stress space

By using the definition of a plane in the principal stress space

(3.111)

where ${\textstyle {\boldsymbol {n}}}$ is the normal to the plane and ${\textstyle \tau _{l}}$ a stress point laying on the plane, ${\textstyle p_{I-II}}$ and ${\textstyle p_{I-III}}$ are expressed as

(3.112)

(3.113)

with

(3.114)

and

(3.115)

the vectors direction of lines ${\textstyle l_{1}}$ and ${\textstyle l_{2}}$ , ${\textstyle {\boldsymbol {r}}_{1}^{p}}$ and ${\textstyle {\boldsymbol {r}}_{2}^{p}}$ the vectors direction of the plastic corrector.

By using geometric arguments, it is possible to identify the four stress regions without defining the planes ${\textstyle p_{II-IV}}$ and ${\textstyle p_{III-IV}}$ . For instance, by considering the parametric equation of a line in the principal stress space

(3.116)

where ${\textstyle t}$ is a parameter with unit of stress and ${\textstyle {\boldsymbol {\tau _{l}}}}$ a stress point on the line, the parametric equations of lines ${\textstyle l_{1}}$ and ${\textstyle l_{2}}$ are

(3.117)

(3.118)

where parameters ${\textstyle t_{1}}$ and ${\textstyle t_{2}}$ are defined in a way that at the apex point ${\textstyle t_{1}=t_{2}=0}$ ; thus, when the condition ${\textstyle t_{1}>0}$ and ${\textstyle t_{2}>0}$ are both satisfied the predictor stress falls in Region IV. Below, in Table 3.1 the conditions, which determine the stress regions and their corresponding return mapping, are listed.

Table. 3.1 Identification of stress regions
Conditions	Region	Type of return
${\textstyle p_{I-II}\geq {0}}$ and ${\textstyle p_{I-III}\leq {0}}$	I	$f({\boldsymbol {\tau }}=0)$
${\textstyle p_{I-II}<0}$ and ${\textstyle p_{I-III}<0}$	II	$l_{1}$
${\textstyle p_{I-II}>0}$ and ${\textstyle p_{I-III}>0}$	III	$l_{2}$
${\textstyle t_{1}>0}$ and ${\textstyle t_{2}>0}$	IV	$apex$

3.3.2 Stress update in principal stress space

In a general three-dimensional framework the development of the return mapping derivation and implementation in presence of singularities may result tedious. However, if isotropic yield criteria are considered, it is possible to reduce the dimension of the problem from six to three. Further, by taking advantage of considering an isotropic linear yield criterion and a perfect plastic law, for the implementation of the implicit integration scheme in principal stress space, the theory presented in [157] is followed. In this work, an efficient return algorithm, based on geometric arguments, is presented for infinitesimal deformation; as previously mentioned, if the specific strain energy function of Equation 3.99 is employed, the algorithmic procedure of [157] can be extended also to the case of finite strains in a straightforward way, as addressed in Appendix A. In the following, the stress update formulas and elasto-plastic tangent tensor are presented for each type of return. These expressions are, then, used in Algorithm 3.2 where the algorithmic procedure is described in the framework of a finite strain plastic model.

3.3.2.1 Return to a plane

In this respect, the evolution equation in finite strains in terms of Hencky strains, derived in Appendix A, is as follows

(3.119)

In addition, in order to obtain the expressions for the return to the yield surface, the following condition is considered

(3.120)

which states that in case of perfect plasticity the strain increment must be tangential to the yield surface.

By firstly substituting Equation 3.119 into Equation 3.103

(3.121)

and, then, Equation 3.121 in Equation 3.120, the following expression is obtained

(3.122)

After rearranging the terms of the last equation, the expression of ${\textstyle \Delta \gamma }$ is found

\Delta \gamma ={\frac {\left({\dfrac {\partial f_{n+1}}{\partial {\boldsymbol {\tau }}}}\right)^{T}:\mathrm {a} :{\boldsymbol {\epsilon }}^{e^{trial}}}{\left({\dfrac {\partial f_{n+1}}{\partial {\boldsymbol {\tau }}}}\right)^{T}:\mathrm {a} :{\dfrac {\partial g_{n+1}}{\partial {\boldsymbol {\tau }}}}}}

(3.123)

By substituting the expression of ${\textstyle \Delta \gamma }$ in Equation 3.121

{\boldsymbol {\tau }}_{n+1}=\mathrm {a} :\left({\boldsymbol {\epsilon }}^{e^{trial}}-{\dfrac {\left({\dfrac {\partial f_{n+1}}{\partial {\boldsymbol {\tau }}}}\right)^{T}:\mathrm {a} :{\boldsymbol {\epsilon }}^{e^{trial}}}{\left({\dfrac {\partial f_{n+1}}{\partial {\boldsymbol {\tau }}}}\right)^{T}:\mathrm {a} :{\dfrac {\partial g_{n+1}}{\partial {\boldsymbol {\tau }}}}}}{\frac {\partial g_{n+1}}{\partial {\boldsymbol {\tau }}}}\right)

(3.124)

it can be defined the elasto-plastic fourth order constitutive tensor ${\textstyle \mathrm {a} ^{ep}}$

\mathrm {a} ^{ep}=\mathrm {a} -{\frac {\left(\mathrm {a} :{\dfrac {\partial g_{n+1}}{\partial {\boldsymbol {\tau }}}}\right)\otimes \left(\mathrm {a} :{\dfrac {\partial f_{n+1}}{\partial {\boldsymbol {\tau }}}}\right)}{\left({\dfrac {\partial f_{n+1}}{\partial {\boldsymbol {\tau }}}}\right)^{T}:\mathrm {a} :{\dfrac {\partial g_{n+1}}{\partial {\boldsymbol {\tau }}}}}}

(3.125)

Moreover, according to Equation 3.124, it is found that the plastic corrector of ${\textstyle {\boldsymbol {\tau }}}$ reads

\mathrm {a} :\left({\frac {{\dfrac {\partial g_{n+1}}{\partial {\boldsymbol {\tau }}}}\left({\dfrac {\partial f_{n+1}}{\partial {\boldsymbol {\tau }}}}\right)^{T}:\mathrm {a} :{\boldsymbol {\epsilon }}^{e^{trial}}}{\left({\dfrac {\partial f_{n+1}}{\partial {\boldsymbol {\tau }}}}\right)^{T}:\mathrm {a} :{\dfrac {\partial g_{n+1}}{\partial {\boldsymbol {\tau }}}}}}\right)={\frac {f^{trial}\mathrm {a} :{\dfrac {\partial g_{n+1}}{\partial {\boldsymbol {\tau }}}}}{\left({\dfrac {\partial f_{n+1}}{\partial {\boldsymbol {\tau }}}}\right)^{T}:\mathrm {a} :{\dfrac {\partial g_{n+1}}{\partial {\boldsymbol {\tau }}}}}}=f^{trial}{\boldsymbol {r}}^{p}

(3.126)

with ${\textstyle {\boldsymbol {r}}^{p}}$ representing the direction of the plastic corrector in principal space and ${\textstyle f^{trial}}$ defined as

(3.127)

3.3.2.2 Return to a line

If it is found that the return has to be performed to a line, the parametric equation which defines a line in the principal stress space has to be considered (see Equation 3.116). By observing Equation 3.116, the direction vector of the line ${\textstyle {\boldsymbol {r}}^{l}}$ is given by the cross product of the perpendicular vector of the two adjacent planes

(3.128)

Similarly, the direction of the plastic potential line ${\textstyle {\boldsymbol {r}}_{g}^{l}}$ is defined by

(3.129)

In the return mapping to a line the plastic strain increment must be perpendicular to the potential line

(3.130)

By considering Equation 3.121, the condition of Equation 3.130 can be expressed as

(3.131)

Since ${\textstyle {\boldsymbol {\tau }}_{n+1}}$ has to belong to the line, the expression of Equation 3.116 is substituted in Equation 3.131

(3.132)

By rearranging the terms of the last equation, the expression for t is obtained

(3.133)

In the Mohr-Coulomb plastic law a plane is delimited by two lines expressed by Equations 3.117 and 3.118 and the corresponding potential directions are

(3.134)

(3.135)

With the definitions of Equations 3.114, 3.115, 3.134 and 3.135 in hand, it is possible to evaluate ${\textstyle t_{1}}$ and ${\textstyle t_{2}}$ with the use of Equation 3.133 and the updated stress ${\textstyle {\boldsymbol {\tau }}_{n+1}}$ laying either on line ${\textstyle l_{1}}$ or ${\textstyle l_{2}}$ , depending on the type of return.

In the definition of the elastoplastic fourth-order constitutive tensor ${\textstyle \mathrm {a} _{l}^{ep}}$ the following observation are made. Firstly, in the case of return to a line the updated stress lays on the line and the stress increment has the same direction of ${\textstyle {\boldsymbol {r}}^{l}}$ , thus, the elastic strain increment must have the direction

(3.136)

This means that ${\textstyle \mathrm {a} _{l}^{ep}}$ has to be singular with respect to the strain directions associated with both the yield planes that define the line, ${\textstyle {\boldsymbol {b}}_{1}}$ and ${\textstyle {\boldsymbol {b}}_{2}}$ and any linear combination of the two

(3.137)

where ${\textstyle \gamma _{1}}$ and ${\textstyle \gamma _{2}}$ are plastic multipliers. After these considerations the following system of equations is defined:

(3.138)

and the solution of it leads to the expression of ${\textstyle \mathrm {a} _{l}^{ep}}$

(3.139)

The expression of Equation 3.139 has only elements related to the principal stresses; in order to consider the shear stiffness, the elasto-plastic constitutive tensor in principal space is modified as follows

(3.140)

where ${\textstyle \mathrm {G} }$ reads

(3.141)

3.3.2.3 Return to a point

In the case of return to the apex, no calculation is needed since ${\textstyle {\boldsymbol {\tau }}={\boldsymbol {\tau }}_{A}}$ , defined as

(3.142)

With respect to the definition of the elasto-plastic fourth order constitutive tensor ${\textstyle \mathrm {a} _{apex}^{ep}}$ , this tensor has to be singular with respect to any direction in the principal stress space, i.e.

(3.143)

and the final expression which consider the shear stiffness, as well, reads

(3.144)

Initial data on material points: ${\boldsymbol {F}}_{n}$ , ${\boldsymbol {b}}_{n}^{e}$

OUTPUT of calculations:

,

UPDATE THE CURRENT CONFIGURATION

Compute the the current configuration: ${\textstyle {\boldsymbol {\varphi }}_{n+1}={\boldsymbol {\varphi }}_{n}+{\boldsymbol {u}}_{n}\circ {\boldsymbol {\varphi }}_{n}}$
Compute the relative deformation gradient: ${\textstyle {\boldsymbol {f}}_{n+1}={\boldsymbol {1}}+\nabla _{{\boldsymbol {x}}_{n}}{\boldsymbol {u}}_{n}}$
Compute the total deformation gradient in updated configuration: ${\textstyle {\boldsymbol {F}}_{n+1}={\boldsymbol {f}}_{n+1}{\boldsymbol {F}}_{n}}$

COMPUTE THE ELASTIC PREDICTOR

Compute the trial elastic left Cauchy-Green tensor of ${\textstyle {{\boldsymbol {b}}^{e}}^{\mathrm {trial} }={\boldsymbol {f}}_{n+1}{{\boldsymbol {b}}^{e}}_{n}{{\boldsymbol {f}}_{n+1}}^{T}}$
Compute the spectral decomposition of ${\textstyle {{\boldsymbol {b}}^{e}}^{\mathrm {trial} }=\sum \limits _{A=1}^{3}\lambda _{A}^{2}{\boldsymbol {m}}^{A}}$
Compute the logarithmic principal stretches ${\textstyle {\boldsymbol {\epsilon }}^{e,\mathrm {trial} }={\mathrm {ln} }(\lambda _{A})}$
Compute the principal trial Kirchhoff stresses ${\textstyle {\boldsymbol {\tau }}^{\mathrm {trial} }=\mathrm {a} {\boldsymbol {\epsilon }}^{e,\mathrm {trial} }}$

CHECK FOR PLASTIC LOADING

Evaluate the Mohr-Coulomb yield condition

a If

, no plastic loading is observed. Thus,

and

b If

, plastic loading is observed.

Define the type of return (see Section 3.3.1)
Applied the corresponding return stress and compute the elasto-plastic tangent moduli (see Section 3.3.2)
Correct ${\textstyle {\boldsymbol {\tau }}}$ : ${\textstyle {\boldsymbol {\tau }}_{n+1}={\boldsymbol {\tau }}^{\mathrm {trial} }-\Delta \gamma \mathrm {a} {\frac {\delta g({\boldsymbol {\tau }}^{\mathrm {trial} })}{\delta {\boldsymbol {\tau }}}}}$
Correct ${\textstyle {\boldsymbol {\epsilon }}^{e}}$ : ${\textstyle {\boldsymbol {\epsilon }}^{e}=\mathrm {a} ^{-1}{\boldsymbol {\tau }}_{n+1}}$

EVALUATE THE STRESS IN CURRENT CONFIGURATION

Transform

and elasto-plastic tangent moduli back to the original coordinate system

UPDATE THE INTERMEDIATE CONFIGURATION

Update

Algorithm. 3.2 Return Mapping algorithm.

3.3.3 The consistent elasto-plastic tangent moduli

As highlighted in [155,156], the exact closed-form linearisation of the return mapping algorithm produces a modified elasto-plastic moduli, referred to as the consistent algorithmic moduli, which compared to the classical elasto-plastic moduli, presented in the earlier sections, is able to restore the quadratic rate of convergence exhibited by Newton-like iterative methods.

Since the consistent tangent operator represents the instantaneous variation of the stress tensor ${\textstyle {\boldsymbol {\tau }}}$ (Equation 3.102) with respect to the strain tensor ${\textstyle {\boldsymbol {\epsilon }}}$ (Equation 3.100) and the updated expression of the stress tensor is a function dependent on the trial deformation state, the consistent tangent moduli has the following form

(3.145)

where the relation

(3.146)

is used.

The expression of the consistent tangent moduli is given by the sum of two terms. In the first term it can be found the elasto-plastic tangent moduli ${\textstyle \mathrm {a} ^{ep}}$ dependent on the specific plastic model and the structure of return mapping algorithm. In the case of a Mohr-Coulomb plastic model, the definition, depending on the type of return, can be found in Equations 3.126, 3.140 or 3.144. On the other hand, the tensor product ${\textstyle {\boldsymbol {m}}^{(A,trial)}}$ and the moduli ${\textstyle \mathrm {c} ^{A,trial}}$ are independent on the plastic model; with regards to ${\textstyle \mathrm {c} ^{A,trial}}$ , this tensor is dependent only on the specific strain energy function and it reflects the changing orientation of the spectral direction of ${\textstyle {\boldsymbol {\tau }}}$ . The closed-form of ${\textstyle \mathrm {c} ^{A,trial}}$ is derived in Appendix B and in the following the final spatial form is presented

(3.147)

where ${\textstyle \left(\mathrm {I} _{b}\right)={\frac {1}{2}}\left(b_{ac}b_{bd}+b_{ad}b_{bc}\right)}$ , ${\textstyle D_{A}=2\lambda _{A}^{4}-I_{1}\lambda _{A}^{2}+I_{3}\lambda _{A}^{-2}}$ , ${\textstyle {D'}_{A}=8\lambda _{A}^{3}-2I_{1}-2I_{3}\lambda _{A}^{-3}}$ , ${\textstyle I_{1}=\mathrm {tr} {\boldsymbol {C}}}$ , ${\textstyle I_{3}=J^{2}}$ and ${\textstyle \lambda _{A}}$ are the principal stretches defined as

(3.148)

with ${\textstyle \lambda _{A}^{2}}$ are the eigenvalues of ${\textstyle {\boldsymbol {b}}^{e^{trial}}}$ , according to Equation 3.101. In Equation 3.147 the superscript ${\textstyle trial}$ has been omitted for the tensor ${\textstyle \mathrm {I} _{b}}$ , ${\textstyle {\boldsymbol {b}}}$ and ${\textstyle {\boldsymbol {m}}^{(A)}}$ .

When the use of a mixed formulation, with mean stress ( ${\textstyle p}$ ) and displacements ( ${\textstyle {\boldsymbol {u}}}$ ) as primary variables, might be required, the reader has to consider the volumetric part of the Kirchhoff stress tensor defined as

(3.149)

and the volumetric part of the spatial constitutive tensor expressed by Equation 3.51.

4 Irreducible formulation

The simulation of granular flow problems, which involve large deformation and complex history-dependent constitutive laws, is of paramount importance in several industrial and engineering processes. Particular attention has to be paid to the choice of a suitable numerical technique such that reliable results can be obtained. In Chapter 2 a review of several numerical techniques is presented in order to individuate the most suited methods for the numerical analysis of granular flows under both quasi-static and inertial regime. For the achievement of such purpose, it is found that the Material Point Method (MPM) and the Galerkin Meshfree Method (GMM) might be two good candidates, as previously shown. In this Chapter, firstly, an irreducible formulation is presented. The displacement-based formulation, defined in a Update Lagrangian framework under finite strain regime, is implemented in both the MPM and GMM strategy. Afterwards, these two numerical strategies, already presented in Chapter 2, are verified against classical benchmarks in solid and geo-mechanics. The aim is to assess their validity in the simulation of cohesive-frictional materials, both in static and dynamic regimes and in problems dealing with large deformations. The vast majority of MPM techniques in the literature is based on some sort of explicit time integration. The techniques proposed in the current work, on the contrary, are based on implicit approaches which can also be easily adapted to the simulation of static cases. Although both methods are able to give a good prediction, it is observed that, under very large deformation of the medium, GMM lacks in robustness due to its meshfree nature, which makes the definition of the meshless shape functions more difficult and expensive than in MPM. On the other hand, the mesh-based MPM demonstrates to be more robust and reliable for extremely large deformation cases.

4.1 Governing equations

Let us consider the body ${\textstyle {\mathcal {B}}}$ which occupies a region ${\textstyle \Omega }$ of the three-dimensional Euclidean space ${\textstyle {\mathcal {E}}}$ with a regular boundary ${\textstyle \partial \Omega }$ in its reference configuration. A deformation of ${\textstyle {\mathcal {B}}}$ is defined by a one-to-one mapping

(4.1)

that maps each point p of the body ${\textstyle {\mathcal {B}}}$ into a spatial point ${\textstyle {\boldsymbol {x}}}$

(4.2)

which represents the location of p in the deformed configuration of ${\textstyle {\mathcal {B}}}$ . The region of ${\textstyle {\mathcal {E}}}$ occupied by ${\textstyle {\mathcal {B}}}$ in its deformed configuration is denoted as ${\textstyle \varphi \left(\Omega \right)}$ .

The problem is governed by mass and linear momentum balance equations

${\frac {D\rho }{Dt}}+\rho \nabla \cdot {\boldsymbol {v}}=0\qquad {\textrm {in}}\quad \varphi (\Omega )$	(4.3.a)
$\rho {\boldsymbol {a}}-\nabla \cdot {\boldsymbol {\sigma }}=\rho {\boldsymbol {b}}\qquad {\textrm {in}}\quad \varphi (\Omega )$	(4.3.b)

where ${\textstyle \rho }$ is the mass density, ${\textstyle {\boldsymbol {a}}}$ is the acceleration, ${\textstyle {\boldsymbol {v}}}$ is the velocity, ${\textstyle {\boldsymbol {\sigma }}}$ is the symmetric Cauchy stress tensor and ${\textstyle {\boldsymbol {b}}}$ is the body force. Acceleration and velocity are, by definition, the material derivatives of the velocity, ${\textstyle {\boldsymbol {v}}}$ , and the displacement, ${\textstyle {\boldsymbol {u}}}$ , respectively. For a compressible material the conservation of mass is satisfied by

(4.4)

where ${\textstyle \rho _{0}}$ is the density in the undeformed configuration and ${\textstyle det({\boldsymbol {F}})}$ is the determinant of the total deformation gradient ${\textstyle {\boldsymbol {F}}:=d{\boldsymbol {x}}/d{\boldsymbol {X}}}$ with ${\textstyle \mathbf {x} }$ and ${\textstyle \mathbf {X} }$ representing the current and initial position, respectively. Equation 4.4 holds at any point, and in particular at the sampling points where the equation is written, e.g. the material points. Thermal effects are not considered in the present work, so the energy balance is considered implicitly fulfiled.

The balance equations are solved numerically in a three-dimensional region ${\textstyle \Omega \subseteq {\mathcal {R}}^{3}}$ , in the time range ${\textstyle t\in [0,T]}$ , given the following boundary conditions on the Dirichlet ( ${\textstyle \varphi (\partial \Omega _{D})}$ ) and Neumann boundaries ( ${\textstyle \varphi (\partial \Omega _{N})}$ ), respectively

${\boldsymbol {u}}={\boldsymbol {\overline {u}}}\quad {\textrm {on}}\quad \varphi (\partial \Omega _{D})$	(4.5.a)
${\boldsymbol {\sigma }}\cdot {\boldsymbol {n}}={\boldsymbol {\overline {t}}}\quad {\textrm {on}}\quad \varphi (\partial \Omega _{N})$	(4.5.b)

where ${\textstyle {\boldsymbol {n}}}$ is the unit outward normal.

In order to fully define the Boundary Value Problem a stress-strain relation, like those ones defined in Chapter 3, is needed.

4.2 Weak form

In Section 4.1 the strong form of the problem has been defined. In this section, the weak form is derived, following the formulation explained in [148], a displacement-based finite element procedure.

Let the displacement space ${\textstyle {\mathcal {V}}\in [\mathbf {\mathbf {H} } ^{1}({\mathcal {B}})]^{d}}$ be the space of vector functions whose components and their first derivatives are square-integrable, the integral form of the problem is

(4.6)

where ${\textstyle {\boldsymbol {w}}}$ is an arbitrary test function, such that ${\textstyle {\boldsymbol {w}}=\left\lbrace {\boldsymbol {w}}\in {\mathcal {V}}\mid {\boldsymbol {w}}={\boldsymbol {0}}\,{\textrm {on}}\,\varphi (\partial \Omega _{D})\right\rbrace }$ , ${\textstyle dv}$ is the differential volume and ${\textstyle da}$ the differential boundary surface. By integrating by parts, applying the divergence theorem and considering the symmetry of the stress tensor, the following expression is obtained

(4.7)

Under the assumption that the stress tensor is a function of the current strain only

(4.8)

the problem is reduced to find a kinematically admissible field ${\textstyle {\boldsymbol {u}}}$ that satisfies

(4.9)

where ${\textstyle G}$ is the virtual work functional defined as

(4.10)

4.3 Linearisation of the spatial weak formulation

In this work the boundary value problems (BVP) is characterized by both geometrical and material non-linearity. When a non-linear BVP is considered, the discretisation of the weak form results in a system of non-linear equations; for the solution of such a system, a linearisation is, therefore, needed. The most used and known technique is the Newton-Raphson's iterative procedure, which makes use of directional derivatives to linearise the non-linear equations. The virtual work functional of Equation 4.10 is linearised with respect to the unknown ${\textstyle {\boldsymbol {u}}}$ , using an arbitrary argument ${\textstyle {\boldsymbol {u}}^{*}}$ , which is chosen to be the last known equilibrium configuration. The linearised problem is to find ${\textstyle \delta {\boldsymbol {u}}}$ such that

(4.11)

where ${\textstyle L}$ is the linearised virtual work function and

(4.12)

is the directional derivative of ${\textstyle G}$ at ${\textstyle {\boldsymbol {u}}^{*}}$ in the direction of ${\textstyle \delta {\boldsymbol {u}}}$ , given by

(4.13)

Under the assumption of conservative external loads, only the terms related to the internal and inertial forces are dependent on the deformation. Using the following definitions

(4.14)

where ${\textstyle {\boldsymbol {\epsilon }}^{*}=\nabla ^{S}({\boldsymbol {u}}^{*})}$ is the strain field at ${\textstyle {\boldsymbol {u}}^{*}}$ and ${\textstyle {\boldsymbol {u}}(\gamma )={\boldsymbol {u}}^{*}+\gamma \delta {\boldsymbol {u}}}$ , the directional derivative ${\textstyle DG({\boldsymbol {u}}^{*},{\boldsymbol {w}})[\delta {\boldsymbol {u}}]}$ reduces to

(4.15)

which can be split in a static and dynamic contribution.

Under the assumption of finite strains and adopting an Updated Lagrangian kinematic framework, the expression of the directional derivative (Equation 4.15) should be derived in spatial form. A common way to do that consists in linearising the material weak form and in doing a push-forward operation to recover the spatial form [148]. Therefore, the linearisation of the weak form derived with respect to the initial configuration reads:

(4.16)

where ${\textstyle \nabla _{X}}$ and ${\textstyle \nabla _{x}}$ are the material and spatial gradient operator, respectively, ${\textstyle {\boldsymbol {S}}}$ is the Second Piola Kirchhoff stress tensor, ${\textstyle \mathbb {C} }$ is the fourth order incremental constitutive tensor and ${\textstyle dV}$ is the differential volume element in the underformed configuration. The linearisation of the weak form with respect to the current configuration can be derived by pushing-forward the linearisation of Equation 4.16. The first term can be directly written in terms of the Kirchhoff stress ${\textstyle {\boldsymbol {\tau }}={\boldsymbol {F}}{\boldsymbol {S}}{\boldsymbol {F}}^{T}}$ as

(4.17)

and using this standard identity ${\textstyle \nabla _{x}a=\nabla _{X}a{\boldsymbol {F}}^{-1}}$ , Equation 4.17 can be written as

(4.18)

The second integral of Equation 4.16 can be re-written as:

(4.19)

adopting the transformation of the fourth order incremental constitutive tensor ${\textstyle \mathbb {C} }$ in Voigt notation [148]:

(4.20)

where lowercase indexes are referred to the incremental constitutive tensor relative to the Kirchhoff stress, while uppercase indexes to the incremental constitutive tensor relative to the Second Piola Kirchhoff stress.

With these transformations, the linearisation of the static contribution at the current configuration is

(4.21)

Considering the definition of the determinant of the deformation gradient:

(4.22)

the following relations hold

(4.23)

(4.24)

where ${\textstyle {\boldsymbol {\sigma }}}$ and ${\textstyle {\boldsymbol {\tau }}}$ are the Cauchy and Kirchhoff stress tensor, respectively, and ${\textstyle {\overline {\widehat {\mathbb {C} }}}}$ is the incremental constitutive tensor relative to the Cauchy stress. Equation 4.16 can now be re-written in the current configuration as

(4.25)

Equation 4.25 represents the linearisation of the spatial weak formulation, also known as the Updated Lagrangian formulation, since the deformation state ${\textstyle {\boldsymbol {u}}^{*}}$ is continuously updated during the non-linear incremental solution procedure, e.g. the Newton Raphson's method.

4.4 Spatial Discretisation

Let ${\textstyle {\mathcal {V}}_{h}}$ be a finite element space to approximate ${\textstyle {\mathcal {V}}}$ . The problem is now finding ${\textstyle \mathbf {u} _{h}\in {\mathcal {V}}_{h}}$ such that

(4.26)

or using Equation 4.25

(4.27)

Let us assume to discretise the continuum body ${\textstyle {\mathcal {B}}}$ by a set of ${\textstyle n_{p}}$ material points and to assign a finite volume of the body ${\textstyle \Omega _{p}}$ to each of those material points. Thus, the geometrical representation ( ${\textstyle {\mathcal {B}}_{h}}$ ) of ${\textstyle {\mathcal {B}}}$ reads

(4.28)

and with this approximation the integrals of the weak form can be written as

(4.29)

For the computation of the linearised system of equations, an integration is necessary over the volume occupied by each material point ${\textstyle \Omega _{p}}$ . By using the spatial discretisation defined in Equation 4.28, the linearised system of equations (see Equation 4.27) is rewritten as

\bigcup _{p=1}^{n_{p}}\int _{\Omega _{p}}\left(\left\lbrace \nabla _{x}\delta {\boldsymbol {u}}_{h}{\boldsymbol {\sigma }}\cdot \nabla _{x}{\boldsymbol {w}}_{h}+\nabla ^{S}{\boldsymbol {w}}_{h}:{\overline {\widehat {\mathbb {C} }}}[\nabla ^{S}\delta {\boldsymbol {u}}_{h}]+\rho {\frac {d{\boldsymbol {a}}_{h}}{d{\boldsymbol {u}}_{h}}}\cdot {\boldsymbol {w}}_{h}[\delta {\boldsymbol {u}}_{h}]\right\rbrace \right)d\Omega _{p}

(4.30)

and by exploiting the finite element approximation with particle integration the final discretised form is obtained

\bigcup _{p=1}^{n_{p}}\sum _{I=1}^{n}\sum _{K=1}^{n}{\boldsymbol {w}}_{I}^{T}\left(\left(\nabla _{x}N_{I}\right)^{T}{\boldsymbol {\sigma }}\left(\nabla _{x}N_{K}\right)\mathbf {I} +\mathbf {B} _{I}^{T}\mathbf {D} \mathbf {B} _{K}+{\frac {N_{I}\rho N_{K}}{\beta \Delta t^{2}}}\mathbf {I} \right)V_{p}\delta {\boldsymbol {u}}_{K}

(4.31)

where ${\textstyle I}$ and ${\textstyle K}$ are the indexes of the finite element's nodes, ${\textstyle \nabla _{x}N_{I}}$ is the spatial gradient of the shape function evaluated at node ${\textstyle I}$ , ${\textstyle \mathbf {D} }$ is the matrix form of the incremental constitutive tensor ${\textstyle {\overline {\widehat {\mathbb {C} }}}}$ , ${\textstyle V_{p}}$ is the volume relative to a single material point, ${\textstyle A_{l}}$ is the surface and ${\textstyle \mathbf {B} _{I}}$ is the deformation matrix relative to node ${\textstyle I}$ , expressed here for a 2D problem as:

\mathbf {B} _{I}={\begin{bmatrix}{\displaystyle {\frac {\partial N_{I}}{\partial x}}}&0\\0&{\displaystyle {\frac {\partial N_{I}}{\partial y}}}\\{\displaystyle {\frac {\partial N_{I}}{\partial y}}}&{\displaystyle {\frac {\partial N_{I}}{\partial x}}}\end{bmatrix}}

(4.32)

The left hand side of Equation 4.31 is given by three addends multiplied by the increment of the unknowns. The first one is commonly known as the geometric stiffness matrix

(4.33)

while the second term is known as the material stiffness matrix

(4.34)

and their sum represents the static contribution to the tangent stiffness matrix

(4.35)

The dynamic component is given by

(4.36)

Finally the tangent stiffness matrix is given by

(4.37)

and represents the submatrix relative to one node of the discretisation with dimension ${\textstyle \left[n_{dof}\times n_{dof}\right]}$ , where ${\textstyle n_{dof}}$ is the number of degrees of freedom of a single node. This matrix can be considered as the Jacobian matrix of the right hand side of Equation 4.31, i.e., the residual ${\textstyle \mathbf {R_{I}} }$ . Equation 4.31 can be rewritten in compact form as

(4.38)

4.5 Numerical verification

In this section three benchmark tests are considered for the comparison of the MPM and GMM formulations. Firstly, the static analysis of a 2D cantilever beam subjected to its self-weight is analysed and a mesh convergence study is performed. Secondly, the rolling of a rigid disk on inclined plane is studied. Finally, a cohesive-soil column collapse is analysed. All the numerical experiments have been performed on a PC with one Intel(R) Core(TM) i7-4790 CPU at 3.60GHz.

4.5.1 2D cantilever beam. Static analysis

The static analysis of a 2D cantilever beam subjected to its self-weight under the assumption of plain strain is presented. The cantilever beam has a length ${\textstyle l=8m}$ and a square cross section of unit side ( ${\textstyle b=h=1m}$ ) (Figure 4.1). The beam is modelled with a hyperelastic material (presented in Section 3.1): the density is ${\textstyle \rho =1000kg/m^{3}}$ , the Young's modulus is ${\textstyle E=90MPa}$ and the Poisson's ratio is ${\textstyle \nu =0}$ . The results obtained with the MPM and GMM algorithms are compared with a standard FEM code using the same UL formulation.

Figure 4.1: Static 2D cantilever analysis: geometry

A mesh convergence study is carried out adopting five different mesh sizes, ${\textstyle h}$ = 0.5, 0.25, 0.125, 0.0625 and 0.01 ${\textstyle m}$ , respectively. Quadrilateral elements are used in FEM, MPM and GMM with four integration points per cell (in the case of MPM and GMM the integration points coincide with the material points). In GMM, the mesh is only initially used for the creation of the material points and then deleted. Regarding the spatial search and the evaluation of the shape functions in GMM, a search radius ${\textstyle R={\sqrt {2h^{2}}}}$ , dilation parameters ${\textstyle R_{eff}=R/2}$ and ${\textstyle \gamma =1.8}$ are adopted in GMM-MLS and GMM-LME, respectively. Under the assumption of linear regime, the vertical deflection at point A of the free edge can be evaluated analytically according to Timoshenko [158] as:

(4.39)

where ${\textstyle g}$ is the gravity acceleration, ${\textstyle I={\dfrac {bh^{3}}{12}}}$ the inertia of the beam section and ${\textstyle A_{s}={\dfrac {5}{6}}A}$ the reduced cross section area due to the shear effect. However, the solution is computed under the assumption of non-linearity and, as benchmark solution, the deflection evaluated through the finest mesh is considered. This value is ${\textstyle \delta =-0.67433m}$ and is equally reached by all the methods.

Figures 4.2 and 4.3 compare the solutions obtained with an Updated Lagrangian FEM, MPM, GMM-MLS and GMM-LME code, respectively, in terms of vertical displacement and Cauchy stress along the horizontal direction. One can observe that the results are in good agreement for all the methods.


(a) FEM code	(b) MPM code

(c) GMM-MLS code	(d) GMM-LME code
Figure 4.2: Static cantilever. Displacement along y-direction


(a) FEM code	(b) MPM code

(c) GMM-MLS code	(d) GMM-LME code
Figure 4.3: Static cantilever. Cauchy stress along x-direction

A convergence study is performed to analyse the accuracy of MPM and GMM in comparison with the UL-FEM. The error is evaluated as

(4.40)

where ${\textstyle u_{num}}$ is the numerical solution measure at point A (see Figure 4.1). Figure 4.4 depicts the error evolution in function of the inverse of the mesh size ${\textstyle h}$ . It is demonstrated that all the methods have a quadratic rate of convergence. In particular, the UL-FEM, MPM and GMM-MLS error curves coincide. Regarding the error, evaluated with the GMM-LME algorithm, the quadratic rate is maintained, but the curve is shifted a bit upwards, which makes this technique less accurate than GMM-MLS in the benchmark case studied.

Figure 4.4: Static cantilever. Convergence analysis

4.5.2 Rolling of a rigid disk on an inclined plane

The second benchmark test is a rigid disk rolling without slipping on an inclined plane. The geometry of the problem is depicted in Figure 4.5. The disk is made of a hyperelastic material (presented in Section 3.1): the density is ${\textstyle \rho =7800kg/m^{3}}$ , the Young's modulus is ${\textstyle E=200MPa}$ and the Poisson's ratio is ${\textstyle \nu =0.3}$ .

Draft Samper 987121664 5803 rolling geometry.png

Figure 4.5: Rolling disk. Geometry

This test is chosen for an objective assessment of the robustness of the MPM and GMM algorithm. The rolling on the plane implies a contact between the nodes belonging to the inclined plane and the nodes belonging to the disk. In a UL-FEM code a contact algorithm would be necessary to set this boundary condition. On the contrary, by using either MPM or GMM, the contact is implicitly caught. The analytical acceleration ( ${\textstyle a}$ ) can be computed imposing the equilibrium of momentum at the contact point ${\textstyle P}$

(4.41)

where ${\textstyle g}$ is gravity and ${\textstyle \theta }$ the angle of the inclined plane. Integrating over time the acceleration, velocity and displacement projected on the x-axis can be obtained as a function of time

(4.42)

(4.43)

For the study of this test case, the analytical solution of Equation 4.43 is used for the assessment of the absolute error obtained with MPM, GMM-MLS and GMM-LME, evaluated as

(4.44)

where ${\textstyle t_{i}}$ is the time where the numerical result is calculated. As a mesh-based and a meshless techniques are compared in a dynamic test, for a more objective comparison, the error is analysed along with the total computational time, needed to finalize the simulation.

A triangular mesh with mesh size ${\textstyle h=0.01m}$ is used for MPM and GMM simulations. In both techniques the same initial distribution of material points is used, which counts for three initial particles for cell. Regarding the GMM-MLS the approximants are constructed by adopting a search radius ${\textstyle R=1.5{\sqrt {2h^{2}}}}$ and a dilation parameter ${\textstyle R_{eff}={\sqrt {2h^{2}}}}$ . In GMM-LME the basis functions are evaluated using a search radius ${\textstyle R={\sqrt {2h^{2}}}}$ and three values of dilation parameter ${\textstyle \gamma =0.8,1.8,2.8}$ . All the numerical tests are repeated for three different time steps with ${\textstyle \Delta t_{1}=2\Delta t_{2}=4\;\Delta t_{3}}$ .

Table 4.1 shows the results of the analysis, in terms of errors and computational times, performed through MPM, GMM-MLS and GMM-LME.

Table. 4.1 Rolling disk. Absolute errors and computational times.
	$\Delta t_{1}$		$\Delta t_{2}$		$\Delta t_{3}$
	$error[m]$	$t_{comp}[s]$	$error[m]$	$t_{comp}[s]$	$error[m]$	$t_{comp}[s]$
MPM	2.34	232.72	1.27	472.92	0.91	894.91
GMM-MLS	0.84	264.78	0.26	517	0.20	981.82
GMM-LME ${\textstyle \gamma =0.8}$	1.37	234.32	0.07	460.73	0.06	971.29
GMM-LME ${\textstyle \gamma =1.8}$	0.9	237.22	0.23	460.50	0.07	1005.21
GMM-LME ${\textstyle \gamma =2.8}$	0.52	232.42	0.07	466.63	0.06	1038.70

Regarding the absolute errors, it can be observed that, for a given computational cost, GMM is generally more accurate than MPM, because of the use of smooth basis functions which provide a better approximation of the unknown variables. In particular GMM-LME presents smaller errors in comparison to GMM-MLS. In all the three cases considered (with ${\textstyle \gamma =0.8,1.8,2.8}$ ) the errors converge to a unique value at the same computational time, while in the case of GMM-MLS, the advantage of using higher order elements is lost for the smallest delta time. Regarding MPM, it is established that to achieve the same order of accuracy of GMM a higher computational time must be expected, due to either a finer discretisation in space or in time. However, it is worth highlighting that GMM is much more time consuming than MPM, showing an increment of computational time of ${\textstyle 10\%}$ in the case of GMM-MLS and from ${\textstyle 8.5\%}$ up to ${\textstyle 16\%}$ in the case of GMM-LME.

In this example, some essential conclusions can be drawn. In Section 4.5.1 it was observed that in a static case the rate of convergence is the same for all the methods under analysis, but the accuracy of GMM-MLS and MPM is better than the GMM-LME. On the contrary, in a dynamic case with a contact problem, the result is overturned. In fact a better behaviour is noted if LME approximants are employed.

In Figures 4.6a, 4.6b and 4.6c the distribution of the module of velocity field within the disk is shown for the test case solved by means of the MPM, GMM-MLS and GMM-LME, respectively. As expected, the minimum velocity is localized in the region of the disk close to the contact point with the inclined plane; while maximum velocity is observed on the opposite part of the disk.

(a)

(b)

(c)

Figure 4.6: Rolling disk. Absolute velocity field in test case solved with MPM (a), GMM-MLS (b) and GMM-LME with

(c)

4.5.3 Cohesive soil column collapse

The third example is the simulation of a soil column collapse. The column is modelled with a cohesive-frictional material, defined by a cohesion ${\textstyle c=5kPa}$ , a friction angle ${\textstyle \phi =25^{\circ }}$ , an elastic bulk modulus ${\textstyle K=1.5MPa}$ and a density ${\textstyle \rho =1850kg/m^{3}}$ . In the current work the Mohr-Coulomb plastic law in finite strains with implicit integration scheme in principal stress space, presented in Section 3.3, is employed.

This test has been chosen for the assessment of the robustness of MPM and GMM when the body undergoes really large deformation. The results are compared with the work of [56], where a Smooth Particle Hydrodynamics method (SPH) is applied to geotechnical problems.

The initial geometry and the boundary conditions are described by Figure 4.7.

Figure 4.7: Granular column collapse. Geometry

Quadrilateral elements with an initial distribution of four material points per cell are used in the simulations. Two different mesh sizes are considered: ${\textstyle h_{1}=0.05\;m}$ (Mesh1) and ${\textstyle h_{2}=0.025\;m}$ (Mesh2). In GMM the basis functions are evaluated using an initial search radius ${\textstyle R={\sqrt {2h^{2}}}}$ , a dilation parameter ${\textstyle R_{eff}=0.5{\sqrt {2h^{2}}}}$ and ${\textstyle \gamma =1.8}$ , in GMM-MLS and GMM-LME, respectively. In this particular case, the procedure for the evaluation of the basis functions in MLS and LME technique has been modified to avoid the creation of a non-convex hull of nodes which might lead to an incorrect set of approximants. This is required because the column is subjected to extremely large deformations. While in the previous examples a constant radius was used for the definition of the cloud of nodes surrounding a material point, in the current example a variable radius is adopted to guarantee a minimum number of nodes in each connectivity. In the case of LME, as a Newton iterative procedure is used for the evaluation of the shape functions, a measure of the goodness of the solution is represented by the condition number ${\textstyle k(A)}$ of the Hessian matrix ${\textstyle A}$ , defined in [145]. If ${\textstyle k(A)}$ exceeds a user-defined tolerance, the LME algorithm is repeated considering the old connectivity plus an additional node, chosen as the next node closer to the material point. In the case of MLS, it has been sufficient to impose a minimum number of six nodes in each cloud of nodes. In Figure 4.8 a comparison of the column deformation at different representative time instants is shown. The SPH model taken from [56] predicts a higher final run-out of the column collapse, while the final configurations at time ${\textstyle 2.0s}$ of MPM, GMM-LME and GMM-MLS are almost coincident using Mesh1 and Mesh2. It is worth highlighting that GMM-MLS and MPM results of Figure 4.8 are always in good agreement. However, this is not the case if the evolution of the equivalent plastic strains is observed (see Figures 4.9 and 4.10). In the case of GMM-LME, an improvement of the results is noted by using the finer mesh (Mesh2) in terms of displacements (Figure 4.8b) and equivalent plastic strains distribution (Figure 4.11b). Regarding MPM, it is proved that a good approximation can be obtained using both meshes.


(a) Mesh 1	(b) Mesh 2
Figure 4.8: Soil column collapse. Configurations of the column at different representative time instants.


(a) Mesh 1	(b) Mesh 2
Figure 4.9: Soil column collapse. Distribution of equivalent plastic strains for different representative time instants in MPM results.


(a) Mesh 1	(b) Mesh 2
Figure 4.10: Soil column collapse. Distribution of equivalent plastic strains for different representative time instants in GMM-MLS results.


(a) Mesh 1	(b) Mesh 2
Figure 4.11: Soil column collapse. Distribution of equivalent plastic strains for different representative time instants in GMM-LME results.

In this example, the capability of handling history-dependent materials, such as cohesive-frictional materials, is verified for both methods. It is noted that, in MPM large deformations can be naturally tracked without modifying the algorithm and accurate results are obtained also using the coarser mesh. In GMM, despite the remarkable features highlighted in the previous benchmark tests, when the continuum undergoes extremely large deformations, special care should be taken in the definition of the MLS and LME approximants. In this regard a lack of robustness of the GMM algorithm is observed due to the impossibility of guaranteeing a correct evaluation of the shape functions during the whole deformation process without an ad hoc modification of the procedure for the definition of the connectivity. Thus, for the solution of this example a correction of the algorithm has been performed and verified to work properly, albeit an increase in the computational time is registered. The establishment of a more general procedure is left for future work.

4.6 Discussion

In this Chapter, two particle methods: a Material Point Method and a Galerkin Meshless Method are tested and compared to assess their capabilities in solving large displacement and large deformation problems. A variational displacement-based formulation, based on an Updated Lagrangian description, is presented and its derivation is described in detail.

A comparison of MPM and GMM is performed through three benchmark tests and the methods are assessed in terms of accuracy, computational time and robustness. The first example is a static cantilever beam. A convergence analysis is performed and all the techniques have a quadratic convergence rate (compared to a FEM code). Secondly, the dynamic test of a rolling disk on an inclined plane is considered. The robustness of MPM and GMM in dealing with contact between two rigid bodies is tested and an analysis in terms of computational time and error is performed. It is found that GMM, in dynamic cases, has a higher accuracy than MPM, despite a higher computational cost. This is because in MPM linear basis functions are considered, while in GMM smooth basis functions are computed allowing to obtain a superior approximation of the unknown variables. As a last example, a cohesive soil column collapse is analysed. In this case, it is assessed the robustness of both methods when the continuum undergoes extremely large deformation. Firstly, it is demonstrated that MPM and GMM can be easily coupled with local plastic laws. Furthermore, it is noted that MPM leads to more accurate results and the algorithm does not need to be modified in a large deformation case. On the contrary, in GMM, a modification of the algorithm has to be considered to avoid the formation of non-convex hull of nodes when the connectivity is defined. Nonetheless, in spite of this modification, a discrepancy in the results is noted, by using either the MLS or the LME technique.

In conclusion, the standard version of MPM represents a good choice to handle problems involving history-dependent materials and large deformations. Regarding GMM, the accuracy of the solution strictly depends on the chosen basis functions. If large deformation of the continuum is not taken into account, this method could be preferred to MPM due to its remarkable feature in getting accurate results at a limited computational time. However, under finite strains regime, independently on the material to model, the construction of a connectivity in the meshless method becomes more complex and, at least to the authors’ knowledge, a general methodology is still missing to properly define a correct connectivity under any deformation condition. Thus, despite the promising features of this approach, an improvement in the robustness of the GMM algorithm is needed to obtain more accurate and reliable solutions in large deformation and failure problems, leaving the MPM, currently, the most suited numerical strategy for the analysis of granular flows.

5 Mixed formulation

In the field of granular flow modelling, there might be some cases where the granular matter undergoes undrained conditions. This corresponds to a situation where the bulk modulus K is much higher than the shear modulus G, and shearing deformations will be more important than dilation or compression in the overall response of the body. In this case, the behaviour of the body is usually approximated by assuming it to be incompressible. In the field of Finite Element Analysis (FEA) it is well established that results may suffer from volumetric locking issues, which can be detrimental for the solution itself when an irreducible formulation is employed. In this Chapter, the numerical strategy of a stabilized mixed formulation for the solution of non-linear solid mechanics problems in nearly-incompressible conditions is presented. The proposed mixed formulation, with displacement and pressure as primary variables, is implemented in the implicit MPM strategy, whose algorithm has been previously described in Chapter 2. The mixed formulation is tested through classical benchmarks in solid mechanics where a hypereleastic Neo-Hookean and a J2-plastic laws are employed. Further, the stabilized mixed formulation is compared with a displacement-based formulation, described in Chapter 4 to demonstrate how the proposed approach gets better results in terms of accuracy, not only when incompressible materials are simulated, but also in the case of compressible ones.

5.1 Introduction

The solution of solid mechanics problems in large displacement and large deformation regime, dealing with incompressible or nearly incompressible materials, is a topic of paramount importance in the computational mechanics community since many engineering problems present such conditions. It is well known that overly stiff numerical solutions appear when Poisson's ratio ${\textstyle \nu }$ tends to 0.5 or when plastic flow is constrained by the volume conservation condition. In these cases, a standard Galerkin displacement-based formulation (u formulation) fails [159,146] due to the inability to evaluate the correct strain field. In the literature, many possible solutions can be found. For instance, Simo and Rifai introduced the Mixed Enhanced Element for small deformation problems [160]. This is a special three-field mixed finite element method in which the space of discrete strains is augmented with local functions. It is worth mentioning that also the class of B-bar methods [161] and the classical incompatible modes formulation [162] fall under this theory. For general purposes, some variants of this procedure are analysed in [163]. Alternative procedures suitable for geometrically non-linear regimes, are given by the F-BAR method [164], a technique based on the concept of multiplicative deviatoric/volumetric split in conjunction with the replacement of the compatible deformation gradient field, the non-linear B-bar method [165] and the family of enhanced elements [166], which represents an extension to the non-linear regime of the procedures exposed in [161] and [162], respectively. Though the good performance of all the aforementioned methods, none of such techniques is, however, suitable for application on simplicial meshes [167,146,168]. In this regards, among the successful strategies for the fulfilment of the incompressibility constraint, it is worth mentioning the group of the Mixed Variational Methods. Different researchers worked on mixed finite element formulations with displacement and mean stress as primary variables [169,170,171,172,173]; Cervera and coworkers, for instance, proposed a strain/displacement mixed formulation in the context of compressible and incompressible plasticity [174,175]; Simo et al. introduced a non linear version of a three-field Hu-Washizu Variational principle, where displacement, pressure and the Jacobian of the deformation gradient are independent field variables [176]. The use of Mixed Variational Methods and the difficulties encountered when applying them with different elements have been largely discussed in the 1970s. In [177,178,179,180] the need to satisfy the stability condition, the so-called inf-sup condition, is demonstrated and the instability and ineffectiveness of elements with equal-order interpolations for all the primary variables are proved. This has motivated the development of a series of stabilization techniques, which allow the employment of low order Galerkin finite elements in computational fluid dynamics and solid mechanics problems [181,182,183,184,185,186,187,188].

The treatment of the incompressibility constraint is relatively new in the context of the Material Point Method (MPM). Most MPM formulations deal with compressible materials, avoiding the issues arising from the imposition of the incompressibility constraint. However, some procedures for the treatment of locking issues can be found in the literature. For instance, in [189] an approach for the solution of kinematic (shearing and volumetric) locking is proposed. The authors identified the employment of linear shape functions in conjunction with a regular, rectangular grid, as cause of the locking. The mixed formulation, employed in such work, is derived from the definition of a three-field Hu-Washizu potential, with stress, strain and displacement considered as primary variables. In [190] the formulation presented makes use of the Chorin's projection [191], a popular fractional step formulation solved implicitly for fluid mechanics problems and in [192] a similar strategy, based on a splitting operator technique for solving the momentum equation, is proposed for the treatment of the incompressibility constraint.

In this Chapter, the computational strategy proposed in [139] for the solution of solid mechanics problems characterized by plastic incompressibility in large displacement and large deformation regime, is described in detail and applied to some representative test examples. A mixed u-p formulation, where the displacement and mean stress are considered as primary variables, is implemented within the framework of the implicit MPM strategy, developed in the Kratos Multiphysics open-source platform [29,30]. A monolithic solution strategy, which allows not to impose "spurious" pressure boundary conditions on the Neumann boundary, as done in [190,192], is used. In the current work, only simplicial elements are considered and a stabilization technique is adopted for the satisfaction of the inf-sup condition. The stabilization, based on the Polynomial Pressure Projection (PPP), presented in [193], is chosen for its ease of implementation and good performance demonstrated in previous works [194,195]. The proposed approach is validated through a series of benchmark examples, where an elastic Neo-Hookean and a J2 plastic material are employed. Further, for each test, the results obtained through a displacement-based (u) and the stabilized mixed (u-p) formulation are compared.

In what follows, the u-p formulation is derived in matrix form. Afterwards, the numerical examples are illustrated and the results are discussed.

5.2 The mixed formulation

In this section the mixed (u-p) formulation is briefly introduced and derived in matrix form.

5.2.1 Governing equations in strong form

Let us consider the body ${\textstyle {\mathcal {B}}}$ which occupies a region ${\textstyle \Omega }$ of the three-dimensional Euclidean space ${\textstyle {\mathcal {E}}}$ with a regular boundary ${\textstyle \partial \Omega }$ in its reference configuration. A deformation of ${\textstyle {\mathcal {B}}}$ is defined by a one-to-one mapping

(5.1)

that maps each point p of the body ${\textstyle {\mathcal {B}}}$ into a spatial point ${\textstyle \mathbf {x} }$

(5.2)

which represents the location of p in the deformed configuration of ${\textstyle {\mathcal {B}}}$ . The region of ${\textstyle {\mathcal {E}}}$ occupied by ${\textstyle {\mathcal {B}}}$ in its deformed configuration is denoted as ${\textstyle \varphi \left(\Omega \right)}$ .

The boundary value problem of finite elastostatics consists in finding a displacement field ${\textstyle {\boldsymbol {u}}:\varphi \left(\Omega \right)\rightarrow {\mathcal {E}}}$ such that the equilibrium equations and the kinematic conditions are satisfied

\left\{{\begin{array}{rcll}-\nabla \cdot {\boldsymbol {\sigma }}&=&{\boldsymbol {b}}&{\textrm {in}}\quad \varphi \left(\Omega \right)\\{\boldsymbol {\sigma }}\cdot {\boldsymbol {n}}&=&{\boldsymbol {\overline {t}}}&{\textrm {on}}\quad \varphi (\partial \Omega _{N})\\{\boldsymbol {u}}&=&{\boldsymbol {\overline {u}}}&{\textrm {on}}\quad \varphi (\partial \Omega _{D})\end{array}}\right.

(5.3)

where ${\textstyle {\boldsymbol {\sigma }}}$ is the Cauchy stress tensor, ${\textstyle {\boldsymbol {b}}}$ denotes the body forces and ${\textstyle \varphi (\partial \Omega _{N})}$ and ${\textstyle \varphi (\partial \Omega _{D})}$ the boundaries of ${\textstyle \varphi \left(\Omega \right)}$ , where both the normal tension ( ${\textstyle {\boldsymbol {\overline {t}}}}$ ) (being ${\textstyle {\boldsymbol {n}}}$ the outer normal) and the displacements ( ${\textstyle {\boldsymbol {\overline {u}}}}$ ) are prescribed.

As described in [159], the mixed formulation can be obtained expressing the system of Equations (5.3) in function of two primary variables: the displacement ${\textstyle {\boldsymbol {u}}}$ and the mean stress ${\textstyle p}$ by splitting the stress tensor in its volumetric and deviatoric part ${\textstyle {\boldsymbol {\sigma }}^{\mathrm {dev} }}$ . Thus, the system can be rewritten as

\left\{{\begin{array}{rcll}-\nabla \cdot \left({\boldsymbol {\sigma }}^{\mathrm {dev} }+p{\boldsymbol {I}}\right)&=&{\boldsymbol {b}}&{\textrm {in}}\quad \varphi \left(\Omega \right)\\p-\left({\frac {1}{3}}{\boldsymbol {I}}:{\boldsymbol {\sigma }}\right)&=&0&{\textrm {in}}\quad \varphi \left(\Omega \right)\\\left({\boldsymbol {\sigma }}^{\mathrm {dev} }+p{\boldsymbol {I}}\right)\cdot {\boldsymbol {n}}&=&{\boldsymbol {\overline {t}}}&{\textrm {on}}\quad \varphi (\partial \Omega _{N})\\{\boldsymbol {u}}&=&{\boldsymbol {\overline {u}}}&{\textrm {on}}\quad \varphi (\partial \Omega _{D})\end{array}}\right.

(5.4)

being ${\textstyle {\boldsymbol {I}}}$ the second order identity tensor. We can observe that if ${\textstyle {\boldsymbol {u}}}$ is a solution of Equation (5.3), then ${\textstyle \left({\boldsymbol {u}},p\right)}$ , satisfying also ${\textstyle p-\left({\frac {1}{3}}{\boldsymbol {I}}:{\boldsymbol {\sigma }}\right)=0}$ , is a solution of Equation (5.4).

5.2.2 Weak form and linearisation of the weak form in spatial form

According to the standard FEM procedure, the weak form of Equation (5.4) is obtained by employing the Galerkin method and is written in spatial configuration, adopting an Updated Lagrangian framework.

For sake of clarity the weak form Equation 5.3, previously derived in Chapter 4, is provided below.

(5.5)

using the notation ${\textstyle \mathbf {A} ^{s}={\frac {1}{2}}\left(\mathbf {A} +\mathbf {A} ^{T}\right)}$ .

With regard to the mixed formulation, linear interpolation finite elements both for displacement and pressure (u-p) are considered. The weak form of the balance of the linear momentum (Equation (5.5)) can be rewritten as

(5.6)

where the Cauchy stress tensor ${\textstyle {\boldsymbol {\sigma }}}$ is decomposed in its deviatoric and volumetric component, denoted as ${\textstyle {\boldsymbol {\sigma }}^{\mathrm {dev} }}$ and ${\textstyle p}$ , respectively. The weak form of the pressure continuity equation is obtained by performing a ${\textstyle L_{2}}$ inner product of the second equation of (5.4) with an arbitrary test function ${\textstyle q\in {\mathcal {Q}}}$ , where ${\textstyle {\mathcal {Q}}}$ is the space of virtual pressure. Finally the weak form of the pressure continuity equation is expressed as

(5.7)

In this work a Newton-Raphson's iterative procedure is employed for the solution of problems characterized by material and geometrical non-linearities. The non-linear weak forms of Equations (5.6) and (5.7) have to be linearised through an expansion in Taylor's series, evaluated at the last known equilibrium configuration ${\textstyle {\boldsymbol {u}}^{*}}$ and ${\textstyle p^{*}}$ .

In this way the solution system of linearised equations can be derived and expressed in matrix form as

(5.8)

where ${\textstyle \mathbf {R} _{\boldsymbol {u}}=G({\boldsymbol {u}},p,{\boldsymbol {w}})}$ and ${\textstyle {\boldsymbol {R}}_{p}=G({\boldsymbol {u}},p,q)}$ are the components of the residual vector, ${\textstyle \delta {\boldsymbol {u}}}$ and ${\textstyle \delta {\boldsymbol {p}}}$ are the vector of unknown displacements and unknown mean stresses, respectively. The components of the matrix on the left hand side (lhs) of Equation (5.8) are given by the tangent stiffness matrix ${\textstyle ^{m}\mathbf {K} ^{\mathrm {tan} }=D_{\boldsymbol {u}}G({\boldsymbol {u}},p,{\boldsymbol {w}})}$ , which can be seen as the sum of the material stiffness matrix

(5.9)

being ${\textstyle \mathbb {I} }$ the fourth order identity tensor, and the geometric stiffness matrix

(5.10)

Furthermore, ${\textstyle \mathbf {M} =D_{p}G({\boldsymbol {u}},p,q)}$ is

(5.11)

and the mixed terms ${\textstyle \mathbf {B} =D_{p}G({\boldsymbol {u}},p,{\boldsymbol {w}})}$ and ${\textstyle \mathbf {B^{*}} =D_{\boldsymbol {u}}G({\boldsymbol {u}},p,q)}$ , are defined, respectively, as

(5.12)

(5.13)

where ${\textstyle D_{\boldsymbol {u}}\left({\frac {1}{3}}{\boldsymbol {I}}:{\boldsymbol {\sigma }}\right)}$ can be derived once determined the volumetric stress as function of the strain field.

One can observe that ${\textstyle ^{m}\mathbf {K} ^{M}}$ and ${\textstyle ^{m}\mathbf {K} ^{G}}$ are distinguished from ${\textstyle \mathbf {K} ^{M}}$ and ${\textstyle \mathbf {K} ^{G}}$ , defined for the irreducible formulation (Equations (4.34) and (4.33)). In the mixed case, the deviatoric part of ${\textstyle \mathbf {D} }$ and ${\textstyle {\boldsymbol {\sigma }}}$ is separated by the volumetric one and an evaluation of the latter is done, not using the material response of the constitutive law, but the interpolation of the nodal pressure field on the material points, i.e., the integration points.

5.2.3 The stabilized mixed formulation

For the treatment of the incompressibility constraint, the Polynomial Pressure Projection (PPP), introduced by Dohrmann and Bochev [193], is used. This stabilization procedure is obtained by modifying the mixed variational equation by using a ${\textstyle L^{2}}$ polynomial pressure projection. If ${\textstyle k}$ is the order of the continuous polynomial shape functions used to approximate ${\textstyle p}$ , the pressure projection is performed into a polynomial space with order of ${\textstyle k-1}$ . As in the current work linear shape functions are used for the pressure, the ${\textstyle L^{2}}$ polynomial pressure projection is made in a discontinuous space and, consequently, it can be performed at the element level as

(5.14)

being ${\textstyle {\tilde {p}}}$ the best approximation of ${\textstyle p}$ in ${\textstyle ({\mathcal {Q}}^{0})}$ and ${\textstyle {\tilde {q}}\in {\mathcal {Q}}^{0}}$ an arbitrary test function, where ${\textstyle {\mathcal {Q}}^{0}}$ is the space of polynomial functions with zero degree in each coordinate direction. Unlike other stabilization techniques, the pressure stabilization is accomplished without the use of the residual of the momentum equation; thus, the calculation of higher-order derivatives and the specification of a mesh-dependent stabilization parameter are avoided. Moreover, it is demonstrated that symmetry of the mixed formulation is retained.

In the case of simplicial elements, as in the current work, the stabilization of the unstable mixed formulation requires only the addition of the bilinear form

(5.15)

to Equation (5.7), where ${\textstyle \alpha }$ is a parameter to be selected for stability and ${\textstyle G}$ the shear modulus. The weak form of the pressure continuity equation (Equation (5.7)) can be rewritten as

(5.16)

and the matrix system (Equation (5.8)) becomes

{\begin{bmatrix}^{m}\mathbf {K} ^{\mathrm {tan} }&\mathbf {B} \\\mathbf {B} ^{*}&-\mathbf {M} -\mathbf {M} ^{\mathrm {stab} }\end{bmatrix}}{\begin{bmatrix}\delta {\boldsymbol {u}}\\\delta {\boldsymbol {p}}\end{bmatrix}}=-{\begin{bmatrix}{\boldsymbol {R}}_{\boldsymbol {u}}\\{\boldsymbol {R}}_{p}+{\boldsymbol {R}}_{p}^{\mathrm {stab} }\end{bmatrix}}

(5.17)

where

(5.18)

and

(5.19)

5.3 The MPM algorithm in the framework of a mixed formulation

If a mixed (u-p) formulation is used in the framework of the MPM, it is important to highlight that some changes have to be considered in the initialization and convective phase of standard algorithm, described in Section 2.4.1. In the initialization phase, initial nodal pressure values ${\textstyle p_{I}^{n}}$ , related to the previous time ${\textstyle t^{n}}$ , have to be evaluated, in addition to the mass, velocity and acceleration ones, using the following expression:

(5.20)

where ${\textstyle N_{I}}$ is the shape function of node ${\textstyle I}$ evaluated at the position of the ${\textstyle p-th}$ material point, and ${\textstyle m_{p}}$ and ${\textstyle p_{p}^{n}}$ are the mass and the pressure of the material point, respectively. The nodal pressure evaluated in Equation (5.20) is used in the predictor step of the Newmark scheme. Once the solution is iteratively computed using the linearised system of Equations (5.17), the convective phase is performed, as explained in detail in Section 2.4.1. The pressure on the material points is updated in addition to the material point displacement, velocity and acceleration, through an interpolation of the converged nodal pressure values ${\textstyle p_{I}^{n+1}}$ on the material point position

(5.21)

The Algorithm 2.1, previously provided in Section 2.4.1, is below presented for a static case, together with the modifications aforementioned.

(we will use $(\bullet )^{n}=(\bullet )(t_{n})$ ), Material DATA: E, $\nu$ , $\rho$

Initial data on material points:

,

and

Initial data on nodes: NONE - everything is discarded in the initialization phase

OUTPUT of calculations:

INITIALIZATION PHASE

Clear nodal info and recover undeformed grid configuration
Calculation of initial nodal conditions.

(a) for p = 1:

Calculation of nodal data

${\textstyle l_{I}^{n}=\sum _{p}N_{I}\,m_{p}p_{p}^{n}}$
${\textstyle m_{I}^{n}=\sum _{p}N_{I}m_{p}}$

(b) for I = 1:

${\textstyle {\widetilde {p}}_{I}^{n}={\dfrac {l_{I}^{n}}{m_{I}^{n}}}}$

Newmark method: PREDICTOR. Evaluation of ${\textstyle ^{it+1}\Delta \mathbf {u} _{I}^{n+1},^{it+1}p_{I}^{n+1}}$ by using Equations ${\textstyle ^{it+1}\Delta {\boldsymbol {u}}_{I}^{n+1}=0.0}$ and ${\textstyle ^{it+1}p_{I}^{n+1}={\widetilde {p}}_{I}^{n}}$

UL-FEM PHASE

for p = 1: ${\textstyle N_{p}}$

(a) Evaluation of local residual (

) (RHS of Equation 5.17)

(b) Evaluation of local Jacobian matrix of residual (

) (LHS of Equation 5.17)

(c) Assemble rhs and lhs to the global vector

and global matrix

(see the matricial system of Equation 5.17))

Solving system ${\textstyle (^{it+1}\delta {\boldsymbol {u}}_{I}^{n+1},^{it+1}\delta p_{I}^{n+1})}$
Newmark method: CORRECTOR by using the Equations ${\textstyle ^{it+1}\Delta {\boldsymbol {u}}_{I}^{n+1}=^{it}\Delta {\boldsymbol {u}}_{I}^{n+1}+^{it+1}\delta {\boldsymbol {u}}_{I}^{n+1}}$ (Equation 2.13) and ${\textstyle ^{it+1}p_{I}^{n+1}=^{it+1}\delta p_{I}^{n+1}}$
Check convergence

(a) NOT converged: go to Step 2

(b) Converged: go to Step 3

CONVECTIVE PHASE

Update the solution on the material points by means of an interpolation of nodal information by using the Equations ${\textstyle \Delta {\boldsymbol {u}}_{p}^{n+1}=\sum _{n=1}^{n_{n}}N_{I}\Delta {\boldsymbol {u}}_{I}^{n+1}}$ (Equation 2.14) and ${\textstyle p_{p}^{n+1}=\sum _{I}N_{I}p_{I}^{n+1}}$ (Equation 5.21)

Save the stress ${\textstyle {\boldsymbol {\sigma }}_{p}^{n+1}}$ , strain ${\textstyle {\boldsymbol {\epsilon }}_{p}^{n+1}}$ , pressure ${\textstyle p_{p}^{n+1}}$ and total deformation gradient ${\textstyle \mathbf {F} _{p}^{n+1}}$ on material points (the latter by ${\textstyle \mathbf {F} _{p}^{n+1}=\Delta \mathbf {F} _{p}\cdot \mathbf {F} _{p}^{n}}$ )

Algorithm. 5.1 MPM algorithm in the framework of a mixed formulation.

5.4 Numerical Examples

In this section, two numerical examples are presented for the validation of the mixed formulation. Firstly, the well-known benchmark test of a Cook's elastic membrane is considered and a mesh convergence study is performed. The stability of the mixed formulation is assessed in a quasi-incompressible elastic case. Secondly, a plane strain tension test of a J2-plastic plate in compressible and incompressible state is analysed. In this example, the performance of the irreducible u and the mixed u-p formulations are compared in the case of incompressible plastic flow. The results obtained with the u and u-p formulations are compared and used to demonstrate that a mixed MPM formulation can provide more accurate and reliable results, not only under the assumption of elastic and plastic incompressibility, but even in compressible situations.

In this work, a stabilization parameter ( ${\textstyle \alpha }$ ) with value of 1 has been used. The direct solver SuperLU is employed for the solution of the system of linearised equations, both in the case of u and u-p formulations.

5.4.1 Cook's membrane problem

As a first numerical example, we consider the well known Cook's membrane test, proposed for the first time by Cook [196]. This test is often used as a benchmark to check the element formulation under compressible and incompressible conditions. In the literature, the Cook's membrane is commonly tested in infinitesimal deformation assumption and material linearity [171], geometric non-linearity and material linearity [197] and, finally, in geometric and material non-linearities [155,164,170,194]. The geometry and material properties of the problem are shown in Figure 5.1. A clamped trapezoidal plate, subjected to a distributed shear load, whose resultant force is ${\textstyle P=1N}$ , applied along the right side, is analysed. The static case is solved studying the response of a compressible and a quasi-incompressible Neo-Hookean material, whose formulation is presented in Section 3.1. The convergence study is performed using six structured triangular meshes each of which uses an initial value of one material point per element.

Figure 5.1: Cook's membrane. Geometry, material properties and boundary conditions

Since the formulations under study are based on the assumption of finite deformation and material non-linearity, the results relative to a very fine mesh (256 elements per side) of a FEM analsys is considered as reference solution in the compressible case, while the result of [194] is the benchmark solution for the quasi-incompressible case. The reference solution of vertical displacement at point A (Figure 5.1) is found to be 0.323m, in the compressible case, and 0.275m in the quasi-incompressible cases, respectively. The results of u and u-p formulations, with and without stabilization term (UP No Stab and UP Stab) are summarized in Table (5.1) for both the compressible and nearly incompressible cases. The same results can be observed graphically in Figures 5.2 and 5.3.

Table. 5.1 Cook's membrane. Compressible case: vertical displacement at point A obtained with the U, UP formulation without and with stabilization
Elements per side	Compressible case			Quasi-incompressible case
	U	UP No Stab	UP Stab	U	UP No Stab	UP Stab
2	0.089	0.1013	0.1172	0.0723	0.0788	0.1277
4	0.1415	0.1718	0.1953	0.0736	0.1157	0.1932
8	0.2183	0.2511	0.2669	0.0742	0.1821	0.2424
16	0.2771	0.2952	0.3025	0.075	0.2356	0.2648
32	0.30386	0.3119	0.315	0.0775	0.2606	0.2725
64	0.3133	0.3176	0.319	0.0862	0.2702	0.275

The u formulation is less accurate than the u-p formulation both for the UP No Stab and UP Stab cases, not only for the nearly incompressible condition, as expected, but also for the compressible one. However, the discrepancy is clearly visible in the quasi-incompressible problem (Figure 5.3), where the capability of the u formulation to predict the displacement field is compromised due to volumetric locking.

Figure 5.2: Cook's membrane. Compressible case: vertical displacement at point A

Figure 5.3: Cook's membrane. Quasi-incompressible case: vertical displacement at point A

Regarding the mixed approaches, from Figure 5.3 it is possible to infer that even not using a stabilization term the solution is not affected by volumetric locking. However, through the stabilized u-p formulation it is also possible to prevent pressure oscillation issues in the mean stress field, as can be observed in Figure 5.4, where the pressure values of Figure 5.4a are all out of the threshold defined by the solution of Figure 5.4b.


(a) u-p without stabilization	(b) u-p with stabilization
Figure 5.4: Cook's membrane. Quasi-incompressible case: Pressure counter fill. The mixed formulation without any stabilization (a) fails to predict the pressure field, while it is correctly evaluated using the PPP stabilization (b). Black contour colour should be intended as out of range.

5.4.2 2D tension test

As second numerical example, a plane strain tension problem is considered to test the mixed formulation in an elasto-plastic regime. A 2D plate, clamped at the bottom of the specimen, is subjected to a prescribed vertical displacement on the upper side. Both geometry and material properties are taken from [173] and are depicted in Figure 5.5. The plate is made by a hyperelastic perfectly-plastic material which is simulated using a J2 plastic law, whose formulation is presented in Section 3.2. An unstructured triangular background mesh with a mesh size of 0.001m and an initial distribution of 12 material points per cell, which is found to give the optimal trade-off between accuracy of the results and computational cost in both the compressible and incompressible cases, are adopted.

Figure 5.5: Tension test. Geometry, material properties and boundary conditions

The results of the compressible case are shown in Figures 5.6, 5.7, 5.8 and 5.9, where the displacement along x and y-direction, the equivalent plastic strains and the vertical Cauchy stresses are shown. Volumetric locking is not affecting the numerical results, as the plate is working under compressible conditions. However, the u-p formulation is more accurate than the u one, not only in the evaluation of the stress field, but also of the displacement field. Moreover, the goodness of the solution can be appreciated looking at Figure 5.8b: the equivalent plastic strains are distinctly distributed along a cross shape, while the result of Figure 5.8a revokes the same shape, but without the same order of precision. In conclusion, even if a compressible material is simulated, the results obtained with the u-p formulation present a higher order of accuracy, by using the same mesh size and the same number of material points per element.

The results of the incompressible case are shown in Figures 5.10, 5.11, 5.12 and 5.13. In this case, the u formulation fails in the simulation of the tension test. As expected, the displacement and stress fields are affected by volumetric locking and the plastic deformations are incorrectly localized. On the other hand, Figures 5.10b, 5.11b, 5.12b and 5.13b show that the u-p formulation is able to evaluate correctly the displacement and stress field under incompressible conditions. The results are similar to those depicted in Figures 5.6b, 5.7b, 5.8b and 5.9b: the cross-shape distribution of the equivalent plastic strains and stresses are recovered. Furthermore, Figures 5.14a, 5.14b, 5.14c and 5.14d show a comparison in the nearly-incompressible case between the reference solution obtained with the formulation proposed in [173] and the results obtained with the MPM u-p formulation presented in the current work. We can observe that there is a good agreement both in the distribution of equivalent plastic strains and pressure fields and in their values range. Finally, the stress - displacement curve, evaluated with the mixed formulation, is shown in Figure 5.15. The results for the compressible and incompressible cases are in good agreement. Both correctly predict the elastic regime and the inception of the plastic flow when the yield stress is reached.

Since a mixed formulation with displacement and pressure as primary variables is adopted, strains are not linearly distributed within the element, but these coincide with a constant function. It is worth highlighting that through this numerical procedure while it is possible to avoid the volumetric locking, the problems related with strain localization are still present. This means that the width of the shear bands still depends on the size of the elements. This problem can be solved by regularization of the element size as proposed, e.g. in [198,174,175] or [156], where the formulations consider the strain field as primary variable and, therefore, its linear distribution can be evaluated, which allows to accurately predict strain localization with mesh independence.


(a) u formulation	(b) u-p formulation
Figure 5.6: Tension test. Compressible case: horizontal displacement


(a) u formulation	(b) u-p formulation
Figure 5.7: Tension test. Compressible case: vertical displacement


(a) u formulation	(b) u-p formulation
Figure 5.8: Tension test. Compressible case: equivalent plastic strain


(a) u formulation	(b) u-p formulation
Figure 5.9: Tension test. Compressible case: Cauchy stress along loading axis. Black contour colour should be intended as out of range.


(a) u formulation	(b) u-p formulation
Figure 5.10: Tension test. Incompressible case: horizontal displacement


(a) u formulation	(b) u-p formulation
Figure 5.11: Tension test. Incompressible case: vertical displacement


(a) u formulation	(b) u-p formulation
Figure 5.12: Tension test. Incompressible case: equivalent plastic strain


(a) u formulation	(b) u-p formulation
Figure 5.13: Tension test. Incompressible case: Cauchy stress along loading axis. Black contour colour should be intended as out of range.


(a) Equivalent plastic strain	(b) Pressure

(c) Equivalent plastic strain	(d) Pressure
Figure 5.14: Tension test. Incompressible case: results evaluated at a total imposed vertical displacement of 0.0001m. a) and b) Results in terms of equivalent plastic strain and pressure using a T1/P1 u-p formulation, taken from [173]. c) and d) Results in terms of equivalent plastic strain and pressure evaluated with the MPM u-p formulation presented in the current study.

Figure 5.15: Tension test. Stress-Displacement curve. Comparison between the compressible case (red curve) and the incompressible curve (green curve).

5.5 Discussion

In this Chapter, a stabilized mixed u-p formulation is presented in its strong and weak forms. The formulation is implemented within the framework of the implicit MPM strategy, able to solve problems which involve large displacements and large deformations. The irreducible u and mixed u-p formulations are tested and compared through a series of benchmark examples. Firstly, the Cook's membrane problem, a bending dominated test, is investigated. Two cases, a compressible and a nearly-incompressible one, are solved through the u and u-p formulations. From the results, it is demonstrated that the u-p formulation always gives the best performance in term of convergence. In the quasi-incompressible case, the volumetric locking issue is overcome and pressure oscillations are avoided if a stabilization term is added to the mixed finite element formulation. In the second example, a J2 plastic plate, subjected to uniform tension on one side and fixed to the other side, is under study and both formulations are tested under an isochoric plastic flow condition. By comparing the displacement-based and mixed formulation it is shown that, even in this case, better results are obtained through the u-p procedure. Indeed, a higher definition of displacement, equivalent plastic strains and vertical Cauchy stress fields is observed. Despite volumetric locking issue is fixed in the case of the u-p formulation, further problems, such as, mesh independence and strain localization, are not addressed in the current work and they would represent interesting topics for a future research.

In conclusion, it is demonstrated that the u-p formulation can evaluate more accurate results in terms of displacement and stress fields, not only under near-incompressible state, avoiding the typical drawback of volumetric locking, but even under compressible conditions.

6 Validation

In Chapter 2 the Material Point Method and its algorithm have been presented. Under the assumption of finite strains an irreducible (see Chapter 4) and a ${\textstyle {\boldsymbol {u}}-p}$ mixed formulation (see Chapter 5) are described and verified by using the constitutive laws, whose algorithms are shown in detail in Chapter 3. In the current Chapter, the MPM numerical strategy, implemented within the Kratos Multiphysics framework, is employed for the solution of typical problems concerning granular flows, involving large displacement and large deformation of the continuum under study.

Firstly, the typical granular column collapse is considered. For the validation of such a case, the comprehensive experimental work of Lube and co-workers [199] is used as a reference. In the second part of the current Chapter, a second example is taken into account: the rigid strip footing test, a typical test in geomechanics for the assessment of the bearing capacity of a soil to an imposed displacement or force. In the examples considered, experimental results will be used for validation, otherwise, solutions of other studies, available in the literature, will be used as a reference.

In this section, the granular collapse on a horizontal plane is considered as a test case for validation. This test has been chosen because, despite the apparent simplicity of the experiment, the description and prediction of the collapse is still a challenge from an experimental, numerical and theoretical point of view [200]. It is a perfect example to test the numerical technique in case of large displacement and large strain. Indeed, inertial granular flows are characterized by unsteady motion, a large variation of the free surface with time and propagation toward the free surface of internal interface separating the static and flowing regions. During the last decades, this test has been object of numerical study of many research groups by using techniques based either on discrete or continuum mechanics. Regarding the use of DEM, Staron and Hinch [201] showed that their results have good agreement with the experimental results in terms of run-out distance, but they did not provide a physical argument able to explain the relation between the initial aspect ratio and the final run-out. Moreover, they focused on the final deposition profiles without paying attention to the influence of material properties and the collapse mechanism. In Lacaze et al. [202], DEM simulations are performed providing good results; they focused on both flow behaviour and final deposition of the collapse. In Kumar [203] DEM simulations are carried out with an analysis on the initial grain properties, which is demonstrated that can influence the structure of internal flow and the kinematic of failure mechanism. The use of DEM, in the context of micro-mechanical analysis, is really helpful in providing some insights. However, when extending to upper scales, the method suffers from extremely demanding computational cost, which is detrimental for its usage to practical application in the engineering and industrial framework. Due to this aspect, a high interest of the computational community is sparked in solving granular flow problems with techniques based on continuum mechanics, which have the advantage to reduce tremendously the computational cost. Some attempts have been carried out by using an ALE FEM [204] or SPH [205,206]. However, as pointed out in Chapter 2, these methods suffer from some issues which do not make them particularly suitable for the modelling of granular flows. The granular column collapse has been also modelled with the MPM [118,207]. In [118] a validation is performed by employing a Mohr-Coulomb plastic law, while in [207] an analysis of different constitutive laws implemented in a MPM code is carried out and interesting insights are provided on the choice of a suitable constitutive model regarding the modelling of granular material flows.

In this section, the results of the comprehensive experimental work of Lube and co-worker [199] are used as a reference for the validation of the MPM code implemented in the Kratos Multiphysics framework. In their works, the authors observed that the final run-out, the final maximum height and the corresponding time, indicated with ${\textstyle d_{\infty }}$ , ${\textstyle h_{\infty }}$ and ${\textstyle t_{\infty }}$ , respectively, to be consistent with the reference works, are found to be independent on the different grains and roughness of the lower boundary and only the initial geometry, the initial aspect ratio, could affect those results. The experimental test consists in a granular column inside a channel, wide enough in order to avoid the wall influence, at one side sustained by a fixed wall and on the other side by a moving wall. At the beginning, the column is at rest and the experiment starts when the moving wall is removed and the granular material is free to collapse.

In this validation work, granular columns of different aspect ratios, ${\textstyle a={\frac {h_{i}}{d_{i}}}=1.2,3,5,7}$ , are considered. It is well known that the first failure surface, which generates after the opening of the moving wall, is very similar in all the sample independently of the initial geometry [199,207]. Hence, the amount of mass which starts moving from the static zone increases with the initial aspect ratios and, consequently, different flow regimes might take place depending on the initial geometry of the column. For taller aspect ratios ( ${\textstyle a>2.8}$ ), the flow regime is mainly dominated by the inertia (Regime I); in the case of low aspect ratios, the flow behaviour is more dominated by the friction and energy dissipation (Regime II) which takes place at the bottom layer, at the interface between static and dynamic zone and in the moving mass, as well. With regards to the kinematic of the granular column collapse, it has been experimentally observed that in Regime I three transient stages can be distinguished: a constant acceleration phase at ${\textstyle 0.75g}$ , a constant velocity and a final deceleration stage. The duration of the second stage decreases with ${\textstyle a}$ and for aspect ratios lower than 1.5 does not appear. Thus, in Regime II only two stages of initial acceleration and final deceleration take place.

In this test case, the Mohr-Coulomb plastic law, presented in Section 3.3, is employed. It is well known that one of the limitations of this constitutive model lies on the inability to express the dissipation due to friction between grains during the transition from static to flowing regime and vice-versa. As done in [118], also in this work of validation a numerical dissipation is added to the matricial formulation to be solved, presented in Chapter 4. It is found that the Rayleigh damping alpha coefficient with a value of 1.5 can replicate with good accuracy the reference solution, as shown in the following paragraphs.

The power-laws deduced in the experimental study [199] are used for the prediction of ${\textstyle d_{\infty }}$ , ${\textstyle h_{\infty }}$ and ${\textstyle t_{\infty }}$ as

(6.1)

(6.2)

(6.3)

In this validation study three different mesh discretisations are employed with a cell size of ${\textstyle 0.0065m,0.005m}$ and ${\textstyle 0.0025m}$ , defined as Mesh 1, Mesh 2, Mesh 3, respectively, and in what follows, the effect of mesh refinement on the kinematic of granular column collapse is analysed. A number of initial material points per element is set to 9 in all the analyses and the material properties are listed in Table table: granular column material properties.

Table 6.1. Granular column collapse: material properties
Young Modulus	Poisson ratio	Density	Internal friction angle	Dilatancy angle	Cohesion
0.84e6 Pa	0.3	2600 kg/mc	31 deg	1 deg	0 Pa

The numerical results are summarized in Tables 6.2, 6.1 and 6.1, and the results, evaluated according to the power laws of Equations 6.1, 6.2 and 6.3, in Table 6.1. It is found that, in general, by using more fine mesh the values of ${\textstyle d_{\infty }}$ , ${\textstyle h_{\infty }}$ and ${\textstyle t_{\infty }}$ approach the empirical values of Table 6.1.

Table 6.2. Granular column collapse: numerical results Mesh 1
Mesh 1
a	$d_{\infty }$	$h_{\infty }$	$t_{\infty }$
1.2	0.332	0.106	0.47
3	0.566	0.149	0.64
5	0.709	0.162	0.71
7	0.811	0.191	0.82

Table 6.3. Granular column collapse: numerical results Mesh 2
Mesh 2
a	$d_{\infty }$	$h_{\infty }$	$t_{\infty }$
1.2	0.33	0.104	0.47
3	0.55	0.145	0.6
5	0.699	0.165	0.71
7	0.795	0.196	0.83

Table 6.4. Granular column collapse: numerical results Mesh 3
Mesh 3
a	$d_{\infty }$	$h_{\infty }$	$t_{\infty }$
1.2	0.31	0.102	0.4
3	0.531	0.144	0.59
5	0.664	0.162	0.71
7	0.766	0.17	0.87

Table 6.5. Granular column collapse: empirical results
a	$d_{\infty }$	$h_{\infty }$	$t_{\infty }$
1.2	0.264	0.108	0.35
3	0.505	0.14	0.55
5	0.673	0.172	0.71
7	0.819	0.197	0.838

In order to carry out a comparison between the experimental and numerical results with regards to the kinematic of the failure mechanism, in Figures 6.1 and 6.2 experimental and numerical normalized distance-time data of the flow front for the test with aspect ratios 3, 5 and 7 are depicted. The normalize distance is evaluate as ${\textstyle d/d_{\infty }}$ , while the normalized time has the following expression

(6.4)

where it is used the empirical law ${\textstyle t_{\infty }=3.3{\sqrt {h_{i}/g}}}$ for high aspect ratios, obtained in [199], which states that the time for the column to spread depends only on the initial height ${\textstyle h_{i}}$ .

In Figure 6.1 the results obtained with the finer mesh (Mesh 3) are depicted. It is found that the sample with the aspect ratio of 3 is the numerical test which fully matches the yellow region, within which all the experimental normalized distance-time data collapse. The higher the aspect ratio, the higher the mismatch between numerical and experimental results. By observing Figures 6.2a, 6.2b and 6.2c, it is found that the numerical results in terms of kinematic of the moving front do not show a sensitivity to the different spatial discretisations. Moreover, the typical three stages of the failure mechanism: the initial constant acceleration, the constant velocity and the final deceleration step, are recognized for all aspect ratios. If in Figure 6.2a it is shown that the numerical results fall in the yellow zone, for higher values of aspect ratio (see Figures 6.2b and 6.2c), the curves slightly deviate from the yellow zone. It is deduced that the velocity, in the second stage, is higher than what experimentally observed. This implies that in the numerical simulations the amount of energy dissipated is underestimated, even if a damping term is added to the matricial system to solve.

Figure 6.1: Granular column collapse: comparison between normalized distance-time data for different aspect ratios with Mesh 3 and experimental results [199].

(a) Case a = 3

(b) Case a = 5

(c) Case a = 7

Figure 6.2: Granular column collapse: comparison between normalized distance-time data of the flow front for different aspect ratios a.

In addition, for the test case with ${\textstyle a=7}$ the configurations of the granular column at different times of the analysis are compared with the experimental ones in Figure 6.4. From the comparison it is found that the numerical results have a good match with the experimental one, except for the maximum height of the column at the position ${\textstyle x=0m}$ in all the time instants considered, most probably due to the adoption of a constitutive law unable to predict the initial failure surface and, thus, the correct flowing behaviour.

The results of Figure 6.4, to some extent, can be considered accurate, providing a good description not only of the final configuration, but also of the dynamic flowing behaviour at different times. Nevertheless, as pointed out in Fern [207], numerical damping aims to mitigate numerical oscillations by reducing the out-of-balance force and is, hence, reducing the dynamic effects. The use of numerical damping to reduce the run-out distance is rather a modification of the dynamic problem than a proper energy dissipating mechanism. In [207] it is confirmed that the initial geometry of the column plays an important role in the dynamics of the failure mechanism, since it establishes the initial potential energy available in the system. From the modelling point of view of such test case an accurate representation of the real triggering mechanism and the flow behaviour it is on the constitutive law which it is found to have two main roles. It defines the first failure surface, thus, the amount of potential energy to be transformed in kinetic energy and to be dissipated, and the way energy is dissipated in the system of the moving mass. As a last observation, in [207] they recognized the initial density as important variable to be considered in the model, able to influence the constitutive model enhancing the mechanical response. It influences the dilatancy characteristics and consequently the failure angle. The enhancement of the angle of failure by density influences, in turn, the volume of the mobilised mass, its potential energy and, in some cases, the dissipation of that energy. They noticed that the initial density affects more the results in low aspect ratios column and this could explain the difference between Regime I and Regime II from a physical point of view. The observations made in the work of Fern and Soga give a critical view of some constitutive models, for instance, the Mohr-Coulomb plastic law adopted in the current work, which are not able to predict the real energy dissipation that should have taken place during the failure mechanism (observed in the current work mainly for higher aspect ratios) and the evolution of density before reaching the critical state.


(a) t1	(b) t2

(c) t3	(d) t4

(e) t5	(f) t6

(g) t7	(h) t8
Figure 6.3: Granular column collapse: Contour fill of the numerical results and the experimental shape of the column collapse [199] in different time instants.

(i) t9

(j) t10

(k) t11

Figure 6.4: Granular column collapse: Contour fill of the numerical results and the experimental shape of the column collapse [199] in different time instants.

6.2 Plain strain rigid footing on undrained soil

The second test of validation is a plain strain rigid strip footing for the evaluation of the bearing capacity of the soil in undrained conditions, underneath the foundation. The soil is modelled as a purely cohesive weightless elastic-perfectly plastic Mohr-Coulomb material with associative flow rule, which is presented in Section 3.3. The geometry, the boundary conditions and material properties are represented in Figure 6.5, where for symmetry only half of the domain is considered.

Figure 6.5: Rigid strip footing. Geometry, material properties, boundary conditions and initial material points density. 12 material points (MP) per element are used in the vicinity of the footing while only 4 are used in the rest of the domain.

In the geomechanics community this is a classical benchmark for the validation of the constitutive law and of the numerical method adopted for its simulation. In the literature, the rigid strip footing has been studied by many authors. In [208] Nazem and coworkers solved this example in three different kinematics frameworks: a Total Lagrangian (TL), an Updated Lagrangian (UL) and an Arbitrary Lagrangian Eulerian (ALE) Finite Element Methods. They show that for large deformations an ALE method is more suitable than UL and TL strategies, avoiding mesh distortion with a remeshing technique. Even if the remeshing could smear a stress concentration and compromise the strain localization, they found that the load-displacement curve is comparable with the numerical solutions available in the literature. In [209] the technique of [208] is generalized to the case of higher order elements. The same test example has been also used to prove that the MPM represents an ideal numerical approach since it naturally tracks large deformations without the need of remeshing procedures. For instance, in [118] this example is successfully solved exploiting the capability of MPM to track large deformation and large displacement of the solid. However, the work of [118] is limited to the infinitesimal strain assumption.

For the validation of this test case, the stabilized mixed formulation, valid under geometric and material non-linearities, presented in Chapter 5, is employed, generalizing the approach used in [118]. The simulation is performed using displacement control with steps of incremental vertical displacement ${\textstyle \Delta {\boldsymbol {u}}=-0.001m}$ . The total displacement has been imposed in 2000 time steps which corresponds to twice the foundation width ${\textstyle B}$ . The discretisation of the computational domain is performed through a unstructured triangular background mesh with a mesh size of 0.05m. At the interface between the foundation and the soil, where the largest deformations take place, a higher initial number of material points per element is used for a better resolution of the results (Figure 6.5).

In Figures 6.6, 6.7 and 6.8 the displacement and stress fields obtained with the u and u-p formulations are compared. As expected, the latter shows to be more reliable and accurate than the first one. It can be noted that the final deformation is accurately described and an improvement is registered if the final deformation is compared with the numerical results of [208] and [209] which are more similar to the final configuration obtained through the displacement-based formulation. The need for a mixed formulation is evident when evaluating the vertical stress field. In Figure 6.8a the displacement-based formulation fails to evaluate a reliable stress response, as the magnitude of the vertical Cauchy stress is out of the expected range in the area where the foundation buries itself. On the other hand, the mixed formulation is able to evaluate a continuous stress field and using such result it is possible to evaluate the normalised load-displacement response of the foundation, which is used for the validation of the current example.


(a) u formulation	(b) u-p formulation
Figure 6.6: Rigid strip footing. Horizontal displacement


(a) u formulation	(b) u-p formulation
Figure 6.7: Rigid strip footing. Vertical displacement


(a) u formulation	(b) u-p formulation
Figure 6.8: Rigid strip footing. Vertical Cauchy stress

Since the problem has no analytical solution, the numerical result of [210], obtained through a sequential limit analysis formulation, is taken as reference solution. The problem is solved under the assumption of large deformations, hence, the bearing capacity of the soil is expected to be higher than the value of ${\textstyle 2+\pi }$ which corresponds to the small deformation case, for a given footing displacement. Under this hypothesis, the mobilized soil resistance does not reach an asymptotic value, but gradually increases, as explained in [210]. In Figure 6.9, the result obtained through the u-p formulation in terms of normalized bearing capacity of the soil, as a function of the normalized settlement, is depicted and compared with the benchmark solution. It can be observed that, the obtained curve is in good agreement with the reference solution [210]. The discrepancy that is observed for the initial values of the settlement is the consequence of the chosen material elastic properties. The Young Modulus ${\textstyle E}$ and the Poisson's ratio ${\textstyle \nu }$ have values which correspond to an undrained bulk modulus of ${\textstyle K_{u}=3,33\cdot 10^{5}Pa}$ , which gives a ratio ${\textstyle K_{u}/c_{u}=3,33\cdot 10^{3}}$ . In [118], the influence of this ratio on the normalised load-displacement curve is studied: the elastic response of the soil becomes less or more important and the bearing capacity of the soil can increase or decrease, for higher or lower values of this ratio, respectively. For this reason the numerical results plotted in Figure 6.9 have an important elastic response and are deviating during the initial phase of the simulation from the perfectly rigid behaviour of the benchmark solution.

Figure 6.9: Rigid strip footing. Normalised load-displacement curve: comparison between reference solution taken from [210] and the u-p formulation solution presented in this work.

The example of the rigid footing on undrained soil has been validated using a stabilized mixed MPM formulation. The soil bearing capacity is well predicted and comparable with accurate numerical results from the literature. Moreover, a good description of the final deformation of the soil is achieved by using the MPM and its capability of solving large displacement and large deformation problems is equivalent, if not superior, to other techniques proposed in the literature [208,209].

6.3 Discussion

In this Chapter, two test cases are proposed for the validation of the MPM strategy, by using an irreducible and a mixed formulation. The first test is represented by the granular column collapse, a classical example which is quite often considered for the validation of both the constitutive laws and the numerical techniques. Despite its simplicity, with this test it is possible to make an assessment of the robustness of the numerical model and understand to what extent the constitutive law can provide reliable results. It has been shown that the MPM code, object of study, is able to track with accuracy the configuration of the collapse at different time frames and the constitutive law, employed in this study, can provide good enough results, even if neither softening and density variable are considered as additional terms in the dissipation plastic process. In this regard, as future work, further constitutive models should be taken into account, in order to improve the prediction capability of this numerical strategy. As a second example for validation, the rigid footing on undrained soil is considered. In this case, the good performance of the u-p formulation are tested also under the finite deformation regime: a higher accuracy of the displacement and stress fields are confirmed. Moreover, evaluating the bearing capacity as a function of the footing displacement, the load-displacement curve is obtained and used as a validation tool to be compared with a reference solution.

7 Application to an industrial case

In this Chapter, our Material Point Method formulation is applied in an industrial framework. Several laboratory tests, carried out at the Nestlé Laboratories aiming at the characterization of flowability of different sugar powders, are reproduced numerically. The practical objective of this work lies on the assessment of flow performance of sugar powders, which can play an important role in product development. In case ingredient properties are not optimized a significant variability can be observed during the operation of filling of jars or sachets, which may be detrimental for the production line. In this study, we propose to discuss the performance of crystalline sugar. If the experimental investigation of food materials process is essential in this context, the capability of numerically modelling particulate processes might represent a complementary tool for a better and a more realistic understanding of the process at a pilot or industrial scale. It is experimentally found that the flow quality is strongly deteriorated below a critical particle size. On the other hand, in the numerical study the MPM strategy is employed and it is demonstrated how material parameters, such as, internal friction angle, dilatancy angle and apparent cohesion are important factors in the prediction of the macroscopic behaviour of a granular material and its flowability performance.

7.1 Introduction

Food Industry, as many other sectors dealing with powders, needs to keep a close look at the flowability properties of the raw materials or final powder mixes they develop. Especially, a good flow performance ensures a smooth movement of materials during operations, from raw materials reception to the final packing. For instance, dosing a few grams of granular matter in a small sachet is a challenge if the factory wants to keep a constant mass and ratio of all ingredients. For that reason, the variations of flowability performance with key materials properties must be mastered by product developers in order to know where to act in case of a problem. Food powders often contain several ingredients, such as, crystalline particles or amorphous particles. Sucrose is a good example of a common crystalline material when processed in standard conditions. A common observation is the deterioration of flowability when increasing the quantity of fine particles [211,212,213,214,215]. Some authors report a threshold size below which the deterioration of performance occurs [211,213]. Unfortunately, this value is shown to be strongly dependent on the products tested in those two papers, where the authors considered only diameters from 50 ${\textstyle \mu m}$ to 600 ${\textstyle \mu m}$ . The effect of size is related to the increase of surface area per unit of mass, leading to more contact points and adhesion force between the particles [215]. Cohesive forces acting between particles are mainly due to van der Waals and capillary forces associated with liquid bridging [215,216]. The threshold size can be seen as a limit above which cohesive forces start having a decreasing effect [211]. The shape of the particle is also an important factor affecting the flow [211,217,218]. More elongated particles and particles with fewer corners tend to flow more difficultly due to higher friction forces. The shape effect is observed to be more relevant above the size threshold [211].

The importance of assuring a continuous industrial production process, e.g. without interruptions, is the main reason not only for a full experimental characterization of the material properties to be processed, as previously mentioned, but, during the last decades, also for a numerical investigation of the process from the laboratory to the real plant scale. Indeed, in the literature, many examples of several numerical techniques, applied in the industrial context, can be found. The most common numerical procedure, employed for the simulation of industrial granular flows, is represented by the Discrete Element Method (DEM) [24], which since the beginning of its definition has been used for industrial oriented applications, such as, mixing and milling [219,220,221,222,36], transport [36] and hopper discharge [223,224,225]. During the last decades, the potential of this technique to simulate more realistic systems evolved hand in hand with the increase of computers power [36], achieving the capability of modelling three dimensional large scale systems with DEM. Despite the popularity of DEM, several aspects limit its usage at real scale. Firstly, it is well established that a long calibration procedure has to be performed in order to define all the DEM micro-parameters characterizing the material under study. Last, but not least, the simulation time of large systems of particles might be prohibitive, unless the analysis is performed in High Performance Computing (HPC) mode. For this reason, some attempts of numerically modelling particulate processes, which use continuous technique, i.e., the Finite Element Method (FEM), can be found in the literature, as well. For instance, in [226] the simulation of the discharge silos is realized by using a FEM-based code, defined in an Eulerian framework. However, since the typical processes, interesting from the industrial world perspective, imply large displacement and large deformation of the medium and due to the ambiguous solid-like and fluid-like nature of granular materials, Lagrangian techniques might be preferable rather than Eulerian methods. In this regard, it is worth mentioning the Particle Finite Element Method (PFEM) and the Material Point Method (MPM), both already employed in the past for the simulation of industrial granular flows, such as, silos discharge [95,227,117] and tumbling ball milling [95]. In this study, we propose to compare the flow performance of different sugar powders observed through laboratory experiments and numerical tests performed using the MPM code, whose algorithm has been presented, verified and validated in the previous Chapters (see Chapter 2, 4 and 6). In the next sections, firstly, the experimental analysis is presented, followed by the numerical simulations and, finally, some conclusions are drawn.

7.2 Experimental study

7.2.1 Material

Four sucrose powders are selected for their differences in particles size and origin: Sugar White Fine Bulk (Schweizer Zucker AG, Switzerland), Sugar White Fine Special (Schweizer Zucker AG, Switzerland), Sugar EGII Fine (Agrana, Austria) and Sugar Icing (Central Sugar Refinery, Malaysia). For ease of reading, those powders are, respectively, named S1, S2, S3 and S4. Their particle size distribution is characterized by means of laser diffraction (Malvern Mastersizer 3000 fitted with Aero dispersion module set at a dispersion pressure of 2 bars). The particle size distribution gives access to different parameters, such as, ${\textstyle d_{10}}$ , ${\textstyle d_{50}}$ or ${\textstyle d_{90}}$ , respectively, e.g., the diameters at which 10%, 50% or 90% of the volume of particles is below this value. The span ${\textstyle s=(d_{90}-d_{10})/d_{50}}$ describes the width of the particle size distribution. In order to obtain a wider range of particle sizes ${\textstyle d_{50}}$ and span s, ten additional samples are generated by mixing or sieving the initial powders. Table 7.1 summarizes the preparation method and size characteristics of the different sucrose powders used in this study.

Table. 7.1 Name, preparation method, $d_{10}$ , $d_{50}$ , $d_{90}$ and *span s* of the sucrose powders used in this study. *Courtesy of Nestlé*
Reference name	Preparation method	$d_{10}[\mu m]$	$d_{50}[\mu m]$	$d_{90}[\mu m]$	s
S1	Commercial powder	244	533	977	1.38
${\textstyle S1_{a}}$	S1 > 800 ${\textstyle [\mu m]}$	666	994	1590	0.93
${\textstyle S1_{b}}$	500 ${\textstyle [\mu m]}$ > S1 > 800 ${\textstyle [\mu m]}$	423	633	935	0.81
${\textstyle S1_{c}}$	S1 < 500 ${\textstyle [\mu m]}$	203	372	602	1.07
S2	Commercial powder	81	182	310	1.26
S3	Commercial powder	71	194	352	1.45
${\textstyle S3_{a}}$	S3 < 200 ${\textstyle [\mu m]}$	142	261	425	1.09
${\textstyle S3_{b}}$	S3 > 200 ${\textstyle [\mu m]}$	55	154	269	1.39
S4	Commercial powder	10	54	190	3.36
${\textstyle S4_{a}}$	S4 > 100 ${\textstyle [\mu m]}$	38	146	280	1.66
${\textstyle S4_{b}}$	S4 < 100 ${\textstyle [\mu m]}$	7	32	84	2.42
S1S3	Mix 50% S1 - 50% S1	98	279	789	2.47
S2S3	Mix 50% S2 - 50% S3	81	195	345	1.35
S1S4	Mix 30% S1 - 70% S4	12	86	592	6.74

7.2.2 Measurements procedure

All sucrose powders (Si) are characterized in terms of flowability using several techniques. At minimum, the measurements are performed twice for each technique.

Firstly, the free-flow density, tap density and Carr Index are measured. In order to estimate the free-flow density, a 500 mL - stainless steel cylinder (Figure 7.1a) is filled with powder using a funnel to increase pouring repeatability. After eliminating the excess powder by levelling the top of the cylinder with the flat blade of a pharmaceutical spatula, the free-flow density ${\textstyle D_{free}}$ is directly deduced by dividing the weight of the powder by its volume. Tap density ${\textstyle D_{tap}}$ is then obtained with the same procedure but after the powder has been subjected to a fixed number of taps. For this operation, an extension of the cylinder is used to start with more powder and the receptacle is placed on an electrical jolting density meter (100 jolts, total time 30 seconds, amplitude 8.5 mm). Finally, the Carr Index or compressibility ${\textstyle I_{C}}$ is calculated with the equation: ${\textstyle I_{C}=(100(D_{tap}-D_{free}))/D_{tap}}$ .

Second, the angle of repose is evaluated. A tailor-made apparatus (Figure 7.1b) is used, inspired from ISO norm 4324, but adapted to allow the measurement of both fluid and sticky powders. The determination of the angle of repose of a powder cone is obtained by passing 150 mL of the product through a special funnel placed at a fixed height (85 mm) above a completely flat and level surface. This surface is materialized by a 25 mm-high and 100 mm wide plastic cylinder that ensures the cone always has the same base diameter. Angle of repose (AR) is then calculated from the measurement of the cone height.

Then, the flow behaviour through apertures of different sizes is quantified. For that purpose, the GranuFall apparatus (Aptis, Belgium, Figure 7.1c) is used. It consists of a hollow cylinder with an internal diameter of 36.3 mm in which we introduce 200 mL of powder. At the bottom of the cylinder, a plate prevents the powder from falling. At the beginning of the experiment, a plate with a centred orifice is moved below the cylinder allowing powder to flow out of the vessel. A height detector located at the top of the cylinder measures the distance of the powder-air interface as a function of time, allowing flow rate calculation in mL/s. Hole diameter ${\textstyle d_{h}}$ can be varied from 4 mm to 34 mm, with steps of 2 mm. In this study, for sugar powders the flow ${\textstyle f_{18}}$ at ${\textstyle d_{h}=18mm}$ and the last diameter where flow occurs ${\textstyle d_{h}^{min}}$ were measured.

Finally, the avalanche properties of powders are obtained using the Revolution powder analyzer (PS Prozesstechnik GmbH, Switzerland, Figure 7.1d). It consists in observing the avalanche movement of a powder (100 mL) in a rotating cylinder with glass walls. Rotation speed is set to 0.3 rpm. The image analysis of the pictures acquired by the CCD camera allows extracting information about powder flowability properties such as avalanche energy ${\textstyle E_{a}}$ , avalanche time ${\textstyle t_{a}}$ , avalanche angle ${\textstyle A_{a}}$ , rest angle ${\textstyle R_{a}}$ or surface linearity ${\textstyle S_{a}}$ by averaging those values over 150 avalanche events.


(a)	(b)	(c)

(d)
Figure 7.1: Flowability methods used in this study. (a) Free-flow density cylinder. (b) Angle of repose apparatus. (c) GranuFall. (d) Revolution powder analyzer. Courtesy of Nestlé

7.2.3 Experimental results

7.2.3.1 Carr Index and bulk density

Density measurements on powder samples allowed collecting compressibility data. Bulk densities and Carr indexes are summarized in Figure 7.2. For sucrose, ${\textstyle D_{free}}$ is found to increase with particle size. A range from 500 g/L at small diameters to approximately 850g/L at larger diameter is observed in Figure 7.2. Samples with larger span tend to have larger bulk densities as demonstrated by the sample S1S3 (outlier at 280 ${\textstyle \mu m}$ ). ${\textstyle I_{C}}$ follows a 1/x trend and reaches the value of 7 at large diameters which corresponds to good flowability according to literature [228]. Then, the compressibility value starts to increase around 250 ${\textstyle \mu m}$ and reaches the maximum value of 30, which corresponds to poor flow.

Figure 7.2: Free-flow density and Carr Index results. Black and red solid lines are guides to the eye, respectively for density and Carr Index. Courtesy of Nestlé.

Table. 7.2 Name, Carr Index and Bulk density [g/L] measured in this study. *Courtesy of Nestlé*.
Reference name	Carr Index	Bulk density [g/L]
S1	7.1	877
${\textstyle S1_{a}}$	6.8	826
${\textstyle S1_{b}}$	7.4	838
${\textstyle S1_{c}}$	7.8	848
S2	15	746
S3	12.6	804
${\textstyle S3_{a}}$	8.7	834
${\textstyle S3_{b}}$	14.7	768
S4	30.7	585
${\textstyle S4_{a}}$	16.9	727
${\textstyle S4_{b}}$	29	494
S1S3	9.4	912
S2S3	13.8	775
S1S4	29.2	695

7.2.3.2 Angle of repose

The values of the angle of repose (AR) measured for sugars are summarized in Figure 7.3, where the angle is plotted as a function of the particle size (represented here with ${\textstyle d_{50}}$ ). It is observed that the smaller the particle size, the higher the angle of repose. A plateau-like part appears at large diameters around 35 ${\textstyle ^{\circ }}$ , which correspond to a good flowability according literature [229], while a strong increase of the AR is observed at small diameters. The increase starts around 250-350 ${\textstyle \mu m}$ and leads to 55 ${\textstyle ^{\circ }}$ at the smallest diameters, corresponding to poor flow.

Figure 7.3: Angle of repose results. Dashed lines are guides to the eye. Courtesy of Nestlé.

7.2.3.3 GranuFall

GranuFall experiments for sugar powders are summarized in Figure 7.4. We plot ${\textstyle f_{18}}$ and ${\textstyle d_{h}^{min}}$ as a function of the particle size ${\textstyle d_{50}}$ . It is observed that the powder does not flow for fine powders while the flow suddenly steps up to 70 mL/s around ${\textstyle d_{50}}$ =200 ${\textstyle \mu m}$ before slowly diminishing down to 60 mL/s at larger diameters. One outlier with smaller flow is observed (sample S1S3), the same than in the density graph. Measurements of the critical diameter for flow show an opposite behavior. For fines, ${\textstyle d_{h}^{min}}$ is larger than 34 mm (represented by an arbitrary point at 36 mm since powder did not flow at the maximum available hole diameter). Then, ${\textstyle d_{h}^{min}}$ decreases with particle size probably reaching a minimum value below 4 mm which could not be determined with the apparatus (4 mm is the minimum available hole diameter).

Figure 7.4: GranuFall measurements on sugar powders. Flow at 18 mm diameter and minimum diameter to flow, versus particle size

. Dashed lines are guides to the eye. Courtesy of Nestlé.

7.2.3.4 Revolution Powder Analyzer

Finally, the last technique for flow characterization allowed obtaining the results presented in Figure 7.5. In this case, the results are presented in terms of angle of avalanche ${\textstyle A_{a}}$ , corresponding to the angle between the powder-air interface and the horizontal plane before the avalanche occurrence, and the rest angle, measured immediately after the avalanche. Typically, ${\textstyle A_{a}}$ varies from 30 ${\textstyle ^{\circ }}$ to 100 ${\textstyle ^{\circ }}$ , from free flow to very poor flow. By observing Figure 7.5, in the case of sucrose, it is evident that both the avalanche and the rest angle are strongly influenced by the particle size below 300 ${\textstyle \mu m}$ ; while above this critical value a plateau-like trend is noted for both the parameters under study.

Figure 7.5: Revolution powder analyzer data. Sugar avalanche and rest angle, as a function of particle diameter

. Dashed lines are guides to the eye, respectively for avalanche and rest angle. Courtesy of Nestlé.

In Table 7.3, the main experimental results in term of angle of repose, flow rate ${\textstyle f_{18}}$ and avalanche angle are resumed. All the data are listed in descending order of ${\textstyle d_{50}}$ .

Table. 7.3 Name, $d_{50}[\mu m]$ , Angle of repose [ $^{\circ }$ ], $f_{18}[mL/s]$ and Avalanche angle [ $^{\circ }$ ] measured in this study. *Courtesy of Nestlé*.
Reference name	$d_{50}[\mu m]$	Angle of repose [ ${\textstyle ^{\circ }}$ ]	flow rate ${\textstyle f_{18}[mL/s]}$	Avalanche angle [ ${\textstyle ^{\circ }}$ ]
${\textstyle S1_{a}}$	994.4	36.1	59.55	41.2
${\textstyle S1_{b}}$	632.9	34.8	64.55	40.7
S1	532.8	34.4	64.00	39.4
${\textstyle S1_{c}}$	372.3	35.0	69.35	40.8
S1S3	279.3	38.5	57.90	43.6
${\textstyle S3_{a}}$	260.8	38.8	70.05	40.0
S2S3	194.6	42.6	69.45	54.7
S3	193.9	42.3	64.15	51.3
S2	182	41.3	39.55	57.3
${\textstyle S3_{b}}$	154.1	43.5	62.65	55.7
${\textstyle S4_{a}}$	145.5	42.5	0.00	61.2
S1S4	86.1	54.1	0.00	67.1
S4	53.6	54.7	0.00	69.1
${\textstyle S4_{b}}$	31.8	53.8	0.00	61.8

7.3 Numerical study

As it can be observed in Section 7.2, the experimental results of the laboratory tests, performed for the assessment of the flow performance of sucrose, show a strong relation with the particle size. A clear trend is outlined: decreasing the size of particles, the flow quality tends to be more poor. In a continuum mechanics approach, using a standard Mohr-Coulomb law (see Chapter 3), the set of material parameters needed does not include micro-scale parameters, such as, e.g., the particle size. In the validation study, macro-parameters are considered, instead, and used for the numerical investigation of flowability properties of different types of sugar.

It is observed that ${\textstyle d_{50}=200\mu m}$ represents a critical value which markedly characterizes the flowability properties of sucrose. In this respect, in Table 7.3 a classification of the experimental results can be found; three groups are individuated depending on whether the corresponding ${\textstyle d_{50}}$ is above (CASE 1), around (CASE 2) or below (CASE 3) the value of ${\textstyle 200\mu m}$ . The classification is made by considering only those samples showing the behaviour in line with the previously observed trends. For this reason, the outliers are not taken into account. In what follows, the groups, listed in Table 7.3, are used as guideline to the numerical study of the different behaviours observed.

Table. 7.4 Classification of flowability behaviours

CASE

Bulk density

Repose angle

Avalanche

angle

CASE 1

CASE 2

CASE 3

Bulk density, ${\textstyle \rho _{bulk}}$ (Figure 7.2), angle of repose ${\textstyle \phi }$ (Figure 7.3) and angle of avalanche ${\textstyle A_{a}}$ (Figure 7.5) are monotonic with ${\textstyle d_{50}}$ . The flow through the orifice with a diameter of 18 mm ${\textstyle f_{18}}$ (Figure 7.4) is monotonic with ${\textstyle d_{50}}$ , as well. However, it can be seen that the evolution of the flow rate looks like a step-wise function, varying between 0 mL/s for values of ${\textstyle d_{50}}$ below ${\textstyle 200\mu m}$ and 70 mL/s for values of ${\textstyle d_{50}}$ just above the critical value of a few ${\textstyle \mu m}$ . In this regard, for example, by comparing the samples ${\textstyle S3_{b}}$ and ${\textstyle S4_{a}}$ , which have similar Particle Size Distribution, but a slightly different ${\textstyle d_{10}}$ ( ${\textstyle 55.3\mu m}$ and ${\textstyle 38\mu m}$ , respectively), it is observed that ${\textstyle f_{18}=62mL/s}$ in the first case while no flow occurs in the second one. There might be the possibility that, when ${\textstyle d_{50}\simeq 200\mu m}$ , other parameters turn out to be leading factors in the determination of the flow rate in a flat bottom silo, such as, the ${\textstyle d_{10}}$ or the particle shape. Thus, for a better understanding of the physical mechanisms and physical parameters which can determine the flowability of a certain granular material, further experimental tests should be planned and performed, as future work, which include other factors not considered in the current study. Moreover, since a remarkable difference is observed in the flow rate values for sugar samples with a ${\textstyle d_{50}}$ close to ${\textstyle 200\mu m}$ , only the samples with a flow rate which ranges between ${\textstyle 60mL/s}$ and ${\textstyle 70mL/s}$ fall under CASE 2.

7.3.1 Numerical models

In this section, the computational models of angle of repose apparatus, GranuFall and Revolution Powder Analyzer, are presented. By referring to the description of the devises in Section 7.2.2, models which use axi-symmetry and plane strain assumptions are introduced. For the solution of the axisymmetric problem, the formulation presented in Appendix C is employed, while in the plane strain case, the formulation presented in Chapter 4 is considered.

7.3.1.1 Angle of repose

With respect to the angle of repose device, described in Section 7.2.2, the geometry of its numerical model is depicted in Figure 7.6a; while boundary conditions are depicted in Figure 7.6b where the fixed displacement along x-direction is represented in blue colour and in red the one along both x and y-directions. As it can be seen, the 3D geometry is reduced to a 2D axisymmetric model, in order to reduce the computational cost of the numerical simulation. The gray coloured area of Figure 7.6a refers to the domain initially occupied by the material points, while in yellow is the remaining area of the computational domain, where the particles are free to move. For the solution of this case an unstructured triangular mesh with element size of 1mm is adopted. The time step length can range between ${\textstyle 10^{-4}s}$ and ${\textstyle 10^{-5}s}$ , depending on the material properties adopted in each test case, and 6 material points are initially located in each element of the grid, which is found to be an optimal trade-off between accuracy of the results and computational cost.


(a)	(b)
Figure 7.6: Angle of repose apparatus: geometry (a) and boundary conditions (b)

This test is the most frequently used for different sizes and combinations of funneling methods (e.g., internal flow funnel and external flow funnel or a combination of both). In this case, with the method described in Section 7.2.3.2 it is measured the so-called external or poured angle of repose. In the literature, the angle of repose is often assumed to be equal to the residual internal friction angle or the constant volume angle in a critical state [230]. However, this assumption is valid under very restricted conditions and assumptions, as shown in [231], such as, uniform density, moisture content and particles size.

It is important to highlight that, in the experimental characterization, a device, which allows the measurement of both fluid and sticky powders, is employed. In the numerical analysis of cohesive powders, if the original geometry is adopted (see Figure 7.6a), with a very narrow outlet diameter, some phenomena of arching, which do not allow to complete the analysis, are observed. In order to overcome this aspect, the minimum diameter of the opening is found and used for the evaluation of the angle of repose.

7.3.1.2 GranuFall

In this section, the numerical model of the second device, used in the experimental study, is presented. By taking into consideration the description made in Section 7.2.2, the geometry is shown in Figure 7.7a; while the boundary conditions are depicted in Figure 7.7b where the fixed displacement along x-direction is represented in blue colour and in red the one along both x and y-directions. As done for the angle of repose model of Figure 7.6a, the 3D geometry is reduced to a 2D axisymmetric model. For the solution of this case an unstructured triangular mesh with element size of 1mm is adopted, the time step length can range between ${\textstyle 10^{-4}s}$ and ${\textstyle 10^{-5}s}$ , depending on the material properties adopted in each test case, and 6 material points are initially located in each element of the grid, which is found to be an optimal trade-off between accuracy of the results and computational cost.


(a)	(b)
Figure 7.7: Granufall apparatus: geometry (a) and boundary conditions (b)

7.3.1.3 Revolution Powder Analyzer

Finally, the model of the Revolution Powder Analyzer is introduced. According to the description of the device in Section 7.2.2, the three-dimensional geometry is reduced to a 2D plane strain model, represented in Figure 7.8. The area coloured in gray refers to the domain initially occupied by the material points, while in yellow is the remaining area of the computational domain, where the particles are free to move. At the boundary of the rotating drum, prescribed displacement along the x and y direction ( ${\textstyle u_{x}}$ and ${\textstyle u_{y}}$ ) is applied in order to impose the rotational motion at an angular velocity of ${\textstyle \omega =0.3[rpm]={\frac {\pi }{100}}[rad/s]}$ :

(7.1)

with ${\textstyle \theta =\omega t[rad]}$ , ${\textstyle \alpha ={\frac {\pi -\theta }{2}}[rad]}$ and ${\textstyle \phi =2\arcsin({\frac {chord}{2R}})[rad]}$ , where R is the radius of the drum and ${\textstyle chord}$ the distance between the point where the displacement is calculated (point P) and a reference point (point O), as depicted in Figure 7.8. For the solution of this case an unstructured triangular mesh with element size of 2mm is adopted, with a time step of ${\textstyle 10^{-5}s}$ and 6 material points are initially located in each element of the grid, which is found to be an optimal trade-off between accuracy of the results and computational cost.

Figure 7.8: Revolution Powder Analyzer: geometry.

Depending on angular velocity, diameter of the cylinder, filling level, friction between particles and the wall, and particle material characteristics, different regimes of granular flow are observed which have been termed slipping, slumping, rolling, cascading, cataracting, and centrifuging (see Figure 7.9). Except for slipping where the granulate assumes a solid state and slides against the wall, these regimes can be observed in dependence on rotation velocity while all other parameters are fixed. For low angular velocity, as in the present case, the flow is non-stationary in a slumping mode. Accordingly, the free surface of the granulate forms a plane whose tilt fluctuates between the avalanche angle and the rest angle, e.g., between the state of maximum and minimum potential energy, respectively.

Figure 7.9: Modes of motion in a Revolution Powder Analyzer [232].

7.3.2 Numerical results

In this section, a validation study is presented by comparing the numerical and experimental results in terms of angle of repose, flow rate and avalanche angle, for CASE 1, CASE 2 and CASE 3. The problem is solved at the macroscale and a study of the influence of the Mohr-Coulomb parameters, such as, the internal friction angle ${\textstyle \phi }$ , the dilatancy angle ${\textstyle \psi }$ and the apparent cohesion ${\textstyle c}$ , on the sugar flowability performance is provided according to the classification previously performed. The internal friction angle describes the bulk friction during the incipient flow of a powder and it is determined from the linearised yield locus, as shown in Figure 3.2; while the apparent cohesion is the intercept from the linearised yield locus and it represents the strength of a powder under zero confining pressure. Even if these material parameters are representative of the behaviour of the bulk, in reality, they can be related to micro-scale parameters, such as, the inter-particle friction and the inter-particle cohesion, respectively. It is known that the inter-particle friction is due to the interlocking generated by the shape of the particles and the surface roughness; while the inter-particle cohesion, in the case of dry powder, is due to the van der Waals forces [233]. The dilatancy is representative of the volume change of a granular material when it undergoes a shear deformation, before reaching the critical state. If the material is compacted, the grains are interlocked and do not have the freedom to move; however, when the sample is stressed, a lever motion occurs between the particles in contact and a bulk expansion of the material is generated.

Since all the experimental results are single-valued function of ${\textstyle d_{50}}$ , the particle size corresponding to 50% of the sample's mass sieved, a classification is made according to this parameter. CASE 1 corresponds to samples with the highest values of ${\textstyle d_{50}}$ , typical of the common granulated sugar type. On the other hand, much lower values of ${\textstyle d_{50}}$ fall under CASE 3 and these samples are associated with a powdered sugar type. Finally, with CASE 2 the case of transition, where flowability properties of sucrose drastically deteriorate in the transition from CASE 1 to CASE 3, is individuated.

In what follows, only the influence of the internal friction angle, dilatancy angle and apparent cohesion, is investigated. The ranges of bulk density value are provided by the experimental results (see Table 7.2), while other material parameters, such as, the Young modulus and Poisson's ratio, are considered to be the same in all three cases. Their values have been found in the literature [234].

7.3.2.1 Case 1: d₅₀ > 200 μm

CASE 1 represents all those samples which are characterized by a ${\textstyle d_{50}}$ much higher than ${\textstyle 200\mu m}$ . The granular material, object of study, is in dry condition and in this case the amount of fine particles is so low that cohesive forces do not appear, and the bulk behaviour results to be totally cohesionless [234]. Thus, a zero apparent cohesion is used in all the investigated scenarios. On the other hand, internal friction angle and dilatancy angle may play a role in the flowability performance. In this regards, different cases have been investigated varying the values of ${\textstyle \phi }$ and ${\textstyle \psi }$ . For CASE 1, a bulk density value of ${\textstyle 830kg/mc}$ , for all the scenarios to be investigated, is chosen following the classification made in Table 7.4.

The first experimental test to be analysed is the GranuFall test. Different values of ${\textstyle \phi }$ and ${\textstyle \psi }$ are considered: the internal friction angle ranges between ${\textstyle 35^{\circ }}$ and ${\textstyle 46^{\circ }}$ , while the dilatancy angle between ${\textstyle 0^{\circ }}$ and ${\textstyle 5^{\circ }}$ . In Figure 7.10 it can be observed that by increasing the dilatancy angle by a few degrees, the flow rate through an orifice of 18mm drastically decreases. In particular, it is found that, for values of ${\textstyle \psi }$ between ${\textstyle 1^{\circ }}$ and ${\textstyle 3^{\circ }}$ , values of volumetric flow rate in the range of the experimental measurements are computed. These latter ones are depicted by the dot lines (Figure 7.10), that, hereinafter, they are used to represent the inferior and superior limit of the experimental range, as indicated in Table 7.4.

Figure 7.10: Case 1. GranuFall test: volumetric flow.

The second model to be investigated, in this validation study, is represented by the test of the angle of repose. The values of internal friction angle range between ${\textstyle 35^{\circ }}$ and ${\textstyle 46^{\circ }}$ ; while the values of dilatancy angle are constrained to the values of ${\textstyle 1^{\circ }}$ , ${\textstyle 2^{\circ }}$ and ${\textstyle 3^{\circ }}$ . In Figure 7.11, the results in terms of angle of repose are presented. In this case, it is found that the dilatancy angle has not an important influence on the results; while a remarkable difference is given by the internal friction angle. It is observed that to higher values of ${\textstyle \phi }$ corresponds higher values of angle of repose and, as previously discussed, it is found that the angle of repose does not coincide with the values of internal friction angle. Further, it is seen that the numerical values are close to the experimental ones, depicted by the dot lines, representing the inferior and superior limit, as indicated in Table 7.4, when the value of ${\textstyle \phi }$ ranges between ${\textstyle 42^{\circ }}$ and ${\textstyle 46^{\circ }}$ .

Figure 7.11: Case 1. Angle of repose test: angle of repose.

The last test is represented by the Revolution Powder Analyzer, described in Section 7.3.1.3. In this case, only the scenarios with a dilatancy angle of ${\textstyle 1^{\circ }}$ , ${\textstyle 2^{\circ }}$ and ${\textstyle 3^{\circ }}$ and an internal friction angle which ranges between ${\textstyle 42^{\circ }}$ and ${\textstyle 46^{\circ }}$ are considered. The results in terms of avalanche and rest angle are shown in Figures 7.12a and 7.12b, respectively. It is found that the internal friction angle has some influence on the avalanche angle, while it is not observed a strong relation with the rest angle, since, in all the scenarios investigated, the value falls in a very narrow range: between ${\textstyle 31^{\circ }}$ and ${\textstyle 33^{\circ }}$ . With regards to the dilatancy angle, this parameters has less influence than the internal friction angle on the numerical results. Thus, for the range of ${\textstyle \psi }$ under study it is not possible to define any relation.


(a)	(b)
Figure 7.12: Case 1. Revolution Powder Analyzer: avalanche angle results (a) and rest angle results (b)

By looking at the results obtained with the three different models, the set of material parameters, which provide numerical results which fall into the range of the experimental data for CASE 1, is found and listed in Table 7.5

Table. 7.5 Case 1. Set of material properties.

Young

Modulus

Poisson

ratio

Density

Internal friction

angle

Dilatancy

angle

Cohesion

1e6 Pa

0.3

830 kg/mc

44 deg

2-3 deg

0 Pa

In what follows, the results obtained by using the set of material parameters of Table 7.5 are shown. In Figures 7.13a, 7.13b and 7.13c the repose angle, the avalanche and the rest angle, evaluated in the Revolution Powder Analyzer, are depicted. It can be noted that the surfaces of the heap formed by the angle of repose of Figure 7.13a and the material at rest after the avalanche of Figure 7.13c are very smooth. With respect to the Revolution Powder Analyzer, an avalanching shear layer is interacting with a quasi-static region. In Figure 7.14, it is possible to observe that the surface flowing layer has a reduced depth in comparison to the quasi-static one, and, consequently, the mass moving during the collapse, as well.


(a)	(b)	(c)
Figure 7.13: Case 1. Numerical results of repose angle test (a) and Revolution Powder Analyzer: before collapse (avalanche angle) (b) and after collapse (rest angle) (c). Black dot lines indicate the angle formed by the inclined surfaces.


(a) t=24.6s	(b) t=24.7s

(c) t=24.8s	(d) t=24.9s
Figure 7.14: Case 1. Revolution Powder Analyzer: velocity field at different time instants.

7.3.2.2 Case 2: d₅₀ ≃200 [μm]

CASE 2 represents the transition between CASE 1 and CASE 3, where the first signs of degradation of the flowability performance are observed. As it is observed in the experimental characterization, described in Section 7.2, these types of sugar are characterized by a median particle size, ${\textstyle d_{50}}$ , which is around the critical value of ${\textstyle 200\mu m}$ . In this case, it is assumed that the internal friction angle is still a material parameter which can affect the material behaviour; on the other hand, unlike CASE 1, a zero dilatancy angle is considered due to the reduced particle size (as shown in [234]). Moreover, a reduction of the particle size generates an increase of inter-particle cohesion and, consequently, of the apparent cohesion [233]. Thus, ${\textstyle c}$ is included in the set of parameters of the numerical study. For CASE 2, a bulk density value of ${\textstyle 740kg/mc}$ , for all the scenarios to be investigated, is chosen following the classification made in Table 7.4. With respect to the internal friction angle and the apparent cohesion, the ranges ${\textstyle 35^{\circ }-47^{\circ }}$ and ${\textstyle 2Pa-18Pa}$ are considered, respectively.

The first model, object of study, is the GranuFall test. Different scenarios are analysed and the numerical results in terms of volumetric flow are shown in Figure 7.15. In this case, only the results, which fall into the range defined by the experimental measurements, are depicted. As can be observed, the greater the apparent cohesion, the lower the volumetric flow. Moreover, increasing the internal friction angle, the curve which describes the relation between the volumetric flow and the apparent cohesion, is shifted to the left part of the chart. Namely, this result represents the competition between the inter-particle cohesion and the inter-particle friction in the bulk friction behaviour of those samples which belong to CASE 2.

Figure 7.15: Case 2. GranuFall test: volumetric flow.

The same competition between the internal friction angle and the apparent cohesion is observed in Figure 7.16, where the results of the angle of repose test, which fall in the range of the experimental data, are shown. It is found that the value of the angle of repose is affected by both ${\textstyle \phi }$ and ${\textstyle c}$ : it increases when higher values of ${\textstyle \phi }$ and ${\textstyle c}$ are adopted. Unlike CASE 1, in this case it has been necessary to modify the model, described in Section 7.3.1.1, in order to avoid phenomena of arching, which do not allow to complete the analyses. In CASE 2, a variation of the opening diameter of a few ${\textstyle mm}$ is made, with the aim of reproducing the same dynamics during the pouring of the granular material, as in the experimental test.

Figure 7.16: Case 2. Angle of repose test: angle of repose.

Finally, the model of Revolution Powder Analyzer is analysed. The results in terms of avalanche and rest angle are shown in Figures 7.17a and 7.17b, respectively. It is found that higher avalanche angles correspond to higher values of internal friction angle and apparent cohesion. Moreover, these results linearly depends on the apparent cohesion and the angular coefficient increases with the ${\textstyle \phi }$ . The same observations can be done concerning the rest angle. In both the charts of Figure 7.17, one can note that the same final results can be achieved with different sets of ${\textstyle \phi }$ and ${\textstyle c}$ . Indeed, in this case there is not a unique set of material parameters which allows to predict the experimental data, but rather two sets are individuated and they are listed in Table 7.6.


(a)	(b)
Figure 7.17: Case 2. Revolution Powder Analyzer: avalanche angle results (a) and rest angle results (b)

Table. 7.6 Case 2. Sets of material properties.

Young

Modulus

Poisson

ratio

Density

Internal friction

angle

Dilatancy

angle

Cohesion

1e6 Pa

0.3

740 kg/mc

39 deg

0 deg

16 Pa

1e6 Pa

0.3

740 kg/mc

47 deg

0 deg

10 Pa

In addition, the numerical results obtained with an internal friction angle of ${\textstyle \phi =47^{\circ }}$ and an apparent cohesion of ${\textstyle c=10[Pa]}$ are shown in Figures 7.18a, 7.18b and 7.18c, where the angle of repose, the avalanche and rest angle, evaluated in the Revolution Powder Analyzer, are represented. It can be observed that the surface smoothness in Figure 7.18a is retained; indeed, it is quite evident and easy to recognize the angle of repose, formed by the heap. On the other hand, in Figure 7.18c the surface is not completely smooth due to the cohesive behaviour shown in CASE 2. In addition, with respect to the Revolution Powder Analyzer, by observing Figure 7.19 the depth of the avalanching shear layer has increased, along with the amount of mass which is flowing during the collapse.


(a)	(b)	(c)
Figure 7.18: Case 2. Numerical results of repose angle test (a) and Revolution Powder Analyzer: before collapse (avalanche angle) (b) and after collapse (rest angle) (c). Black dot lines indicate the angle formed by the inclined surfaces.


(a) t=26.4s	(b) t=26.6s

(c) t=26.8s	(d) t=27s
Figure 7.19: Case 2: Results of rotating drum test. Velocity field at different time instants.

7.3.2.3 Case 3: d₅₀ < 200 μm

CASE 3 is representative of all those samples which show a very poor flowability performance. These granular materials are characterized by a ${\textstyle d_{50}<200\mu m}$ and by a high amount of fine particles, which make them very sticky and cohesive. When the particles are small, the inter-particle cohesion dominates the flow behaviour and enhances the shear resistance. In addition, these dry cohesive interactions result in the formation of clusters, which generate many voids within the bulk, thus, at the macroscale, resulting in a low bulk density. If the bulk density is low, there are free spaces for the particles to move and the geometrical inter-locking does not play an important role in this case. For this reason, a zero dilatancy angle is considered due to the reduced particle size (as shown in [234]). On the other hand, several scenarios are investigated where a different set of internal friction angle and apparent cohesion are considered. For CASE 3, a bulk density value of ${\textstyle 600kg/mc}$ , for all the scenarios to be investigated, is chosen in accordance with the classification made in Table 7.4. With respect to the internal friction angle and apparent cohesion, values which range between ${\textstyle 35^{\circ }-45^{\circ }}$ and ${\textstyle 0Pa-40Pa}$ are considered, respectively.

As done already in the previous cases, the first test case to be considered is the GranuFall test. According to the experimental results listed in Table 7.3, in this case no volumetric flow takes place. In this regard, from the numerical simulation it is observed that the minimum value of apparent cohesion for which no flow is observed corresponds to ${\textstyle 20Pa}$ . With regard to the internal friction angle, no influence is noted in the numerical solutions.

The second model, under study, is represented by the angle of repose test. In this case, the value of apparent cohesion is high enough that it is not possible to perform the simulation with the original model, described in Section 7.3.1.1. This is due to the fact that the material is very cohesive and the opening diameter is too narrow in order to see the material starting flowing. In the simulation of this case, the diameter has to be increased of several times its original dimension; however, it is observed that the material is flowing down at a very high velocity generating a mismatch with the dynamics, which have taken place during the experimental test. This can create a variation in the final numerical solution and, for this reason, it is decided that the results obtained from this model are not reliable and representative of the physical process, and, consequently, they are not included in this validation study.

Finally, the third model, to be considered, is the Revolution Powder Analyzer. In this case, the internal friction angle ranges between ${\textstyle 35^{\circ }}$ and ${\textstyle 45^{\circ }}$ , while the apparent cohesion between ${\textstyle 20Pa}$ and ${\textstyle 40Pa}$ . In Figures 7.20a and 7.20b the numerical results in terms of avalanche and rest angle are represented, respectively. With regard to the avalanche angle, it is seen that the internal friction angle has more influence in correspondence of lower values of cohesion; while the apparent cohesion has a strong influence in all the scenarios considered and a linear relation is observed, with an angular coefficient inversely proportional to the internal friction angle value. With respect to the rest angle, it seems that ${\textstyle \phi }$ does not have any influence on the final results; on the other hand, a linear relation between the rest angle and the apparent cohesion is noted, with an angular coefficient common to all the investigated cases.


(a)	(b)
Figure 7.20: Case 3. Revolution Powder Analyzer: avalanche angle results (a) and rest angle results (b)

As can be observed in Figure 7.20, the set of material parameters, which are able to provide results in the ranges defined by the experimental data, represented by the dot lines, are listed in Table 7.7. It is found that, in CASE 3, the behaviour of the samples are not affected by the internal friction angle, but are mostly influenced by the apparent cohesion.

Table. 7.7 Case 3. Set of material properties.

Young

Modulus

Poisson

ratio

Density

Internal friction

angle

Dilatancy

angle

Cohesion

1e6 Pa

0.3

600 kg/mc

35-45 deg

0 deg

40 Pa

In what follows, the results obtained by using the set of material parameters of Table 7.7 are shown. In Figures 7.21a and 7.21b the avalanche and the rest angle, evaluated in the Revolution Powder Analyzer, are depicted. It can be noted that the results of CASE 3 are not characterized by the surface smoothness, observed in the previous cases, due to the sticky behaviour of the material. Moreover, in Figure 7.22 it is possible to observe a flowing layer with a depth comparable to the quasi-static one and a mass which is moving in a very cohesive way.


(a)	(b)
Figure 7.21: Case 3. Numerical results of Revolution Powder Analyzer: before collapse (avalanche angle) (a) and after collapse (rest angle) (b). Black dot lines indicate the angle formed by the inclined surfaces.


(a) t=30.8s	(b) t=30.9s

(c) t=31s	(d) t=31.1s
Figure 7.22: Case 1: Results of rotating drum test. Velocity field at different time instants.

7.4 Discussion

In this Chapter, an application of the MPM strategy in the industrial framework is presented. The main aim of this study is the experimental and numerical characterization of flowability of different sugar powders. It is experimentally found that the flowability performance strictly depends on the particle size distribution, and, in particular, a strong relation with ${\textstyle d_{50}}$ is shown. In the numerical study, a macroscopic approach is followed. The phenomenological Mohr-Coulomb plastic law is employed and the study is performed in order to find a correlation between the behaviour of different types of sugar and the macroscopic material parameters of internal friction angle, dilatancy angle and apparent cohesion. The numerical solutions have shown that, depending on the class of flowability, the bulk friction behaviour can be influenced by different parameters. In those samples characterized by a good flowability and high values of ${\textstyle d_{50}}$ (CASE 1), it is numerically found that the parameters, which mainly affect the shear resistance, are represented by the internal friction angle and dilatancy angle; this is confirmed by those mechanisms which have been observed to take place at the microscale: the shear resistance is due to interlocking generated by the shape of the particles and their surface roughness. For those samples with very poor flowability and a sticky behaviour (CASE 3), a ${\textstyle d_{50}}$ much lower than ${\textstyle 200\mu m}$ is experimentally estimated. In this case, it is numerically demonstrated that the apparent cohesion is the parameter which mostly governs the macroscopic behaviour of the material; while, at the microscale, due to the high amount of fine particles, cohesive forces appear between adjacent granules and, consequently, the inter-particle cohesion is the main mechanism in the shear resistance. In the case of transition (CASE 2), where the first signs of a deterioration of the flowability performance are visible, it is numerically observed that the results are affected in equal measure by the internal friction angle and apparent cohesion. This can be traduced, at the microscale, in a competition between inter-particle friction and inter-particle cohesion; this might be due to the presence of fine particles, even if their quantity is limited. In conclusion, all the numerical solutions, presented in the numerical study of Section 7.3, as demonstrated, are representative of the mechanisms which are observed to take place at the particle level and the observations made are representative and totally in line with the experimental results of Section 7.2.

8 Conclusions and future work

In this Chapter, the conclusions of the monograph are presented and an overview of the future lines of research is made.

8.1 Concluding remarks

The main aim of this work was the development of a numerical strategy for the simulation of quasi-static and dense granular flows in the industrial and engineering framework. These kinds of problems are characterized by a non-linear behaviour of the material and by large deformation of the continuum during the whole flow process. In order to perform a numerical investigation, a strategy, which is able to consider these non-linearities, is needed. It is found that, among all the numerical methods, mostly used for the solution of granular flows problems, the Material Point Method (MPM) is the one which has shown the most suited capabilities for the cases targeted to study in this monograph (Chapter 2). In the current work, an implicit MPM has been developed by the author in the multi-disciplinary Finite Element codes framework Kratos Multiphysics [29,30,31] and in order to obtain a verified and validated numerical strategy the following points have been performed (Chapters 3, 4, 5 and 6):

Three phenomenological constitutive laws, implemented within the MPM numerical strategy, are introduced and their formulations are derived. In particular, a hyperelastic Neo-Hookean, a hyperelastic-plastic J2 and Mohr-Coulomb plastic laws, along with their limits of applicability, are discussed. All the constitutive materials are defined under the assumption of isotropy and finite strains. With the object of the current monograph in hand, the focus is mainly on the Mohr-Coulomb plastic law, where in this work, to the knowledge of the author, the return mapping is used for the first time under finite strain assumption. This phenomenological law is commonly adopted in the modelling of granular materials, since it shows a pressure-dependent behaviour. This constitutive law is used in the verification, validation and application of the MPM strategy.

A variational displacement-based formulation, based on an Updated Lagrangian description, is presented and its derivation is described in detail. A verification of the MPM code is performed through some benchmark tests, typical in solid and geo-mechanics. Moreover, in the verification analysis, a comparison of the MPM code is done against the Galerkin Meshfree Method (GMM), a continuum particle-based technique. Since both the methods are implemented in the Kratos Multiphysics platform, it has been possible to perform a more objective comparison, which allows to better appreciate the main differences between these two techniques. In GMM the Eulerian background grid is replaced by a Lagrangian one and, unlike MPM, the shape functions are evaluated once the cloud of nodes of each material point is defined. The comparison is made with the aim of assessing the accuracy and robustness of the two methods in the simulation of cohesive-frictional materials, both in static and dynamic regimes and in problems dealing with large deformations. It is found that MPM leads to more accurate results and its robustness is proven. On the other hand, it is observed that the accuracy of GMM strictly depends on the choice of the basis functions and a modification of the algorithm has to be considered in large displacement cases. After this study, it is demonstrated that the MPM strategy represents a good choice to handle problems involving history-dependent materials and large deformations.

A variational displacement and pressure-based ${\textstyle {\boldsymbol {u}}-p}$ formulation, based on an Updated Lagrangian description, is presented and its derivation is described in detail. The ${\textstyle {\boldsymbol {u}}-p}$ mixed formulation is developed with the aim of solving granular flow problems which undergo nearly-incompressible conditions. To the knowledge of the author, the treatment of the incompressibility constraint is relatively new in the context of MPM and this formulation represents an original solution among the works that one can find in the literature. A verification of the MPM code is performed through some benchmark tests, typical in solid mechanics. In the verification analysis, the results obtained through the mixed formulation are compared with those evaluated by means of the displacement-based one. As expected, it is noted that, in nearly-incompressible conditions, the typical issue of volumetric locking is overcome and pressure oscillations are avoided with the ${\textstyle {\boldsymbol {u}}-p}$ formulation. In addition, it is found that also in compressible cases the ${\textstyle {\boldsymbol {u}}-p}$ formulation provides more accurate results than the irreducible one. However, despite a higher accuracy in terms of displacement, equivalent plastic strains and vertical Cauchy stress fields, the mixed formulation, presented in this work, is not able to fix some other issues, such as, mesh independence and strain localization.

The MPM strategy, developed in the Kratos Multiphysics platform, is employed to solve typical problems of geo-mechanics. Two problems are considered in the validation study: the first one is represented by the typical test of column collapse of a dry cohesionless granular material and the second one by the evaluation of the bearing capacity of an undrained soil in the rigid strip footing test. In the first case, the irreducible formulation is employed and the numerical results are compared against experimental results. Different geometries of the column are investigated. It is both experimentally and numerically observed that the dynamic of the collapse strictly depends on the initial geometry. It is seen that for lower aspect ratio the MPM code is able to provide results in agreement with the experimental ones. On the other hand, in the case of higher aspect ratio, the MPM strategy underestimates the dissipation which takes place during the collapse of the column. In this regard, the disagreement might be mostly due to the employed constitutive law: the Mohr-Coulomb plastic law adopted in the current work is not able to predict the real energy dissipation that should have taken place during the failure mechanism. Indeed, as indicated in some similar works available in the literature, the evolution of some factors, such as, density and dilatancy, which play important role in defining the first failure surface, and, hence, the total mass that will move, are not here considered. In the second test, the ${\textstyle {\boldsymbol {u}}}$ and ${\textstyle {\boldsymbol {u}}-p}$ formulations are both employed. It is observed that the mixed formulation is able to provide better results in terms of displacement and stress field. By sampling the computed stress field at the edge, where the imposed displacement of the rigid footing is applied, it has been possible to define the curve which describes the bearing capacity of the soil. This result is compared with a curve obtained from a sequential limit analysis and a good agreement is observed between the two solutions.

Finally, in Chapter 7 the MPM strategy is employed in an industrial framework, in the context of a collaboration with Nestlé. The numerical results are compared against unpublished experimental measurements performed for the assessment of the flowability performance of different types of sucrose. It is experimentally observed that the flowability performance is strictly dependent on the particle size distribution of the granular material and, in particular, on the median particle size ${\textstyle d_{50}}$ . On the other hand, the numerical study is performed by following a macroscopic approach and the flowability is studied according to macro-parameters, such as, the internal friction angle, the dilatancy angle and the apparent cohesion. In this study, the MPM strategy has been successfully employed and the advantage of its usage, as a complementary tool for a better understanding of the granular flow process, is demonstrated.

8.2 Future work

From the observation made and the conclusions drawn, the future lines of research are provided. It has been demonstrated that the MPM is not only a robust numerical tool for the simulation of problems involving large displacement and large deformation, but it is also an optimal platform for the implementation of complex constitutive laws. In this work, it is observed that the implemented Mohr-Coulomb plastic law is not able to predict the real energy dissipation that should have taken place during the failure mechanism. In the context of material modelling, a research should be focused on implementing constitutive laws with features, able to improve the prediction in terms of triggering of the collapse and amount of energy dissipated during the flow process. During the PhD, a collaboration with the Multiscale Mechanics group (MSM, University of Twente), lead by Prof. Stefan Luding, has started with the aim of implementing in the MPM strategy framework an elastic isotropic micro-based constitutive model for granular materials. This constitutive law, valid under quasi-static/elastic regime, has been developed by the researchers of the MSM group [235,236,237,238,239], by performing a series of DEM simulations of isotropic compression and pure shear tests. By averaging some microscopic quantities, that can be directly retrieved from the DEM simulations, macro parameters like stress and fabric ¹ tensors can be evaluated. Then, by applying the definition of bulk and shear moduli, their expressions as a function of micro parameters, such as the volume fraction, pressure and coordination number, are obtained. Thus, the elastic mechanical properties are not considered constant, as it is usually assumed in a phenomenological model, but they vary accordingly with the evolution of the micro-structure.

With regard to the numerical formulation, other mixed variational formulations, which allow to overcome the issues of mesh dependence and strain localization, can be considered to improve the accuracy of the results.

In addition, in this work the applicability of the developed MPM strategy is limited to the simulation of dry granular flows. Other applications, where MPM is still a suited choice, is represented by granular flows interacting with rigid or deformable structures. Some examples can be found in the field of environmental engineering, such as, landslides interacting with systems of protective barriers or the quantitative risk assessment in landslide prone-area. For the development of the numerical strategy, a coupling between the MPM code and a FEM code is needed. In this regard, a work, mainly focused on the imposition of boundary conditions and contact algorithm, is still missing. However, some promising results have been already done within the Kratos team and published in [240,241], where algorithms for the imposition of non-conforming boundary conditions and frictional contact are presented and tested.

Last but not least, the important aspect of parallelisation of the code, which is not addressed in the current monograph, has to be developed as future work. In the current state, the MPM strategy uses an OpenMP method, which is able to guarantee a sustainable computational cost for the simulation of real scale systems. However, in order to fully exploit the MPM capabilities and make the application competitive with other commercial and open-source software, a modification of the code in favour of a MPI parallelisation should be addressed.

(¹) The fabric tensors is a tensor able to provide information which characterize the anisotropic architecture of the microstructure in a porous material

Appendix A. Plastic flow rule in finite strains regime

In this section a general plastic flow rule within the framework of multiplicative plasticity is defined and it is demonstrated that if the specific strain energy function is expressed in terms of Hencky strains it is possible to recover the small strains format return mapping [155]. In order to formulate a plastic flow rule in finite strains regime, it is convenient to introduce the kinematic quantities of rate of plastic deformation ${\textstyle {\boldsymbol {D}}^{p}}$ and the plastic spin tensor ${\textstyle {\boldsymbol {W}}^{p}}$ , defined as

(A.1)

(A.2)

where ${\textstyle {\boldsymbol {L}}^{p}\equiv {\dot {\boldsymbol {F}}}^{p}\left({\boldsymbol {F}}^{p}\right)^{-1}}$ is the plastic part of the velocity gradient ${\textstyle {\boldsymbol {L}}\equiv \nabla _{x}{\boldsymbol {v}}}$ . Since ${\textstyle {\boldsymbol {D}}^{p}}$ is a kinematic variables defined in the intermediate configuration, it is useful to perform the following rotation of ${\textstyle {\boldsymbol {D}}^{p}}$ in order to express it in the spatial configuration:

(A.3)

with ${\textstyle {\boldsymbol {R}}^{e}}$ is the orthogonal tensor used in the polar decomposition

(A.4)

where ${\textstyle {\boldsymbol {V}}^{e}}$ is the left stretch tensor.

Given a general plastic potential ${\textstyle g({\boldsymbol {\tau }})}$ defined in terms of the Kirchhoff stress tensor ${\textstyle {\boldsymbol {\tau }}}$ and the rate of the plastic multiplier ${\textstyle {\dot {\gamma }}}$ , the evolution of the plastic deformation gradient is defined by the following constitutive equation:

(A.5)

by postulating a zero plastic spin ${\textstyle {\boldsymbol {W}}^{p}}$ , which is compatible with the assumption of plastic isotropy. With Equation A.3 in hand and the following property ${\textstyle {\boldsymbol {R}}^{e^{T}}={\boldsymbol {R}}^{e^{-1}}}$ , it is possible to rewrite the evolution equation as

(A.6)

In the definition of the plastic problem an isotropic perfectly plastic constitutive model is assumed. In this regard, the model is defined by postulating

a specific strain energy function ${\textstyle \Psi }$ , from which the hyperelastic law is derived;
a yield function ${\textstyle f}$ which defines when plastic flow starts;
a plastic potential ${\textstyle g}$ , from which the plastic flow rule is derived.

Thus, the basic constitutive initial value problem states: given an initial value of ${\boldsymbol {F}}^{p}$ at $t_{0}$ and the history of the deformation gradient ${\boldsymbol {F}}(t)$ for $t\in \left[t_{0},T\right]$ , find the functions ${\boldsymbol {F}}^{p}(t)$ and ${\dot {\gamma }}(t)$ which satisfy the flow rule

(A.7)

and the Kuhn-Tucker loading/unloading conditions

(A.8)

with ${\textstyle f({\boldsymbol {\tau }})}$ the yield function and ${\textstyle {\boldsymbol {\tau }}(t)}$ defined as

(A.9)

where ${\textstyle {\boldsymbol {\epsilon }}^{e}(t)}$ is the elastic Hencky strain.

The algorithmic procedure to be established in order to solve the plastic problem is based on the discrete form of the evolution equation (see Equation A.7). Accordingly, a time stepping algorithm is performed by applying a backward exponential integrator on Equation A.7, which leads to the updated formula for the plastic deformation gradient:

(A.10)

Equation A.10 can be written in terms of the current elastic deformation gradient ${\textstyle {\boldsymbol {F}}_{n+1}^{e}}$ as follows

(A.11)

where the expressions of the incremental deformation gradient ${\textstyle {\boldsymbol {f}}_{n+1}\equiv {\boldsymbol {F}}_{n+1}{\boldsymbol {F}}_{n}^{-1}}$ and Equation 3.53 are employed.

In what follows, few steps are performed in order to express Equation A.11 in terms of Hencky strains and it is demonstrated that the final form has the same format of the elastic strain update formula of the return mapping algorithm defined under the assumption of infinitesimal strains [155].

By post-multiplying Equation A.11 by ${\textstyle {\boldsymbol {R}}^{e^{T}}}$

(A.12)

and moving the exponential to the left side of the equation

(A.13)

Equation A.13 is, then, multiplied on both sides by its transpose

(A.14)

and by rearranging the terms in Equation A.14 and taking the square root of it gives

(A.15)

The final form is obtained by applying the tensor logarithm on both side of Equation A.15

(A.16)

In conclusion, Equation A.16 represents the Hencky strain updated formula of the return mapping in finite strains.

Appendix B. Derivatives of the rank-one matrices principal direction

In this section the expression of the spatial form of ${\textstyle \mathrm {C} ^{A,trial}}$ is derived

(B.1)

where ${\textstyle {\boldsymbol {n}}^{A}}$ and ${\textstyle {\boldsymbol {m}}^{A}}$ denote the eigenvectors and eigenbases associated with the eigenvalues ${\textstyle {\boldsymbol {\lambda }}_{A}}$ of the left Cauchy-Green deformation tensor ${\textstyle {\boldsymbol {b}}^{e}}$ , respectively, and ${\textstyle {\boldsymbol {g}}}$ , the metric tensor in the current configuration. In order to find the expression of the closed-form, firstly, the relation of Equation B.1 is transformed in material description by operating a pull-back transformation

(B.2)

with ${\textstyle {\boldsymbol {C}}}$ being the right Cauchy-Green strain tensor and ${\textstyle {\boldsymbol {M}}^{A}}$ the eigenbases associated to ${\textstyle {\boldsymbol {C}}}$ . The spectral decomposition of the right Cauchy-Green strain tensor ${\textstyle {\boldsymbol {C}}}$ which reads

(B.3)

where ${\textstyle \lambda _{A}}$ and ${\textstyle {\boldsymbol {N}}^{(A)}}$ denotes eigenvalues and associated eigenvectors, is considered and from Serrin's representation [242] it is possible to express ${\textstyle {\boldsymbol {N}}^{(A)}\otimes {\boldsymbol {N}}^{(A)}}$ in a closed-form in terms of ${\textstyle {\boldsymbol {C}}}$ as

(B.4)

where ${\textstyle I_{1}}$ and ${\textstyle I_{3}}$ are the first and third principal invariants of ${\textstyle {\boldsymbol {C}}}$ . From Equation B.4 it follows that the expression of ${\textstyle {\boldsymbol {M}}^{A}}$ reads

(B.5)

With the expression of ${\textstyle {\boldsymbol {M}}^{A}}$ in hand, its derivative with respect to ${\textstyle {\boldsymbol {C}}}$ is

{\frac {\partial {\boldsymbol {M}}^{A}}{\partial {\boldsymbol {C}}}}={\frac {1}{D_{A}}}{\frac {\partial \left({\boldsymbol {C}}-\left(I_{1}-\lambda _{A}^{2}\right){\boldsymbol {I}}+I_{3}\lambda _{A}^{-2}{\boldsymbol {C}}^{-1}\right)}{\partial {\boldsymbol {C}}}}+{\frac {\partial D_{A}^{-1}}{\partial {\boldsymbol {C}}}}\left({\boldsymbol {C}}-\left(I_{1}-\lambda _{A}^{2}\right){\boldsymbol {I}}+I_{3}\lambda _{A}^{-2}{\boldsymbol {C}}^{-1}\right)

(B.6)

In order to obtain the final expression the following relations have been used:

(B.7)

(B.8)

(B.9)

(B.10)

By performing a push-forward operation on the final result of Equation B.6, it is possible to obtain its spatial counterpart

(B.11)

Appendix C. Irreducible formulation in axisymmetric problems

In this section the matrix formulation for an axi-symmetrical finite element, undergoing finite deformations with respect to the spatial configuration, is presented.

This formulation can be used in the case of 3D bodies, which have rotational symmetry, as depicted on the left side of Figure C.1, and, thus, can be reduced to a 2D axi-symmetrical model, right side of the picture.

Hereinafter, it is assumed that the axis of symmetry coincides with the coordinate ${\textstyle X_{2}}$ .

Figure C.1: Axisymmetric representation of a 3D body with rotational symmetry.

Additionally, to the strains in the plane ${\textstyle X_{1},X_{2}}$ , hoop strains (along ${\textstyle X_{3}}$ direction) occur in case of axi-symmetrical deformations. The deformation gradient, is, then, given, as:

{\boldsymbol {F}}:=d{\boldsymbol {x}}/d{\boldsymbol {X}}={\begin{bmatrix}\displaystyle {\frac {\partial x_{1}}{\partial X_{1}}}&\displaystyle {\frac {\partial x_{1}}{\partial X_{2}}}&0\\\displaystyle {\frac {\partial x_{2}}{\partial X_{1}}}&\displaystyle {\frac {\partial x_{2}}{\partial X_{2}}}&0\\0&0&\displaystyle {\frac {x_{1}}{X_{1}}}\\\end{bmatrix}}

(C.1)

By using linear shape functions, the symmetric gradient of the test functions reads

(C.2)

and this leads to ${\textstyle \mathbf {B} _{I}^{A}}$ , the deformation matrix relative to node ${\textstyle I}$ , expressed here for a 2D axis-symmetrical problem as:

\mathbf {B} _{I}^{A}={\begin{bmatrix}\displaystyle {\frac {\partial N_{I}}{\partial x_{1}}}&0\\0&\displaystyle {\frac {\partial N_{I}}{\partial x_{2}}}\\\displaystyle {\frac {N_{I}}{x_{1}}}&0\\\displaystyle {\frac {\partial N_{I}}{\partial x_{2}}}&\displaystyle {\frac {\partial N_{I}}{\partial x_{1}}}\\\end{bmatrix}}

(C.3)

Introduction of the Cauchy stress, written in a vector form ${\textstyle {\boldsymbol {\sigma }}^{T}=\left[\sigma _{11},\sigma _{22},\sigma _{33},\sigma _{12}\right]}$ , and multiplied by ${\textstyle \nabla ^{S}{\boldsymbol {w}}}$ , expressed by Equation C.2 leads to

(C.4)

By using the results of Equation C.4, the virtual internal work of one element ${\textstyle \Omega ^{e}}$ is

(C.5)

It is observed that the integration has to be performed over the coordinates ${\textstyle X_{1}}$ and ${\textstyle X_{2}}$ as in circumferential direction. Due to that, the coordinate ${\textstyle x_{1}}$ appears in Equation C.5. For the node ${\textstyle I}$ of element ${\textstyle \Omega ^{e}}$ the residual is defined as

(C.6)

and its linearization yields to the tangent matrix, sum of the geometric and material stiffness matrix, expressed as in Equation 4.35.

The geometric stiffness matrix, whose definition is expressed by the first term of the integral of Equation 4.25, in its discretized form reads

(C.7)

where ${\textstyle I}$ and ${\textstyle K}$ are the indexes of the finite element's nodes, ${\textstyle \nabla _{x}N_{I}}$ is the spatial gradient of the shape function evaluated at node ${\textstyle I}$ , ${\textstyle V_{p}}$ is the volume relative to a single material point.

In case of axi-symmetrical deformations, matrix form of the gradient of the test function ${\textstyle {\boldsymbol {w}}}$ takes the form:

\nabla _{x}{\boldsymbol {w}}=\left\{{\begin{matrix}{\boldsymbol {w}}_{1,1}\\{\boldsymbol {w}}_{1,2}\\{\boldsymbol {w}}_{3,3}\\{\boldsymbol {w}}_{2,1}\\{\boldsymbol {w}}_{2,2}\end{matrix}}\right\}=\sum _{I=1}^{n}{\begin{bmatrix}N_{I,1}&0\\N_{I,2}&0\\{\frac {N_{I}}{x_{1}}}&0\\0&N_{I,1}\\0&N_{I,2}\\\end{bmatrix}}\left\{{\begin{matrix}{\boldsymbol {w}}_{1}\\{\boldsymbol {w}}_{2}\\\end{matrix}}\right\}=\sum _{I=1}^{n}{\overline {G}}_{I}{\boldsymbol {w}}_{I}

(C.8)

Using this relation of Equation C.8, together with the expression of the Cauchy stress in the following matricial form:

{\boldsymbol {\widehat {\sigma }}}={\begin{bmatrix}\sigma _{11}&\sigma _{12}&0&0&0\\\sigma _{21}&\sigma _{22}&0&0&0\\0&0&\sigma _{33}&0&0\\0&0&0&\sigma _{11}&\sigma _{12}\\0&0&0&\sigma _{21}&\sigma _{22}\\\end{bmatrix}}

(C.9)

the geometric stiffness matrix in discretized form can be written as

(C.10)

As explained in [243], it is possible to defined the term ${\textstyle {\overline {G}}_{I}^{T}{\boldsymbol {\widehat {\sigma }}}{\overline {G}}_{K}^{T}}$ in an explicit way as follows:

(C.11)

if the terms

(C.12)

(C.13)

(C.14)

are employed.

With respect to the material stiffness matrix, the discretized form of the 3D case is expressed by Equation 4.34. In the case of axi-symmetrical deformations, its expression reads

(C.15)

by taking into account the relations of ${\textstyle \mathbf {B} _{I}^{A}}$ (Equation C.3) and ${\textstyle \mathbf {D} ^{A}}$ , matrix form of the incremental constitutive tensor of Equation 4.24 for axi-symmetric case.

Finally, by considering the contribution of the terms ${\textstyle \mathbf {K} ^{A,G}}$ and ${\textstyle \mathbf {K} ^{A,M}}$ , the tangent stiffness matrix ${\textstyle \mathbf {K} ^{A,tan}}$ , referred to the spatial configuration, can be expressed as follows:

(C.16)

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[239] Kievitsbosch, R. and Smit, H. and Magnanimo, V. and Luding, S. and Taghizadeh, K. (2017) "Influence of dry cohesion on the micro-and macro-mechanical properties of dense polydisperse powders & grains", Volume 140. EDP Sciences. EPJ Web of Conferences 08016

[240] Chandra, B. and Larese, A. and Iaconeta, I. and Wüchner, R. (2019) "Soil-structure interaction simulation of landslides impacting a structure using an implicit material point method". Proceedings of the 2nd International Conference on the Material Point Method (MPM 2019)

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[243] Wriggers, P. (2008) "Nonlinear Finite Element Methods". Springer

@@ Line 1: / Line 1: @@
-<!-- metadata commented in wiki content
+''"Non quia difficilia sunt non audemus, sed quia non audemus difficilia sunt."''<br />
+Seneca (CIV,26)
+=Acknowledgement=
-<math display="inline">\mathbf{a} — th</math>
+It has been a long and challenging way, but, fortunately, during these years I have met many people, who encouraged me to go ahead and taught me not to give up. I am very grateful for this.
--->
-=Abstract=
+I would like to thank my advisor, Prof. Eugenio Oñate, for giving me the opportunity to study in CIMNE. During the years spent at this research center, I could find many sources of knowledge and inspiration, that I will take with me wherever I will be in the future. A special thanks goes to my co-advisor, Prof. Antonia Larese, for introducing me to the field of computational mechanics and for supporting me in the most important moments of this experience. I would like to thank those colleagues of CIMNE, who with great patience, helped me dedicating part of their time, Riccardo, Stefano, Charlie, Pablo, Lorenzo, Jordi and Alessandro.
-Abstract  1.15 The physics lying behind rotordynamics is complex to model, so that in many cases numerical processing is the only feasible approach. Being rotordynamics a field of great interest in the aerospace industry, the efforts devoted to its understanding are increasing day by day. Following this tendency, the aim of the present study is to develop a simplified elastodynamic model for the case of rotating structures such that can be addressed through numerical tools, built using the finite element method. For the purpose of analysing the vibration phenomena, modal decomposition and numerical integration have been taken advantage from. In this context, it has been found that the singular value decomposition could be applied in structural analysis to extract dominant displacement fluctuations, allowing the unfolding of global properties of the dynamic response. In the present report, the singular value decomposition has been applied to cantilever beams undergoing a single rotation, giving rise to reasonably satisfactory results.
+I would like to thank Prof. Massimiliano Cremonesi for his supervision and kind welcome during my visit at the Department of Civil and Environmental Engineering, at Politecnico di Milano. I want to express my gratitude to Dr. Fausto Di Muzio for his kindness and warming welcome during my brief stay at Nestlé. This experience allowed me to look at the research world from a different perspective and to recognize how important is to conjugate fundamental research with more practical applied aspects. In this respect, I would like to thank Dr. Julien Dupas for his collaboration and support in providing some experimental data.
-'''Keywords :'''  ''blades, FEM, rotordynamics, SVD, vibration ''
+I have been enough fortunate to be part of the European project T-MAPPP, which greatly contributed not only  to the development of my technical knowledge, but also of my soft skills. I would like to thank all the researchers involved in the project. In particular, Prof. Stefan Luding and Prof. Vanessa Magnanimo, for the valuable discussions and the interest shown in my work; Yousef, Kostas, Kianoosh, Behzad, Somik, Sasha and Niki, because without them it would have not been the same.
-=List of Symbols=
+A special thank you goes to all the people who made my day with a smile and let me see the positive side in any situation. Thank you Josie, Manu, Edu, Vicente, Nanno, Giulia, Eugenia and Alessandra.
-List of Symbols  The next list describes several symbols that will be later used within the body of the document.
+Last but not least, all my gratitude goes to my family: Marta, Gianni and Claudia, for their wholehearted support and love.
-==Abbreviations==
+The research was supported by the Research Executive Agency through the T-MAPPP project (FP7 PEOPLE 2013 ITN-G.A.n607453).
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-==Mathematical notation==
+=Resumen=
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-==General variables==
+El manejo, el transporte y el procesamiento de materiales granulares y en polvo son operaciones fundamentales en una amplia gama de procesos industriales y de fenómenos geofísicos. Los materiales particulados, que pueden encontrarse en la naturaleza, generalmente están caracterizados por un tamaño de grano, que puede variar entre varios órdenes de magnitud: desde el nanómetro hasta el orden de los metros. En función de las condiciones de fracción volumétrica y de deformación de cortante, los materiales granulares pueden tener un comportamiento diferente y a menudo pueden convertirse en un nuevo estado de materia con propiedades de sólidos, de líquidos y de gases. Como consecuencia, tanto el análisis experimental como la simulación numérica de medios granulares es aún una tarea compleja y la predicción de su comportamiento dinámico representa aun hoy día un desafío muy importante. El principal objetivo de esta monografía es el desarrollo de una estrategia numérica con la finalidad de estudiar el comportamiento macroscópico de los flujos de medios granulares secos en régimen cuasiestático y en régimen dinámico.  El problema está definido en el contexto de la mecánica de medios continuos y las leyes de gobierno están resueltas mediante un formalismo Lagrangiano.  El Metodo de los Puntos Materiales (MPM), método basado en el concepto de discretización del cuerpo en partículas, se ha elegido por sus características que lo convierten en una técnica apropiada para resolver problemas en grandes deformaciones donde se tienen que utilizar complejas leyes constitutivas. En el marco del MPM se ha implementado una formulación  irreducible que usa una ley constitutiva de Mohr-Coulomb y que tiene en cuenta no-linealidades geométricas. La estrategia numérica está verificada y validada con respecto a tests de referencia a resultados experimentales disponibles en la literatura. También, se ha implementado una formulación mixta para resolver los casos de flujo granular en condiciones no drenadas. Por último, la estrategia MPM desarrollada se ha utilizado y evaluado con respecto a un estudio experimental realizado para caracterizar la fluidez de diferentes tipologías de azúcar. Finalmente se presentan unas observaciones y una discusión sobre las capacidades y las limitaciones de esta herramienta numérica  y se describen las bases para una investigación futura.
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-==Coordinate systems==
+=Abstract=
-*
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-==Kinematics==
+Bulk handling, transport and processing of granular materials and powders are fundamental operations in a wide range of industrial processes and geophysical phenomena. Particulate materials, which can be found in nature, are usually characterized by a grain size which can range across several scales: from nanometre to the order of metre. Depending on the volume fraction and on the shear strain conditions, granular materials can have different behaviours and often can be expressed as a new state of matter with properties of solids, liquids and gases. For the above reasons, both the experimental and the numerical analysis of granular media is still a difficult task and the prediction of their dynamic behaviour still represents, nowadays, an important challenge.  The main goal of the current monograph is the development of a numerical strategy with the objective of studying the macroscopic behaviour of dry granular flows in quasi-static and dense flow regime. The problem is defined in a continuum mechanics framework and the balance laws, which govern the behaviour of a solid body, are solved by using a Lagrangian formalism. The Material Point Method (MPM), a particle-based method, is chosen due to its features which make it very suitable for the solution of large deformation problems involving complex history-dependent constitutive laws. An irreducible formulation using a Mohr-Coulomb constitutive law, which takes into account geometric non-linearities, is implemented within the MPM framework. The numerical strategy is verified and validated against several benchmark tests and experimental results, available in the literature. Further, a mixed formulation is implemented for the solution of granular flows that undergo undrained conditions. Finally, the developed MPM strategy is used and tested against the experimental study performed for the characterization of the flowability of several types of sucrose. The capabilities and limitations of this numerical strategy are observed and discussed and the bases for future research are outlined.
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-==Elasticity==
+=List of Symbols=
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-==Finite element method==
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-==Modal analysis==
+{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
-*
+|-
-*
+| <math display="inline">t</math>:
-*
+| time;
-*
+|-
-*
+| <math display="inline">\phi </math>:
+| internal friction angle;
-==Numeric integration and SVD==
+|-
-*
+| <math display="inline">d </math>:
-*
+| particle diameter;
-*
+|-
-*
+| <math display="inline">\rho _p</math>:
-*
+| particle density;
-*
+|-
+| <math display="inline">\dot{\boldsymbol{\gamma }}</math>:
+| shear rate;
+|-
+| <math display="inline">\Omega ^0</math>:
+| undeformed configuration;
+|-
+| <math display="inline">\varphi \left(\Omega \right)^n</math> or <math display="inline">\varphi _n</math>:
+| deformed configuration at time <math display="inline">t^n</math>;
+|-
+| <math display="inline">\varphi \left(\Omega \right)^{n+1}</math> or <math display="inline">\varphi _{n+1}</math>:
+| deformed configuration at time <math display="inline">t^{n+1}</math>;
+|-
+| <math display="inline">\chi _p</math>:
+| characteristic function in the Generalized Interpolation Material Point Method;
+|-
+| <math display="inline">V_p </math>:
+| particle volume;
+|-
+| <math display="inline">\Omega _p</math>:
+| current particle domain;
+|-
+| <math display="inline">\Omega </math>:
+| current domain occupied by the continuum;
+|-
+| <math display="inline"> N_I(\boldsymbol{x})</math>:
+| shape function relative to node <math display="inline">I</math> evaluated at position at position <math display="inline">\boldsymbol{x}</math>;
+|-
+| <math display="inline">\nabla N_I(\boldsymbol{x})</math>:
+| gradient shape function relative to node <math display="inline">I</math> evaluated at position at position <math display="inline">\boldsymbol{x}</math>;
+|-
+| <math display="inline">\widehat{\nabla N_I}(\boldsymbol{x})</math>:
+| gradient shape function from node-based calculation;
+|-
+| <math display="inline">\overline{\nabla N_I}(\boldsymbol{x})</math>:
+| gradient shape function evaluated in Dual Domain Material Point Method;
+|-
+| <math display="inline">\left(\cdot \right)_p</math>:
+| <math display="inline">\left(\cdot \right)</math> relative to the material point <math display="inline">p</math>;
+|-
+| <math display="inline">\left(\cdot \right)_I</math>:
+| <math display="inline">\left(\cdot \right)</math> relative to the node <math display="inline">I</math>;
+|-
+| <math display="inline">\left(\cdot \right)_I^n</math>:
+| <math display="inline">\left(\cdot \right)</math> relative to the node <math display="inline">I</math> at time <math display="inline">t^n</math>;
+|-
+| <math display="inline">^{it}\left(\cdot \right)_I^n</math>:
+| <math display="inline">\left(\cdot \right)</math> relative to the node <math display="inline">I</math> at time <math display="inline">t^n</math> and iteration <math display="inline">it</math>;
+|-
+| <math display="inline">\boldsymbol{u}</math>:
+| displacement vector;
+|-
+| <math display="inline">\Delta \boldsymbol{u}</math>:
+| increment displacement vector;
+|-
+| <math display="inline">\boldsymbol{v}</math>:
+| velocity vector;
+|-
+| <math display="inline">\boldsymbol{a}</math>:
+| acceleration vector;
+|-
+| <math display="inline">p</math>:
+| pressure;
+|-
+| <math display="inline">\boldsymbol{q}</math>:
+| momentum vector;
+|-
+| <math display="inline">\boldsymbol{f}</math>:
+| inertia vector;
+|-
+| <math display="inline">m</math>:
+| mass;
+|-
+| <math display="inline">\left(\lambda , \zeta \right)</math>:
+| stability parameters of Bossak method;
+|-
+| <math display="inline">\delta \boldsymbol{u}</math>:
+| vector of unknown incremental displacement;
+|-
+| <math display="inline"> \mathbf{K}^{tan} </math>:
+| tangent matrix of the linearised system of equation;
+|-
+| <math display="inline"> \mathbf{R}</math>:
+| residual vector of the linearised system of equation;
+|-
+| <math display="inline">\left(\xi _p, \eta _p\right)</math>:
+| local coordinates relative to a material point;
+|-
+| <math display="inline">\beta </math>:
+| on-negative locality coefficient in Local Maximum Entropy technique;
+|-
+| <math display="inline">h</math>:
+| measure of nodal spacing in Local Maximum Entropy technique;
+|-
+| <math display="inline">\gamma </math>:
+| dimensionless parameter in Local Maximum Entropy technique;
+|-
+| <math display="inline">\boldsymbol{F}</math>:
+| total deformation gradient;
+|-
+| <math display="inline">J</math>:
+| determinant of the total deformation gradient;
+|-
+| <math display="inline">\boldsymbol{f}</math>:
+|  incremental deformation gradient;
+|-
+| <math display="inline">\boldsymbol{P}</math>:
+|  First Piola-Kirchhoff stress tensor;
+|-
+| <math display="inline">\boldsymbol{S}</math>:
+|  Second Piola-Kirchhoff stress tensor;
+|-
+| <math display="inline">\boldsymbol{\tau }</math>:
+|  Kirchhoff stress tensor;
+|-
+| <math display="inline">\boldsymbol{\sigma }</math>:
+|  Cauchy stress tensor;
+|-
+| <math display="inline">\Psi </math>:
+|  specific strain energy function;
+|-
+| <math display="inline">\boldsymbol{Q}</math>:
+|  rotation tensor;
+|-
+| <math display="inline">\boldsymbol{R}</math>:
+|  rotation tensor from polar decomposition of <math display="inline">\boldsymbol{F}</math>;
+|-
+| <math display="inline">\boldsymbol{U}</math>:
+|  right stretch tensor;
+|-
+| <math display="inline">\boldsymbol{V}</math>:
+|  left stretch tensor;
+|-
+| <math display="inline">\boldsymbol{C}</math>:
+|  right Cauchy-Green tensor;
+|-
+| <math display="inline">\boldsymbol{I_C}</math>:
+|  first principal invariant of <math display="inline">\boldsymbol{C}</math>;
+|-
+| <math display="inline">\boldsymbol{I_C^*}</math>:
+|  first invariant of <math display="inline">\boldsymbol{C}</math>;
+|-
+| <math display="inline">\boldsymbol{b}</math>:
+|  left Cauchy-Green tensor;
+|-
+| <math display="inline">\left(\lambda , \mu \right)</math>:
+|  Lame constant;
+|-
+| <math display="inline">K</math>:
+|  bulk modulus;
+|-
+| <math display="inline">G</math>:
+|  shear modulus;
+|-
+| <math display="inline">\mathrm{C}^{CE}</math>:
+|  material incremental constitutive tensor;
+|-
+| <math display="inline">\mathrm{C}^{\tau }</math>:
+|  spatial incremental constitutive tensor;
+|-
+| <math display="inline">\boldsymbol{I}</math>:
+|  symmetric second order unit;
+|-
+| <math display="inline">\mathrm{I}</math>:
+|  fourth order identity tensor;
+|-
+| <math display="inline">\mathrm{I}_s</math>:
+|  fourth order symmetric identity tensor;
+|-
+| <math display="inline">\mathrm{I}_d</math>:
+|  fourth order deviatoric projector tensor;
+|-
+| <math display="inline">\boldsymbol{d}</math>:
+|  symmetrical spatial velocity gradient;
+|-
+| <math display="inline">\boldsymbol{D}</math>:
+|  rate of deformation tensor;
+|-
+| <math display="inline">\boldsymbol{W}</math>:
+|  spin tensor;
+|-
+| <math display="inline">\boldsymbol{L}</math>:
+|  velocity gradient tensor;
+|-
+| <math display="inline">\left(\cdot \right)_{vol}</math>:
+|  volumetric part of <math display="inline">\left(\cdot \right)</math>;
+|-
+| <math display="inline">\left(\cdot \right)_{dev}</math>:
+|  deviatoric part of <math display="inline">\left(\cdot \right)</math>;
+|-
+| <math display="inline">dev\left(\cdot \right)</math>:
+|  deviatoric part of <math display="inline">\left(\cdot \right)</math>;
+|-
+| <math display="inline">\left(\cdot \right)^e</math>:
+|  elastic part of <math display="inline">\left(\cdot \right)</math>;
+|-
+| <math display="inline">\left(\cdot \right)^p</math>:
+|  plastic part of <math display="inline">\left(\cdot \right)</math>;
+|-
+| <math display="inline">\left(\overline{\cdot }\right)</math>:
+|  volume preserving part of <math display="inline">\left(\cdot \right)</math>;
+|-
+| <math display="inline">\sigma _Y</math>:
+|  flow stress;
+|-
+| <math display="inline">H</math>:
+|  isotropic hardening;
+|-
+| <math display="inline">\alpha </math>:
+|  hardening parameter;
+|-
+| <math display="inline">\gamma </math>:
+|  plastic multiplier;
+|-
+| <math display="inline">\boldsymbol{n}</math>:
+|  unit vector of <math display="inline">\boldsymbol{\tau }_{dev}</math>;
+|-
+| <math display="inline">\left(\cdot \right)^{trial}</math>:
+| <math display="inline">\left(\cdot \right)</math> in trial state;
+|-
+| <math display="inline">tr\left(\cdot \right)</math>:
+|  trace of <math display="inline">\left(\cdot \right)</math>;
+|-
+| <math display="inline">\mathrm{g}</math>:
+|  metric tensor in current configuration;
+|-
+| <math display="inline">\mathrm{C}^{ep}</math>:
+|  spatial algorithmic elastoplastic moduli;
+|-
+| <math display="inline">c</math>:
+|  cohesion;
+|-
+| <math display="inline">\psi </math>:
+|  dilation angle;
+|-
+| <math display="inline">\boldsymbol{\sigma }_n</math>:
+|  normal stress;
+|-
+| <math display="inline">\boldsymbol{\epsilon }^e</math>:
+|  principal Hencky strain;
+|-
+| <math display="inline">\mathrm{a}</math>:
+|  Hencky elastic constitutive tensor;
+|-
+| <math display="inline">\mathrm{a}^{ep}</math>:
+|  elasto-plastic fourth order constitutive tensor;
+|-
+| <math display="inline">\mathrm{C}^{cep}</math>:
+|  consistent elasto-plastic tangent;
+|-
+| <math display="inline">\boldsymbol{\lambda }</math>:
+|  eigenvalues vector of <math display="inline">\boldsymbol{b}</math>;
+|-
+| <math display="inline">\boldsymbol{n}</math>:
+|  eigenvector vector of <math display="inline">\boldsymbol{b}</math>;
+|-
+| <math display="inline">\boldsymbol{m}</math>:
+|  eigenbases vector of <math display="inline">\boldsymbol{b}</math>;
+|-
+| <math display="inline">\boldsymbol{N}</math>:
+|  eigenvector vector of <math display="inline">\boldsymbol{C}</math>;
+|-
+| <math display="inline">\boldsymbol{M}</math>:
+|  eigenbases vector of <math display="inline">\boldsymbol{C}</math>;
+|-
+| <math display="inline">  \mathcal{B} </math>:
+|  body;
+|-
+| <math display="inline"> \mathcal{E} </math>:
+|  3D Euclidean space;
+|-
+| <math display="inline">\mathcal{R}^3 </math>:
+|  real coordinate space in 3D;
+|-
+| <math display="inline">\rho </math>:
+|  mass density;
+|-
+| <math display="inline">\rho _0</math>:
+|  mass density in underformed configuration;
+|-
+| <math display="inline">\boldsymbol{b}</math>:
+|  body force;
+|-
+| <math display="inline">g</math>:
+|  gravity;
+|-
+| <math display="inline">\varphi (\partial \Omega _N)</math>:
+|  Neumann boundary in deformed configuration;
+|-
+| <math display="inline">\varphi (\partial \Omega _D)</math>:
+|  Dirichlet boundary in deformed configuration;
+|-
+| <math display="inline"> \boldsymbol{\overline{t}} </math>:
+|  prescribed normal tension on Neumann boundary;
+|-
+| <math display="inline"> \boldsymbol{\overline{u}} </math>:
+|  prescribed displacement on Dirichlet boundary;
+|-
+| <math display="inline">\boldsymbol{w}</math>:
+|  displacement weight function;
+|-
+| <math display="inline">q</math>:
+|  pressure weight function;
+|-
+| <math display="inline">\mathcal{V}</math>:
+|  displacement space;
+|-
+| <math display="inline">\mathcal{V}_{h}</math>:
+|  displacement finite element space;
+|-
+| <math display="inline">\left(\cdot \right)_h</math>:
+| <math display="inline">\left(\cdot \right)</math> in the finite element space;
+|-
+| <math display="inline">\mathcal{B}_h</math>:
+|  geometrical representation of <math display="inline">\mathcal{B}</math>;
+|-
+| <math display="inline"> \Omega _p </math>:
+|  finite volume assigned to a material point;
+|-
+| <math display="inline">\mathbf{\mathbf{H}}^{1}(\mathcal{B})</math>:
+|  Hilbert space;
+|-
+| <math display="inline">dv</math>:
+|  differential volume in deformed configuration;
+|-
+| <math display="inline">da</math>:
+|  differential boundary surface in deformed configuration;
+|-
+| <math display="inline">dV</math>:
+|  differential volume in undeformed configuration;
+|-
+| <math display="inline">\nabla _X\left(\cdot \right)</math>:
+|  material gradient operator;
+|-
+| <math display="inline">\nabla _x\left(\cdot \right)</math>:
+|  spatial gradient operator;
+|-
+| <math display="inline">\nabla \left(\cdot \right)^s</math>:
+|  symmatric part of <math display="inline">\left(\cdot \right)</math> gradient;
+|-
+| <math display="inline"> \mathbb{C} </math>:
+|  fourth order incremental constitutive tensor relative to <math display="inline">\boldsymbol{S}</math>;
+|-
+| <math display="inline"> \widehat{\mathbb{C}} </math>:
+|  fourth order incremental constitutive tensor relative to <math display="inline">\boldsymbol{\tau }</math>;
+|-
+| <math display="inline"> \overline{\widehat{\mathbb{C}}} </math>:
+|  fourth order incremental constitutive tensor relative to <math display="inline">\boldsymbol{\sigma }</math>;
+|-
+| <math display="inline">\mathbf{D}</math>:
+|  matrix form of <math display="inline"> \overline{\widehat{\mathbb{C}}} </math>;
+|-
+| <math display="inline"> \mathbf{B}</math>:
+|  deformation matrix;
+|-
+| <math display="inline"> \mathbf{K}^{G} </math>:
+|  geometric stiffness matrix;
+|-
+| <math display="inline"> \mathbf{K}^{M} </math>:
+|  material stiffness matrix;
+|-
+| <math display="inline"> \mathbf{K}^{static} </math>:
+|  static part of <math display="inline"> \mathbf{K}^{tan} </math>;
+|-
+| <math display="inline"> \mathbf{K}^{dynamic} </math>:
+|  dynamic part of <math display="inline"> \mathbf{K}^{tan} </math>;
+|-
+| <math display="inline"> \mathcal{Q} </math>:
+|  pressure space;
+|-
+| <math display="inline">\delta p</math>:
+|  vector of unknown pressure;
+|-
+| <math display="inline"> ^m\mathbf{K}^{G} </math>:
+|  geometric stiffness matrix in mixed formulation;
+|-
+| <math display="inline"> ^m\mathbf{K}^{M} </math>:
+|  material stiffness matrix in mixed formulation;
+|-
+| <math display="inline"> \boldsymbol{R}_{\boldsymbol{u}} </math>:
+|  residual vector relative to the momentum balance equation;
+|-
+| <math display="inline"> \boldsymbol{R}_p </math>:
+|  residual vector relative to the pressure continuity equation;
+|-
+| <math display="inline"> \boldsymbol{R}_p^{\mathrm{stab}} </math>:
+|  residual vector relative to the stabilization term;
+|-
+| <math display="inline"> \mathbf{M} </math>:
+|  mass matrix;
+|-
+| <math display="inline"> \mathbf{M}^{\mathrm{stab}} </math>:
+|  mass matrix relative to the stabilization term;
+|-
+| <math display="inline"> R_p^{\mathrm{stab}} </math>:
+|  residual vector relative to the stabilization term;
+|-
+| <math display="inline">\alpha </math>:
+|  stabilization parameter;
+|-
+| <math display="inline">\left(\mathbf{B},\mathbf{B}^*  \right)</math>:
+|  mixed terms in the mixed formulation
+|-
+| <math display="inline">d_{10}</math>:
+|  diameter at which 10% of the sample's mass is comprised of particles with a diameter less than this value;
+|-
+| <math display="inline">d_{50}</math>:
+|  diameter at which 50% of the sample's mass is comprised of particles with a diameter less than this value;
+|-
+| <math display="inline">d_{90}</math>:
+|  diameter at which 90% of the sample's mass is comprised of particles with a diameter less than this value;
+|-
+| <math display="inline">\mathbf{D}^A</math>:
+|  matrix form of <math display="inline"> \overline{\widehat{\mathbb{C}}} </math> in axisymmetric case;
+|-
+| <math display="inline">\mathbf{K}^{A, tan}</math>:
+|  tangent stiffness matrix in axisymmetric case;
+|-
+| <math display="inline">\mathbf{K}^{A, G}</math>:
+|  geometric stiffness matrix in axisymmetric case;
+|-
+| <math display="inline">\mathbf{K}^{A, M}</math>:
+|  material stiffness matrix in axisymmetric case;
+|-
+|
+|}
 =1 Introduction=
-The aim of this section is to briefly introduce the reader to the field of '''rotordynamics''' and outline the main models used nowadays to tackle these kind of problems. An overview of the report structuring will also be presented, as well as a review on the actual possibilities to develop the present work. This introduction follows the key concepts presented in <span id='citeF-1'></span>[[#cite-1|[1]]], <span id='citeF-2'></span>[[#cite-2|[2]]] and <span id='citeF-3'></span>[[#cite-3|[3]]] .
+Bulk handling, transport and processing of particulate materials, such as, granular materials and powders, are fundamental operations in a wide range of industrial processes <span id='citeF-1'></span>[[#cite-1|[1]]] or geophysical phenomena and hazards, such as, landslides, debris flows, etc. <span id='citeF-2'></span>[[#cite-2|[2]]]. Particulate systems are difficult to handle and they can show an unpredictable behaviour, representing a great challenge in the industrial production, concerning both design and functionality of unit operations in plants, but also in the research community of Powders and Grains <span id='citeF-3'></span><span id='citeF-4'></span>[[#cite-3|[3,4]]]. Granular materials and powders consist of discrete particles such as, e.g., separate sand-grains, agglomerates (made of several primary particles), natural solid materials like sandstone, ceramics, metals or polymers sintered during additive manufacturing. The primary particles can be as small as nano-metres, micro-metres, or millimetres <span id='citeF-5'></span>[[#cite-5|[5]]] covering multiple scales in size and a variety of mechanical and other interaction mechanisms, such as, friction and cohesion <span id='citeF-6'></span>[[#cite-6|[6]]], which become more and more important the smaller the particles are. All those particle systems have a particulate, usually disordered, possibly inhomogeneous and often anisotropic micro-structure; nowadays, the research community is working actively in order to have a deeper understanding and aware knowledge of bulk behaviour affected by micro-scale parameters. Indeed, particle systems as bulk show a completely different behaviour as one would expect from the individual particles. Collectively, particles either flow like a fluid or rest static like a solid. In the former case, for rapid flows, granular materials are collisional, inertia dominated and compressible similar to a gas. In the latter case, particle aggregates are solid-like and, thus, can form, e.g., sand piles or slopes that do not move for long time. Between these two extremes, there is a third flow regime, dense and slow, characterized by the transitions (i) from static to flowing (failure, yield) or vice-versa (ii) from fluid to solid (jamming). At the particle and contact scale, the most important property of particle systems is their dissipative, frictional, and possibly cohesive nature. In this context, dissipation shall be understood as kinetic energy, at the particle scale, which converts into heat, for instance, due to plastic deformations. The transition from fluid to solid can be caused by dissipation alone, which tends to slow down motion. The transition from solid to fluid (start of flow) is due to failure and instability when dissipation is not strong enough and the solid yields and transits to a flowing regime.
-==1.1 Historical frame and background==
+In this Chapter, the granular flow theory is presented more in detail and the main attempts, available in the literature, for the modelling of granular matter behaviour in the different regimes are discussed. Afterwards, objectives and layout of the current monograph are presented.
-Since the invention of the potter's wheel, almost 6000 years from now, humanity has relied on rotating machinery for a wide range of activities and processes. This idea of a symmetrical object turning on an axis is the key behind complex inventions such as mills, water wheels and even engines and jet turbines.
+==1.1 The granular flows==
-Behind those great inventions lies a great understanding of the physics describing this rotational motion. '''Greeks''' were the first who tackled machinery in a systematic and even mathematical point of view, bringing new inventions such as Archimedes' screw and Hero's ''aelopile'', the first reaction turbine — which unfortunately could not produce useful work.
+The heavy involvement of particle materials in many different industrial processes makes the granular matter, nowadays, a remarkable object of study. Particulate materials exist in large quantity in nature and it is established that most of the industrial processes, such as, pharmaceutical, agricultural, chemical, just to cite a few, deal with materials that are particulate in structure. In the industrial field, dealing with processes at large scales and huge quantities of raw material, any issue, encountered in the production line, may cause losses in terms of productivity, and, thus, of money. During the last decades, it has been documented that also in processes of granular matters, a lack of knowledge implies non-optimal production quality.  In older industrial surveys, Merrow <span id='citeF-7'></span>[[#cite-7|[7]]] found that the main factor causing long start-up delays in chemical plants is represented by the processing granular materials, especially due to the lack of reliable predictive models and simulations, while Ennis et al. <span id='citeF-8'></span>[[#cite-8|[8]]] reported that 40<math display="inline">%</math> of the capacity of industrial plants is wasted because of granular solid phenomena. More recently, Feise <span id='citeF-9'></span>[[#cite-9|[9]]] analysed the changes in chemical industry and predicted an increase in particulate solids usage along with new challenges due to new concepts like versatile multi-purpose plants, and fields like nano and bio-technology.  For these reasons, it is clear that it is fundamental to have a better understanding of particulate materials behaviour under different conditions and to be able to improve the production quality through experimental campaigns and numerical modelling works. However, due to the wide variety of intrinsic properties of particulate materials a unified constitutive description, under any condition, has not been established yet.
-<div id='img-1.1'></div>
+With granular flow, we refer to motions where the particle-particle interactions play an important role in determining the flow properties and the flow patterns which are quite different from those of conventional fluids. The most evident differences between granular systems and simple fluids affecting the macroscopic properties of the flow, as pointed out in <span id='citeF-10'></span>[[#cite-10|[10]]], are:
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+* The ''size of grains'' is typically approximately <math display="inline">10^{18}</math> more massive and voluminous than a water molecule. As both fluid and particle motions can be studied according to the laws of classical mechanics, this is not a fundamental difference, but could represent an important factor while evaluating the applicability of continuum hypothesis, as explained below;
+* In granular systems, when granules collide, a loss of ''kinetic energy converted in true heat'' is observed. This difference determines the main feature which deeply distinguishes the granular flows modelling from the fluid flows one;
+* In nature, ''grains are not identical'': particle shape, particle roughness and solid density are only some of the particle properties which characterize a grain from the other. As real particles are not exactly spherical and typically the surface is rough, in grain-grain interactions frictional forces and torque are created and grains rotate during the collision.
+The above comparison is useful to emphasise that the main assumptions on the basis of fluid flow modelling can not coincide with those of granular flow modelling. Further, this comparison can also be useful to set the bounds of the continuum assumptions enforceability. Three length scales have to be considered for the definition of these hypotheses. The first one is related to the particle size. Typically the value of density in grain systems is much smaller than in molecular fluids; this means that, for instance, in a cubic mm of fluids the number of molecules is much higher than the number of grains. If a macroscopic quantity changes significantly over a 1 mm of length, the variation over the molecules is small, but in the granular materials, if the number of particles in a 1 mm is low, a bigger variation is registered falling out of the continuum assumptions. The second length is related to the container confining the system and the third one to the inelasticity in grain-grain collisions. The latter can be defined as the radius of the pulse, related to a degradation of factor <math display="inline">e^-</math> of the total kinetic energy in a system of grains after a localized input of energy. If the inelasticity is not small this length covers just a few particles. This implies substantial changes in macroscopic quantities over distances measured over a small number of grain diameters, which do not allow to respect the continuum assumptions.
+When one wants to study the flow of granular materials has to bear in mind that the bulk in motion is represented by an assembly of discrete solid particles interacting with each other. Depending on the intrinsic properties of the grains and the macroscopic characteristic of the system (i.e. geometry, density, velocity gradient), the internal forces can be transmitted in different ways within the granular material. Depending on this, three main flow patterns can be observed experimentally and numerically <span id='citeF-11'></span>[[#cite-11|[11]]]. At large solids concentration and low shear rate, the stresses are not evenly distributed, but are concentrated along networks of particles, called ''force chains''. The force chains are dynamic structures, which rotate, become unstable and, finally, collapse as a result of the shear motion. When granular material fails it is observed that the failure occurs along narrow planes, within the material, which have not infinitesimal thickness, but are zones of the order of ten particles across called ''shear bands''. Within the shear bands, the stresses <math display="inline">\boldsymbol{\tau }</math> are still distributed along the force chains and the shear <math display="inline"> \tau _{xy} </math> and normal stresses <math display="inline"> \tau _{yy} </math> are related in non-cohesive material as
+<span id="eq-1.1"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-Hero.png|300px|Replica of Hero's ''aelopile'' <span id='citeF-4'></span>[[#cite-4|[4]]]]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 1.1:''' Replica of Hero's ''aelopile'' <span id='citeF-4'></span>[[#cite-4|[4]]]
+|-
+| style="text-align: center;" | <math>\frac{\tau _{xy}}{\tau _{yy}} = \mathrm{tan}\phi   </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (1.1)
 |}
-Water wheels became an important source of energy in ancient Chinese civilisation, and later on in the Roman empire as a substitute for manual and animal work. It was used for a wide range of applications, including iron casting and grain grinding. Wind mills took longer to appear, the first practical wind-powered machines dating from the <math display="inline">9^{th}</math> century in Persia, and reaching Europe three centuries later. These kind of rotating machinery were, however, quite basic, and needed from better scientific background in order to be meaningfully developed.
+where <math display="inline">\phi </math> is the internal friction angle. Observing Equation [[#eq-1.1|1.1]], it is noted that <math display="inline">\mathrm{tan}\phi </math> depends on the geometry of the force chains; thus, the friction-like response of the bulk is a result of the internal structure of force chains, as well.
-The world had to wait until the Renaissance Period to see substantial contributions to this field. Leonardo da Vinci is credited for some fundamental contributions to solid mechanics, which were improved by Galileo and later on by Euler and Bernoulli, who formulated the correct equations for simple bending. In the zenith of '''scientific revolution''', Newton's development of calculus unfold a new range of applications of mathematical methods to engineering. It was thanks to the latter that Daniel Bernoulli could formulate the differential equation of motion for a vibrating beam, to be later accepted by Euler when investigating beams under different loading conditions.
+This flow pattern takes the name of ''Quasi-static regime'' because the rate of formation of the force chain divided by their lifetime is independent on the shear rate <math display="inline">\dot{\boldsymbol{\gamma }}</math> and with this also the generated stresses. If on one hand, the shear rate does not affect the response of the system, on the other hand, it is worth highlighting that the inter-particle stiffness ''k'' plays an important role, as the stresses within the force chains show a linear dependence with ''k''. The inter-particle stiffness, in turn, can be expressed as a linear function of the Young modulus ''E'' of the material and also depends on the local radius of curvature, thus, on the geometry of the contact. Bathurst and Rothenburg <span id='citeF-12'></span>[[#cite-12|[12]]] have derived an expression for the bulk elastic modulus ''K'', which linearly depends on the stiffness and can be used in the definition of the sound speed in a static granular material. As shown in the data of Goddard <span id='citeF-13'></span>[[#cite-13|[13]]] and Duffy <math display="inline">\& </math> Mindlin <span id='citeF-14'></span>[[#cite-14|[14]]], the wave velocity is dependent on the pressure applied to the bulk assembly. Thus, increasing the confining pressure, the elastic bulk modulus increases along with the inter-particle stiffness ''k'' and the ''force chains'' lifetime.
-The Euler-Bernoulli beam theory was improved by Rayleigh (1877) and later on by Timoshenko (1921), adding the effect of shear. This beam theory, valid even today in the <math display="inline">21^{st}</math> century, is the backbone of all rotor analysis. The two-dimensional study of structures was carried by Germain and Kirchhoff (1850), and lucid discussions about vibrations of elastic systems were carried by Lord Rayleigh.
+Until the shear rates <math display="inline">\dot{\boldsymbol{\gamma }}</math> are kept low, the inertia effects are small and force chains are the only mechanism available to balance the applied load. Increasing the shear rate, the particles are still locked in force chains, but the forces generated have to take into account the inertia introduced in the system. Further increasing <math display="inline">\dot{\boldsymbol{\gamma }}</math>, the inertial component of the internal forces linearly increases with the shear rate and when these are comparable with the static forces the flow transition to the ''Inertial regime'' takes place. The ''Inertial regime'' encompasses flows where force chains cannot form and the momentum is transported largely by particle inertia. In this regime, the shear stresses are independent of the stiffness, but dependent on the second power of the shear rate, as expressed in the Bagnold scaling <math display="inline">\tau _{xy}/\rho d^2\dot{\boldsymbol{\gamma }}^2</math> <span id='citeF-15'></span>[[#cite-15|[15]]], where <math display="inline">d</math> and <math display="inline">\rho </math> are the particle diameter and particle density, respectively. Even if force chains are not present, multiple simultaneous contacts between the particles still coexist allowing longer contact period <math display="inline">t_c</math>. By defining with <math display="inline">T_{bc}</math> the binary contact time, i.e., the duration of a contact between two freely colliding particles, the ratio <math display="inline">t_c/T_{bc} > 1</math>. If this ratio has the value of 1, the dominant particle collisions are binary and instantaneous and the flow is defined with the name of ''Rapid Granular Flow'' <span id='citeF-16'></span>[[#cite-16|[16]]], which can be considered as an asymptotic case of the ''Inertial Regime''. This flow path is controlled by the property of granular temperature, which represents a measure of the unsteady components of velocity. The granular temperature is generated by the shear work and it drives the transport rate in two principal modes of internal (momentum) transport: a collisional and a streaming mode. In the first case, the granular temperature provides the relative velocity that drives particle to collide; while, in the second case, it generates a random velocity that makes the particles move relatively to the velocity gradient. In the ''Rapid Granular Flow'' the coexistence of contact and streaming stresses can be observed; obviously, the collisional mode dominates at high concentrations, while the streaming mode at low concentrations. It is generally assumed that, at small shear rates, a flow behaves quasi-statically, and that by increasing the shear rate, one will eventually end up in the ''Rapid Flow regime''. As pointed out by Campbell <span id='citeF-11'></span>[[#cite-11|[11]]], the transition through the regimes is regulated by the volume fraction and the shear rate. However, by fixing the first or the second field, the transition may take place in a different way.
-Thanks to the contribution of these and many other scientist and mathematicians,the fundamentals of the '''Theory of Elasticity''' could be established during the scientific revolution. These deformable bodies were referred as structures by engineers. As the industrial revolution began and the first steam and internal combustion engines appeared, engineers needed from approximate methods, based on energy principles, to carry out accurate studies for solid structures with intricate geometry.
+==1.2 Granular flow modelling==
-By then, as the required speed of the machinery was rather slow, reciprocating machinery could be used instead of rotating devices. In fact, little was known about the mechanical behaviour of beams under rotating forces. A first study on the topic was carried by Rankine (1869), proposing that a critical speed exists above which rotation becomes impossible. This was later denied by '''De Laval''' (1883), a pioneer in the design of steam turbines.
+Despite the prevalence of granular materials in most of the industrial applications, there is still a large discrepancy between results predicted by analytical or numerical solutions and their real behaviour <span id='citeF-17'></span>[[#cite-17|[17]]]. Thus, structures and facilities for dealing with particulate material handling are not functioning efficiently and there is always a probability of structural failure and an unexpected arrest of the production line. Due to the intrinsic nature of granular materials, the prediction of their dynamic behaviour represents nowadays an important challenge for two main reasons. Firstly, the characteristic grain size has an excessively wide span: from nanoscale powders (such as colloids with a typical size of nanometre) to large blocks of coal extracted from mines. This feature gives rise to some difficulties in defining a unique model able to properly work across many scales. Secondly, although these materials are solid in nature, they behave differently in various circumstances and often changes in a new state of matter with properties of solids, liquids and gases <span id='citeF-17'></span>[[#cite-17|[17]]]. Indeed, as with solids, they can withstand deformation and form heap; as with liquids, they can flow; as with gases, they can exhibit compressibility. This second aspect makes the modelling of granular matter even more difficult to define, as the macroscopic behaviour is affected by a set of microscopic parameters which often are not directly measurable from laboratory tests. Nevertheless, these challenges encouraged the research community to work actively in the particle technology field, developing and improving several numerical and experimental techniques for the characterization of granular materials.
-Once the physics of rotordynamics were well understood and steam could be used as motive force, there was a great improvement in the capacity of power generation. The first reaction turbine came up in 1884 at the hands of Charles Parsons. Later on, great development was achieved in the field of  rotordynamics due to the invention of the Dynamo in 1878, which led to an outstanding expansion of the steam turbine.
+As explained in Section [[#1.1 The granular flows|1.1]], a granular flow can undergo three main regimes in different domains of volume fraction and shear rate.  When the grains have very little kinetic energy, the assembly of the particle is dense, and if the structure is dominated by the force chains the response of the bulk is independent on the shear rate. In this case, the flow pattern is known as ''Quasi-static regime'' and the behaviour is well described by classical models used in soil mechanics <span id='citeF-18'></span>[[#cite-18|[18]]]. On the other hand, if a lot of energy is brought to the grains, the system is dilute, granular materials are collisional, inertia-dominated and compressible similar to a gas. In this case, the stresses vary as the square of the shear rate <math display="inline">\dot{\boldsymbol{\gamma }}</math> <span id='citeF-15'></span>[[#cite-15|[15]]] and the flow falls under the ''Rapid granular flow'' theory. The principal approach, provided in the literature, for modelling granular flow under these conditions is represented by the Kinetic Theory of granular gases <span id='citeF-16'></span>[[#cite-16|[16]]]. In the definition of such a model, the formalism of gas kinetic theory is used with the constraint to consider the particles perfectly rigid and the kinetic theory formalism leads to a set of Navier-Stokes equations. Between these two regimes, we can find the dense and slow flow regime, characterized by the presence of multiple particles contact, but also by the absence of force chains. For the modelling of granular flows under this regime, in <span id='citeF-19'></span>[[#cite-19|[19]]] the constitutive relation of a viscoplastic fluid is proposed, commonly known with the name of <math display="inline">\mu (I)</math> rheology. The idea comes from the analogy observed with Bingham fluids, characterized by a yield criterion and a complex dependence on the shear rate. By assuming the particles perfectly rigid and a homogeneous and steady flow, a set of Navier-Stokes equations is provided. Despite it has been demonstrated that the model can successfully reproduce the results of some experimental tests <span id='citeF-19'></span>[[#cite-19|[19]]], the model can only qualitatively predict the basic features of granular flows. In fact, some phenomena, such as, the formation of shear bands, flow intermittence and hysteresis in the transition solid to fluid and vice-versa, cannot be modelled through the <math display="inline">\mu (I)</math> law.
-Little time had to pass for rotating machinery to be used as source of '''propulsion'''. The first turbojet-powered aircraft was the ''Heinkel HE-178'', dating from 1939. Since then, high-speed complex rotating machinery has taken an important role in many industrial and aeronautical fields. However, they require from precise estimation to work properly, as they are submitted to high thermal and stress loads. Elasticity theories and energy methods are relatively easy to apply to bars, beams, discs, plates and even membranes, but when it comes of complex three-dimensional structures, no analytical solution exists.
+Even if there are models able to predict the flow behaviour in single regimes, a comprehensive rheology, able to gather together all three regimes, is still missing in the literature. Many attempts have been done during the last years. To cite a few of them, it is worth mentioning the contribution of Vescovi and Luding <span id='citeF-20'></span>[[#cite-20|[20]]] where a homogeneous steady shear flow of soft frictionless particles is investigated; both fluid and solid regimes are considered and merged into a continuous and differentiable phenomenological constitutive relation, with a focus on the volume fraction close to the jamming value. Also Chialvo and coworkers <span id='citeF-21'></span>[[#cite-21|[21]]] turn the attention to the interface between the quasi-static and the inertial regime in the context of a jamming transition, still neglecting any time dependency (under the assumption of steady-state flow), but considering soft friction particles. Other proposals are based on the relaxation of some hypotheses at the base of the <math display="inline">\mu (I)</math> rheology, such as, for instance, the constitutive laws provided by Kamrin et al. <span id='citeF-22'></span>[[#cite-22|[22]]] where the non-local effects are considered and by Singh et al. <span id='citeF-23'></span>[[#cite-23|[23]]] where the particle stiffness influence is included in the model.
-In this context, '''numerical analysis''' arose. Until the half of the past century, analysis of complex structures was already been carried in a numerical approach using the matrix method.  However, even tough it gave good results for simple structures under static conditions, the emergence of aeronautics uncovered some of its main drawbacks, such as the impossibility to describe the flutter phenomena.  <div id='img-1.2'></div>
+In order to perform a numerical investigation of the granular flow problem, one has to keep in mind that not only a constitutive model is needed for its accomplishment, but also a numerical technique used to solve the system of algebraic equations which govern the problem. Numerical methods can be distinguished according to the kinematic description adopted and to the spatial and time scales that balance laws are based on. As previously observed, particulate materials can be studied at different scales and depending on this the selection of the numerical technique may change.
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-He-178-1.png|330px|Picture of the ''Heinkel HE-178'' <span id='citeF-5'></span>[[#cite-5|[5]]]]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 1.2:''' Picture of the ''Heinkel HE-178'' <span id='citeF-5'></span>[[#cite-5|[5]]]
-|}
-With the release of <math display="inline">NASTRAN</math> by the end of the sixties and further publications pointing its mathematical basis, the '''finite element method''' gained popularity among physicists and engineers for its wide range of application. Nowadays, its main drawback is the computational cost, as the computation time increases with the number of elements to simulate.
+Granular materials have a discrete nature and it is of paramount importance to have a clear description at the microscale for a deep understanding of the physics behind the bulk behaviour. In recent decades, the most common and used numerical tool for such investigation is represented by the Discrete Element Method (DEM) <span id='citeF-24'></span>[[#cite-24|[24]]], which considers a finite number of discrete interacting particles, whose displacement is described by the Newton's equations related to translational and rotational motions.
-This problem has already been addressed by applying modal decomposition analysis, which provides great results with less computation cost for dynamics systems under harmonic excitations. Unfortunately, physics describing the vibration phenomena in rotating structures are highly nonlinear, and thus alternative paths are to be sought. The '''singular value decomposition''' seems to be a good candidate that, in combination with modal analysis and numerical integration, could unfold predominant patters of the structure. If applied to rotordynamics, some phenomena of aeronautical interest could be studied, including the vibrations generated in helicopters’ and wind turbines’ blades. Other effects caused by the rotation conditions could also be addressed, such as the centrifugal stiffening and resonance phenomena. The latter is, in fact, one of the fields of investigation that lead to the foundation of the finite element method.
+In the research community, granular materials are studied by using a continuum approach, as well. For instance, all the aforementioned constitutive models, proposed under the condition which range between the ''Quasi-static regime'' and ''Rapid granular flow'', are based on a continuous description of the granular matter behaviour. On one hand these approaches, obviously, do not allow to predict the material response on the point, where two particles collide, i.e., at the microscale, but are able to provide a mesoscale response, constant on a representative elemental volume (REV), where the continuum assumptions are still valid <span id='citeF-10'></span>[[#cite-10|[10]]].
-==1.2 State of the art==
+Other methods, available in the literature, are used to scale-up from the micro-level. In this regard, it is worth mentioning the Population Balance Method (PBM) <span id='citeF-25'></span>[[#cite-25|[25]]], able to describe the evolution of a population of particles from the analysis of single particle in local conditions. For instance, PBM is widely used to track the change of particle size distribution during processes where agglomeration or breakage of particles are involved, by using information, such as, impact velocity distribution, provided by DEM analysis <span id='citeF-26'></span><span id='citeF-27'></span>[[#cite-26|[26,27]]].
-This section reviews some of the basic topics of rotordynamics and introduces the reader to some essential but important concepts and models related with the dynamics of rotating structures.
+==1.3 Objectives==
-===1.2.1 Introduction to rotordynamics===
+In the present work, we focus on the macroscale analysis of granular flows. More specifically, the current monograph aims at providing a verified and validated numerical model able to predict the behaviour of highly deforming bulk of granular materials in their real scale systems. Some examples, objects of this study, are represented by hopper flows, the collapse of granular columns and measurement of bearing capacity of a soil undergoing the movement of a rigid strip footing. These tests are characterized by some common features which are essential for the definition of the numerical tool to be developed. All the examples of granular flow mentioned above are densely packed with solid concentrations well above 50% of the volume and they can be considered dense granular flow where forces are largely generated by inter-particle contacts. This implies that a collisional state of the matter, where the principal mechanism of momentum transport is based on binary particle contacts, will never be reached. Moreover, in all cases an elastic/quasi-static regime usually coexists with a plastic/flowing regime. In order to be able to model the quasi-static and the flowing behaviours simultaneously in different parts of the material domain, a constitutive law which accounts for both the elastic and plastic regime is needed. The use of viscous fluid materials, as those described by the <math display="inline">\mu (I)</math> rheology and its different versions, can predict the granular flow behaviour under the inertial regime. However, the good reliability of these models is still limited to the steady case and to volume fractions whose values never exceed the jamming point. Further, if a viscous material is chosen, some difficulties might be encountered in the evaluation of internal forces where a zero strain rate is present.  With the picture described above, the numerical model is conceived in a continuum mechanics framework in order to optimize the high, not to say prohibitive, computational cost which might be induced by the high value of density in the grain system, if a discrete technique is, then, selected. Moreover, the transition between the solid-like and fluid-like behaviour induces large displacement and huge deformation of the continuum which, from the numerical viewpoint, is well established to be tough to handle with standard techniques, such as the well-known Finite Element Method (FEM). Last but not least, not only geometric, but also material non-linearities should be considered. To address to this concern, the choice falls on those elasto-plastic laws, defined within the solid mechanics framework, whose stress response depends on the total strain history and historical parameters characteristic of the material model. The numerical model to adopt has to be based not only on a continuum mechanics framework, but also has to be able to track with high accuracy the huge deformation of the medium and the spatial and time evolution of its own material properties. After a search focused on the numerical model which closest fits with the features outlined above, it is found that the Material Point Method (MPM) <span id='citeF-28'></span>[[#cite-28|[28]]], a continuum-based particle method, might be a good candidate in solving granular flows problems under multi-regime conditions and an optimal platform for the numerical implementation of new constitutive laws which attempt to include a bridge between different scales (from the particle-particle contact (micro) to the bulk (macro) scale). In the current work, an implicit MPM is developed by the author in the multi-disciplinary Finite Element codes framework ''Kratos Multiphysics'' <span id='citeF-29'></span><span id='citeF-30'></span><span id='citeF-31'></span>[[#cite-29|[29,30,31]]]. Unlike most MPM codes, which make use of explicit time integration, in this monograph it is decided to adopt an implicit integration scheme. The choice is made with the aim of analysing cases characterized by a low-frequency motion and providing results with a higher stability and better convergence properties. Two formulations are implemented within the MPM framework by taking into account the geometric non-linearity, which allows to treat problems of finite deformation, usually not considered in many MPM codes that one can find in the literature. Firstly, an irreducible formulation and a Mohr-Coulomb constitutive law are developed. Further, a mixed formulation is proposed for the analysis of granular flows under undrained conditions, which represents, to the knowledge of the author, an original solution in the context of the MPM technique. The MPM strategy, with both the formulations, is validated by using experimental results or solutions of other studies, available in the literature. Last but not least, as final objective of this monograph, the developed MPM numerical tool is successfully tested in an industrial framework, in the context of a collaboration with Nestlé. A comparison is performed against an unpublished experimental study conducted for the characterization of flowability of several types of sucrose. Advantages and limitations of the numerical strategy provided are observed and discussed.
-Rotordynamics is the branch of applied mechanics dealing with devices in which at least one part, the ''rotor'', rotates with significant angular momentum. Although the definition of rotor states that a set of hinges or bearings constrain the position of the rotation axis, there are some cases in which the rotor is not restricted at all. These are known as ''free rotors'', with a spin speed governed by conservation of angular momentum. Examples of free rotors are spinning projectiles, space vehicles and even celestial bodies such as spinning neutron stars.
+==1.4 T-MAPPP project==
-The aim of this report is, however, to study the so-called ''fixed rotors'', in which the spin axis is somehow fixed and the spin speed imposed by a '''driving device'''.  First rotordynamic studies date from the late <math display="inline">19^{th}</math> century, when many methods to deal with rotatory machinery were developed (see [[#A.2 Classical analysis methods|A.2]]). One of the simplest models that described the behaviour of rotatory structures was the '''Jeffcott rotor''', which can qualitatively explain many important features of real-life rotors. However, this simple model fails in computing precise values and in modelling complex machinery, as cannot predict some phenomena such as the dependence of the natural frequencies on the rotational speed.
+The current work has been funded by the T-MAPPP (Training in Multiscale Analysis of MultiPhase Particulate Processes and Systems, FP7 PEOPLE 2013 ITN-G.A. n60) project. This project has been conceived in order to bring together European organizations leading in their respective fields of production, handling and use of particulate systems. T-MAPPP is an Initial Training Network funded by FP7 Marie Curie Actions with 10 full partners and 6 associate partners. The role of the network is to train the next generation of researchers who can support and develop the emerging inter- and supra-disciplinary community of Multiscale Analysis (MA) of multi Phase Particulate Processes. The goal is to develop skills to progress the field in both academia and industry, by devising new multiscale technologies, improving existing designs and optimising dry, wet, or multiphase operating conditions. One aim of the project is to train researchers who can transform multiscale analysis and modelling from an exciting scientific tool into a widely adopted industrial method; in other words, the establishment of an avenue able to increasingly link academic to real world challenges.
-As time passed by, higher power density, lower weight and faster machinery were demanded. As a consequence, correct quantitative prediction became particularly important. Torque is usually the critical factor when sizing the machine, so in applications related with power generation higher rotation speed is a must. With the increase in speed, power devices became lighter by using stronger but lighter materials. However, stronger doesn't always mean stiffer, an these lighter structures were more prone to '''vibrate'''.
+==1.5 Layout of the monograph==
-Another field that demanded high performing features was propulsion, which needed from high thermodynamic efficiency and power density. That meant that the same amount of heat had to be generated in a smaller volume, and hence with lower thermal capacity. Rotating machines working under these conditions not only have to handle high temperatures, but also stresses due to the thermal gradient.
+The layout of the document is as follows: in '''Chapter 2''', after a brief review of the state of the art in particle methods, the focus is put on those methods which are more consistently used for the prediction of granular flows behaviour, such as, the Discrete Element Method (DEM), the Particle Finite Element Method (PFEM), the Galerkin Meshless Methods (GMM) and the Material Point Method (MPM). The latter is the chosen approach, used and developed in this monograph. The choice is discussed and the details of the proposed formulations are provided. In '''Chapter 3''' the theory of constitutive laws used in the current work is presented with their implementations under the assumption of finite strains. In '''Chapter 4''' and '''Chapter 5''' an irreducible and a mixed stabilized formulation, respectively, are presented and verified with solid mechanics benchmark examples. Then, in '''Chapter 6''' the numerical model of MPM, presented in the previous chapters, is applied and validated (with experimental and numerical results available in the literature) against granular flow examples, such as, the granular column collapse and the rigid strip footing test. In '''Chapter 7''' the MPM strategy is applied in an industrial framework. The numerical results are compared against experimental measurements performed for the assessment of the flowability performance of different types of sucrose. Finally, in '''Chapter 8''' some conclusions are drawn, where observations and limitations of the numerical strategy are provided, and the bases for future research are outlined.
-Although there has been an outstanding progress in the field of rotordynamics during the recent years, it is still a field of very '''active research'''. Attitude control of space vehicles, design of wind turbines, industrial rotating machinery, turbojets, electric motors, power generators, helicopter's rotors and space launchers (rockets) are only a few of all the engineering fields that could benefit from those investigations in the near future.
+=2 Particle Methods=
-===1.2.2 Linear rotordynamics===
+Computer modelling and simulation are now an indispensable tool for resolving a multitude of scientific and challenging problems in science and engineering. During the last decades the importance of computer-based science has exponentially grown in the engineering field and, nowadays, it is widely adopted in the study of different processes because of its advantages of ''low cost'', ''safety'' and ''efficiency'' over the experimental modelling. The numerical simulation of solid mechanics problems involving history-dependent materials and large deformations has historically represented one of the most important topics in computational mechanics. Depending on the way deformation and motion are described, existing spatial discretisation methods can be classified into Lagrangian, Eulerian and hybrid ones. Both Lagrangian and Eulerian methods have been widely used to tackle different examples characterized by extreme deformations. In this chapter, firstly, the most common numerical techniques used in the modelling of granular flows are presented. Then, the focus is put on the Material Point Method, which is the object of the present study.
-'''
+==2.1 Lagrangian and Eulerian approaches==
-====1.2.2.1 Rigid body equations===='''
+In continuum mechanics two fundamental descriptions of the kinematic and the material properties of the body, under analysis, are possible. The first one is represented by the Lagrangian approach. In this case the description is made as ''the observer'' were attached to a material point forming part of the continuum. Lagrangian algorithms, traditionally employed in structural mechanics, make use of a moving deforming mesh dependent on the motion of the body and are distinguished by the ease with which the material interfaces can be tracked and the boundary conditions can be imposed.  According to <span id='citeF-32'></span>[[#cite-32|[32]]] three Lagrangian formulations can be defined:
-Imagine a rigid body rotating and moving around, with mass <math display="inline">m</math> and principal moments of inertia <math display="inline">J_{\varsigma }, J_{\varrho }, { and } J_{\varpi }</math> referred to a frame <math display="inline">\varsigma \varrho \varpi </math> fixed to the body and coincident with its principal axes of inertia. The equations describing its motion are quite complex, with non-linearities in the rotational degrees of freedom. Being <math display="inline">xyz</math> the inertial frame of reference, the equations of motion under the effect of a moment <math display="inline">\mathbf{M}</math> and force <math display="inline">\mathbf{F}</math> are:
+* the ''Total Lagrangian'' formulation, where all the variables are written with reference to the undeformed configuration <math display="inline">\Omega ^0</math> at the initial time <math display="inline">t_0</math>
+* the ''Updated Lagrangian'' formulation, where all the variables are written with reference to the deformed configuration <math display="inline">\varphi \left(\Omega \right)^n</math> at the previous time <math display="inline">t_n</math>
+* the ''Updated Lagrangian'' formulation, where all the variables are written with reference to the deformed configuration <math display="inline">\varphi \left(\Omega \right)^{n+1}</math> at the current time <math display="inline">t_{n+1}</math>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+Moreover, history-dependent constitutive laws can be readily implemented and, since there is not advection between the grid and the material, no advection term appears in the governing equations. In this regard, Lagrangian methods are more simple and more efficient than Eulerian methods.  The greatest drawback of this approach is represented by the high distortion of the mesh and element entanglement when the material undergoes really large deformation, which makes more difficult to obtain a stable solution with an explicit integration scheme. The second approach lies on an Eulerian description, i.e., ''the observer'' is located at a fixed spatial point. Thus, Eulerian techniques, mostly employed in fluid mechanics, are characterized by the use of a fixed grid and no mesh distortion or element entanglement are observed neither in the case of very large deformation. On the other hand, due to its intrinsic nature, it is difficult to identify the material interfaces and the definition of history-dependent behaviour is computationally intensive if compared with Lagrangian methods. As can be seen, each of the two approaches has advantages and drawbacks; thus, depending on the problem to solve, one technique is preferable over the second one.
+In the framework of granular flow modelling the Lagrangian viewpoint presents, in this context, a rather obliged choice, since the adoption of such a framework greatly simplifies the constitutive modelling and the tracking of the entire deformation process. In the case of mesh-based methods, the natural limitation of the Lagrangian approach is related to the deformation of the underlying discretisation, which tends to get tangled as the deformation increases. Massive remeshing procedures have proved to be capable of further extending the realm of applicability of Lagrangian approaches, effectively extending the limits of the approach well beyond its original boundaries.  Nevertheless, while, on one hand, it is possible to alleviate the distortion of the mesh, on the other hand, additional numerical errors arise from the remeshing and the mapping of state variables from the old to the new mesh. In this regard, the Arbitrary Lagrangian–Eulerian method (ALE) <span id='citeF-32'></span>[[#cite-32|[32]]], a generalization of the two approaches described earlier, has been developed in the attempt to overcome the limitation of the Total Lagrangian (TL) and Updated Lagrangian (UL) techniques when severe mesh distortion occurs by making the mesh independent of the material, so that the mesh distortion can be minimized. However, for very large deformation severe computational errors are introduced by the distorted mesh. Furthermore, the convective transport effects can lead to spurious oscillations that need to be stabilized by artificial diffusion or by other stabilization techniques. Such disadvantages make the ALE methods less suitable than other techniques which can be found in the literature.
+In the current work, the Lagrangian framework is considered, but the focus is on the so-called ''particle methods'', a series of techniques which represent a natural choice for the solution of granular flow problems involving large displacement, large deformation and history-dependent materials. The next section introduces a brief state of the art of the most common particle methods with their distinguished features and fields of application.
+==2.2 Particle methods. A review of the state of the art==
+Particle methods are techniques which have in common the discretisation of the continuum by only a set of nodal points or particles. According to <span id='citeF-33'></span>[[#cite-33|[33]]], they can be classified based on two different criteria: physical principles or computational formulations. For those methods classified according to physical principles a further distinction is made if the model is deterministic or probabilistic; while according to the computational formulations, the particle methods can be distinguished in two subcategories, those serving as approximations of the strong forms of the governing partial differential equations (PDEs), and those serving as approximations of their weak forms. In Tables [[#table-2.1|2.1]] and [[#table-2.2|2.2]] the classification is graphically shown with a list of the main approaches which fall under each category.
+{|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-2.1'></span>Table. 2.1 Physical principles based particle methods.
+|-style="font-size: 85%;"
+| colspan='2' style="text-align: center;" | ''Physical principles''
 |-
+| colspan='1' style="text-align: center;" | Deterministic models
+| colspan='1' style="text-align: center;" | Probabilistic models
+|-style="font-size: 85%;"
+|  Discrete Element Method (DEM)
+| Molecular Dynamics
+|-style="font-size: 85%;"
 |
-{| style="text-align: left; margin:auto;width: 100%;"
+| Monte Carlo methods
-|-
+|-style="font-size: 85%;"
-| style="text-align: center;" | <math>\mathbf{F} = \mathrm{m} \frac{{d\;\mathbf{v}}}{d\mathrm{t}} = m\;\mathbf{a}\;\;\;,\;\;\;\mathbf{J}{\dot{\boldsymbol{\Omega } }} + \boldsymbol{\Omega }\times (\mathbf{J}{\boldsymbol{\Omega } }) = \mathbf{M} </math>
+|
+| Lattice Boltzmann Equation method
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.1)
+{|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-2.2'></span>Table. 2.2 Computational formulations based particle methods.
+|-style="font-size: 85%;"
+| colspan='2' style="text-align: center;" | ''Computational formulations''
+|-style="font-size: 85%;"
+| colspan='1' style="text-align: center;" | Approximations of the strong form
+| colspan='1' style="text-align: center;" | Approximations of the weak form
+|-style="font-size: 85%;"
+|  Smooth Particle Hydrodynamics
+| Meshfree Galerkin Method:
+|-style="font-size: 85%;"
+| Vortex Method
+| - ''Diffusive Element Method''
+|-style="font-size: 85%;"
+| Generalized finite Difference Method
+| - ''Element Free Galerkin Method''
+|-style="font-size: 85%;"
+| Finite Volume PIC
+| - ''Reproducing Kernel Method''
+|-style="font-size: 85%;"
+|
+| - ''h-p Cloud Method''
+|-style="font-size: 85%;"
+|
+| - ''Partition of Unity Method''
+|-style="font-size: 85%;"
+|
+| - ''Meshless Local Petrov-Galerkin Method''
+|-style="font-size: 85%;"
+|
+| - ''Free Mesh Method''
+|-style="font-size: 85%;"
+|
+| Mesh-based Galerkin Method
+|-style="font-size: 85%;"
+|
+| - ''Material Point Method''
+|-style="font-size: 85%;"
+|
+| - ''Particle Finite Element Method''
 |}
-Which correspond to Newton's second law and Euler's equation for rigid body dynamics, being <math display="inline">\boldsymbol{\Omega }</math> the angular velocity of the body and <math display="inline">\mathbf{a}</math> its linear acceleration. Splitting into components, yields to:
-<span id="eq-1.2"></span>
+In the following sections, a bibliographic review of the most common and widely used particle methods in granular flow modelling is presented. The first method to be presented is the Discrete Element Method. Then, the Smooth Particle Hydrodynamics, the Meshfree Galerkin Method, the Particle Finite Element Method and the Particle Finite Element Method 2 are briefly introduced. For each of those advantages and disadvantages are discussed. Finally, the Material Point Method and a meshless variation of it and their algorithms are extensively described.
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+===2.2.1 The Discrete Element Method (DEM)===
+The numerical approach which considers the problem domain as a conglomeration of independent units is known as Discrete Element Method (DEM), developed by Cundall and Strack in 1979 <span id='citeF-24'></span>[[#cite-24|[24]]]. DEM was initially used for studying of rock mechanics problems using deformable polygonal-shaped blocks. Later, it has been widely utilized to study geomechanics, powder technology and fluid mechanics problems. Each particle is identified separately having its own mass, velocity and contact properties and, during the calculation, it is possible to track the displacement of particles and evaluate the magnitude and direction of forces acting on them. The main distinction between DEM and continuum approaches is the assumption on material representation; in DEM every particles represents a physical entity, e.g., the single grains in the granular system, while in a continuum method particles take the place of material points, which have instead just a numerical purpose in the computation of the solution.
+According to <span id='citeF-24'></span>[[#cite-24|[24]]] the time step must be chosen in a way that disturbances from an individual particle cannot propagate further than their neighbours. Usually, in order to avoid significant instability in the granular system the time step should be smaller than a critical time step, called the Rayleigh time step.
+DEM is a good example of numerical technique that treats the bulk solid as a system of distinct interacting bodies. Thus, with DEM it is possible to simulate interaction at the particle level (at a spatial scale which ranges from <math display="inline">10^{-6}m</math> to <math display="inline">10^{-1}m</math>, depending on the size of the grains) and, at the same time, to obtain an insight into overall response, bulk properties such as stresses and mean velocities <span id='citeF-34'></span>[[#cite-34|[34]]]. Therefore, it can provide a clear explanation on particle-scale behaviour of granular solids to characterize bulk mechanical responses, as it is done in several contributions <span id='citeF-35'></span><span id='citeF-5'></span>[[#cite-35|[35,5]]]. Moreover, this technique is really useful and interesting in the research field of granular matter since DEM can be seen as a tool for performing numerical experiments that allow contact-less measurements of microscopic quantities that are usually impossible to quantify using physical experiments. Discrete element modelling has been also used extensively to analyse various handling and processing systems that deals with multiple bulk solids <span id='citeF-36'></span><span id='citeF-37'></span><span id='citeF-38'></span>[[#cite-36|[36,37,38]]].  However, the extremely high computational cost, proportional to the number of particles, leads to the limitation of considering relatively small system sizes and idealized geometries.
+The schematic flowchart, which has to be followed in order to execute a DEM calculation <span id='citeF-39'></span>[[#cite-39|[39]]] at each time step, is displayed in Figure [[#img-2.1a|2.1a]]. Even if the algorithm looks to be straightforward to run and easy to implement, the computational cost is proportional to the number of discrete elements and to their shapes. Thus, simplified assumptions have been made in the mathematical models in order to reduce the computational efforts. The primary idealized factor in DEM simulations is the shape of particles which is considered as spheres to simplify the contact detection process, which is the most time consuming step in DEM simulations. Among diverse physical properties of individual particles in particulate materials, the shape and morphology play important roles in shear strength and flowability of the bulk. In order to improve this aspect several approaches have been utilized in DEM, such as, clumped spheres <span id='citeF-40'></span>[[#cite-40|[40]]], polyhedral shapes <span id='citeF-41'></span>[[#cite-41|[41]]], super-quadric function <span id='citeF-42'></span>[[#cite-42|[42]]]. However, in order to obtain accurate results the computational cost may arise significantly. Another important step in DEM is to realistically simulate the physical impact between particles. This is usually approximated by defining spring and dashpots between contacting surfaces, as it is done in the linear-spring dash-pot models <span id='citeF-43'></span>[[#cite-43|[43]]] or the Hertzian visco-elastic models. In the literature other contact models can be found, such as meso-scale models <span id='citeF-35'></span><span id='citeF-44'></span><span id='citeF-45'></span>[[#cite-35|[35,44,45]]] or realistic contact models <span id='citeF-46'></span><span id='citeF-47'></span><span id='citeF-48'></span>[[#cite-46|[46,47,48]]], which can provide a high accuracy both at the particle and bulk level, but valid only for the limited class of materials they are particularly designed for.
+Moreover, for a proper understanding of a process and to study realistic behaviour through DEM simulations, the input parameters, listed in Figure [[#img-2.1b|2.1b]], play a vital role. The input parameters are often assumed without careful assessment or calibration which often leads to unrealistic behaviours and erroneous results. Designing of equipment or of a process route with an un-calibrated DEM model may lead to serious handling and processing operations such as segregations, unexpected wear, irregular density of products, flow blockages and etc. Thus, a correct definition of the input parameters by experimental characterization and/or calibration, using particle-level tests, directly affect the reliability of the final response at the bulk level. However, this might result in an extremely time and cost consuming procedure, that not always it is possible to perform for a lack of time and/or money.
+In conclusion, DEM is a valid and useful numerical tool for research in the granular matter field and development of new contact models. However, the drawbacks aforementioned in the current section make the DEM still not an easy and limited tool to use in the engineering and industrial framework, mainly when real scale systems are under study.
+<div id='img-2.1a'></div>
+<div id='img-2.1b'></div>
+<div id='img-2.1'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;"
 |-
-|
+|[[Image:Draft_Samper_987121664-monograph-dem_algorithm2.png|282px|]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
+| (a)
 |-
-| style="text-align: center;" | <math>\left\{\begin{align}m \ddot{x} &=F_{x} \\ m \ddot{y} &=F_{y} \\ m \ddot{z} &=F_{z} \end{align}\right. \qquad  \left\{\begin{align}M_{\varsigma } &=\dot{\Omega }_{\varsigma }\, J_{\varsigma }+\Omega _{\varrho } \,\Omega _{\varpi }\,\left(J_{\varpi } - J_{\varrho }\right)\\ M_{\varrho } &=\dot{\Omega }_{\varrho }\, J_{\varrho }+\Omega _{\varsigma }\, \Omega _{\varpi }\,\left(J_{\varsigma } - J_{\varpi }\right)\\ M_{\varpi } &=\dot{\Omega }_{\varpi }\, J_{\varpi }+\Omega _{\varsigma }\, \Omega _{\varrho }\,\left(J_{\varrho } - J_{\varsigma }\right)\end{align}\right. </math>
+|[[Image:Draft_Samper_987121664-monograph-input_parameters_dem.png|390px|]]
-|}
+|- style="text-align: center; font-size: 75%;"
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.2)
+| (b)
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 2.1:''' Dem algorithm (a), with <math>t</math> the time instant and <math>\Delta t</math> the time interval, and input parameters in DEM model (b).
 |}
-As can be spotted, Euler equations are '''nonlinear''' in the angular velocity. Moreover, the previous equations only describe the behaviour of a rigid body and elasticity plays no role whatsoever. Thus, one can state that the equations describing the behaviour of an structure under rotational conditions will be highly nonlinear.
+===2.2.2 The Smoothed Particle Hydrodynamics===
-However, a linearised model describing the basic features of rotating systems can be obtained  embracing some '''simplifications''' :
+Among the methods which serve as approximations of the strong of PDEs, the Smoothed Particle Hydrodynamics (SPH) <span id='citeF-49'></span><span id='citeF-50'></span>[[#cite-49|[49,50]]] is one of the earliest particle methods in computational mechanics. It was initially designed for solving hydrodynamics problems <span id='citeF-51'></span><span id='citeF-52'></span>[[#cite-51|[51,52]]], such as astrophysical applications <span id='citeF-53'></span><span id='citeF-54'></span><span id='citeF-55'></span>[[#cite-53|[53,54,55]]]; later, SPH has been also applied to solid mechanics problems involving impact, penetration and large deformation of geomaterials <span id='citeF-56'></span><span id='citeF-57'></span><span id='citeF-58'></span><span id='citeF-59'></span>[[#cite-56|[56,57,58,59]]] or compressible and incompressible flow problems <span id='citeF-60'></span><span id='citeF-61'></span>[[#cite-60|[60,61]]]. The PDE is usually discretized by a specific collocation technique: the essence of the method is to choose a smooth kernel which not only smoothly discretized a PDE, but also furnishes an interpolant scheme on a set of moving particles. Even if the method is widely used by the computational mechanics community, it is well established in the literature that the SPH suffers of some pathologies such as tensile instability <span id='citeF-62'></span><span id='citeF-63'></span>[[#cite-62|[62,63]]], lack of interpolation consistency <span id='citeF-64'></span><span id='citeF-65'></span>[[#cite-64|[64,65]]], zero-energy mode <span id='citeF-66'></span>[[#cite-66|[66]]] and difficulty in enforcing essential boundary condition <span id='citeF-67'></span><span id='citeF-68'></span>[[#cite-67|[67,68]]]. To solve the aforementioned fundamental issues several improvements have been provided through the years. To mitigate the tensile instability and the zero-mode issues the stress point approach <span id='citeF-63'></span><span id='citeF-69'></span>[[#cite-63|[63,69]]] has been proposed. To correct completeness, or consistency, closely related to convergence, the use of corrective kernels are considered; for instance Liu et al. <span id='citeF-64'></span>[[#cite-64|[64]]] proposed new interpolants named the Reproducing Kernel Particle Method and many other approaches can be found in the literature <span id='citeF-70'></span><span id='citeF-71'></span><span id='citeF-72'></span>[[#cite-70|[70,71,72]]] addressing this shortcoming. To enforce essential boundary conditions it is worth mentioning the contribute of Randles and Libersky <span id='citeF-67'></span>[[#cite-67|[67]]], where the so-called ghost particle approach is proposed.
-* The rotor has a rotation axis that coincides with one of its baricentrical principal axes of inertia (the rotor is perfectly balanced).
+===2.2.3 The Meshfree Galerkin Methods===
-* The deviations from the previous condition are assumed to be small.
-* Displacements and velocities (linear and angular) are assumed to be small, with the exception of the rotation angle and angular velocity about the spin axis, which in many cases will be imposed by the driving system.
-'''
+The Meshfree Galerkin Methods, unlike the SPH, were mainly developed only in the early of 1990s. The first meshless methods appeared in the literature are represented by the Diffusive Element Method <span id='citeF-73'></span>[[#cite-73|[73]]], where moving least square (MLS) interpolants <span id='citeF-74'></span>[[#cite-74|[74]]] are employed, and the Reproducing Kernel Particle Method (RKPM) <span id='citeF-64'></span><span id='citeF-75'></span>[[#cite-64|[64,75]]], defined in the attempt to provide a corrective SPH. Later, other techniques were proposed, such as the Element Free Galerkin Method (EFGM) <span id='citeF-76'></span>[[#cite-76|[76]]], in which the MLS interpolants are for the first time used in a Galerkin procedure, or the Partition of Unity Method <span id='citeF-77'></span>[[#cite-77|[77]]], where a partition of unity is taken and multiplied by any independent basis. Usually most meshfree interpolants do not satisfy the ''Kronecker delta property'' <span id="fnc-1"></span>[[#fn-1|<sup>1</sup>]] and the impossibility of a correct imposition of the essential boundary conditions represents one of the principal bottleneck of these approaches. Some remedies for the enforcement of the EBCs are given by the Lagrange Multipliers and Penalty method <span id='citeF-78'></span>[[#cite-78|[78]]], the Transformation method <span id='citeF-79'></span>[[#cite-79|[79]]], the Boundary singular kernel method <span id='citeF-74'></span>[[#cite-74|[74]]] and the Coupled finite element and particle approach <span id='citeF-80'></span>[[#cite-80|[80]]].  Most Meshfree Galerkin Methods make use of background grid to locate the quadrature points to integrate the weak form. From this aspect some problems in terms of accuracy as well as invertibility of the stiffness matrix may arise, due to the arbitrariness in locating the Gauss quadrature points. If these points are not enough in a compact support or are not evenly distributed spurious modes may also occur. In order to completely eliminate quadrature points some approaches have been proposed in the literature, e.g., the one proposed by Chen et al. <span id='citeF-81'></span>[[#cite-81|[81]]] based on a stabilized nodal integration method. Despite the typical drawbacks of Meshfree Galerkin Methods, e.g., the aforementioned issue of quadrature integration and the higher computation cost in comparison with standard FEM, during the last decades meshless methods have been increasingly used to solve applied mechanics problem due to some key advantages which distinguish them from other techniques. For instance to mention some of them, in these methods the connectivity changes with time as they do not have a fixed topological data structure, the accuracy can be controlled easily given a h-adaptivity procedure and the meshfree discretisation can provide accurate representation of geometric object. Initially, meshfree methods have been used to address the challenging field of computational fracture mechanics. In this regards, the EFGM and the Partition of Unity Methods have been applied to crack growth and propagation problems <span id='citeF-82'></span><span id='citeF-83'></span><span id='citeF-84'></span>[[#cite-82|[82,83,84]]]. The great advantage of not using a remeshing procedure has been also exploited in the application of large deformation problems; in particular, it is worth mentioning the use of the RPKM to metal forming, extrusion <span id='citeF-85'></span>[[#cite-85|[85]]] and soil mechanics problems <span id='citeF-86'></span><span id='citeF-87'></span>[[#cite-86|[86,87]]]. Meshfree methods have been also extensively applied to simulation of strain localization problems <span id='citeF-88'></span>[[#cite-88|[88]]] since meshfree interpolants can successfully reduce the mesh alignment sensitivity in the formation of the shear bands.
-====1.2.2.2 Equation of motion===='''
+<span id="fn-1"></span>
+<span style="text-align: center; font-size: 75%;">([[#fnc-1|<sup>1</sup>]])  Let us define the Lagrange polynomials of degree <math>n-1</math>, <math>L_k(x)\, (k=1,....,n)</math>. <math>L_k(x)</math> satisfy the ''Kronecker delta property'' if
-If the previous simplifications are taken into account, a rotor with '''constant''' spin velocity <math display="inline">\Omega </math> can be modelled  discreetly by using the adequate elastic model, and the following dynamic equilibrium equation is obtained:
-<span id="eq-1.3"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 236: / Line 668: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M} \,\ddot{\mathbf{q}}(\mathbf{t})+(\mathbf{D}+\mathbf{G})\, \dot{\mathbf{q}}(t)+(\mathbf{K}+\mathbf{H}) \,\mathbf{q}(t)=\mathbf{f}(t) </math>
+| style="text-align: center;" | <math> L_k(x)=\begin{cases} 1 & \mbox{ at } x=x_k,\\ 0 & \mbox{ at } x=x_i \mbox{ for } i\neq k \end{cases} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.3)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.1)
 |}
-In the previous equation, some of the present terms may look familiar to the reader: <math display="inline">\mathbf{M}</math>, <math display="inline">\mathbf{K}</math> and <math display="inline">\mathbf{D}</math> stand for the symmetric mass, stiffness and damping matrices,  while <math display="inline">\mathbf{q(t)}</math> is a vector containing the generalised coordinates referred to the inertial frame <math display="inline">xyz</math> and <math display="inline">\mathbf{f(t)}</math> a time-dependant vector that accounts for the  effect of external forces. However, in the context of rotordynamics, new matrices appear: <math display="inline">\mathbf{G}</math> and <math display="inline">\mathbf{H}</math> are the skew-symmetric gyroscopic and circulatory matrices, respectively.
+i.e., <math>L_k(x_i)=\delta _{ik}</math>. With regards to the shape functions, these ones lack of the ''Kronecker delta property'' when the weight function <math>w_j(\boldsymbol{x})</math> associated with the nodal points <math>\boldsymbol{x}_j\, \mbox{for }\left(j\neq i \right)</math> is not zero at the location of nodal point of interest <math>\boldsymbol{x}_i</math>.  </span>
-The '''gyroscopic''' matrix contains inertial and conservative terms, linked with the gyroscopic moments that act on the rotating elements of the structure. If the equations are referred to the rotating and thus noninertial frame of reference, the '''Coriolis''' effect is included in the gyroscopic matrix. The '''circulatory''' matrix, on the other hand, contains the nonconservative effects related with internal damping of the rotating parts and, in some cases, accounts for the effect of the surrounding fluids. In many cases, it is found that in the field of rotordynamics, both <math display="inline">\mathbf{G}</math> and <math display="inline">\mathbf{H}</math> are proportional to the spin velocity <math display="inline">\Omega </math>, and when the latter tends to zero, the gyroscopic and circulatory skew-symmetric elements vanish and the system reduces to that of a static structure.
+===2.2.4 The Particle Finite Element Method===
-It is important to notice that in this context, <math display="inline">\mathbf{K}</math> and <math display="inline">\mathbf{D}</math> may not have the same expression they had in the case of still strictures. In fact, they can be split into two matrices: one corresponding to static structure and another which depends on the rotating speed, often on <math display="inline">\Omega ^2 </math>.
+The Particle Finite Element Method (PFEM) is a particle method which falls under the category of mesh-based Galerkin approaches. In PFEM the domain is modelled using an Updated Lagrangian formulation and the continuum equations are solved by means of a FEM approach on the mesh built up from the underlying node, also called <math display="inline">particles</math>. The main feature of the method is based on the employment of a fast remeshing procedure to relieve the typical issue of high distortion of the mesh and a boundary recognition method, i.e., the alpha-shape technique <span id='citeF-89'></span>[[#cite-89|[89]]], needed to define the free surfaces and the boundaries of the material domain. Given an initial mesh, the remeshing procedure can be used arbitrarily at every time step <span id='citeF-90'></span>[[#cite-90|[90]]] or when the mesh starts affecting the accuracy of the numerical solution, as in the case of explicit formulations <span id='citeF-91'></span>[[#cite-91|[91]]]. This Lagrangian technique was first developed for the simulation of free surface flows and breaking waves <span id='citeF-92'></span><span id='citeF-93'></span>[[#cite-92|[92,93]]], and then successfully adapted for structural mechanics problems involving large deformations <span id='citeF-94'></span><span id='citeF-95'></span>[[#cite-94|[94,95]]], for the simulation of viscoplastic materials <span id='citeF-96'></span><span id='citeF-91'></span><span id='citeF-97'></span><span id='citeF-98'></span><span id='citeF-99'></span><span id='citeF-100'></span>[[#cite-96|[96,91,97,98,99,100]]], in geomechanics <span id='citeF-101'></span><span id='citeF-102'></span>[[#cite-101|[101,102]]] and Fluid-Structure Interaction (FSI) applications <span id='citeF-103'></span><span id='citeF-104'></span><span id='citeF-105'></span>[[#cite-103|[103,104,105]]]. Although the method has broad capabilities, some disadvantages come from the use of remeshing procedures <span id='citeF-106'></span><span id='citeF-107'></span>[[#cite-106|[106,107]]]. In the practice, although possible, the application of PFEM in problems with elastic or elasto-plastic behaviour faces difficulties related to the storage of historical variables, since information on the integration points is not preserved and needs to be remapped at the moment of remeshing. Moreover the alpha-shape technique, and the remeshing itself, lead to intrinsic conservation problems related to the arbitrariness of the reconnection patterns <span id='citeF-108'></span>[[#cite-108|[108]]]. Last but not least, the characteristics of the remeshing approach at the base of the PFEM make it very hard to parallelize, thus, limiting the possibility of the method in terms of computational efficiency.
-'''
+===2.2.5 The Particle Finite Element Method 2===
-====1.2.2.3 Complex coordinates===='''
+A second generation of PFEM (PFEM2) has been recently introduced <span id='citeF-109'></span><span id='citeF-110'></span><span id='citeF-111'></span><span id='citeF-112'></span>[[#cite-109|[109,110,111,112]]], which tries to repair to the shortcomings observed in the previous version. PFEM2 is a hybrid particle method, which exploits the combination of both the Eulerian and Lagrangian approaches, as in the Material Point Method. The method has been tested in the analysis of interaction between different materials, such as, incompressible multifluids and fluid-structure interaction <span id='citeF-113'></span>[[#cite-113|[113]]], and simulation of landslides and granular flows problems <span id='citeF-114'></span>[[#cite-114|[114]]]. This technique is based on the use of a set of Lagrangian particles, in order to track properties of the continuum, and by a fixed finite element mesh, employed for the solution of the governing equations. Moreover, a projection of the data between the two spaces is performed during the calculation of a time step in order to keep updated the kinematic information between the particles and the nodes of the Eulerian mesh. One of the main differences which distinguishes PFEM2 from the Material Point Method lies on the definition of Lagrangian particle itself. In the case of PFEM2, the particles represent material points without a fixed amount of mass; in order to guarantee a good particle distribution in the computational domain, the number of particle might change during the simulation time. These Lagrangian entities do not represent integration points, but are just used with the purpose of convecting all the historical material properties and kinematic information through the simulation. This feature makes PFEM2 particularly adapted for the modelling of incompressible fluids, with Newtonian and non-Newtonian rheology and FSI problems.
-In some cases, the simplifications made in the previous page cannot be assumed, and the study of the rotor becomes very complicated. In this situations, however, an model resembling equation [[#eq-1.3|1.3]] can be obtained, although with reference to a '''noninertial frame'''. Moreover, most flexible rotors can be assumed to behave as beams. With these hypotheses, the lateral behaviour of the rotor can be uncoupled from its axial and torsional, simplifying thereby the resulting equations. Mathematically, this is done by using complex coordinates.
+==2.3 The Materal Point Method==
+An alternative particle method proposed in the literature is represented by the Material Point Method (MPM) <span id='citeF-115'></span><span id='citeF-28'></span>[[#cite-115|[115,28]]], which is object of study of this monograph.  The Material Point Method (MPM) is a particle-based method, whose origin goes back to the paper of <span id='citeF-116'></span>[[#cite-116|[116]]], where the particle-in-cell method (PIC), a technique for the solution of fluid flow problems, was proposed for the first time. Some decades after, in the works of <span id='citeF-115'></span><span id='citeF-28'></span>[[#cite-115|[115,28]]], the PIC method is redefined within the solid mechanics framework, and after that, it is known to the computational community with the name of Material Point Method. MPM combines a Lagrangian description of the body under analysis, which is represented by a set of particles, the so-called ''material points'', with the use of a computational mesh, named ''background grid'', as can be observed in Figure [[#img-2.2|2.2]].
+<div id='img-2.2'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding-top:10px;|[[Image:Draft_Samper_987121664-monograph-GMPM.png|300px|''MPM''. The shape functions on the material point p<sub>i</sub> are evaluated using FE shape function of element I-J-K.]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;| '''Figure 2.2:''' ''MPM''. The shape functions on the material point <math>p_i</math> are evaluated using FE shape function of element I-J-K.
+|}
-If the problem can be reduced into two-dimensional, that is, spinning around the <math display="inline">z</math> axis and displacements only in the <math display="inline">xy</math> plane, the latter can be expressed in terms of a complex number <math display="inline">r(t) = x(t)+iy(t)</math>, where <math display="inline">i</math> stands for the imaginary unit <math display="inline">\sqrt{-1}</math>. This formulation is equivalent to representing displacements in vectorial form, but allows a more convenient analytical form to solve the rotor equation of motion (eq. [[#eq-1.3|1.3]]).
+This distinctive feature allows one to track the deformation of the body and retrieve the history-dependent material information at each time instant of the simulation, without committing mapping information errors, typical of methods which make use of remeshing techniques. This makes the method particularly attractive for the solution of problems, characterized by very large deformations and by the use of complex constitutive laws <span id='citeF-117'></span><span id='citeF-118'></span>[[#cite-117|[117,118]]]. For instance, the method has been extensively used for geotechnical problems <span id='citeF-119'></span><span id='citeF-120'></span><span id='citeF-118'></span>[[#cite-119|[119,120,118]]] for its capabilities in tracking extremely large deformation while preserving material properties of the material points.
-The key of this method is that symmetric matrices are real when the equations are written in terms of complex coordinates, whereas skew-symmetric matrices become symmetric but with imaginary terms. Using the complex number formulation, the equation of motion for a axisymmetrical  rotor results in:
+In the key works of Sulsky and co-workers <span id='citeF-115'></span><span id='citeF-28'></span>[[#cite-115|[115,28]]], the MPM has been applied for the first time in the solid mechanics framework. Even if through the original MPM it was possible to solve complex problems involving, for instance, contact <span id='citeF-121'></span>[[#cite-121|[121]]], interaction between different materials <span id='citeF-122'></span><span id='citeF-123'></span>[[#cite-122|[122,123]]] and the use of history-dependent material laws <span id='citeF-115'></span>[[#cite-115|[115]]], it was observed that the first version of MPM suffers from some intrinsic shortcomings. Indeed, due to the use of piece-wise linear shape functions, the latter are only locally defined and their gradients are discontinuous. This implies that a material point on the cell boundary would not be covered by the local shape functions defined within the respective cells around the particle. This would produce a noise in the numerical solution, which is called ''cell-crossing error''.  Recently, many improvements to the original MPM have been provided to alleviate the cell-crossing noise and to have a more efficient and algorithmically straightforward evaluation of grid node integrals in the weak formulation. The Generalized Interpolation Material Point method (GIMP) <span id='citeF-124'></span>[[#cite-124|[124]]] represents the first attempt to provide an improved version of the original MPM. The essence of this method is based on the definition of a characteristic function <math display="inline">\chi _p(x)</math> which has to satisfy the partition of unity criterion, i.e.,
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 262: / Line 704: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M}^{\prime }\, \ddot{\mathbf{q}}^{\prime }(t)+\left(\mathbf{D}^{\prime }+i \mathbf{G}^{\prime }\right)\,  \dot{\mathbf{q}}^{\prime }(t)+\left(\mathbf{K}^{\prime }+i \mathbf{H}^{\prime }\right)\,  \mathbf{q}^{\prime }(t)=\mathbf{f}^{\prime }(t) </math>
+| style="text-align: center;" | <math>\sum _p \chi _p(\boldsymbol{x}) = 1 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.4)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.2)
 |}
-Separating real from imaginary parts:
+The particle characteristic function defines the spatial volume occupied by the particle <math display="inline">V_p</math> as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 274: / Line 716: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left[\begin{array}{cc}{\mathbf{M}^{\prime }} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{M}^{\prime }}\end{array}\right]\ddot{\mathbf{q}}(t) + \left(\left[\begin{array}{cc}{\mathbf{D}^{\prime }} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{D}^{\prime }}\end{array}\right]+ \left[\begin{array}{cc}{\mathbf{0}} & {\mathbf{-G'}} \\ {\mathbf{G'}} & {\mathbf{0}}\end{array}\right]\right)\dot{\mathbf{q}}(t)+ \left(\left[\begin{array}{cc}{\mathbf{K}^{\prime }} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{K}^{\prime }}\end{array}\right]+ \left[\begin{array}{cc}{\mathbf{0}} & {\mathbf{-H'}} \\ {\mathbf{H'}} & {\mathbf{0}}\end{array}\right]\right){\mathbf{q}}(t) = {\mathbf{f}}(t) </math>
+| style="text-align: center;" | <math>V_p = \int _{\Omega _p \bigcap \Omega } \chi _p(\boldsymbol{x}) dV </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.5)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.3)
 |}
-The solution of the previous system can be written in terms of the complex frequency <math display="inline">\;\;\mathbf{s=\sigma{+}i \omega }</math>: the imaginary part <math display="inline">\omega </math> stands for the frequency of the free motion (''whirl'' frequency) while the real part <math display="inline">\sigma </math>  is the decay rate changed in sign.
+where <math display="inline">\Omega _p</math> and <math display="inline">\Omega </math> are the current particle domain and the current domain occupied by the continuum, respectively. Moreover, since a material property <math display="inline">f(\boldsymbol{x})</math> can be approximated by its particle value <math display="inline">f_p</math> as
-The main advantage of this method is that allows the system to be analysed using '''modal decomposition''', which in conventional coordinates required from all the matrices to be square and symmetrical. Further details on the modal analysis and computation of eigenvectors and eigenvalues can be found in section [[#5.1 Modal decomposition analysis|5.1]].
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>f(\boldsymbol{x}) = \sum _p f_p \chi _p(\boldsymbol{x}) </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.4)
+|}
-'''
+<math display="inline">\chi _p(x)</math> acts as a smoothing of the particle properties and it determines the smoothness of the spatial variation. The full version of GIMP requires integration over the current support of <math display="inline">\chi _p(x)</math>, which deforms and rotates according to the deformation of the background grid. To do that, a tracking of the particle shape is mandatory, but in a multi-dimensional problem this could be very difficult to accomplish. Thus, an alternative version of the GIMP is represented by the uniform GIMP (uGIMP), where shear deformation and rotation of the particles are neglected. The uGIMP assumes that the sizes of particles are fixed during the material deformation. In this way, the particle characteristic function, whose support may overlap or leave gaps for very large deformation, is no longer able to satisfy the partition of unity criterion, and, thus, the ability of computing rigid body motion is lost. Therefore, the uGIMP is unable to completely eliminate the ''cell-crossing error''.
-====1.2.2.4 The Campbell diagram===='''
+In the attempt to improve the issues left by the GIMP, the Convected Particle Domain Interpolation technique (CPDI) <span id='citeF-125'></span>[[#cite-125|[125]]] is proposed. In the CPDI the particle has an initial parallelogram shape and a constant deformation gradient is assumed over the particle domain. This technique is a first-order accurate approximation of the particle domain <math display="inline">\Omega _p</math>. Even if in the CPDI a more accurate approximation of <math display="inline">\Omega _p</math> is obtained, the issues of overlaps and gaps are not overcome. Only with the second-order Convected Particle Domain Interpolation (CPDI2) <span id='citeF-126'></span>[[#cite-126|[126]]], an enhanced CPDI, which provides a second-order approximation of the particle domain, these issues are totally corrected. It is also worth mentioning the Dual Domain Material Point Method (DDMPM) <span id='citeF-127'></span>[[#cite-127|[127]]], an alternative technique which is able to definitely eliminate the ''cell-crossing error''. Unlike the GIMP or CPDI, the DDMPM does not make use of particle characteristic functions and the issue of tracking the particle domain through the whole simulation is avoided. The essence of this technique relies on the use of modified gradient of the shape function, defined as follows
-As has been seen, the spin speed of the structure can appear explicitly in the equation of motion. Thus, a dependence of the natural frequencies on this speed is expected. If that is the case, it is quite common to '''plot''' the natural frequencies <math display="inline">\omega </math> as functions of <math display="inline">\Omega </math>. In many cases, frequencies characterising exciting forces also depend on <math display="inline">\Omega </math> and so are reported in the same graph, which is commonly known as  ''Campbell diagram''. Clearly, it is symmetrical for both <math display="inline">\omega </math> and <math display="inline">\Omega </math> axis, so only the first quadrant is represented. The intersections with the <math display="inline">\omega </math>  axis correspond to the natural frequencies at standstill of the system (<math display="inline">\Omega </math> = 0). It is important to note that the Campbell diagram can only be graphed if the equation of motion is linearised, because only in this case the concept of natural frequencies applies.
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\overline{\nabla N_I}(\boldsymbol{x}) = \alpha (\boldsymbol{x})\nabla N_I(\boldsymbol{x}) + (1-\alpha (\boldsymbol{x}))\widehat{\nabla N_I}(\boldsymbol{x}) </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.5)
+|}
-If the exciting forces are time-dependent with an harmonic time story, they can be plotted in the Campbell diagram. For instance, that is the case of the forces resulting from the unbalance of the rotor, which can be modelled with frequency equal to the rotation speed <math display="inline">\Omega </math>. If the shape of the force function is less regular, it can be split into harmonic terms using Fourier decomposition. The complete response of the system will be the sum of the responses of the system under those harmonic components.
+where <math display="inline">\nabla N_I(\boldsymbol{x})</math> is the gradient of the shape function evaluated as in the original MPM, <math display="inline">\widehat{\nabla N_I}(\boldsymbol{x})</math> is the gradient from the node-based calculation as used in FLIP ( FLuid-Implicit Particle)<span id='citeF-128'></span>[[#cite-128|[128]]].
-Quite often, the relation between the frequency of the forcing function and <math display="inline">\Omega </math> is of simple proportionality, and thus can be represented in the Campbell diagram by a straight line. If the proportion is one to one, the line <math display="inline">\omega  =\Omega </math> is graphed, and the excitation is said to be '''synchronous'''.   The spin speeds <math display="inline">\Omega </math> at which a forcing function has a frequency coinciding with a natural frequency of the system at that same <math display="inline">\Omega </math> are known as '''critical speeds'''. They can be found in the diagram by looking for the intersections between natural and forcing frequencies.
+Most MPM codes make use of explicit time integration, due to the ease of the formulation and implementation <span id='citeF-115'></span><span id='citeF-129'></span><span id='citeF-130'></span>[[#cite-115|[115,129,130]]]. Explicit methods are preferable to solve transient problems, such as impact or blast, where high frequencies are excited in the system <span id='citeF-131'></span><span id='citeF-132'></span><span id='citeF-121'></span>[[#cite-131|[131,132,121]]]. However, when only the low-frequency motion is of interest, the adoption of an implicit time scheme may reduce the computational cost in comparison to the employment of an explicit time scheme <span id='citeF-133'></span>[[#cite-133|[133]]]. Some implicit versions of MPM can be found in the literature. For instance, Guilkey <span id='citeF-134'></span>[[#cite-134|[134]]] exploits the similarities between MPM and FEM in an implicit solution strategy. Beuth <span id='citeF-135'></span>[[#cite-135|[135]]] proposes an implicit MPM formulation for quasi-static problems using high order elements and a special integration procedure for partially filled boundary elements. Sanchez <span id='citeF-136'></span>[[#cite-136|[136]]] presented an implicit MPM for quasi-static problems using a Jacobian free algorithm and in <span id='citeF-137'></span>[[#cite-137|[137]]] a GIMP method is used together with an implicit formulation.
-As one may suspect, the critical speeds bring up '''resonance''', and thus may be avoided as great displacements may be encountered. If that is the case, the rotor cannot operate near this speed without encountering strong vibrations and even catastrophic failure.  However, not all critical speeds are equally dangerous, and that will depend on the main modes of the structure. Flexural critical speed are those linked with the '''flexural''' natural frequencies, and use to be particularly dangerous. They can be detected in the Campbell diagram by looking for the intersection with the line <math display="inline">\omega  =\Omega </math>. In addition to these flexural speeds, torsional critical speeds can also be very dangerous.
+In order to assess the features of MPM, as reference for a comparison, a standard Lagrangian FEM is chosen. In Table [[#table-2.3|2.3]], the two methods are compared and a list of differences is made,  according to the basic formulation, computational efficiency and computational accuracy. It is observed that in the small deformation range the MPM has a lower accuracy and efficiency than a Lagrangian FEM. Nevertheless, the FEM procedure shows its advantageous use only in a narrow range of strain magnitude, established by a critical deformed configuration for which the element quality is seriously compromised, which may cause a drastic deterioration of accuracy or even the end of the computation. In this regard, it is evident that MPM finds its natural field of application in large deformation problems. However, it is important to highlight that an extra computational cost is expected in MPM compared to FEM. This is due to additional steps in the MPM algorithmic procedure, in order to be able to track the kinematic and historical variables through the deformation process, and to a number of ''material points'' higher than the number of Gauss points normally employed in a FEM simulation.
-The range starting from zero to the first critical speed is referred as subcritical range, and above it, supercritical range starts. In the past, it was thought that operation above the first critical speed was impossible, but it was later denied by De Laval, Stodola and many other rotordynamists. It is easy to mix up natural and critical speeds, mostly due that in many cases their value coincides. Even if that happens, the two physical phenomena are very different, particularly concerning the stressing of the rotor.
-<div id='img-1.3'></div>
+{|  class="floating_tableSCP wikitable" style="text-align: right; margin: 1em auto;width:80%;"
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+|+ style="font-size: 75%;" |<span id='table-2.3'></span>Table. 2.3 Comparison between the Finite Element Method and the Material Point Method
-|-
+|- style="border-top: 2px solid;font-size: 85%;"
-|[[Image:Draft_Samper_987121664-monograph-Campbell_example.png|450px|Example of Campbell diagram <span id='citeF-6'></span>[[#cite-6|[6]]]]]
+| colspan='1' style="text-align: center;border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | '''STANDARD LAGRANGIAN FEM'''
-|- style="text-align: center; font-size: 75%;"
+| colspan='1' style="text-align: center;border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | '''MPM'''
-| colspan="1" | '''Figure 1.3:''' Example of Campbell diagram <span id='citeF-6'></span>[[#cite-6|[6]]]
+|- style="border-top: 2px solid;font-size: 85%;"
+| colspan='2' style="text-align: center;border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | BASIC FORMULATION
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |  <math>\bullet</math> Gauss quadrature
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |  <math>\bullet</math> Particle quadrature
+|- style="font-size: 85%;"
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> The Lagrangian computational mesh is attached to the continuum during the whole solution process
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> No fixed mesh connectivity is required
+|-style="font-size: 85%;"
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> Higher accuracy and efficiency for small deformations
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> Lower accuracy and efficiency of the MPM for small deformations
+|-style="font-size: 85%;"
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> For large deformations accuracy rapidly deteriorates and computational cost increases dramatically due to mesh distortion and the need for remeshing
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> It naturally deals with large deformation problems
+|-style="font-size: 85%;"
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> Contact between different bodies can be modelled only by applying a contact technique
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> Unphysical material interpenetration and non-slip contact constraint are inherent in the MPM
+|- style="border-top: 2px solid;font-size: 85%;"
+| colspan='2' style="text-align: center;border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | COMPUTATIONAL EFFICIENCY
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |  <math>\bullet</math> Mass and momentum are carried by the mesh nodes and they are calculated only at the beginning of the analysis. Further Gauss points move according to the mesh such that it is not necessary to update their positions and velocities
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | Additional steps have to be performed such as mapping of particle info (mass and momentum) on the grid and the update of particle info at the end of the Lagrangian step
+|-style="font-size: 85%;"
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> Less Gauss points per element
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> Minimum number of particles per element higher than number of Gauss points per element
+|-style="font-size: 85%;"
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> In explicit time scheme, the critical time step decreases with the element deformation
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> In MPM the characteristic element length does not change, thus, the critical time step size in MPM is constant
+|- style="border-top: 2px solid;font-size: 85%;"
+| colspan='2' style="text-align: center;border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | COMPUTATIONAL ACCURACY
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |  <math>\bullet</math> The Gauss quadrature can integrate accurately the weak form
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math>\bullet</math> The particle quadrature is less accurate than the Gauss one for integrating the weak form
+|- style="border-bottom: 2px solid;font-size: 85%;"
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |<math>\bullet</math> For large deformations, in order to avoid element entanglement, a remeshing technique has to be adopted. However, this can lead to conservation issues of mass, momentum and energy. Further, the remapping of material properties of history-dependent material will result in significative errors
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |<math>\bullet</math> The original MPM suffers from a cell crossing instability
 |}
-To illustrate a little bit this section, figure [[#img-1.3|1.3]] shows the Campbell diagram of a compressor rotor blade. Bold curves represent natural frequencies of the structure, while the straight lines show the exciting frequencies. Each intersection between them is a resonance point. Here, rotation introduces an '''stiffening''' effect: natural frequencies increase with the spin speed.
-It is important to remember that the concept of critical speeds and the Campbell diagram only apply to linear systems, although a more arbitrary definition of these speeds can be used in the case of nonlinear systems, referring the speed at which greater amplitudes are found.    '''
+As earlier discussed, the MPM is a particle method, whose advantages are evident in applications at large strain and displacement regime. Moreover, the MPM is characterized by some features which make this technique able to overcome all the typical disadvantages of other particle methods, listed in the previous sections. MPM does not employ any kind of remeshing procedure, the calculation is performed always at a local level, allowing an easy adaptation of the code to parallel computation and a good conservation of the properties. A conservation of the mass is also guaranteed during the whole simulation time, as the total mass is distributed between the material points representing the volume of the entire continuum under study. A remapping of the state variables is avoided and the employment of complex time dependent constitutive laws can be used without committing any mapping error. In addition, since this technique is a grid-based method all the issues, related to the meshless methods, such as, lack of interpolation consistency and difficulties in enforcing the Essential Boundary conditions are avoided. Last but not least, MPM is a technique defined in the continuum mechanics framework, thus, it can be easily applied to real scale problems at a not prohibitive computational cost.  Given the fulfillment of the aforementioned features, MPM represents a suitable choice for the solution of real scale large deformation problems and particularly attractive for the modelling of granular flow problems.
-====1.2.2.5 Frequency and time domains===='''
+==2.4 The Material Point Method in ''Kratos Multiphysics''==
-Linear rotordynamics can be analysed in the frequency domain, a widely used approach that brings up an insight on how the rotor behaves under different conditions. These kind of solutions allow  the engineers to perform parametric design and optimization studies, as natural frequencies and vibration modes are easily obtained using this frame of analysis. When the system is linear and operating in steady-state condition, the most popular tool to analyse its behaviour is modal decomposition, which reduces the system of equations to an eigenvalue problem.
+In the current work, an implicit MPM is developed in the multi-disciplinary Finite Element codes framework ''Kratos Multiphysics'' <span id='citeF-29'></span><span id='citeF-30'></span><span id='citeF-31'></span>[[#cite-29|[29,30,31]]]. Two formulations are investigated: a displacement-based <span id='citeF-138'></span>[[#cite-138|[138]]] and a mixed displacement-pressure (''u-p'') <span id='citeF-139'></span><span id='citeF-140'></span>[[#cite-139|[139,140]]] formulations, presented in Chapters [[#4 Irreducible formulation|4]] and [[#5 Mixed formulation|5]], respectively. In both the numerical strategies, the original version of the MPM is implemented, i.e., a particle integration is adopted, where the particle mass is assumed to be concentrated only on one spatial point, the particle position. As earlier discussed, this may lead to the so-called ''cell-crossing error''; however, it is demonstrated that GIMP method is not able to definitely fix this issue and only other more computationally expensive techniques, such as, the CPDI2 and the DDMPM can remarkably mitigate this inherent error. In this work, it is made the choice to focus on the capabilities of the method in the modelling of granular flows under large deformation and large displacement regime and to leave, as future work, the exploration and investigation of other versions of MPM, which can improve the accuracy of the numerical results. The MPM in ''Kratos Multiphysics'' is developed in an Updated Lagrangian finite deformation framework and the matrix system to be solved is built-up from taking into account the contribution of each ''material point'', to be considered as ''integration point'', as well. In the initialization of the solving process, the initial position of the ''material points'' is chosen to coincide with the Gauss points of a FE grid and the mass, which remains constant during the simulation, is equally distributed between the material points, falling, initially, in the same element. At each time step, the governing equations are solved on the computational nodes, while history dependent variables and material information are saved on the particles during the entire deformation process. Thus, in the MPM the ''material points'' shall be understood as the integration points of the calculation, each carrying information about the material and kinematic response. Each material point represents a computational element with one single integration point (the material point itself), whose connectivity is defined by the nodes of the elements in which it falls. In Figure [[#img-2.2|2.2]], for instance, the case of the ''i-th'' material point, which falls in a triangular element with a connectivity of nodes <math display="inline">I, J</math> and <math display="inline">K</math>, is depicted. In the evaluation of the FEM integrals, the shape functions are evaluated at the material point location on the basis of the grid element the material point falls into (Figure [[#img-2.2|2.2]]). At the end of every time step, in order to prevent mesh distortion, the undeformed background grid is recovered, i.e., the nodal solution is deleted.
-This is achieved, however, when the rotating structure is idealised. If the system is studied in the transient state, no frequency analysis can be achieved and only '''numerical integration''' in time is possible. Although integrating numerically the equations of motion is a widely used method nowadays to tackle complex problems, it only allows to solve very particular cases without bringing up an overall insight on the phenomena.  Moreover, there is a high computational cost associated to this kind of numerical methods which is not present when performing a frequency analysis.
+In what follows, the algorithm of MPM for an implicit time scheme discretization is presented in detail.
-===1.2.3 Nonlinear and nonstationary rotordynamics===
+===2.4.1 MPM Algorithm===
-The previous section presented a linearised model for a rotating structure. This is, however, only an idealisation as real rotors always deviate from the lineal model. Rotors are often designed to behave linearly in their nominal conditions. However, some sources of nonlinearity such as bearings, dampers and seals may disturb the system from its ideal behaviour.  Typical approaches are to consider those elements as rigid bodies, or to linearise its behaviour, in order to simplify the resulting equations. This, however, leads to '''unsatisfactory''' approximations.
-If the structure is highly nonlinear, sub- and super-harmonics play an important role, as well as chaotic oscillations. Chaos has been spotted in nonlinear rotor models, and numerical results indicate that they are present in turbojet engines. If the whirl amplitude grows in time, nonlinearities emerge until the system reaches a '''limit cycle'''.
-As is to expect from nonlinear systems, sudden jumps from an equilibrium configuration to another may happen. Moreover, hysteresis may appear, making the spin-up of the system different from the deceleration, even if the speeds are the same.  When the system to study becomes that complex, the only way to solve the equation is by means of numerical integration.
+Traditionally, the  MPM algorithm is composed of three different phases <span id='citeF-28'></span>[[#cite-28|[28]]], as graphically represented in Figure [[#img-2.3|2.3]] and below described:
-The previous linear analysis required from constant spin velocity <math display="inline">\Omega </math> to be valid. It is, in fact, a common simplification in rotordynamics analysis. This model do not explains, however, the accelerating and decelerating transient states, or quick changes in the forcing functions. Even if the rotor is assumed to be linear, angular accelerations may introduce new terms in the matrices of equation  [[#eq-1.3|1.3]] which will also become time-dependant. As a consequence, the system becomes complicated and the equations of motion are no longer solvable analytically. As an integrating numerical scheme will be required to solve the nonstationary system, there is little advantage in previously linearising the equation of motion.
+<ol style='list-style-type:lower-alpha;'>
-===1.2.4 The Jeffcott rotor===
+<li>''Initialization phase (Fig.[[#img-2.3a|2.3a]])'': at the beginning of the time step the connectivity is defined for each material points and the initial conditions on the FE grid nodes are created by means of a projection of material points information obtained at the previous time step <math display="inline"> t_{n} </math>; </li>
+<li>''UL-FEM calculation phase (Fig.[[#img-2.3b|2.3b]])'': the local elemental matrix, represented by the left-hand-side (<math display="inline"> lhs </math>) and the local elemental force vector, constituted by the right-hand-side (<math display="inline"> rhs </math>) are evaluated in the current configuration according to the formulation presented in Chapters [[#4 Irreducible formulation|4]] and [[#5 Mixed formulation|5]]; the global matrix <math display="inline"> LHS </math> and the global vector <math display="inline"> RHS </math> are obtained by assembling the local contributions of each material point, as in a classical FEM approach, and, finally, the solution system is iteratively solved. During the iterative procedure, the nodes are allowed to move, accordingly to the nodal solution, and the material points do not change their local position within the geometrical element until the solution has reached  convergence;   </li>
+<li>''Convective phase (Fig.[[#img-2.3c|2.3c]])'': during the third and last phase the nodal information at time <math display="inline"> t_{n+1} </math> are interpolated back to the material points. The position of the material points is updated and, in order to prevent mesh distortion, the undeformed FE grid is recovered.  </li>
+</ol>
+<div id='img-2.3a'></div>
+<div id='img-2.3b'></div>
+<div id='img-2.3c'></div>
+<div id='img-2.3'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;"
+|-
+| style="padding:10px;" |[[Image:Draft_Samper_987121664-monograph-ALGORITHM_MPM1_init.png|220px|Initialization phase]]
+|style="padding:10px;" |[[Image:Draft_Samper_987121664-monograph-ALGORITHM_MPM1_ulfem.png|300px|Updated Lagrangian FEM phase]]
+|- style="text-align: center; font-size: 75%;"
+| (a) Initialization phase
+| (b) Updated Lagrangian FEM phase
+|-
+| colspan="2" style="padding:10px;" |[[Image:Draft_Samper_987121664-monograph-ALGORITHM_MPM1_conv.png|358px|Convective phase]]
+|- style="text-align: center; font-size: 75%;"
+|  colspan="2" | (c) Convective phase
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding-bottom:10px;padding-top:10px;" |'''Figure 2.3:''' MPM phases.
+|}
-The analytical study of rotors dates back to the late nineteenth century, when Föppl, Stodola and Belluzzo described the behaviour of a simplified rotor.  This model was deeply studied by Jeffcott (1919), and thus received its name. The model consists of a '''point mass''' attached to a massless shaft, being one of the simplest models used to describe the flexural behaviour of rotors. Yet, this is an '''oversimplification''' and one has to understand its limitations, although it may bring up qualitative insight into important phenomena of rotordynamics.
-The model covered in this section considers an axisymmetrical fixed-free rotor, in which damping plays no role whatsoever. The restoring force is provided by an overall stiffness ''<math>k</math>'', which will be considered the stiffness of the shaft.  Let's consider an inertial frame <math display="inline">xy</math> and rotating noninertial frame <math display="inline">\varsigma \varrho </math>. In addition, small displacements are assumed, and the point mass is always to be contained in the ''<math>xy</math>'' plane. As in classic rotordynamics, the spin speed <math display="inline">\Omega </math> is assumed to be constant.
+Many features of the MPM are connected to the Finite Element Method <span id='citeF-115'></span>[[#cite-115|[115]]]. Indeed, phase ''b'' coincides with the calculation step of a standard non-linear FE code, while phases ''a'' and ''c'' define the MPM features. At the beginning of each time step (<math display="inline">t_n</math>), during phase ''a'', the degrees of freedom and the variables on the nodes of the fixed mesh are defined  gathering the information from the material points (Figure [[#img-2.3a|2.3a]]).
-Previously to the study of the Jeffcott rotor, let's consider an even simpler model consisting in a point mass constrained to move only in the axial direction <math display="inline">\varsigma </math>. This is the kind of model '''Rankine''' used to develop his rotor analysis. The Lagrangian to the system yields to:
+For the sake of clarity, hereinafter, <math display="inline"> p </math> and <math display="inline">I</math> subscripts are used to refer to variables attributed to material points and computational nodes, respectively, while <math display="inline">n</math> superscript refers to the time instance in which the variable is defined.  The momentum <math display="inline"> \boldsymbol{q}_{p} </math> and inertia <math display="inline"> \boldsymbol{f}_{p} </math> on the material points, which are expressed as functions of mass <math display="inline"> m_p </math>, velocity <math display="inline">\boldsymbol{v}_{p}</math> and acceleration <math display="inline">\boldsymbol{a}_{p}</math>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 338: / Line 849: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathfrak{L}=\frac{1}{2} m\left(\dot{\varsigma }^{2}+\Omega ^{2} \varsigma ^{2}\right)-\frac{1}{2} k \varsigma ^{2} </math>
+| style="text-align: center;" | <math>\boldsymbol{q}_{p}^n := \boldsymbol{v}_{p}^n m_{p} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.6)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.6)
 |}
-And the equation of motion results in
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 350: / Line 859: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>m \ddot{\varsigma }+\left(k-m \Omega ^{2}\right)\varsigma=0 </math>
+| style="text-align: center;" | <math>\boldsymbol{f}_{p}^n := \boldsymbol{a}_{p}^n m_{p} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.7)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.7)
 |}
-This equations shows a decrease in the stiffness known as centrifugal softening, as the term multiplying the displacement is <math display="inline">k-m \Omega ^{2}</math>, which is clearly smaller than <math display="inline">k</math>. Moreover, this system becomes unstable above the critical speed <math display="inline">\omega = \sqrt{(k / m)}</math>, prohibiting operation in the supercritical range.
+are projected on the background grid by evaluating in a first step, the global values of mass <math display="inline"> m_I </math>, momentum <math display="inline"> \boldsymbol{q}_{I} </math> and inertia <math display="inline"> \boldsymbol{f}_{I} </math> on node <math display="inline">I</math> as described in Algorithm [[#algorithm-2.1|2.1]].
-<div id='img-1.4'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-rankine.png|300px|Rankine rotor model <span id='citeF-3'></span>[[#cite-3|[3]]] ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 1.4:''' Rankine rotor model <span id='citeF-3'></span>[[#cite-3|[3]]]
-|}
-If the guide is removed, displacement in the transversal direction <math display="inline">\varrho </math> becomes possible, and a '''Jeffcott''' rotor is obtained. The equation of motion in the rotating coordinates  results in
+Once <math display="inline"> m_I </math>, <math display="inline"> \boldsymbol{q}_{I} </math> and <math display="inline"> \boldsymbol{f}_{I} </math> are obtained, it is possible to compute the values of nodal velocity <math display="inline">\widetilde{\boldsymbol{v}}_{I}^n</math> and nodal acceleration <math display="inline">\widetilde{\boldsymbol{a}}_{I}^n</math> of the previous time step as
+<span id="eq-2.8"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 372: / Line 874: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>m \left[\begin{array}{cc}{1} & {0} \\ {0} & {1}\end{array}\right]\left\{\begin{array}{l}{\ddot{\varsigma }} \\ {\ddot{\varrho }}\end{array}\right\}-2 m \Omega \left[\begin{array}{cc}{0} & {1} \\ {-1} & {0}\end{array}\right]\left\{\begin{array}{c}{\dot{\varsigma }} \\ {\dot{\varrho }}\end{array}\right\}+\left[\begin{array}{cc}{k-m \Omega ^{2}} & {0} \\ {0} & {k-m \Omega ^{2}}\end{array}\right]\left\{\begin{array}{l}{\varsigma } \\ {\varrho }\end{array}\right\}=0 </math>
+| style="text-align: center;" | <math>\widetilde{\boldsymbol{v}}_{I}^n = \frac{\boldsymbol{q}_{I}^n}{m_{I}}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.8)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.8)
 |}
-This model now takes into consideration '''Coriolis''' forces, which push the mass out of is axis, allowing self-centring and thus operation in the supercritical range.  The equation of motion can be written in the inertial frame using a simple change of coordinates consisting in a rotation of angle <math display="inline">\Omega t</math>.  The equation of motion of the Jeffcott rotor in the inertial frame reads as
+<span id="eq-2.9"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 384: / Line 885: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{array}{l}{m \ddot{x}+k x=0} \\ {m \ddot{y}+k y=0}\end{array} </math>
+| style="text-align: center;" | <math>\widetilde{\boldsymbol{a}}_{I}^n = \frac{\boldsymbol{f}_{I}^n}{m_{I}}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.9)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.9)
 |}
-The equation above predicts no softening effect nor instability in the inertial frame, and a whirl frequency <math display="inline">\omega _n=\pm \sqrt{k/m}</math>, equal to the flexural critical speed <math display="inline">\Omega _{cr}</math>. For the case of a simply supported shaft, transverse stiffness <math display="inline">k</math> is equal to <math display="inline">\frac{48 E I}{L^{3}}</math> <span id='citeF-7'></span>[[#cite-7|[7]]], where <math display="inline">E</math> stands for the Young’s modulus, <math display="inline">I</math> for the second moment of area of the shaft cross-section and <math display="inline">L</math> for the span of the shaft. The above equation predicts that, if the initial conditions are such that the mass moves into circular orbits, the system is not affected by any kind of vibration but describes whirling motion instead. In the inertial frame, angular frequency may not be constant, but the period will be equal to <math display="inline">2\pi / \omega _n</math>.
+It is worth mentioning that, the initial nodal conditions are evaluated at each time step using material point information in order to have initial values even on grid elements empty at the previous time step (<math display="inline">t_{n-1}-t_{n}</math>).
-The natural frequency of the Jeffcott rotor does not depend on <math display="inline">\Omega </math>, and thus the Campbell diagram consists in horizontal straight lines. This is the simplest way of tackling the Jeffcott rotor, although more '''realistic''' variations of the model are widely used, taking into account forcing functions, damping, shaft bow, unbalance and out-of-plane displacements.
+The MPM makes use of a predictor/corrector procedure, based on the Newmark integration scheme.  The prediction of the nodal displacement, velocity and acceleration reads
-===1.2.5 Gyroscopic effect===
-In the previous section, the Jeffcott model considered a rotor consisting in a point mass, and consequently, its moments of inertia were not taken into account. This is, however, a huge simplification. In reality, Campbell diagrams are not horizontal lines, but natural frequencies of bending modes depend on the spin speed. If one aims to account for this behaviour, has to consider the influence of '''inertia moments'''. Strictly speaking, the system has six degrees of freedom, and so six generalised coordinates should be defined in order to study its dynamic behaviour.
-This behaviour was first noticed by Rayleigh, who identified the effect of the rotatory inertia of a disk mounted on a shaft. Stodola also considered this effect, and identified it as the responsible of giving rise to the split of natural frequencies. The study of critical speeds under the gyroscopic effect was carried by Green, and later studied using energy methods by Carnegie.
-One of the simplest models follows the main assumptions of the Jeffcott rotor, but with a '''rigid body''' with nonvanishing moments of inertia instead of a point mass. In the undeformed position, one of the principal axes of inertia coincides with <math display="inline">z</math>. Principals moments of inertia will be referred as polar moment of inertia <math display="inline">J_p</math>, about the rotation axis, and transversal moment of inertia <math display="inline">J_t</math>, about any axes in the rotation plane. In matrix form:
+<span id="eq-2.10"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 406: / Line 900: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{J}=\left[\begin{array}{ccc}{J_{t}} & {0} & {0} \\ {0} & {J_{t}} & {0} \\ {0} & {0} & {J_{p}}\end{array}\right] </math>
+| style="text-align: center;" | <math>^{it+1}\Delta \boldsymbol{u}_{I}^{n+1}= 0.0  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.10)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.10)
 |}
-<div id='img-1.5'></div>
+<span id="eq-2.11"></span>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-gyro_diagram.png|570px|Massless shaft modes (lumped mass and disk models)]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 1.5:''' Massless shaft modes (lumped mass and disk models)
+|-
+| style="text-align: center;" | <math>^{it+1}\boldsymbol{v}_{I}^{n+1} = \frac{\lambda }{\zeta \Delta t}\left[^{it+1}\Delta \boldsymbol{u}_{I}^{n+1}\right]- \left(\frac{\lambda }{\zeta } - 1 \right)\widetilde{\boldsymbol{v}}_{I}^{n} - \frac{\Delta t}{2}\left(\frac{\lambda }{\zeta }-2\right) \widetilde{\boldsymbol{a}}_{I}^{n}  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.11)
 |}
-In order to illustrate the problem, imagine a beam and its two first structural modes (figure [[#img-1.5|1.5]]). The first mode introduces a kinetic energy term, as the mass and the disk have been displaced by the deflection of the shaft. However, if the shaft is massless, there is no kinetic energy associated to the second mode, as the point mass has not changed its position. On the contrary, if the disk rotation given by the slope <math display="inline">(\partial v / \partial x)</math> is taken into account, kinetic energy associated to the angular velocity emerges and has the following expression:
+<span id="eq-2.12"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 425: / Line 922: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{1}{2} \; J_{T}\left\{\frac{d}{d t}\left(\frac{\partial v}{\partial x}\right)\right\}^{2} </math>
+| style="text-align: center;" | <math>^{it+1}\boldsymbol{a}_{I}^{n+1} = \frac{1}{\zeta \Delta t^2}\left[^{it+1}\Delta \boldsymbol{u}_{I}^{n+1}\right]- \frac{1}{\zeta \Delta t} \widetilde{\boldsymbol{v}}_{I}^{n} - \left(\frac{1}{2\zeta }-1\right) \widetilde{\boldsymbol{a}}_{I}^{n}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.11)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.12)
 |}
-The above value of rotation energy is clearly zero if the beam is considered to be a point mass with no inertia, but when dealing with disks and rigid bodies, the inertia term cannot be neglected and neither can the rotation energy associated. This was a very known problem in static structures, known as rotatory inertia of beams. However, when the shaft rotates, the body acts like a spin top and '''gyroscope''', having a significant effect in rotordynamics.
+where the upper-left side index <math display="inline">it </math> indicates the iteration counter, while the upper-right index <math display="inline">n</math> the time step. <math display="inline">\lambda </math> and <math display="inline">\zeta </math> are the Newmark's coefficients equal to 0.5 and 0.25, respectively.
-To understand the physics behind this phenomena, one has to remember Euler's equation for rigid bodies, which are clearly nonlinear with the angular velocities (see equation [[#eq-1.2|1.2]]). Imagine a freely spinning disk (for instance, figure [[#img-1.6|1.6]]) with angular velocity <math display="inline">\Omega </math>, rotating around an axis perpendicular to its non-deformed symmetry plane. Consider now that a whirling motion is introduced now around the <math display="inline">x</math> axis, due to any kind of perturbation or imbalance. <div id='img-1.6'></div>
+Once the nodal velocity and acceleration are predicted (Equations [[#eq-2.10|2.10]]-[[#eq-2.12|2.12]]), the system of linearised governing equations is formulated, as in classic FEM, and the local matrix <math display="inline"> \mathbf{K}^{tan} </math> and the residual <math display="inline">\mathbf{R}_I</math> are evaluated and assembled, respectively (phase ''b'', Figure [[#img-2.3b|2.3b]]).
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-freedisk.png|300px|Free spinning disk <span id='citeF-1'></span>[[#cite-1|[1]]] ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 1.6:''' Free spinning disk <span id='citeF-1'></span>[[#cite-1|[1]]]
-|}
-Because of whirling, precessional motions linked to the shaft slopes at the disk centre are introduced. The disk then moves similar to a spin top, and the gyroscopic effect is introduced. Moreover, precession can happen in both <math display="inline">xz</math> and <math display="inline">yz</math> planes, leading to two different gyroscopic torques at which inertia torques (<math display="inline"> J_{T}\cdot \ddot{\theta }_{i}</math> and <math display="inline">J_{T}\cdot \ddot{\phi }_{i} </math>) are added. In this context, the moment relations across the disc <math display="inline">i</math> are written as
+The solution in terms of increment of nodal displacement is found iteratively solving the residual-based system. Once the solution <math display="inline"> ^{it+1}\delta \boldsymbol{u}_I^{n+1} </math> is obtained, a correction of the nodal increment of displacement is performed
+<span id="eq-2.13"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 447: / Line 939: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{array}{l}{M_{y \;i}=J_{p}\cdot \Omega \cdot \dot{\phi }_{i}+J_{T}\cdot \ddot{\theta }_{i}} \\ {M_{z \;i}=-J_{p} \cdot \Omega \cdot \dot{\theta }_{i}+J_{T}\cdot \ddot{\phi }_{i}}\end{array} </math>
+| style="text-align: center;" | <math>^{it+1}\Delta \boldsymbol{u}_{I}^{n+1} =^{it}\Delta \boldsymbol{u}_{I}^{n+1} +^{it+1}\delta \boldsymbol{u}_I^{n+1}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.12)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.13)
 |}
-This relations have to be taken into account when modelling the dynamics of the system. In the model seen in equation [[#eq-1.3|1.3]], these effects are included in the gyroscopic matrix <math display="inline">\mathbf{G}</math>. A further simplification can be achieved, '''uncoupling''' rotational and translational motions, leading only to four degrees of freedom. If that is the case, one can define the non-dimensional term <math display="inline">\delta =\frac{J_{p}}{J_{t}}</math>, and study the behaviour of the rotor as a function of <math display="inline">\delta </math>.
+Velocity and acceleration are corrected according to Equations [[#eq-2.11|2.11]] and [[#eq-2.12|2.12]], respectively. This procedure has to be repeated until convergence is reached.
-The Jeffcott model, for instance,  corresponds to <math display="inline">\delta = 0</math>, in which case curves in the Campbell diagram are horizontal lines. If <math display="inline">\delta > 1</math>, the rotor is disk-like, and as there is no intersection of the curves in the Campbell diagram with the line <math display="inline">\omega = \Omega </math>, no critical speed exist. On the contrary, if <math display="inline">\delta < 1</math>, the rotor is considered to be very long, and a critical speed linked with conical motion exists. Finally, if <math display="inline">\delta </math> is exactly one, the rotor is spherical and although there is no intersection with the line <math display="inline">\omega = \Omega </math>, it tends asymptotically to it as <math display="inline">\Omega </math> increases. Regarding the frequency of whirling, it decreases in absolute value as <math display="inline">\Omega </math> increases for the case of backward whirling, and increases with an inclined asymptote <math display="inline">\omega = \delta \Omega </math> in the case of forward whirling.
+Unlike a FEM code, the nodal information is available only during the calculation of a time step: at the beginning of each time step a reset of all the nodal information is performed and the accumulated displacement information is deleted. The computational mesh is allowed to deform only during the iterative procedure of a time step, avoiding the typical element tangling of a standard FEM. When  convergence is achieved, the position of the nodes is restored to the original one (phase c, Figure [[#img-2.3c|2.3c]]). Before restoring the undeformed configuration of the FE grid, the solution in terms of nodal displacement, velocity and acceleration is interpolated on the material points, as
-===1.2.6 Centrifugal softening and stiffening===
+<span id="eq-2.14"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-In previous sections, the concepts of centrifugal softening and stiffening have been used to refer the decrease  and increase, respectively, of the natural frequencies of a rotating structure, compared with the natural frequencies of the same structure but standstill. While centrifugal stiffening is a very well-known effect, centrifugal softening is ''"an alleged and elusive phenomena, and some doubts can be cast on the mathematical models causing it to appear"'' <span id='citeF-3'></span>[[#cite-3|[3]]]. A model of this type is used for instance in <span id='citeF-1'></span>[[#cite-1|[1]]] (see page 285), trying to account for the softening effect in a solid rotor. The resulting equation is:
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\Delta \boldsymbol{u}_{p}^{n+1} = \sum _{n = 1}^{n_n} N_I\left(\xi _{p},\eta _{p} \right)\Delta \boldsymbol{u}_{I}^{n+1}  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.14)
+|}
+<span id="eq-2.15"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 465: / Line 965: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M}\,\ddot{\mathbf{q}}+(\mathbf{K}-\Omega ^{2} \mathbf{M})\,{\mathbf{q}}=0 </math>
+| style="text-align: center;" | <math>\boldsymbol{a}_{p}^{n+1} = \sum _{n = 1}^{n_n} N_I\left(\xi _{p},\eta _{p} \right)\boldsymbol{a}_{I}^{n+1}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.13)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.15)
 |}
-There is a term proportional to the mass matrix and to the square of the spin velocity, which is subtracted from <math display="inline">\mathbf{K}</math> and thus natural frequencies wane. However, when flexibility and inertia of the disks and blades are accounted for, centrifugal '''stiffening''' plays a crucial role. It can be explained as the result of the stressing caused by tensile forces, which produces a virtual increase of stiffness proportional to the centrifugal force and thus to <math display="inline">\Omega ^2</math>.  The simplest way to model a system accounting for both softening and stiffening effects is to use a '''1'''<math display="inline">\mathbf{\frac{1}{2}}</math> '''dimensional''' approach, which consists in modelling shafts as beams  and using axisymmetrical elements.  The linearised homogeneous equation of motion <span id='citeF-3'></span>[[#cite-3|[3]]] for the free vibration, taking into account axial, torsional, out-of-plane and in-plane bending behaviour reads as
+<span id="eq-2.16"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 477: / Line 976: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M \,\ddot { q }}+\Omega \,\mathbf{G}\, \dot{\mathbf{q}}+\left(\mathbf{K}+\mathbf{K}_{\Omega } \,\Omega ^{2}-\mathbf{M}_{n} \,\Omega ^{2}\right)\,\mathbf{q}=\mathbf{0} </math>
+| style="text-align: center;" | <math>\boldsymbol{v}_{p}^{n+1} = \boldsymbol{v}_{p}^{n} + \frac{1}{2} \Delta t \left(\boldsymbol{a}_{p}^{n} + \boldsymbol{a}_{p}^{n+1}\right)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (1.14)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.16)
 |}
-<math display="inline">\mathbf{M}</math>, <math display="inline">\mathbf{G}</math> and <math display="inline">\mathbf{K}</math> are the mass, gyroscopic and stiffness matrices. <math display="inline">\mathbf{K_{\Omega }}</math> is the centrifugal stiffening matrix, a geometric matrix whose effects are proportional to <math display="inline">\Omega ^2</math>. On the other hand, <math display="inline">\mathbf{M_{n}}</math> is the centrifugal softening matrix, responsible for causing a decrease of the natural frequencies. The key is to assess whether natural frequencies actually decrease or not as <math display="inline">\Omega </math> increases. To check that, it must be known whether the effect of <math display="inline">\mathbf{M_{n}}</math> is stronger than <math display="inline">\mathbf{K_{\Omega }}</math> or not.
+where <math display="inline"> n_n </math> is the total number of nodes per geometrical element, <math display="inline"> (\xi _{p},\eta _{p}) </math> are the local coordinates of material point <math display="inline">p</math> and <math display="inline"> N_I\left(\xi _{p},\eta _{p} \right)</math> is the shape function evaluated at the position of the material point <math display="inline">p</math>, relative to node <math display="inline">I</math>.
-The effect of <math display="inline">\mathbf{K_{\Omega }}</math>-<math display="inline">\mathbf{M_{n}}</math> is dual: a stiffening effect due to tensile stresses caused by rotation and a softening due to the effects of the '''noninertial''' frame in which the equations are written. It can be demonstrated (see <span id='citeF-3'></span>[[#cite-3|[3]]]) that that matrix is positive defined, and thus the overall net effect is of stiffening.  The only models that, for its simplicity, are not able to capture the stiffening effect are the Rankine and Jeffcott rotors, which neither account for the gyroscopic effect. Moreover, the Rankine model is the only one that predicts a strong softening effect, which is known to lead to '''incorrect''' results.
+Finally  the current position of the material points is updated as
-==1.3 Report outline==
+<span id="eq-2.17"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\boldsymbol{x}_{p}^{n+1} = \boldsymbol{x}_{p}^{n} + \Delta \boldsymbol{u}_{p}^{n+1}  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.17)
+|}
-As presented in the previous section, the dynamic behaviour of rotormachinery is complex and the reigning equations can easily become convoluted, leading to a set of equations whose solving is far out of the scope of this report. Hence, instead of modelling the structure with complex and accurate equations, simpler models will be considered. The goal is not to get credible results, but to study on the '''methods''' used to solve the equations. For the cases where <math display="inline">\Omega </math> is not constant, the equations cannot be solved in terms of modal decomposition and only numerical integration is possible. These numerical schemes take a lot of time and computation cost, and give only particular solutions of the system. The key of this approach is to develop a numeric method based on the '''SVD''' that has the advantage of the modal analysis of capturing global properties of the system —such as vibration modes and natural frequencies— when applied to rotors with angular acceleration and other nonlinearities.
+The details of the MPM algorithm are presented in Algorithm [[#algorithm-2.1|2.1]].
-Regarding the written '''organization''' of the document, it is worth noting that an appendix is included at the end of the report (sections enumerated with capital Latin letters). Particular topics that could be from interest to the reader are presented there, as well as some figures related with the simulations carried out in chapters 6 and 7.
-Focusing on the report, the chapters into which is divided are:
+<span id="eq-2.14"></span>
+{| style="margin: 1em auto;border: 1px solid darkgray;width:80%;"
+|-
+|
+:
+(we will use <math>(\bullet )^n = (\bullet )(t_{n}) </math>), Material DATA: E, <math> \nu </math>, <math> \rho </math>
+|-
+| Initial data on material points: <math> m_p </math>, <math> \mathbf{x}_p^n</math>, <math> \Delta t </math>, <math> \mathbf{u}_p^n, \mathbf{v}_p^n, \mathbf{a}_p^n,  \mathbf{F}_p^n = \displaystyle{ \sum _I\frac{\partial N_I}{\partial \mathbf{x}_I^{0}}}\cdot \mathbf{x}_I^{n}</math> and <math> \Delta \mathbf{F}_p =\displaystyle{ \sum _I\frac{\partial N_I}{\partial \mathbf{x}_I^{n}}}\cdot \mathbf{x}_I^{n+1} </math>
+|-
+| Initial data on nodes: '''NONE - everything is discarded in the initialization phase'''
+|-
+| OUTPUT of calculations: <math> \Delta \mathbf{u}^{n+1}_I, \boldsymbol{\sigma }^{n+1}_p </math>
+|-
+|
-'''Chapter 2.'''  The elastic model that describes the vibration behaviour of rotors is formulated. The strong form of the elastodynamic equation is obtained and analytically solved for the 1D stationary case.
+<ol>
-'''Chapter 3.'''  The weak form of the elastodynamic equations is obtained, posed in terms of the finite element method and formulated in the elemental frame. General concepts regarding the FEM methods are reviewed in this section.
+<li>'''INITIALIZATION PHASE'''
+{|
+|-
+| </li>
+:* Clear nodal info and recover undeformed grid configuration
+|-
+|
+:* Calculation of initial nodal conditions.
+{|
+|-
+|
+::(a) for p = 1:<math display="inline"> N_p </math>
+{|
+|-
+|
+:::* Calculation of nodal data
+{|
+|-
+|
+::::* <math display="inline"> \mathbf{q}_I^n = \sum _p N_I \, m_p \mathbf{v}_p^n </math>
+|-
+|
+::::* <math display="inline"> \mathbf{f}_I^n = \sum _p N_I m_p \mathbf{a}_p^n </math>
+|-
+|
+::::* <math display="inline"> m_I^n = \sum _p N_I m_p </math>
+|-
+|
+|}
+|}
-'''Chapter 4.'''  The general steps of every FEM simulation — preprocessing, solver and postprocessing — are reviewed. The 1D stationary case is solved using the FEM.
+::(b) for I = 1:<math display="inline"> N_I </math>
+{|
+|-
+|
+:::* <math display="inline"> \widetilde{\mathbf{v}}_{I}^n = \dfrac{\mathbf{q}_I^n}{m_I^n}</math>
+|-
+|
+:::* <math display="inline">\widetilde{\mathbf{a}}_{I}^n = \dfrac{\mathbf{f}_I^n}{m_I^n}</math>
+|-
+|
+|}
+|}
-'''Chapter 5.'''  Different methods to tackle the dynamic case are compared: modal decomposition analysis, numerical integration and singular value decomposition.
+:* Newmark method: PREDICTOR. Evaluation of <math display="inline">^{it+1}\Delta \mathbf{u}_I^{n+1} , ^{it+1}\mathbf{v}_I^{n+1}</math> and <math display="inline">^{it+1}\mathbf{a}_I^{n+1}</math> using Equations (MPM disp predictor)&#8211;(MPM acc predictor)
+|-
+|
+|}
+<li>'''UL-FEM PHASE'''
+{|
+|-
+| </li>
+:* for p = 1:<math display="inline"> N_p </math>
+{|
+|-
+|
+::(a) Evaluation of local residual (<math display="inline">rhs</math>)
+|-
+|
+::(b) Evaluation of local Jacobian matrix of residual (<math display="inline">lhs</math>)
+|-
+|
+::(c) Assemble <math display="inline">rhs</math> and <math display="inline">lhs</math> to the global vector <math display="inline">RHS</math> and global matrix <math display="inline">LHS</math>
+|-
+|
+|}
+:* Solving system <math display="inline"> (\Delta \mathbf{u}_I^{n+1}) </math>
+|-
+|
+:* Newmark method: CORRECTOR (Equations (MPM vel predictor)&#8211;(MPM disp corrector))
+|-
+|
+:* Check convergence
+{|
+|-
+|
+::(a) NOT converged: go to Step 2
+|-
+|
+::(b) Converged: go to Step 3
+|-
+|
+|}
+|}
+<li>'''CONVECTIVE PHASE'''
+{|
+|-
+| </li>
+:* Update the kinematics on the material points by means of an interpolation of nodal information (Equations (displacement mapping)&#8211;(material point position update))
+|-
+|
+:* Save the stress <math display="inline"> \boldsymbol{\sigma }^{n+1}_p </math>, strain <math display="inline"> \boldsymbol{\epsilon }^{n+1}_p </math> and total deformation gradient <math display="inline"> \mathbf{F}^{n+1}_p </math> on material points (the latter by&nbsp;<math display="inline">\mathbf{F}_p^{n+1}=\Delta \mathbf{F}_p \cdot \mathbf{F}_p^n</math>)
+|}
+</ol>
+|-
+| style="text-align: center; font-size: 75%;"|
+<span id='algorithm-2.1'></span>'''Algorithm. 2.1''' MPM algorithm.
+|}
-'''Chapter 6.'''  The methods reviewed in chapter 5 are used to perform the structural analysis of rotating beams. In particular, the chapter focuses on the application of the SVD to recover predominant information of rotating structures.
+==2.5 The Galerkin Meshless Method==
-'''Chapter 7.''' The dynamic model is further generalised, including aerodynamic forces and rigid-elastic coupling. The numeric solver is used to simulate the starting of an helicopter rotor.
+In Section [[#2.3 The Materal Point Method|2.3]] the Material Point Method and its state of the art is presented. Different versions of the method, proposed with the attempt to overcome the ''cell-crossing error'' issue, are discussed. An alternative to the approaches aforementioned, such as GIMP, CPDI and DDMPM, is represented by the Galerkin Meshless Method (GMM) <span id='citeF-141'></span>[[#cite-141|[141]]]. The GMM can be seen as the MPM, where the Eulerian background grid is replaced by a Lagrangian one, defined by a cloud of nodes. In GMM, the material points move together with the computational nodes and the shape functions are evaluated once the surrounding cloud of nodes is defined (Figure [[#img-2.4a|2.4a]]). In this case, unlike MPM, the nodes preserve ''their history'' through the whole simulation, as in FEM. GMM is a continuum method and it does not make use of a remeshing technique which gives all the advantages of MPM, previously discussed, such as, local level calculation, good conservation properties and an easy adaptation to parallel computing.  The GMM is a truly meshless method, based on a Galerkin formulation. Unlike other methods, such as, the Element-Free Galerkin Method <span id='citeF-76'></span>[[#cite-76|[76]]] or the Reproducing Kernel Particle Method <span id='citeF-64'></span>[[#cite-64|[64]]], this technique does not need element connectivity for integration or interpolation purposes. However, as it belongs to the class of techniques described in Section [[#2.2.3 The Meshfree Galerkin Methods|2.2.3]], it may suffer from the drawbacks which typically affects all the meshless methods (e.g. tension instability, difficulty in enforcing the Essential Boundary Conditions (EBCs), lack of interpolation consistency, etc.). Apart from these shortcomings, GMM with MPM might represent a suitable choice for the solution of real scale large deformation problems and a comparison within a unified framework would be beneficial for an objective evaluation of the capabilities of each method.
-'''Chapter 8.''' The developed work is reviewed and final conclusions are stated.
+<div id='img-2.4a'></div>
+<div id='img-2.4'></div>
-The presented outline may seem confusing for those with little experience in numerical methods. If that is the case, the reader should keep in mind the following scheme that encompasses the different parts of the '''numerical tool''' that is to be developed, as well as its chronological order:
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 40%;"
-<div id='img-1.7'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-initial_scheme.png|600px|Overall numeric scheme <span id="fnc-1"></span>[[#fn-1|<sup>1</sup>]]]]
+|style="padding:10px;" |[[Image:Draft_Samper_987121664-monograph-MMPM.png|300px|''GMM''. The shape functions on the material point p<sub>i</sub> are evaluated using the information on the nodes sufficiently closed to the material point itself.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 1.7:''' Overall numeric scheme <span id="fnc-2"></span>[[#fn-2|<sup>2</sup>]]
+| colspan="1" style="padding-bottom:10px;" | '''Figure 2.4:''' ''GMM''. The shape functions on the material point <math>p_i</math> are evaluated using the information on the nodes sufficiently closed to the material point itself.
 |}
-<span id="fn-1"></span>
+==2.6 The Galerkin Meshless Method in ''Kratos Multiphysics''==
-<span style="text-align: center; font-size: 75%;">([[#fnc-1|<sup>1</sup>]]) </span>
-<span id="fn-2"></span>
+In this section the Galerkin Meshless Method, implemented in ''Kratos Multiphysics'', is described. The algorithm presented in <span id='citeF-141'></span>[[#cite-141|[141]]] to simulate fluid-structure interaction problems is taken as a starting point and adapted to the simulation of deformable solids <span id='citeF-138'></span>[[#cite-138|[138]]].
-<span style="text-align: center; font-size: 75%;">([[#fnc-2|<sup>2</sup>]])  The used software is property of the following trademarks: <math>\mathrm{3D}</math> <math>\mathrm{SOLIDWORKS}^{\mbox{™}}</math>. 2019. Version 2018 SP4. Dassault Systèmes <math>^{\mbox{®}}</math>, Vélizy-Villacoublay, France.
-<math>\mathrm{GiD}^{\mbox{®}}</math>. 2019. Official version 14.0.2. CIMNE <math>^{{\mbox{©}}}</math>, Barcelona, Spain.
+As explained in Section [[#2.5 The Galerkin Meshless Method|2.5]], the GMM is a truly meshless method and it can be seen as the application of the MPM idea extended to the case in which both the nodes and the material points behave as purely Lagrangian through the whole analysis. Thus, it is relatively easy to enforce conservation properties at the integration points, while also maintaining the history of nodal results during all the simulation time, provided that a reliable technique is chosen for the computation of the meshless shape functions. The difficulty is, hence, moved to the construction of such an effective meshless base, which is addressed in Section [[#2.6.2 Calculation of GMM shape functions|2.6.2]]. In what follows, the algorithm used for the implementation of the GMM in ''Kratos Multiphysics'' is described. Differences and analogies with the MPM algorithm procedure are highlighted. Moreover, the construction of effective meshless intepolants is discussed in Section [[#2.6.2 Calculation of GMM shape functions|2.6.2]], where two techniques are presented: the Moving Least Square (MLS) technique and the Local Max-Ent (LME) method. Further, in Chapter [[#4 Irreducible formulation|4]] a comparison between the MPM and the GMM is performed through some benchmark tests and an assessment in terms of computational cost, accuracy and robustness is provided.
-<math>MATLAB^{\mbox{®}}</math>. 2019. Version R2018a. The MathWorks, Inc., Natick, MA, USA.
+===2.6.1 GMM Algorithm===
-</span>
+The GMM algorithm is based on three principal steps (see Figure [[#img-2.5|2.5]]).
-=2 Rotor modelling=
+<ol style='list-style-type:lower-alpha;'>
-The first step in the study of vibrations in rotating structures is to define an adequate model that describes its behaviour. In section [[#1.3 Report outline|1.3]], a justification on why a simple model is preferable to more accurate but complex equations is presented. The aim of this section is to keep going with this justification, presenting the chosen model and its limitations. The equations of motion will also be presented latter in this chapter.
+<li>''Initialization phase (Fig.[[#img-2.5a|2.5a]])'': this is the step which mostly distinguishes GMM from MPM. During this phase the connectivity of each integration point (i.e., each material point) is computed as the "cloud of nodes", centred on the material point, and obtained by a search-in-radius. Such a cloud is then employed for the calculation of the shape functions. Unlike MPM, the Newmark prediction is performed by using the nodal information of the previous time step, as in FEM. Once <math display="inline">N</math> and <math display="inline">\nabla N</math> are suitably defined, MPM and GMM essentially coincide in the following steps; </li>
+<li>''UL-FEM calculation phase (Fig.[[#img-2.5b|2.5b]])'': the local matrix, represented by the left-hand-side (<math display="inline"> lhs </math>) and the local vector, constituted by the right-hand-side (<math display="inline"> rhs </math>) are evaluated in the current configuration according to the formulation presented in Chapter [[#4 Irreducible formulation|4]]; the global matrix <math display="inline"> LHS </math> and the global vector <math display="inline"> RHS </math> are obtained by assembling the local contributions of each material point and, finally, the system is iteratively solved. During the iterative procedure, the nodes are allowed to move, accordingly to the nodal solution, and the material points do not change their local position within the geometrical element until the solution has reached  convergence;   </li>
+<li>''Convective phase (Fig.[[#img-2.5c|2.5c]])'': during the third and last phase the nodal information at time <math display="inline"> t_{n+1} </math> are interpolated back to the material points. The position of the material points is updated. Unlike MPM, the nodal information are not deleted, but used as initial conditions in the next time step.  </li>
-==2.1 Problem formulation and hypotheses==
+</ol>
-As it has already been stated in section [[#1.2 State of the art|1.2]], the study of rotordynamics can easily become complicated, as the understanding of the whole phenomena of rotating structures requires from a deep knowledge in several fields. However, an accurate study of rotatory structures is far from the scope of this project. Instead, more emphasis will be given to the '''methods''' used to solve the resulting equations, rather than on the modelling and simulation of the rotor. That is why the model will be defined under the following conditions and '''hypotheses''':
+The details of the GMM algorithm are presented in Algorithm [[#algorithm-2.2|2.2]].
-* The axis of rotation of the rotor is known and '''fixed'''. Angular velocity and acceleration are known and given by the driving device.
+<div id='img-2.5a'></div>
-* The problem is restricted to '''small''' displacements but great rotations induced by the driving device. Hence, it can be modelled using the linear elastic equations ([[#A.1 Introduction to elasticity|A.1]]).
+<div id='img-2.5b'></div>
-* The rotor will be modelled in terms of the finite element method using solid elements, being nodal displacements the degrees of freedom.
+<div id='img-2.5c'></div>
-* The inertia of the rotor will not be taken into consideration, and thus no '''gyroscopic effect''' is expected.
+<div id='img-2.5'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-ALGORITHM_MPM2_init.png|228px|Initialization phase]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-ALGORITHM_MPM2_ulfem.png|216px|Updated Lagrangian FEM phase]]
+|- style="text-align: center; font-size: 75%;"
+| (a) Initialization phase
+| (b) Updated Lagrangian FEM phase
+|-
+| colspan="2" style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-ALGORITHM_MPM2_conv.png|198px|Convective phase]]
+|- style="text-align: center; font-size: 75%;"
+|  colspan="2" | (c) Convective phase
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 2.5:''' GMM phases.
+|}
-The first point gives an important hint about the nature of the problem. In the inertial frame, rotating velocities and accelerations will be '''known'''. In other words, the inertial kinematic variables are not the unknown, which in fact is the torque needed to deliver the required power to maintain the given angular conditions. One has to take into account that, although the driving engine may introduce '''large rotations''', strains will still be small in a rotating frame of reference static with respect the rotor.
-The strongest '''limitation''' of the model is given by the forth point: no gyroscopic effect will be taken into consideration. This limits the correctness of the model when applied to structures with non-negligible inertia. The gyroscopic effect would induce rotations in different axes, breaking the first hypothesis. With this type of formulation, great part of the phenomena description is lost, as complex models such as the one of equation [[#img-A.2|A.2]] are difficult to model using classical formulations.
+{| style="margin: 1em auto;border: 1px solid darkgray;width:90%;"
+|-
+|
+:<span style="font-size: 75%;">
+Material DATA: E, <math> \nu </math>, <math> \rho </math>
+|-
+| style="padding:5px;"|Initial data on material points: <math> m_p </math>, <math> \mathbf{x}_p^n</math>, <math> \Delta t , \mathbf{F}_p^n = \displaystyle{ \sum _I\frac{\partial N_I}{\partial \mathbf{x}_I^{0}}}\cdot \mathbf{x}_I^{n}</math> and <math> \Delta \mathbf{F}_p =\displaystyle{ \sum _I\frac{\partial N_I}{\partial \mathbf{x}_I^{n}}}\cdot \mathbf{x}_I^{n+1} </math>
+|-
+| style="padding:5px;"|Initial data on nodes: <math> \mathbf{u}_I^n, \mathbf{v}_I^n, \mathbf{a}_I^n </math>
+|-
+| style="padding:5px;"|OUTPUT of calculations: <math> \mathbf{u}^{n+1}_I \boldsymbol{\sigma }^{n+1}_p </math>
+|-
+|
+<ol>
+<li>'''INITIALIZATION PHASE'''
+{|
+|-
+|   </li>
+:* for every material point with position <math display="inline"> \mathbf{x}_p^n</math> gather the cloud of nodes with position <math display="inline"> \mathbf{x}_I^n</math> such that <math display="inline">\left|\mathbf{x}_p-\mathbf{x}_I\right|<R</math>
+:* compute the shape functions <math display="inline">N_I\left(\mathbf{x}_p^n\right)</math> for all nodes <math display="inline">I</math> in the cloud
+:* Newmark method: PREDICTOR<br />
+::for the prediction of displacement, unlike Equation ([[#eq-2.10|2.10]]), <math display="inline"> ^{it+1}\mathbf{u}_{I}^{n+1} = \boldsymbol{u}_{I}^{n} </math>,<br />
+::while for the prediction of nodal velocity and nodal acceleration, see Equations ([[#eq-2.11|2.11]]) and ([[#eq-2.12|2.12]])
+|}
+<li>'''UL-FEM PHASE''' (identical to MPM, see Algorithm [[#algorithm-2.1|2.1]] )
+;  </li>
+<li>'''CONVECTIVE PHASE'''
+{|
+|-
+| </li>
+:* Update position of the material points by means of an interpolation of nodal solution
+::(a) for p = 1:<math display="inline"> N_p </math> <br />
+::<math display="inline"> \mathbf{x}_p^{n+1} = \mathbf{x}_p^n + \sum N_{I} \Delta \mathbf{u}_I^{n+1} </math>
+:* Save state of stress <math display="inline"> \boldsymbol{\sigma }^{n+1}_p </math>, state of strain <math display="inline"> \boldsymbol{\epsilon }^{n+1}_p </math> and total deformation gradient <math display="inline"> \mathbf{F}^{n+1}_p </math> on material points<br />
+::(the latter by <math display="inline">\mathbf{F}_p^{n+1}=\Delta \mathbf{F}_p \cdot \mathbf{F}_p^n</math>)
+|}
+</ol>
+</span>
+|-
+| style="text-align: center; font-size: 75%;"|
+<span id='algorithm-2.2'></span>'''Algorithm. 2.2''' GMM algorithm.
+|}
-The above hypotheses will make the model inherently '''inaccurate'''. The resulting simulations will not describe the vibration behaviour of real rotor blades, although it is a great first step into the actual rotordynamic problem. The key of this approach is to make equations easy to pose, and thus expedite the programming of the model in terms of the FEM.  Hopefully enough, this will allow a deeper study on the methods used to solve the dynamic system.
+===2.6.2 Calculation of GMM shape functions===
-==2.2 Frames of reference and rotation matrix==
+While the computation of the shape functions is trivial for the standard MPM (as in FEM), thanks to the presence of a background grid (Figure [[#img-2.2|2.2]]), the evaluation of the shape functions in GMM is more complex.  From a technical point of view, GMM is based on a conceptually simple operation: given an arbitrary position <math display="inline">x_p</math> in space (which will, in the practice, coincide with the position of the material point) and a search radius <math display="inline">R</math>, one may find all of the Nodes <math display="inline">I</math> such that <math display="inline">||x_I-x_p||<R</math>. Given such a cloud of nodes, one may then compute, at the position <math display="inline">x_p</math>, the shape functions <math display="inline">N_{I}(x_p)</math> (together with their gradients), such that, a given function <math display="inline">u</math>, whose nodal value is <math display="inline">u_I</math>, can be interpolated at the position <math display="inline">x_p</math> as <math display="inline">u(x_p)= \sum _I N_{I}(x_p)\,u_I</math> (Figure [[#img-2.4a|2.4a]]).
-Previous to the posing of the equations describing the kinematics of the system, it is convenient to define the different frames of references that will be used: an inertial frame <math display="inline">xyz</math>, fixed to a static point <math display="inline">O</math>; and a noninertial rotating frame <math display="inline">\varsigma \varrho \varpi </math> fixed to the rotor — thus sharing its angular velocity <math display="inline">\Omega </math> and acceleration <math display="inline">\alpha </math>. The planes defined by <math display="inline">xy</math> and <math display="inline">\varsigma \varrho </math> are meant to be coincident, and thus <math display="inline">z</math> and <math display="inline">\varpi </math> will be the same axis (since both basis are positive definite). From now on, they both will be referred as <math display="inline">z</math>, the rotation axis.
+However, in order to construct a convergent solution, some guarantees must be provided by the shape functions. In particular, they shall comply with the ''Partition of Unity'' (PU) property, as a very minimum at all of the positions <math display="inline">x_p</math> at which the shape functions are evaluated. A number of shape functions exist complying with such property <span id='citeF-142'></span><span id='citeF-143'></span><span id='citeF-33'></span>[[#cite-142|[142,143,33]]]. Among the available options, two appealing class of meshless functions are considered in this work: the first choice is constituted by the so called Moving Least Square (MLS) method and the second one represented by the Local Maximum Entropy (LME) technique.
-To change from one coordinate system to another, one can define a '''rotation matrix''', taking advantage of the fact that the angular displacement is known for every value of time. Following Euler's definition of rotation, it is stated that rotation in space can always be described by a rotation along a certain axis over a certain angle.  In this case, the axis of rotation is <math display="inline">z</math>. One can, for instance, represent the rotation in a two-dimensional space, being <math display="inline">{\mathbf{r}}_i</math> and <math display="inline">{\mathbf{r}}_f</math> the vectors describing initial and final positions, respectively, of a given point. The rotation is represented in figure [[#img-2.1|2.1]]: <div id='img-2.1'></div>
+The first technique is based on the MLS approach, first introduced by Lancaster <span id='citeF-74'></span>[[#cite-74|[74]]] and Belytschko <span id='citeF-76'></span><span id='citeF-143'></span>[[#cite-76|[76,143]]].  The MLS-approximation fulfils the reproducing conditions by construction, so no corrections are needed.  The fundamental principle of MLS approximants is based on a weighted least square fitting of a target solution, sampled at a given, possibly randomly distributed, set of points, via a function of the type
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+<span id="eq-2.18"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-rotation_2.png|240px|Vector rotation in the plane ''xy'']]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 2.1:''' Vector rotation in the plane ''xy''
+|-
+| style="text-align: center;" | <math>P(\mathbf{x})=a_1+a_2x+a_3y+a_4xy+\cdots   </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.18)
 |}
-<span id="eq-2.1"></span>
+where the coordinates <math display="inline">x,y</math> are to be understood as relative to the sampling position.
+The reconstruction of a continuous function <math display="inline">h(\mathbf{x})</math> can be obtained considering the data <math display="inline"> h_{I} </math> be located at points <math display="inline"> \mathbf{x}_{I} </math> and an arbitrary, smooth and compactly supported, weight function <math display="inline"> W_{I}(\mathbf{x}) </math>, such that the <math display="inline"> \mathbf{x}_{I} </math> fall within the support of <math display="inline">W</math>. Assuming now that the reconstructed function (<math display="inline">h^{h}_{x}</math>) is computed as
+<span id="eq-2.19"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 567: / Line 1,240: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathbf{r}_f} = \begin{pmatrix}x_f\\  y_f \end{pmatrix} = \begin{pmatrix}R \cos (\theta +\varphi )\\  R \sin (\theta +\varphi ) \end{pmatrix} = \begin{pmatrix}\overset{x_i}{\overbrace{R \cos (\varphi )}}\cos (\theta ) - \overset{y_i}{\overbrace{R \sin (\varphi )}}\sin (\theta ) \\  \underset{y_i}{\underbrace{R \sin (\varphi )}}\cos (\theta ) - \underset{x_i}{\underbrace{R \cos (\varphi )}} \sin (\theta ) \end{pmatrix}  = \begin{bmatrix}\cos (\theta ) & -\sin (\theta )\\  \sin (\theta )& \cos (\theta ) \end{bmatrix}\begin{bmatrix}x_i\\  y_i \end{bmatrix} = {\mathbf{{{Q}}\;_{(\theta )}}} \; {\mathbf{r}_i} </math>
+| style="text-align: center;" | <math>h(\mathbf{x})\approx h^{h}_{x} =\mathbf{P}^{T}(\mathbf{x})\cdot \mathbf{a}(\mathbf{x})  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.1)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.19)
 |}
-Where <math display="inline">{\mathbf{{{Q}}\;_{(\theta )}}}</math> stands for the rotation matrix, and it is satisfied that <math display="inline"> \left\| {\mathbf{r}}_{i} \right\|= \left\| {\mathbf{r}}_{f} \right\|= R</math>.
+the fitting to <math display="inline"> h(\mathbf{x}) </math> is done by minimizing the error function <math display="inline">J</math>, defined as
-It will be interesting, when posing the kinetic equations, to consider the cross product or '''spin''' of a vector <math display="inline">t</math>, represented with <math display="inline">\widetilde{t}</math> and defined as the linear operator:
+<span id="eq-2.20"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 581: / Line 1,253: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{t} =\begin{pmatrix}t_1\\  t_2\\  t_3 \end{pmatrix}\;  \Rightarrow \; \widetilde{{{t}}} = \begin{bmatrix}0 &-t_3  &t_2 \\  t_3 &  0& -t_1\\  -t_2& t_1 & 0 \end{bmatrix} </math>
+| style="text-align: center;" | <math>J=\sum _{I} (\mathbf{P}_{I}^{T}\cdot \mathbf{a}(\mathbf{x})-h_{I})^{2}W_{I}(\mathbf{x})  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.2)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.20)
 |}
-Using this operator, the rotation can be expressed in terms of the Rodrigues formula <span id='citeF-8'></span>[[#cite-8|[8]]]:
+where <math display="inline"> \mathbf{P}_{I}=\mathbf{P}(\mathbf{x}_{I}) </math>.
+This allows defining a set of ''approximating'' shape functions <math display="inline">N</math> such that
+<span id="eq-2.21"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 593: / Line 1,268: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathbf{{{Q}}\;_{(\theta , {n})}}} = \left[\cos{\theta } \, {{\hbox{I}}} + (1-\cos{\theta }){\hbox{n}}{\hbox{n}}^T +\sin{\theta }\; {\widetilde{\hbox{n}}}\right] </math>
+| style="text-align: center;" | <math>h(\mathbf{x})=\sum _{I}N_{I}(\mathbf{x})h_{I}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.3)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.21)
 |}
-Which for the case <math display="inline">{\mathbf{n}} = \widehat{\mathbf{k}}</math> gives the same result as in equation [[#eq-2.1|2.1]]. What is interesting about the spin operator is the easiness to represent matrices derivatives. Defining the temporal derivative of a function as <math display="inline">\frac{\mathrm{d} \mathbf{f} }{\mathrm{d} t} = \dot{\mathbf{f}}</math>, one can compute the temporal derivative of the three-dimensional rotation matrix using the spin operator:
+where
+<span id="eq-2.22"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 605: / Line 1,281: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{\mathrm{d} {\mathbf{{{Q}}\;_{(\theta )}}} }{\mathrm{d} t} = {\mathbf{{{\dot{Q}}}\;_{(\theta )}}} =  \begin{bmatrix}-\sin (\theta )\dot{\theta } & -\cos (\theta )\dot{\theta } &0\\  \cos (\theta )\dot{\theta }& -\sin (\theta )\dot{\theta } & 0\\  0& 0&0 \end{bmatrix} = {\mathbf{{{Q}}\;_{(\theta )}}}\begin{bmatrix}0 & -\dot{\theta }  & 0\\  \dot{\theta } &0  & 0\\  0& 0 & 0 \end{bmatrix} = {\mathbf{{{Q}}\;_{(\theta )}}} \; {{\widetilde{\dot{\theta }}}} \;=\; {\mathbf{{{Q}}\;_{(\theta )}}} \;\; {{\widetilde{\Omega }}}  </math>
+| style="text-align: center;" | <math>N_{I}=\mathbf{P}^{T}(\mathbf{x})\cdot \mathbf{M}^{-1}(\mathbf{x})\cdot \mathbf{P}_{I}W_{I}(\mathbf{x})  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.4)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.22)
 |}
-The above expression can be generalised to compute matrix derivatives of order <math display="inline">n</math>:
+with <math display="inline">\mathbf{M}</math> defined as
+<span id="eq-2.23"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 617: / Line 1,294: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{\mathrm{d^n} {\mathbf{{{Q}}\;_{(\theta )}}} }{\mathrm{d} t^n} = {\mathbf{{{Q}}\;_{(\theta )}}} \;\; {{\widetilde{\Omega }}}\; ^n </math>
+| style="text-align: center;" | <math>\mathbf{M}(\mathbf{x})=\sum _{I}\mathbf{P}_{I}\mathbf{P}_{I}^{T}W_{I}(\mathbf{x})  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.5)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.23)
 |}
-==2.3 Kinematic equations==
+It can be readily verified that the shape functions are able to reproduce exactly a polynomial up to the order used in the construction. This fact can also be used to prove compliance to the partition of unity property. Namely, if one assumes <math display="inline">h_I=\mathbf{P}_I(\mathbf{x_I})</math> and substitutes into Equation [[#eq-2.21|2.21]] then
-The kinematic description of the rotor is going to be quite simple. Imagine a particle of the rotor, represented by a '''point mass''' with no inertia associated. This point mass is subjected to a rotation given by the driving device. If the body were perfectly rigid, displacements in the rotating frame <math display="inline">\varsigma \varrho </math> would be zero, and positions in the <math display="inline">xy</math> plane would be given by the driving device. This is, however, not the case. Displacements in all <math display="inline">\varsigma </math>, <math display="inline">\varrho </math> and even <math display="inline">z</math> directions may occur.
+<span id="eq-2.24"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\sum _{I}N_{I}(\mathbf{x})\mathbf{P}_{I}^{T}=\mathbf{P}^{T}(\mathbf{x})\cdot \mathbf{M}^{-1}(\mathbf{x})\cdot \sum _{I}\mathbf{P}_{I}\mathbf{P}_{I}^{T}W_{I}(\mathbf{x})</math>
+|-
+| style="text-align: center;" | <math> =\mathbf{P}^{T}(\mathbf{x})\cdot \mathbf{M}^{-1}(\mathbf{x})\cdot \mathbf{M}(\mathbf{x}) </math>
+|-
+| style="text-align: center;" | <math> =\mathbf{P}^{T}(\mathbf{x}) </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.24)
+|}
-To illustrate this effect, imagine a two-dimensional case in which elastic displacements are only possible in <math display="inline">\varsigma </math>. Imagine the mass point starting from an initial position <math display="inline">{\mathbf{r}}_i</math>, which after a certain time holds a generic position <math display="inline">{\mathbf{r}}</math>. The total displacement <math display="inline">{\mathbf{U}}</math> can be split in terms of rigid body <math display="inline">{\mathbf{U}}\;_{rb}</math> and elastic <math display="inline">{\mathbf{U}}\;_{el}</math> displacements:
+Hence, considering the special case of a constant polynomial <math display="inline">\mathbf{P}(\mathbf{x}) = 1</math>, or of a linear variation in <math display="inline">\mathbf{x}</math>, <math display="inline">\mathbf{P}(\mathbf{x})=\mathbf{x}</math> we obtain respectively
-<div id='img-2.2'></div>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-rotation_3.png|240px|Displacement as composition of rotation and deformation]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 2.2:''' Displacement as composition of rotation and deformation
+|-
+| style="text-align: center;" | <math>\sum _{I}N_{I}\equiv{1}</math>
+|-
+| style="text-align: center;" | <math> \sum _{I}\mathbf{x}_{I}N_{I}\equiv \mathbf{x}</math>
+|-
+| style="text-align: center;" |
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.25)
 |}
-The equations describing this motion are easy to deduce. For simplicity, given a vector <math display="inline">v</math>, let's denote as <math display="inline">v_{(x)}</math> the vector described in inertial coordinates, and <math display="inline">v_{\mathbb{(\varsigma )}}</math> the same vector but in the corotational coordinates. For instance, <math display="inline">{\mathbf{r}}_{\;(x)} = \left(x \, y\,  z   \right)^T</math> and <math display="inline">{\mathbf{r}}_{\;(\varsigma ) }= \left(\varsigma \, \varrho \,  \varpi   \right)^T</math>. It can be easily deduced from equation [[#img-2.2|2.2]]:
+A similar reasoning also gives
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 643: / Line 1,339: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathbf{U}}\;_{(x)} = {\mathbf{r}}_{\;(x) }- {\mathbf{r}}_{\;i\;(x) } = {\mathbf{U}}\;_{rb\;(x)}\;  + \; {\mathbf{U}}\;_{el\;(x)} </math>
+| style="text-align: center;" | <math>\sum _{I}\nabla N_{I}\equiv{0}</math>
+|-
+| style="text-align: center;" | <math> \sum _{I}\mathbf{x}_{I}\cdot \nabla N_{I}\equiv 1 </math>
+|-
+| style="text-align: center;" |
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.6)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.26)
 |}
-The following figure helps on visualising the different displacements affecting the point mass, starting from a given time <math display="inline">t_i</math> until a generic <math display="inline">t</math>. The total displacement results from the combination of both motions. Note that, for simplification, only '''axial''' elastic deformations are represented:
+thus proving the compliance with the PU property.
-<div id='img-2.3'></div>
+However, MLS shape functions are not able to guarantee the ''Kronecker delta property'' at the nodes. This implies that two nodal shape functions may be simultaneously non zero at a given nodal position. This has practical implications at the moment of imposing Dirichlet Boundary conditions,  namely, in order to impose <math display="inline">u(x_d) = 0</math> at a given point on the Dirichlet boundary <math display="inline">x_d</math>, one must impose that <math display="inline">\sum (N_I(x_d) u_I) = 0</math>, which constitutes a classical  ''multipoint constraint'' <span id='citeF-144'></span>[[#cite-144|[144]]].
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+Interestingly, the choice of different shape functions could ease this particular problem. An appealing choice could be the use of LME approximants, which guarantees complying with a weak ''Kronecker delta property'' until the cloud of nodes is represented by a convex hull.
+The LME technique is based on the evaluation of the local max-ent approximants <span id='citeF-145'></span>[[#cite-145|[145]]], which represents the solution that exhibits a (Pareto) compromise between competing objectives: the principle of ''max-ent'' subject to the constraints:
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-rotation_4.png|420px|Representation of the rigid body rotation and axial elastic deformation]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 2.3:''' Representation of the rigid body rotation and axial elastic deformation
+|-
+| style="text-align: center;" | <math>\hbox{For fixed}\; \boldsymbol{x}\;\hbox{maximize}\; H(N_1, N_2,..., N_m) = - \sum _I N_I \hbox{ln} N_I,</math>
+|-
+| style="text-align: center;" | <math> \hbox{subject to}\; N_I \geq{0},\quad I = 1,...,n_{node}, </math>
+|-
+| style="text-align: center;" | <math> \sum _I N_I = 1,\quad \sum _I N_I\boldsymbol{x}_I = \boldsymbol{x} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.27)
 |}
-Using the rotation matrix obtained in equation [[#eq-2.1|2.1]], one can develop the following relations regarding the '''displacements''' of the particle:
+and the objective function interpreted as a measure of locality of the shape functions of the Delaunay triangulation
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 665: / Line 1,377: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathbf{U}}\;_{rb\;(x)}\; = \; {\mathbf{{{Q}}_{(\theta )}}} {\mathbf{r}}_{\, i}  \; - \; {\mathbf{r}}_{\, i} = ({\mathbf{{{Q}}_{(\theta )}}} - \hbox{I}){\mathbf{r}}_{\, i} \;,\;\;\;\;\; \; \; {\mathbf{U}}\;_{el\;(x)}\; = \;{\mathbf{{{Q}}_{(\theta )}}}\; {\mathbf{U}}\;_{el\;(\mathbb{\varsigma })} </math>
+| style="text-align: center;" | <math>\hbox{For fixed}\; \boldsymbol{x}\;\hbox{minimize}\; U(\boldsymbol{x},N_1, N_2,..., N_m) = - \sum _I N_I |\boldsymbol{x}-\boldsymbol{x}_I|^2,</math>
+|-
+| style="text-align: center;" | <math> \hbox{subject to}\; N_I \geq{0},\quad I = 1,...,n_{node}, </math>
+|-
+| style="text-align: center;" | <math> \sum _I N_I = 1,\quad \sum _I N_I\boldsymbol{x}_I = \boldsymbol{x} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.7)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.28)
 |}
-For the sake of notational simplification, let's simply denote the total displacement in the inertial frame <math display="inline">{\mathbf{U}}\;_{(x)}</math> as <math display="inline">{\mathbf{U}}</math>, and the elastic deformations in the corotating frame <math display="inline">{\mathbf{U}}\;_{el\;(\mathbb{\varsigma })}</math> as <math display="inline">{\mathbf{u}}</math>. With this notation, the equations describing the particle's position, velocity and acceleration are:
+The solution to the problem can be found minimizing <math display="inline">\beta U(\boldsymbol{x},N_1, N_2,..., N_m)-H(N_1, N_2,..., N_m)</math> subjected to the usual constrains. The optimization problem takes the form
+<span id="eq-2.29"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 677: / Line 1,394: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathbf{U}}= {\mathbf{{{Q}}_{(\theta )}}}  (\, {\mathbf{r}}_{\, i} +  {\mathbf{u}} \, ) -\, {\mathbf{r}}_{\, i}  </math>
+| style="text-align: center;" | <math>\hbox{For fixed}\; \boldsymbol{x}\in \hbox{conv}X,\;\hbox{minimize}\; \sum _I\beta _I N_I |\boldsymbol{x}-\boldsymbol{x}_I|^2 + \sum _I N_I \hbox{ln} N_I,</math>
+|-
+| style="text-align: center;" | <math> \hbox{subject to}\; N_I \geq{0},\quad I = 1,...,n_{node}, </math>
+|-
+| style="text-align: center;" | <math> \sum _I N_I = 1,\quad \sum _I N_I\boldsymbol{x}_I = \boldsymbol{x} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.8)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2.29)
 |}
+with <math display="inline">\beta = \gamma /h^2</math> representing a non-negative locality coefficient, where <math display="inline">\gamma </math> is a dimensionless parameter and <math display="inline">h</math> is a measure of nodal spacing. The value of <math display="inline">\gamma </math> is always chosen in a range between <math display="inline">0.6</math>, relative to spread-out meshfree shape functions, and <math display="inline">4</math>, relative to linear finite elements basis functions. Unlike MLS approximants, the LME basis functions possess the weak ''Kronecker delta property'' at the boundary of the convex hull of the nodes and they are <math display="inline">C^{\infty }</math> function of <math display="inline">\beta </math> in <math display="inline">(0, +\infty )</math>. However, the computation of the LME approximation scheme is more onerous than MLS basis functions, as the problem described by Equation [[#eq-2.29|2.29]] is a convex problem.
+=3 Constitutive Models=
+As explained in the previous Chapters, granular material can be modelled by continuum approach on a macroscopic scale. In the continuum approach, the macroscopic behaviour of granular flow is described by the balance equations (introduced in Chapter [[#4 Irreducible formulation|4]]) facilitated with boundary conditions and constitutive laws. These latter ones characterize the relation between the stress and strain, thus, defining the behaviour of the material. In this Chapter, the constitutive models employed in this monograph for verification, validation and application of the MPM strategy are presented and their numerical implementation is explained.
+The irreducible and mixed formulations, presented and verified in Chapters [[#4 Irreducible formulation|4]] and [[#5 Mixed formulation|5]], respectively, are written in an Updated Lagrangian framework, e.g, the last known configuration is considered to be the reference one, and are valid under the hypothesis of large deformations, since the strain information, used for the evaluation of the material response, is represented by the total deformation gradient <math display="inline"> \boldsymbol{F} := \frac{\partial \phi \left(\boldsymbol{X},t\right)}{\partial \boldsymbol{X}} </math> and not by the symmetric gradient of displacement <math display="inline"> \left[\Delta \boldsymbol{u}\right]^{sym} = \frac{1}{2}\left(\Delta \boldsymbol{u}+\Delta \boldsymbol{u}^T\right)</math>. In this monograph, homogeneous isotropic elasic and elasto-plastic materials are considered. More specifically, a hyperelastic Neo-Hookean, a hyperelastic-plastic J2 and Mohr-Coulomb plastic laws have been implemented in the framework of MPM and they are presented in Sections [[#3.1 Hyperelastic law|3.1]], [[#lb-3.2|3.2]] and [[#3.3 Hyperelastic - Mohr-Coulomb plastic law|3.3]].
+==3.1 Hyperelastic law==
+The first constitutive law to be presented is an elastic law under finite strain regime. In this regard, in the current work a hyperelastic material is considered. Typically, these models are suitable to describe the behaviour of engineering materials which can undergo deformation of several times their initial configuration, such as, rubber-like solids, elastomers, sponges and other soft flexible materials. Hyperelastic materials are non-dissipative and their state of stress solely depends on the current deformation, as the ''Cauchy elastic models'', and do not depend on the path of deformation. For this reason, in this case it is possible to derive the stress-strain relation from a ''specific strain energy function''
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 687: / Line 1,420: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathbf{\dot{U}}}= {\mathbf{{{Q}}_{(\theta )}}} \left[{{\widetilde{\Omega }}}({\mathbf{r}}_{\, i} +{\mathbf{u}} ) + {\mathbf{\dot{u}}} \right] </math>
+| style="text-align: center;" | <math>\Psi = \Psi (\boldsymbol{F}) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.9)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.1)
 |}
-<span id="eq-2.10"></span>
+which relates the specific strain energy to the total deformation gradient as unique state variable, since dissipative related state variables can be neglected and the assumption of isothermal processes is made <span id='citeF-146'></span>[[#cite-146|[146]]].  As stated in <span id='citeF-146'></span>[[#cite-146|[146]]], any constitutive law must satisfy the axioms of ''thermodynamic determinism'', ''material objectivity'' and ''material symmetry''. The compliance of the first axiom implies the stress constitutive equation in terms of the first Piola-Kirchhoff stress tensor <math display="inline">\boldsymbol{P}</math>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 698: / Line 1,432: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathbf{\ddot{U}}}= {\mathbf{{{Q}}_{(\theta )}}} \left[\underset{Centrifugal}{\underbrace{{{\widetilde{\Omega }}}^{\, 2}({\mathbf{r}}_{\, i} + {\mathbf{u}} )} }+ \underset{Tangential}{\underbrace{{{\widetilde{\alpha }}}({\mathbf{r}}_{\, i} + {\mathbf{u}} ) }}+ \overset{Coriolis}{\overbrace{2 {{\widetilde{\Omega }}}{\mathbf{\dot{u}}} }}+\underset{Relative}{\underbrace{{\mathbf{\ddot{u}}}}} \right] </math>
+| style="text-align: center;" | <math>\boldsymbol{P}= \frac{\partial \Psi (\boldsymbol{F})}{\partial \boldsymbol{F}} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.10)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.2)
 |}
-Where <math display="inline">\alpha </math> stands the angular acceleration set by the driving device. Although until now, the figures  accounted only for axial elastic deformations — that is, <math display="inline">{\mathbf{u}}= \left(u\,_\varsigma , 0,  0   \right)^T</math> — the fact is that the previous equations can be used to describe a general deformation in three dimensions: <math display="inline">{\mathbf{u}}= \left(u_\varsigma , u_\varrho ,  u_\varpi  \right)^T</math>. From the previous equations, <math display="inline">\Omega </math> and <math display="inline">\alpha </math> are imposed by the driving device, while <math display="inline">{\mathbf{r}}_{\, i}</math> only depends on the initial condition. The elastic kinematic '''unknowns''' are, then, <math display="inline">{\mathbf{{u}}}</math>, <math display="inline">{\mathbf{\dot{u}}}</math> and <math display="inline">{\mathbf{\ddot{u}}}</math>.
+Accordingly, the stress constitutive relation can be expressed also by the Kirchhoff stress tensor
-==2.4 Elasticity notation==
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\boldsymbol{\tau }\equiv \boldsymbol{P}\boldsymbol{F}^T, </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.3)
+|}
-Previous to the posing of the equilibrium equations, some definitions regarding the used notation should be clarified. Let <math display="inline">n_d</math> denote the number of space '''dimensions''' of the problem under consideration, and let <math display="inline"> \mho  \subset \mathbb{R}^{n_{d}}</math>  be an open set with piecewise smooth boundary <math display="inline">\Gamma </math>. A general point, denoted by <math display="inline"> \boldsymbol{x}</math>, is identified with its position vector emanating from the '''origin'''. Tensors and vectors will be described in terms of Cartesian components. In indicial notation, differentiation is denoted by:
+which for a hyperelastic material is given by
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 714: / Line 1,456: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{\partial }{\partial x_{i}}=\partial _{x_{i}}=\partial _{i}\; , \;\;\; \partial _{i i}=\sum _{i=1}^{n_{d}} \partial _{i i} </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau }(\boldsymbol{F})=\frac{\partial \Psi (\boldsymbol{F})}{\partial \boldsymbol{F}}\boldsymbol{F}^T </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.11)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.4)
 |}
-This time, a general form of the elastic problem is to be developed. The displacement field, referred as <math display="inline">\mathfrak{u}</math>, pretends to represent displacement in a generic case, and should not be confused with the elastic deformations in the corotational frame <math display="inline">{\mathbf{{u}}}</math>, presented in the previous section. Regarding the domain boundary, we shall assume that it does not change it time and admits the following decomposition:
+and the Cauchy stress tensor, <math display="inline">\boldsymbol{\sigma }=\boldsymbol{\tau }/J</math>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 726: / Line 1,468: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\Gamma =\Gamma _{\mathfrak{u}}^{i} \cup \Gamma _{\sigma }^{i}\, , \;\; \Gamma _{\mathfrak{u}}^{i} \cap \Gamma _{\sigma }^{i}=\varnothing  </math>
+| style="text-align: center;" | <math>\boldsymbol{\sigma }(\boldsymbol{F})=\frac{1}{J}\frac{\partial \Psi (\boldsymbol{F})}{\partial \boldsymbol{F}}\boldsymbol{F}^T </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.12)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.5)
 |}
-With the previous decomposition, two different '''boundary conditions''' can be stated. Regarding the prescribed displacements, the '''Dirichlet''' boundary conditions <math display="inline">\mathfrak{{u}}_{i}=\overline{\mathfrak{u}}^{\,i}</math> are defined on <math display="inline">\Gamma _{\mathfrak{u}}^{i} </math>. Note that prescribed displacements may change in time, thus <math display="inline">\overline{\mathfrak{u}}_{i} : \Gamma _{u}^{i} \times ] 0, T[\rightarrow \mathbb{R}</math>, where <math display="inline">] 0, T[</math> is the open time interval of length <math display="inline">T>0</math>. On the other hand, regarding the prescribed tractions, the '''Neumann''' boundary conditions <math display="inline">\overline{\mathbf{t}}^{\,i}=\sigma _{i j} n_{j} </math> are defined on <math display="inline">\Gamma _{\sigma }^{i}</math>, where again <math display="inline">\overline{\mathbf{t}}_{i} : \Gamma _{\sigma }^{i} \times ] 0, T[\rightarrow \mathbb{R}</math>.
+where <math display="inline">J\equiv det\boldsymbol{F}</math>.
-To complete the definition of the boundary value problem, '''initial conditions''' shall be given. Since the governing equation will involve accelerations and thus be  second order in time, both initial displacements and velocities must be specified:
+In order to comply with the second axiom, the ''specific strain energy function'' has to be invariant under changes in the observer. This can be expressed by the following equation
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 740: / Line 1,482: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathfrak{{u}}_i}_{(\boldsymbol{x}, 0)}={\mathfrak{u}_{i}^{0}}_{(\boldsymbol{x})}\,,\;\;\;\;{\dot{\mathfrak{{u}}}}_{i\;(\boldsymbol{x}, 0)}={\dot{\mathfrak{u}}^{0}}_{i \;(\boldsymbol{x})} </math>
+| style="text-align: center;" | <math>\Psi (\boldsymbol{QF})=\boldsymbol{F} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.13)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.6)
 |}
-As stated in the previous section, the key is to find the displacements, which are represented by a vector field <math display="inline">\mathfrak{{u}} : \mho  \rightarrow \mathbb{R}^{n_{d}}</math>. Using infinitesimal theory (see [[#A.1 Introduction to elasticity|A.1]]), the deformation state at a given point <math display="inline"> \boldsymbol{{x}} \in \mho  </math> is characterised by the infinitesimal '''strain tensor''', which is defined to be the symmetric part of the displacement gradients:
+which has to be valid for any rotation tensor <math display="inline">\boldsymbol{Q}</math>. If we consider <math display="inline">\boldsymbol{Q}=\boldsymbol{R}^T</math> with <math display="inline">\boldsymbol{R}</math> the rotation obtained from the polar decomposition <math display="inline">\boldsymbol{F}=\boldsymbol{RU}</math>, the following identity is obtained
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 752: / Line 1,494: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\varepsilon _{i j} :=\left[\nabla ^{s} \mathfrak{{u}}\,\right]_{i j}=\frac{1}{2}\left(\frac{\partial{u}_{i}}{\partial x_{j}}+\frac{\partial {u}_{j}}{\partial x_{i}}\right)\;\;\;  </math>
+| style="text-align: center;" | <math>\Psi (\boldsymbol{F})=\Psi (\boldsymbol{U}) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.14)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.7)
 |}
-For linear elastostatic systems, the relationship between the Cauchy stress tensor and the strain tensor is given in equation [[#eq-A.6|A.6]]. In the dynamic case, however, the stress at a given time is not only proportional to the strain but also to its derivative (linear '''viscoelasticity'''). In indicial form:
+Thus, the ''material objectivity'' or ''frame invariance'' implies that <math display="inline">\Psi </math> depends on <math display="inline">\boldsymbol{F}</math> solely through the right stretch tensor <math display="inline">\boldsymbol{U}</math> and in an equivalent way, <math display="inline">\Psi </math> can be expressed as a function of the right Cauchy-Green strain tensor <math display="inline">\boldsymbol{C}\equiv \boldsymbol{F}^T\boldsymbol{F}=\boldsymbol{U}^2</math>:
-<span id="eq-2.15"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 765: / Line 1,506: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\sigma _{i j}=\mathbf{C}_{i j k l} \varepsilon _{k l}+\overline{\mathbf{D}}_{i j k l} \dot{\varepsilon }_{k l} </math>
+| style="text-align: center;" | <math>\Psi (\boldsymbol{F})=\widehat{\Psi }(\boldsymbol{C})\equiv \Psi (\sqrt{\boldsymbol{C}}) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.15)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.8)
 |}
-Where <math display="inline">\overline{\mathbf{D}}</math> is the tensor  of viscoelasticity coefficients. It is often assumed to be proportional to the elasticity tensor, <math display="inline">\overline{\mathbf{D}} = \overline{\beta }\,\mathbf{C}</math>. For the dynamic case, inertia forces — proportional to acceleration — have to be taken into account, and will be treated as body forces per unit volume. Additionally, a damping term proportional to velocity is also to be included:
+The stress constitutive equations in terms of <math display="inline">\boldsymbol{P}</math>, <math display="inline">\boldsymbol{\tau }</math> and <math display="inline">\boldsymbol{\sigma }</math> can be expressed as
+<span id="eq-3.9"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 777: / Line 1,519: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\overset{static}{\overbrace{{f}_{i} }}\rightarrow \overset{dynamic}{\overbrace{{f}_{i}-\rho \ddot{\mathfrak{{u}}}_{i}-\overline{\mu }\; \dot{\mathfrak{{u}}}_{i} }} </math>
+| style="text-align: center;" | <math>\boldsymbol{P}=\frac{\partial \widehat{\Psi }}{\partial \boldsymbol{C}}:\frac{\partial \boldsymbol{C}}{\partial \boldsymbol{F}}=2 \boldsymbol{F}\frac{\partial \widehat{\Psi }}{\partial \boldsymbol{C}},  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.16)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.9)
 |}
-Where <math display="inline">\rho : \uwave{\overline{\Omega }} \rightarrow \mathbb{R}</math> stands for the material density, and <math display="inline">\overline{\mu }</math> is a damping parameter usually defined as <math display="inline">\overline{\mu }=\overline{\alpha } \rho </math>, where <math display="inline">\overline{\alpha } </math> is known as damping coefficients, often considered to be constant.
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\boldsymbol{\tau }=2\boldsymbol{F}\frac{\partial \widehat{\Psi }}{\partial \boldsymbol{C}}\boldsymbol{F}^T, </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.10)
+|}
-==2.5 Strong form of the elastodynamic equation==
+and,
-Supposing the rotor as perfectly elastic, the equation describing the dynamic behaviour is given by Newton's second law for '''continuum''' media, which in indicial notation reads as:
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\boldsymbol{\sigma }=\frac{2}{J}\boldsymbol{F}\frac{\partial \widehat{\Psi }}{\partial \boldsymbol{C}}\boldsymbol{F}^T </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.11)
+|}
+where the definition of <math display="inline">\frac{\partial C_{ij}}{\partial F_{kl}}=\delta _{il}F_{kj}+\delta _{jl}F_{ki}</math> is used. The ''specific strain energy function'' has to be further constrained by considering the third and last axiom of ''material symmetry''. With a focus on material isotropy, <math display="inline">\Psi </math> must satisfy
-<span id="eq-2.17"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 794: / Line 1,553: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{\partial \sigma _{i j}}{\partial x_{j}}+\left({\boldsymbol{f}}_{i}-\overline{\mu }\; \dot{\mathfrak{{u}}}_{i}\right)=\rho \ddot{\mathfrak{{u}}}_{i} </math>
+| style="text-align: center;" | <math>\Psi (\boldsymbol{FQ})=\Psi (\boldsymbol{F}) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.17)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.12)
 |}
-But to solve the previous equation, the boundary conditions and the elastic model defined in the previous section are required. The '''strong form''' of the elastodynamic boundary value problem is then formulated as:
+for all rotations <math display="inline">\boldsymbol{Q}</math>. By choosing <math display="inline">\boldsymbol{Q}=\boldsymbol{R}^T</math>, it is established that
-<span id="eq-2.18"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 807: / Line 1,565: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" |
+| style="text-align: center;" | <math>\Psi (\boldsymbol{F})=\Psi (\boldsymbol{V}) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.18)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.13)
 |}
-Given <math display="inline">{\boldsymbol{f}}_{i}, \overline{\mathfrak{u}}^{\,i}\,,\;\; \overline{\mathbf{t}}^{\,i}\,,\;\; \mathfrak{u}_{i}^{0}</math> and <math display="inline"> \dot{\mathfrak{u}}_{\,i}^{0} \;</math> , find <math display="inline">\mathfrak{{u}}_{i} : \overline{\mho } \times ] 0, T[\rightarrow \mathbb{R}\;</math> such that
+which states that the ''specific strain energy function'' of an isotropic hyperelastic material must depend only on <math display="inline">\boldsymbol{F}</math> through the left stretch tensor <math display="inline">\boldsymbol{V}</math>. In an equivalent way, <math display="inline">\Psi </math> can also be a function of the left Cauchy-Green strain tensor <math display="inline">\boldsymbol{b}\equiv \boldsymbol{F}\boldsymbol{F}^T=\boldsymbol{V}^2</math> as follows
-<math display="inline"> \frac{\partial \sigma _{i j}}{\partial x_{j}}+\left({\boldsymbol{f}}_{i}-\overline{\mu }\; \dot{\mathfrak{{u}}}_{i}\right)=\rho \; \ddot{\mathfrak{{u}}}_{i}\; </math> in <math display="inline">\; \mho  \times ] 0, T[</math>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\Psi (\boldsymbol{F})=\widehat{\Psi }(\boldsymbol{b})\equiv \Psi (\sqrt{\boldsymbol{b}}) </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.14)
+|}
-<math display="inline">\mathfrak{{u}}_{i}=\overline{\mathfrak{u}}^{\,i}\;</math> on <math display="inline">\; \Gamma _{\mathfrak{u}}^{i}\times ] 0, T[</math>
+By considering both the axiom of ''frame invariance'' and ''material symmetry'' it implies that
-<math display="inline">\sigma _{i j} n_{j}=\overline{\mathbf{t}}^{\,i}\;</math> on <math display="inline">\;\Gamma _{\sigma }^{i} \times ] 0, T[</math>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\Psi (\boldsymbol{F})={\Psi }(\boldsymbol{U})={\Psi }(\boldsymbol{V})=\widehat{\Psi }(\boldsymbol{C})= \widehat{\Psi }(\boldsymbol{b}) </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.15)
+|}
-<math display="inline"> {\mathfrak{{u}}_{i}}_{(\boldsymbol{x}, 0)}=\mathfrak{u}_{i \,(\boldsymbol{x})}^{0}\,</math> and <math display="inline">\;{\dot{\mathfrak{{u}}}_{i \,(\boldsymbol{x}, 0)}}=\dot{\mathfrak{u}}_{i\,(\boldsymbol{x})}^{0}\;</math> with <math display="inline">\; \boldsymbol{x} \in \mho  </math>
+The hyperelastic law, considered in the current work, is a Neo-Hookean model, of the form
-where <math display="inline">\; \sigma _{i j}=\mathbf{C}_{i j k l} \varepsilon _{k l}+\overline{\mathbf{D}}_{i j k l} \dot{\varepsilon }_{k l}</math>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\Psi (\boldsymbol{C}) = \Psi (I_C) </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.16)
+|}
-The previous system has no closed form '''solution''', and an analytical solution is only found for simple domains . For more complex geometries and boundary conditions, approximate methods have to be used. There is no need to point out that the effect of rotation does nothing but adds complexity to the equations. In the inertial frame, equation [[#eq-2.17|2.17]] yields to the following expression when accounting for the acceleration induced by rotation:
+with dependence only on the first invariant of <math display="inline">\boldsymbol{C}</math>, <math display="inline">I_C = \mathrm{tr} \boldsymbol{C}</math>, which exhibits the following ''specific strain energy function'' <span id='citeF-147'></span>[[#cite-147|[147]]]
-<span id="eq-2.19"></span>
+<span id="eq-3.17"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 832: / Line 1,614: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\nabla {{\boldsymbol{\sigma }}}_{\;\boldsymbol{x}} +\; \mathbf{{{Q}}_{(\theta )}}  \left({{\mathbf{f}}}-\overline{\mu }\; \dot{{\mathbf{u}}}\,\right)=\rho \mathbf{{{Q}}_{(\theta )}} \left[{{\widetilde{\Omega }}}^{\, 2}({\mathbf{r}}_{\, i} + {\mathbf{u}} )+ {{\widetilde{\alpha }}}({\mathbf{r}}_{\, i} + {\mathbf{u}} ) + 2 {{\widetilde{\Omega }}}{\mathbf{\dot{u} }}+{\mathbf{\ddot{u}}} \right] </math>
+| style="text-align: center;" | <math>\Psi (\boldsymbol{C}) = \frac{1}{2}\lambda \left(\mathrm{ln} J\right)^2 - \mu \mathrm{ln} J + \frac{1}{2} \mu \left(\mathrm{tr} \boldsymbol{C} -3\right)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.19)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.17)
 |}
-===2.5.1 Transformation of coordinates===
+where <math display="inline">\lambda </math> and <math display="inline">\mu </math> are the Lamé constants.
-One of the problems of the above equation is that the '''stress field''' is difficult to pose in terms of the inertial frame, as the elasticity tensor is only known in the corotating coordinates. Using the rotation matrix, stresses can be transformed from one basis to another:
+According to Equation [[#eq-3.9|3.9]], it is known the stress constitutive relation in terms of the first Piola-Kirchhoff stress tensor and, thus, also in terms of the second Piola-Kirchhoff stress tensor
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 846: / Line 1,628: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{{\sigma }}_{\;\boldsymbol{x}}  = \;\; \mathbf{{{Q}}_{(\theta )}}\;\; {{\sigma }}_{\;\boldsymbol{\varsigma }}\;\; \mathbf{{{Q}}_{(\theta )}}^{\,T} </math>
+| style="text-align: center;" | <math>\boldsymbol{S} = \boldsymbol{F}^{-1}\boldsymbol{P}= 2 \frac{\partial \Psi (\boldsymbol{C})}{\partial \boldsymbol{C}} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.20)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.18)
 |}
-Upon some manipulations, the above expression can be rephrased in '''Voigt''''s notation as follows:
+and the Kirchhoff stress tensor
+<span id="eq-3.19"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 858: / Line 1,641: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left[\begin{array}{c}{\sigma _{x}} \\ {\sigma _{y}} \\ {\sigma _{z}} \\ {\tau _{y z}} \\ {\tau _{x z}} \\ {\tau _{x z}}\end{array}\right]=\underset{\mathbf{{{T}}_{(\theta )}}}{\underbrace{\left[\begin{array}{cccccc}{\cos ^{2} \theta } & {\sin ^{2} \theta } & {0} & {0} & {0} & {-\sin 2 \theta } \\ {\sin ^{2} \theta } & {\cos ^{2} \theta } & {0} & {0} & {0} & {\sin 2 \theta } \\ {0} & {0} & {1} & {0} & {0} & {0}\\ {0} & {0} & {0} & {\cos \theta } & {\sin \theta } & {0} \\ {0} & {0} & {0} & {-\sin \theta } & {\cos \theta } & {0} \\ {0.5(\sin 2 \theta )} & {-0.5(\sin 2 \theta )} & {0} & {0} & {0} & {\cos 2 \theta }\end{array}\right]}} \left[\begin{array}{c}{\sigma _{\varsigma }} \\ {\sigma _{\varrho }} \\ {\sigma _{\varpi }} \\ {\tau _{\varrho \varpi }} \\ {\tau _{\varsigma \varpi }} \\ {\tau _{\varsigma \varrho }}\end{array}\right] </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau } = \boldsymbol{F} \cdot \boldsymbol{S} \cdot \boldsymbol{F}^T = 2\boldsymbol{F}\cdot \frac{\partial \Psi (\boldsymbol{C})}{\partial \boldsymbol{C}}\cdot \boldsymbol{F}^T  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.21)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.19)
 |}
-Notice that <math display="inline">\;\mathbf{{{T}}}^{}_{(\theta )}\; = \mathbf{{{T}}}_{(-\theta )}\;</math>.
+In order to derive the strain energy function of Equation [[#eq-3.17|3.17]] with respect to <math display="inline">\boldsymbol{C}</math>, the following expression needs to be solved
-Then, the relation between the stresses in the inertial <math display="inline">{{\boldsymbol{\sigma }}}_{\;\boldsymbol{x}} </math> and in the corotational frame <math display="inline">{{\boldsymbol{\sigma }}}_{\;\boldsymbol{\varsigma }} </math>, and strains in the inertial <math display="inline">{{\boldsymbol{\varepsilon }}}_{\;\boldsymbol{x}} </math> and in the corotational frame <math display="inline">{{{\boldsymbol{\varepsilon }}}}_{\;\boldsymbol{\varsigma }} </math> in Voigt notation is given by:
+<span id="eq-3.20"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 872: / Line 1,654: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{{\boldsymbol{\sigma }}}_{\;\boldsymbol{x}}\; = \; \mathbf{{{T}}}_{(\theta )}\; {{{\boldsymbol{\sigma }}}}_{\;\boldsymbol{\varsigma }}\;,\;\;\;\;\;\;\;{{\boldsymbol{\varepsilon }}}_{\;\boldsymbol{x}}\; = \; \mathbf{{{T}}}_{(\theta )}^{T}\; {{{\boldsymbol{\varepsilon }}}}_{\;\boldsymbol{\varsigma }} </math>
+| style="text-align: center;" | <math>\frac{\partial \Psi (\boldsymbol{C})}{\partial \boldsymbol{C}}:= \frac{1}{2}\lambda \frac{\partial \left(\mathrm{ln} J\right)^2}{\partial \boldsymbol{C}} - \mu \frac{\partial \mathrm{ln} J}{\partial \boldsymbol{C}} + \frac{1}{2} \mu \frac{\partial \left(\mathrm{tr} \boldsymbol{C} -3\right)}{\partial \boldsymbol{C}}</math>
+|-
+| style="text-align: center;" | <math> = \frac{1}{2}\lambda \frac{\partial \left(\mathrm{ln} J\right)^2}{\partial J}:\frac{\partial J}{\partial \boldsymbol{C}} - \mu \frac{\partial \mathrm{ln} J}{\partial \boldsymbol{C}} + \frac{1}{2} \mu \frac{\partial \left(\mathrm{tr} \boldsymbol{C}\right)}{\partial \boldsymbol{C}} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.22)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.20)
 |}
-An analogous expression can be deduced regarding the elasticity tensor:
+By using these identities
-<span id="eq-2.23"></span>
+<span id="eq-3.21"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 885: / Line 1,669: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{{\mathbf{C}}}_{\;\boldsymbol{x}}\; = \; \mathbf{{{T}}}_{(\theta )}\;\; {{\mathbf{C}}}_{\;\boldsymbol{\varsigma }}\;\;\mathbf{{{T}}}_{(\theta )}^{T}\; </math>
+| style="text-align: center;" | <math>\frac{\partial J}{\partial \boldsymbol{C}}=\frac{1}{2}J \boldsymbol{C}^{-1}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.23)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.21)
 |}
-Having defined these relations, equation [[#eq-2.19|2.19]] can be rewritten in terms of the known elasticity matrix <math display="inline">{{\mathbf{C}}}_{\;(\varsigma )}</math>:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 897: / Line 1,679: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\nabla \left( \mathbf{{{T}}}_{(\theta )}\; {{{\boldsymbol{\sigma }}}}_{\;\boldsymbol{\varsigma }}\right)+\; \mathbf{{{Q}}_{(\theta )}}  \left({{\mathbf{f}}}-\overline{\mu }\; \dot{{\mathbf{u}}}\,\right)=\rho \mathbf{{{Q}}_{(\theta )}} \left[{{\widetilde{\Omega }}}^{\, 2}({\mathbf{r}}_{\, i} + {\mathbf{u}} )+ {{\widetilde{\alpha }}}({\mathbf{r}}_{\, i} + {\mathbf{u}} ) + 2 {{\widetilde{\Omega }}}{\mathbf{\dot{u} }}+{\mathbf{\ddot{u}}} \right] </math>
+| style="text-align: center;" | <math>\frac{\partial I_C}{\partial \boldsymbol{C}}=\frac{\mathrm{tr} \boldsymbol{C}}{\partial \boldsymbol{C}}=\boldsymbol{I} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.24)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.22)
 |}
-If the dynamics of the system are studied in the '''noninertial frame''', the above equation reduces to
+and substituting them into the Equation [[#eq-3.20|3.20]], the final expression is obtained
-<span id="eq-2.25"></span>
+<span id="eq-3.23"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 910: / Line 1,692: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\nabla  {{{\boldsymbol{\sigma }}}}_{\;\boldsymbol{\varsigma }}+\;  \left({{\mathbf{f}}}-\overline{\mu } \; \dot{{\mathbf{u}}}\,\right)=\rho  \left[{{\widetilde{\Omega }}}^{\, 2}({\mathbf{r}}_{\, i} + {\mathbf{u}} )+ {{\widetilde{\alpha }}}({\mathbf{r}}_{\, i} + {\mathbf{u}} ) + 2 {{\widetilde{\Omega }}}{\mathbf{\dot{u} }}+{\mathbf{\ddot{u}}} \right] </math>
+| style="text-align: center;" | <math>\frac{\partial \Psi (\boldsymbol{C})}{\partial \boldsymbol{C}}=\frac{\lambda }{2}\mathrm{ln} J\boldsymbol{C}^{-1}+\frac{\mu }{2}\left(\boldsymbol{I}-\boldsymbol{C}^{-1}\right)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.25)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.23)
 |}
-===2.5.2 One-dimensional stationary solution===
+By considering the result of Equation [[#eq-3.23|3.23]], and substituting it in the expression of Equation [[#eq-3.19|3.19]], the Kirchhoff stress tensor reads
-One of the simplest cases that can be tackled analytically is a one-dimensional stationary approach. In other words, it means that elastic deformations <math display="inline">{\mathbf{u}}</math> are only present in the '''axial''' direction <math display="inline">\varsigma </math>, while the rotation is considered to be stationary with constant angular velocity <math display="inline">\Omega </math>. Moreover, no vibration phenomena is taken into consideration, and so <math display="inline">{\mathbf{\dot{u}}}</math> and <math display="inline">{\mathbf{\ddot{u}}}</math> are neglected. That is to imagine the rotor as a sequence of one-dimensional bars, which can only experience pure traction and compression. For this case where <math display="inline">n_{d}=1</math>, neglecting damping and external forces, equation [[#eq-2.25|2.25]] yields to
-<span id="eq-2.26"></span>
+<span id="eq-3.24"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 925: / Line 1,705: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{d A_{(\varsigma )}\sigma _\varsigma }{\partial \varsigma }=-\rho \,A_{(\varsigma )}\, {{\widetilde{\Omega }}}^{\, 2}({\mathbf{r}}_{\, i} + {\mathbf{u}} )\;\;\; \Rightarrow \;\;\; A\frac{d \left(\mathbf{C}_{\varsigma \varsigma }\; \varepsilon _\varsigma \right)}{d \varsigma }= -\rho A^{\, 2}(\varsigma +{{u}_\varsigma } ) </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau } = 2\boldsymbol{F}\left(\frac{\lambda }{2} \mathrm{ln} J \boldsymbol{C}^{-1} + \frac{\mu }{2} \left(\boldsymbol{I} - \boldsymbol{C}^{-1}\right)\right)\boldsymbol{F}^T  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.26)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.24)
 |}
-Identifying the coefficient <math display="inline">\mathbf{C}_{\varsigma \varsigma }</math> as the Young's Modulus, considered independent on the axial position, and considering constant section <math display="inline">A</math>, the above equation can be rewritten as:
+which, after reminding the definition of the left Cauchy-Green strain tensor <math display="inline">\boldsymbol{b}=\boldsymbol{F}\boldsymbol{F}^T</math> and the identity <math display="inline">\boldsymbol{F}\boldsymbol{C}^{-1}\boldsymbol{F}^T=\boldsymbol{F}(\boldsymbol{F}^{-1}\boldsymbol{F}^{-T})\boldsymbol{F}^T=\boldsymbol{I}</math>, the final expression reads
+<span id="eq-3.25"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 937: / Line 1,718: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>E\,A\frac{d^2 u_\varsigma }{d \varsigma ^2}=-\rho A{\Omega }^{\, 2}(\varsigma +{{u}_\varsigma } )\;\;\; \Rightarrow \;\;\; \frac{d^2 u_\varsigma }{d \varsigma ^2}= -\frac{1}{E}\rho {\Omega }^{\, 2}(\varsigma +{{u}_\varsigma } ) </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau } = \lambda \mathrm{ln} J \boldsymbol{I} + \mu \left(\boldsymbol{b} - \boldsymbol{I}\right)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.27)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.25)
 |}
-Where <math display="inline">\rho </math> stands for the density of the material and <math display="inline">E</math> is the Young's Modulus. The above equation describes the rotation of a 1D structure under constant angular velocity. In the corotational frame, the deformation can be interpreted as the result of applying an axially distributed force, whose modulus is proportional to the radial position (figure [[#img-2.4|2.4]]). Note that the displacement field will not depend on the area of the beam.
+where <math display="inline">\boldsymbol{I}</math> denotes the symmetric second order unit tensor.
-The resulting equation can be identified as a second order ordinary differential equation ('''ODE'''). Thus, two boundary conditions will be required. Supposing a '''fixed-free''' structure, the displacement at the root is to be zero, and the same happens with the stress at the free end, where <math display="inline">\varsigma = R</math> :
+As shown in Chapter [[#4 Irreducible formulation|4]], where the linearisation of the weak form is derived, a rate form of the constitutive equation is needed, and it can be obtained by taking the material derivative of the stress tensor <math display="inline">\boldsymbol{S}</math> <span id='citeF-148'></span>[[#cite-148|[148]]]
+<span id="eq-3.26"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 951: / Line 1,733: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left.\begin{matrix}u \end{matrix}\right|_{\varsigma=0} = 0\;,\;\;\;\;\left.\begin{matrix}\frac{d u_\varsigma }{d \varsigma } \end{matrix}\right|_{\varsigma =R} = 0 </math>
+| style="text-align: center;" | <math>\dot{\boldsymbol{S}}=4\frac{\partial ^2\Psi \left(\boldsymbol{C}\right)}{\partial \boldsymbol{C}\partial \boldsymbol{C}}:\frac{\dot{\boldsymbol{C}}}{2}=\boldsymbol{\mathrm{C}}^{CE}:\frac{\dot{\boldsymbol{C}}}{2}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.28)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.26)
 |}
-To simplify the solution, let's denote <math display="inline">k=\sqrt{\frac{\rho {\Omega }^{\, 2}}{E}}</math>.
+where, with <math display="inline">\boldsymbol{\mathrm{C}}^{CE}</math>, it is denoted the tangent modulus. The push-forward operation of Equation [[#eq-3.26|3.26]], <math display="inline">\boldsymbol{F}\cdot \dot{\boldsymbol{S}}\cdot \boldsymbol{F}^T</math>, yields to the Lie derivative of <math display="inline">\tau </math> in compact form
-<div id='img-2.4'></div>
+<span id="eq-3.27"></span>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-1DBAR.png|390px|1D static problem interpreted as an axially loaded static bar in the rotating frame]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 2.4:''' 1D static problem interpreted as an axially loaded static bar in the rotating frame
+|-
+| style="text-align: center;" | <math>\mathcal{L}_v\boldsymbol{\tau }:=\boldsymbol{F}\cdot \dot{\boldsymbol{S}}\cdot \boldsymbol{F}^T=\boldsymbol{F}\left(4\frac{\partial ^2\Psi \left(\boldsymbol{C}\right)}{\partial \boldsymbol{C}\partial \boldsymbol{C}}:\frac{\dot{\boldsymbol{C}}}{2}\right)\boldsymbol{F}^T=4\boldsymbol{F}\frac{\partial ^2\Psi \left(\boldsymbol{C}\right)}{\partial \boldsymbol{C}\partial \boldsymbol{C}}\boldsymbol{F}^T:\frac{\dot{\boldsymbol{C}}}{2} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.27)
 |}
-The solutions of the ODE are:
+After expressing the rate of <math display="inline">\boldsymbol{C}</math> as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 973: / Line 1,758: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{u_{\varsigma }}_{(\varsigma )} = \frac{\sin (k\varsigma )}{k\cos (kR)}-\varsigma \;,\;\;\;{\sigma _{\varsigma }}_{(\varsigma )} = E\, \left( \frac{\cos (k\varsigma )}{\cos (kR)} -1\right) </math>
+| style="text-align: center;" | <math>\frac{\dot{\boldsymbol{C}}}{2}=\frac{1}{2}\left(\boldsymbol{F}^T\dot{\boldsymbol{F}}+\dot{\boldsymbol{F}^T}\boldsymbol{F}\right)=\boldsymbol{F}^T\frac{1}{2}\left(\dot{\boldsymbol{F}}\boldsymbol{F}^{-1}+(\dot{\boldsymbol{F}}\boldsymbol{F}^{-1})^T \right)\boldsymbol{F}=\boldsymbol{F}^T\frac{1}{2}\boldsymbol{d}\boldsymbol{F} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2.29)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.28)
 |}
-Imagine now an helicopter's rotor, with a constant cross-section and <math display="inline">4\,m</math> radius rotating at <math display="inline">60 \,rad/s</math>.  Let's be idealistic, and suppose that the blade can only deform in the axial direction. Moreover, let's imagine that the conditions are perfectly '''ideal''' and no source of vibration exists. In that case, the previous equations could be used to compute the deformations and stresses across the beam. For an aluminium beam (<math display="inline">E = 70 \,GPa</math> , <math display="inline">\rho = 2700\, kg/m^3</math>), the obtained results are showed in figure [[#img-2.5|2.5]], displaying an overall elongation of the blade of <math display="inline">3\,mm</math>:
+with <math display="inline">\boldsymbol{d}</math> the symmetrical spatial velocity gradient, defined as
-<div id='img-2.5'></div>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-analytic1d2.png|600px|Analytic solution for an axially loaded bar due to centrifugal forces]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 2.5:''' Analytic solution for an axially loaded bar due to centrifugal forces
+|-
+| style="text-align: center;" | <math>\boldsymbol{d} = \frac{1}{2}\left(\boldsymbol{l}+\boldsymbol{l}^T\right) </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.29)
 |}
-For sure, this model cannot be applied to real rotors. However, it is an exact solution to equation [[#eq-2.25|2.25]], and will be used latter on this report to '''validate''' the developed approximated methods.
+and <math display="inline">\boldsymbol{l}=\dot{\boldsymbol{F}}\boldsymbol{F}^{-1}</math>, the Lie derivative can be expressed as
-=3 FEM model=
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\mathcal{L}_v\boldsymbol{\tau }=4\boldsymbol{F}\frac{\partial ^2W\left(\boldsymbol{C}\right)}{\partial \boldsymbol{C}\partial \boldsymbol{C}}\boldsymbol{F}^T:\left(\boldsymbol{F}^T\boldsymbol{d}\boldsymbol{F}\right)=4\left[\left(\boldsymbol{F}\otimes \boldsymbol{F}^T\right):\frac{\partial ^2W\left(\boldsymbol{C}\right)}{\partial \boldsymbol{C}\partial \boldsymbol{C}}:\left(\boldsymbol{F}^T\otimes \boldsymbol{F}\right)\right]:\boldsymbol{d}</math>
+|-
+| style="text-align: center;" | <math> =\mathrm{C}^{\tau }:\boldsymbol{d} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.30)
+|}
-Once the strong form of the elastodynamic equation has been posed, is now time to transform it in terms of the '''finite element method''', in such a way that it can be solved using this computation technique. The aim of the present chapter is to pose the weak form of the equation and give a glimpse on how to solve it. It is taken for granted that the reader has basic notions on the topic of FEM, and thus this chapter will not explain the very basics of the method.
+where <math display="inline">\mathrm{C}^{\tau }</math> denotes the spatial incremental constitutive tensor. In order to define <math display="inline">\mathrm{C}^{\tau }</math>, the second derivative of <math display="inline"> \Psi </math> with respect to <math display="inline">\boldsymbol{C}  </math>, <math display="inline"> \displaystyle \frac{\partial ^2\Psi \left(\boldsymbol{C}\right)}{\partial \boldsymbol{C}\partial \boldsymbol{C}}</math>, has to be computed. By recovering Equation [[#eq-3.23|3.23]], it can be written as
-==3.1 Weak form of the elastodynamic equation==
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\frac{\partial ^2\Psi \left(\boldsymbol{C}\right)}{\partial \boldsymbol{C}\partial \boldsymbol{C}}=\frac{\lambda }{2}\frac{\partial (\mathrm{ln} J\boldsymbol{C}^{-1})}{\partial \boldsymbol{C}}+\frac{\mu }{2}\frac{\partial \left(\boldsymbol{I}-\boldsymbol{C}^{-1}\right)}{\partial \boldsymbol{C}}</math>
+|-
+| style="text-align: center;" | <math> =\frac{\lambda }{2}\left\lbrace \frac{\partial (\mathrm{ln} J)}{\partial J}:\frac{\partial J}{\partial \boldsymbol{C}}\right\rbrace \boldsymbol{C}^{-1} + \frac{\lambda }{2}\mathrm{ln} J\frac{\partial \boldsymbol{C}^{-1}}{\partial \boldsymbol{C}} -\frac{\mu }{2}\frac{\partial \boldsymbol{C}^{-1}}{\partial \boldsymbol{C}} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.31)
+|}
-For the sake of simplicity, let's take advantage of the formulation presented in the previous chapter while developing the strong form of the problem. To start with, the '''test''' and '''trial''' functions will be defined. Those are continuous with square-integrable derivative functions, defined over the problem boundary <math display="inline">\overline{\mho }</math>, with specific boundary values:
+The derivative of <math display="inline">\boldsymbol{C}^{-1}</math> follows from the rule which is valid for any second order tensor <math display="inline">\boldsymbol{A}</math> , which, in Cartesian components, can be expressed as
-<span id="eq-3.1"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,004: / Line 1,810: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{matrix}\mathbf\hbox{Test functions   } \mathcal{}{V} = \left\{v_{i} : \overline{\mho } \rightarrow \mathbb{R}\,,\,\,v_{i}=0 { on }\, \Gamma _{u}^{i}\right\}\\  \mathbf\hbox{Trial functions   }\mathcal{}{S} = \left\{u_{i} : \overline{\mho } \rightarrow \mathbb{R}\,,\,\,u_{i}=\overline{u}_{i} { on }\, \Gamma _{u}^{i}\right\} \end{matrix} </math>
+| style="text-align: center;" | <math>\left(\frac{\partial A^{-1}}{\partial A}\right)_{ijkl}=-\left(A^{-1}\right)_{ik}\left(A^{-1}\right)_{lj} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.1)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.32)
 |}
-Denoting as <math display="inline">f_{ij}</math> a <math display="inline">C_1</math> function, the Gauss' divergence theorem reads as:
+Since <math display="inline">\boldsymbol{C}</math> is a symmetric tensor, only the symmetrical part is needed. This latter is given by the fourth order tensor <math display="inline">\mathrm{I}_{\boldsymbol{C}^{-1}}</math> defined in component form as
-<span id="eq-3.2"></span>
+<span id="eq-3.33"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,017: / Line 1,823: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\int _{{\mho }} \frac{\partial f_{i j}}{\partial x_{j}} d \mho =\int _{\Gamma } f_{i j}\; n_{j} d \Gamma  </math>
+| style="text-align: center;" | <math>\left(\mathrm{I}_{\boldsymbol{C}^{-1}}\right)_{ijkl}=\frac{1}{2}\left(C^{-1}_{ik} C^{-1}_{kl}+C^{-1}_{il} C^{-1}_{jk} \right)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.2)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.33)
 |}
-The weak form is obtained by integrating over the domain the '''product''' of the balance equation (see eq. [[#eq-2.17|2.17]]) in  with the test functions, and by employing the '''Divergence Theorem''' to shift the derivatives on the stresses <math display="inline">\sigma _{ij}</math> to the test functions <math display="inline">v_{i}</math>. For simplicity, let's denote <math display="inline">(f_{i}-\overline{\mu }\; \dot{ \mathfrak{u}}_{i}-\rho \;\ddot{ \mathfrak{u}}_{i})</math> as <math display="inline">\uwave{f_i}</math>.
+According to Equations [[#eq-3.21|3.21]] and [[#eq-3.33|3.33]], it is possible to write again the expression of <math display="inline"> \displaystyle \frac{\partial ^2\Psi \left(\boldsymbol{C}\right)}{\partial \boldsymbol{C}\partial \boldsymbol{C}}</math> as:
-<span id="eq-3.3"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,030: / Line 1,835: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\int _{{\mho }} v_{i}\left(\frac{\partial \sigma _{i j}}{\partial x_{j}}+\uwave{f_i} \right)d \mho \,=\,0 \;\;\Rightarrow \;\; \int _{{\mho }} v_{i} \;\frac{\partial \sigma _{i j}}{\partial x_{j}} d \mho +\int _{{\mho }} v_{i} \;\uwave{f_i} d \mho \,=\,0 </math>
+| style="text-align: center;" | <math>\frac{\partial ^2\Psi \left(\boldsymbol{C}\right)}{\partial \boldsymbol{C}\partial \boldsymbol{C}}=\frac{\lambda }{4}\boldsymbol{C}^{-1}\otimes \boldsymbol{C}^{-1}+\frac{1}{2}\left(\mu - \lambda \mathrm{ln}J\right)\mathrm{I}_{\boldsymbol{C}^{-1}} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.3)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.34)
 |}
-Integrating by parts and applying equation [[#eq-3.2|3.2]] to the first term:
+By performing a push-forward operation of this last expression, which defines the material tangent modulus <math display="inline">\boldsymbol{\mathrm{C}}^{CE}</math> (see Equation [[#eq-3.26|3.26]]), the spatial incremental constitutive tensor is obtained
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,042: / Line 1,847: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\int _{{\mho }} v_{i} \frac{\partial \sigma _{i j}}{\partial x_{j}} d \mho =\int _{\Gamma } v_{i}\left(\sigma _{i j}\; n_{j}\right)d \Gamma{-\int}_{{\mho }} \frac{\partial v_{i}}{\partial x_{j}}\; \sigma _{i j} d \mho  </math>
+| style="text-align: center;" | <math>\mathrm{C}^{\tau }=4\left[\left(\boldsymbol{F}\otimes \boldsymbol{F}^T\right):\frac{\partial ^2\Psi \left(\boldsymbol{C}\right)}{\partial \boldsymbol{C}\partial \boldsymbol{C}}:\left(\boldsymbol{F}^T\otimes \boldsymbol{F}\right)\right]</math>
+|-
+| style="text-align: center;" | <math> =4\left[\left(\boldsymbol{F}\otimes \boldsymbol{F}^T\right):\left(\frac{\lambda }{4}\boldsymbol{C}^{-1}\otimes \boldsymbol{C}^{-1}+\frac{1}{2}\left(\mu - \lambda \mathrm{ln}J\right)\mathrm{I}_{\boldsymbol{C}^{-1}}\right):\left(\boldsymbol{F}^T\otimes \boldsymbol{F}\right)\right]</math>
+|-
+| style="text-align: center;" | <math> =\lambda \left(\boldsymbol{I}\otimes \boldsymbol{I}\right)+2\left(\mu - \lambda \mathrm{ln}J\right)\mathrm{I}_s </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.4)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.35)
 |}
-The first term on the right side of the preceding equation can be expressed as the sum of the integrals over <math display="inline">\Gamma _{u}^{i} { and } \Gamma _{\sigma }^{i}</math>. Taking into account that <math display="inline">v_{i}=0 { on } \Gamma _{u}^{i}</math>,  and <math display="inline">\sigma _{i j} n_{j} = \overline{t}^{\,i}</math> on <math display="inline">\Gamma _{\sigma }^{i}</math> :
+with <math display="inline">\mathrm{I}_s</math> the fourth order symmetric identity tensor.
+The spatial counterparts <math display="inline">\boldsymbol{I}\otimes \boldsymbol{I}</math> and <math display="inline">\mathrm{I}_s</math> of the fourth-order tensors <math display="inline">\boldsymbol{C}^{-1}\otimes \boldsymbol{C}^{-1}</math> and <math display="inline">\mathrm{I}_{\boldsymbol{C}^{-1}}</math>, have been evaluated through the push-forward operation:
+<span id="eq-3.36"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,054: / Line 1,866: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\int _{\Gamma } v_{i}\left(\sigma _{i j}\; n_{j}\right)d \Gamma = \sum _{i=1}^{n_{ d}} \int _{\Gamma _{u}^{i}} v_{i}\left(\sigma _{i j} \;n_{j}\right)d \Gamma{+\sum}_{i=1}^{n_{ d}} \int _{\Gamma _{\sigma }^{i}} v_{i}\left(\sigma _{i j}\; n_{j}\right)d \Gamma = \sum _{i=1}^{n_{d}} \int _{\Gamma _{\sigma }^{i}} v_{i}\; \overline{t}^{\,i} d \Gamma  </math>
+| style="text-align: center;" | <math>\boldsymbol{I}\otimes \boldsymbol{I}=\left[\left(\boldsymbol{F}\otimes \boldsymbol{F}^T\right):\left(\boldsymbol{C}^{-1}\otimes \boldsymbol{C}^{-1}\right):\left(\boldsymbol{F}^T\otimes \boldsymbol{F}\right)\right]  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.5)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.36)
 |}
-Introducing the latter into equation [[#eq-3.3|3.3]]:
+and
+<span id="eq-3.37"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,066: / Line 1,879: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\sum _{i=1}^{n_{d}} \int _{\Gamma _{\sigma }^{i}} v_{i}\; \overline{t}^{\,i} d \Gamma{-\int}_{{\mho }} \frac{\partial v_{i}}{\partial x_{j}}\; \sigma _{i j}\; d \mho +\int _{{\mho }} v_{i} \;\uwave{f_i} \;d \mho =0 </math>
+| style="text-align: center;" | <math>\mathrm{I}_s=\left[\left(\boldsymbol{F}\otimes \boldsymbol{F}^T\right):\left(\mathrm{I}_{\boldsymbol{C}^{-1}}\right):\left(\boldsymbol{F}^T\otimes \boldsymbol{F}\right)\right]  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.6)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.37)
 |}
-As <math display="inline">\sigma _{ij} = \sigma _{ji}</math>, the derivatives of the test function can be expressed in terms of the '''symmetric gradient operator''': <math display="inline"> \frac{\partial v_{i}}{\partial x_{j}} \sigma _{i j} = \left[\nabla ^{s} v\right]_{i j} \sigma _{i j}</math>. It will be convenient to express the problem in '''Voigt's notation''' (see [[#A.1.2 Generalised Hooke's law|A.1.2]]). Thus, the latter expression <math display="inline">\left[\nabla ^{s} v\right]_{i j} \sigma _{i j}</math> transforms into <math display="inline">\left(\boldsymbol{\nabla }^{s} \boldsymbol{v}\right)^{T} \boldsymbol{\sigma }</math>.
+In the case of nearly-incompressible materials, the numerical treatment is not trivial. With regard to the finite element analysis, as it is addressed in Chapter [[#5 Mixed formulation|5]], the use of mixed finite element formulations is required where the mean stress is considered as an additional primary variable, together with the displacements. In the context of the ''specific strain energy function'' and the stress constitutive relation, the isochoric/volumetric split permits to treat in a different way the incompressible part. As follows, the split of the total deformation gradient is exploited
-Taking into account the initial conditions defined in the previous chapter, the '''weak form''' of the elastodynamic problem using Voigt's notation is written as:
-<span id="eq-3.7"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,081: / Line 1,891: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" |
+| style="text-align: center;" | <math>\boldsymbol{F}=\boldsymbol{F}_{vol}\boldsymbol{F}_{dev} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.7)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.38)
 |}
-Given <math display="inline">\mathbf{f}, \;\overline{ \mathfrak{u}}^{\,i}\,,\;\; \overline{\mathbf{t}}^{\,i}\,,\;\;  \mathfrak{u}^{0}</math> and <math display="inline"> \dot{ \mathfrak{u}}^{0} \;</math> , find <math display="inline"> \mathfrak{u}_{(t)} \in \mathcal{}{S}\;,\;\; t  \in [ 0, T]\;</math> such that
+and the expression of the volumetric <math display="inline">\boldsymbol{F}_{vol}</math> and deviatoric <math display="inline">\boldsymbol{F}_{dev}</math> terms respectively read
+<span id="eq-3.39"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,093: / Line 1,904: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\int _{{\mho }} \mathbf{v}^{T} \rho \, \ddot{ \mathfrak{u}}\; d \mho +\int _{{\mho }} \mathbf{v}^{T} \, \overline{\mu }  \,\dot{ \mathfrak{u}} \; d \mho +\int _{{\mho }} \nabla ^{s} \mathbf{v}^{T} \, \overline{D} \, \nabla ^{s} \dot{ \mathfrak{u}} \, d \mho \, +    </math>
+| style="text-align: center;" | <math>\boldsymbol{F}_{vol}\equiv (det \boldsymbol{F})^{1/3} \boldsymbol{I}  </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.39)
 |}
+<span id="eq-3.40"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,102: / Line 1,915: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>+ \int _{{\mho }} \nabla ^{s} \mathbf{v}^{T}  \,\mathbf{C} \, \nabla ^{s}  \mathfrak{u} \, d \mho \;\; = \int _{{\mho }} {\mathbf{v}}^{T} \boldsymbol{f} d \mho +\sum _{i=1}^{n_{ d}} \int _{\Gamma _{\sigma }^{i}} v_{i} \overline{t}^{\,i} d \Gamma \,,\;\forall \mathbf{v} \in \mathcal{}{V}   </math>
+| style="text-align: center;" | <math>\boldsymbol{F}_{dev}\equiv (det \boldsymbol{F})^{-1/3} \boldsymbol{F}  </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.40)
 |}
+Accordingly, it is possible to define the deviatoric left Cauchy-Green strain tensor as
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\boldsymbol{C}_{dev}\equiv (\boldsymbol{F}^T)_{dev}\boldsymbol{F}_{dev}=J^{-\frac{2}{3}}\boldsymbol{C} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.41)
+|}
-<math display="inline"> \mathfrak{u}=\overline{ \mathfrak{u}}^{\,i}\;</math> on <math display="inline">\; \Gamma _{ \mathfrak{u}}^{i}\times [0, T]</math>
+and its first principal invariant
-<math display="inline"> \mathfrak{u}_{(\boldsymbol{x}, 0)}= \mathfrak{u}^{0}_{(\boldsymbol{x})}\,</math> and <math display="inline">\;\dot{ \mathfrak{u}}_{(\boldsymbol{x}, 0)}=\dot{ \mathfrak{u}}^{0}_{(\boldsymbol{x})}\;</math> with <math display="inline">\; \boldsymbol{x} \in \mho  </math>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>I^*_C=\mathrm{tr}\boldsymbol{C}_{dev} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.42)
+|}
-The above expression is known as the variational form of the elastodynamic problem, and can be seen as the '''balance''' of internal and external virtual works, an equilibrium between stresses and inertia, viscous, body and traction forces. The term <math display="inline">  \nabla ^{s} \mathbf{v}^{T}</math> can be thought as virtual strains, while <math display="inline"> \mathbf{v}^{T} </math> as virtual displacements. The major difference with the strong formulation is that the weaker form of the problem demands less '''smoothness''' of the solution, involving only firsts derivatives of the test functions.
+With this last expression at hand, the ''specific strain energy function'' of a Neo-Hookean model is written as
-The formulation of the weak form of the elastodynamic problem is only the first step for the finite element method to be fully developed. The following sections will introduce the required formulation to tackle the problem in terms of the FEM, allowing its programming.
+<span id="eq-3.43"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\tilde{\Psi }(\boldsymbol{C}_{dev})=K\,  U(J) + \frac{\mu }{2}\left(\mathrm{tr}\boldsymbol{C}_{dev} - 3\right)  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.43)
+|}
-==3.2 Global formulation==
+with <math display="inline">K</math> representing the bulk modulus and <math display="inline">U(J)</math> the volumetric part of <math display="inline"> \tilde{\Psi } </math> dependent on <math display="inline">J</math>.
-The key concept of the FEM lies in the domain decomposition into a certain number of '''elements''', finite-sized subdomains. Each element has a given number of ''nodes'', which will depend on the element type. The important fact here is to know that the FEM only computes values of the variable at the nodes. Then, the values of non-nodal points are approximated by '''interpolation''' of the nodal values. The FEM equations are formulated in such a way that continuity of the field variables is ensured. However, interelement continuity of the gradients of the variable does not often exist. The latter is a problem affecting some variables of interest such as strains and stresses, and is solved by increasing the number of elements used.
+It is possible to find the expression of the Kirchhoff stress tensor <math display="inline">\boldsymbol{\tau }</math> by using Equation [[#eq-3.19|3.19]], which in the case of <math display="inline">\tilde{\Psi }</math> is derived:
-Following with the concept of domain decomposition, it is said that the domain <math display="inline">\overline{\mho }</math> is '''discretised''' into element domains <math display="inline">\overline{\mho }\,^e</math>. The geometric shape resulting from the domain division is known as mesh. If the domain can be thought as a polygon, and <math display="inline">n_{el}</math> is the total number of elements in which the domain is split, the decomposition can be written as
+<span id="eq-3.44"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,127: / Line 1,967: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\overline{\mho }=\overline{\mho }^{\,1} \cup \overline{\mho }^{\,2} \cdots \cup \overline{\mho }^{\,n_{e l}}=\bigcup _{e=1}^{n_{e l}} \overline{\mho }^{\,e} </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau }=2\boldsymbol{F}\cdot \frac{\partial \tilde{\Psi }(\boldsymbol{C}_{dev})}{\partial \boldsymbol{C}}\cdot \boldsymbol{F}^T=2\boldsymbol{F}\cdot \left(K\,  U'(J):\frac{\partial J}{\partial \boldsymbol{C}}+\frac{\mu }{2}\frac{\partial \mathrm{tr}\boldsymbol{C}_{dev}}{\partial \boldsymbol{C}}  \right)\cdot \boldsymbol{F}^T</math>
+|-
+| style="text-align: center;" | <math> =2\boldsymbol{F}\cdot \left(K\,  U'(J)\frac{1}{2}J\boldsymbol{C}^{-1}+\frac{\mu }{2}\frac{\partial \mathrm{tr}\boldsymbol{C}_{dev}}{\partial \boldsymbol{C}_{dev}}:\frac{\partial \boldsymbol{C}_{dev}}{\partial \boldsymbol{C}} \right)\cdot \boldsymbol{F}^T</math>
+|-
+| style="text-align: center;" | <math> =2\boldsymbol{F}\cdot \left(K\,  U'(J)\frac{1}{2}J\boldsymbol{C}^{-1}+\frac{\mu }{2}\boldsymbol{I}:\frac{\partial \left(J^{-\frac{2}{3}}  \boldsymbol{C} \right)}{\partial \boldsymbol{C}}\right)\cdot \boldsymbol{F}^T</math>
+|-
+| style="text-align: center;" | <math> =2\boldsymbol{F}\cdot \left(K\,  U'(J)\frac{1}{2}J\boldsymbol{C}^{-1}+\frac{\mu }{2}\boldsymbol{I}:\left(\frac{\partial J^{-\frac{2}{3}}}{\partial J}\frac{\partial J}{\partial \boldsymbol{C}}\otimes \boldsymbol{C} + J^{-\frac{2}{3}}\frac{\partial \boldsymbol{C}}{\partial \boldsymbol{C}}\right)\right)\cdot \boldsymbol{F}^T</math>
+|-
+| style="text-align: center;" | <math> =2\boldsymbol{F}\cdot \left(K\,  U'(J)\frac{1}{2}J\boldsymbol{C}^{-1}+\frac{\mu }{2}\boldsymbol{I}:\left(-\frac{1}{3} J^{-\frac{2}{3}} {\boldsymbol{C}}^{-1}\otimes \boldsymbol{C} + J^{-\frac{2}{3}}\mathrm{I}_s\right)\right)\cdot \boldsymbol{F}^T</math>
+|-
+| style="text-align: center;" | <math> =2\boldsymbol{F}\cdot \left(K\,  U'(J)\frac{1}{2}J\boldsymbol{C}^{-1}+\frac{\mu }{2}J^{-\frac{2}{3}}\boldsymbol{I}:\mathrm{I}_d\right)\cdot \boldsymbol{F}^T</math>
+|-
+| style="text-align: center;" | <math> =K\,  JU'(J)\boldsymbol{I} + \mu J^{-\frac{2}{3}}\boldsymbol{b}:\mathrm{I}_d=J\, p\boldsymbol{I} + \mu \mathrm{dev}(\boldsymbol{b}_{dev}) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.8)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.44)
 |}
-If <math display="inline">n_{pt}</math> is the total number of nodes of the domain, then the FEM has to compute the field variable (in this case, displacement) in each of those nodes. For <math display="inline">n_d</math> dimensions, each nodal displacement will be a <math display="inline">n_d</math>-sized vector. We can define a global vector of displacements '''d'''<math display="inline">\,\in \, \mathbb{R}^{n_{ d} n_{p t}}</math> and variations '''c'''<math display="inline">\,\in \, \mathbb{R}^{n_{ d} n_{p t}}</math> :
+where <math display="inline"> \mathrm{I}_d =\mathrm{I}_s - \frac{1}{3}\boldsymbol{I}\otimes \boldsymbol{I}</math> is the fourth-order deviatoric projector tensor, <math display="inline">\boldsymbol{b}_{dev}=J^{-\frac{2}{3}}\boldsymbol{b}</math> the volume-preserving part of <math display="inline">\boldsymbol{b}</math> and the quantity <math display="inline">p = K U'(J)</math>, the only contribution to the volumetric part of <math display="inline">\boldsymbol{\tau }</math>, which in the case of a nearly-incompressible material is represented by the mean stress primary variable, as addressed in Chapter [[#5 Mixed formulation|5]].
+As it was previously done, in order to find the expression of the spatial incremental constitutive tensor <math display="inline">\mathrm{C}^{\tau }</math>, given by the sum of the volumetric and deviatoric contribution
+<span id="eq-3.45"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,139: / Line 1,994: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{d} :=\left[\begin{array}{c}{\mathbf{d}_{1}} \\ {\mathbf{d}_{2}} \\ {\vdots } \\ {\mathbf{d}_{\,n_{p t}}}\end{array}\right], \quad \mathbf{c} :=\left[\begin{array}{c}{\mathbf{c}_{1}} \\ {\mathbf{c}_{2}} \\ {\vdots } \\ {\mathbf{c}_{\,n_{p t}}}\end{array}\right] </math>
+| style="text-align: center;" | <math>\mathrm{C}^{\tau } =\mathrm{C}^{\tau }_{vol}+\mathrm{C}^{\tau }_{dev},  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.9)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.45)
 |}
-It is important to notice that not all the components of the previous vectors are unknown: <math display="inline">\mathbf{r} \subset \left\{1,2 \ldots n_{d} n_{p t}\right\}</math> is defined as the set of '''constrained''' degrees of freedom (DOF), along which displacement is known. Thus, <math display="inline">\mathbf{d}_{{r}}=\overline{ \mathfrak{u}}</math> and <math display="inline">\mathbf{c}_{{r}}=0</math>. On the other hand, <math display="inline">\mathbf{l}</math> is known as the set of '''unconstrained''' degrees of freedom, the global DOFs along which displacement is unknown.
+the second derivative of <math display="inline">\tilde{\Psi }</math>, <math display="inline"> \displaystyle \frac{\partial ^2\tilde{\Psi }(\boldsymbol{C}_{dev})}{\partial \boldsymbol{C}\partial \boldsymbol{C}}</math>
-As it has been stated in the first paragraph of this section, the solution along non-nodal points is interpolated using the nodal values. The interpolation functions are known as '''shape functions''', and in the '''Galerkin''' method the same functions are used to interpolate both shape and trial functions. Let's define the continuous and with piecewise continuous derivative scalar shape function <math display="inline">N_{A} : \overline{\mho } \rightarrow \mathbb{R}</math>, where <math display="inline">1\leq A\leq n_{pt}</math>.
+<span id="eq-3.46"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\frac{\partial ^2\tilde{\Psi }(\boldsymbol{C}_{dev})}{\partial \boldsymbol{C}\partial \boldsymbol{C}}=\frac{\partial \left(\frac{1}{2}J\, p\boldsymbol{C}^{-1}\right)}{\partial \boldsymbol{C}}+\frac{\partial \left(\frac{\mu }{2}J^{-\frac{2}{3}}\boldsymbol{I}:\mathrm{I}_d\right)}{\partial \boldsymbol{C}}  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.46)
+|}
-In the FEM, nodal shape functions <math display="inline">N_A</math> are defined in such a way that their value is <math display="inline">\mathbf{1}</math> at node <math display="inline">A</math>, and <math display="inline">\mathbf{0}</math> at every other node. In problems tackling with vector fields, a matrix of nodal shape functions is defined. Using the same function to interpolate displacements in all dimensions, the nodal matrix of shape functions for a three-dimensional case is:
+needs to be computed. The first addend of Equation [[#eq-3.46|3.46]] refers to the volumetric component and the second addend to the deviatoric component of the second derivative of <math display="inline">\tilde{\Psi }</math>. With regards to the first term, by performing the derivative with respect to <math display="inline">\boldsymbol{C}</math> leads to
+<span id="eq-3.47"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,155: / Line 2,020: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{N}_{A} :=N_{A}\, \mathbf{I}=N_{A} \left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right]\, , \;\;\mathbf{N}_{A} \in \mathbb{R}^{n_{d} \times n_{d}} </math>
+| style="text-align: center;" | <math>\frac{\partial \left(\frac{1}{2}J\, p\boldsymbol{C}^{-1}\right)}{\partial \boldsymbol{C}}=\frac{1}{2}p\left(\frac{\partial J}{\partial \boldsymbol{C}}\boldsymbol{C}^{-1}+J\frac{\partial \boldsymbol{C}^{-1}}{\partial \boldsymbol{C}}\right)=\frac{1}{2}J\, p\left(\frac{1}{2}\boldsymbol{C}^{-1}\otimes \boldsymbol{C}^{-1} + \frac{\partial \boldsymbol{C}^{-1}}{\partial \boldsymbol{C}}\right)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.10)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.47)
 |}
-A global matrix of shape functions '''N''' <math display="inline">\rightarrow \mathbb{R}^{n_{d} \times n_{d}\,n_{pt}}</math> can be defined, involving all the nodal matrices: <math display="inline">\mathbf{N} :=\left[\begin{array}{llll}{\mathbf{N}_{1}} & {\mathbf{N}_{2}} & {\cdots } & {\mathbf{N}_{n_{p t}}}\end{array}\right]</math>. In the same line, a global matrix <math display="inline">\mathbf{B}</math> involving the gradient of the shape functions is defined: <math display="inline">\mathbf{B} = \boldsymbol{\nabla }^{s}\, \mathbf{N}</math>.
+With respect to the deviatoric term of Equation [[#eq-3.46|3.46]], it is possible to derive it, by firstly rewriting it as
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\frac{\partial \left(\frac{\mu }{2}J^{-\frac{2}{3}}\boldsymbol{I}:\mathrm{I}_d\right)}{\partial \boldsymbol{C}}=\frac{\partial \left(\frac{\mu }{2}J^{-\frac{2}{3}}\boldsymbol{I}:\left(-\frac{1}{3}  {\boldsymbol{C}}^{-1}\otimes \boldsymbol{C} + \mathrm{I}_s\right)\right)}{\partial \boldsymbol{C}}</math>
+|-
+| style="text-align: center;" | <math> =\frac{\mu }{2}\frac{\partial \left(J^{-\frac{2}{3}}\left(-\frac{1}{3}\mathrm{tr}\boldsymbol{C}{\boldsymbol{C}}^{-1}+\boldsymbol{I}\right)\right)}{\partial \boldsymbol{C}} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.48)
+|}
-The next step is to construct finite-dimensional '''approximations''' of the test and trial functions, denoted as <math display="inline">\mathcal{}{V}^h</math> and <math display="inline">\mathcal{}{S}^h</math>, respectively. If one thinks of these approximations as subsets of <math display="inline">\mathcal{}{V}</math> and <math display="inline">\mathcal{}{S}</math>, and if <math display="inline">u^{h} \subset \mathcal{}{S}^{h}</math> and <math display="inline">v^{h} \subset \mathcal{}{V}^{h}</math>, then the approximated functions will meet the boundary conditions defined in [[#eq-3.1|3.1]]. The '''finite''' dimensional space of test and trial functions is defined as the space including all the functions of the form
+and, then, by deriving with respect to <math display="inline">\boldsymbol{C}</math>
-<span id="eq-3.11"></span>
+<span id="eq-3.49"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,170: / Line 2,047: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{matrix}\mathcal{}{V}^h \rightarrow \mathbf{v}^{h}_{\,(\boldsymbol{x})}=\mathbf{N}_{\,(\boldsymbol{x})} \,\mathbf{c}, { with } \mathbf{c}_{\mathrm{r}}=0 \\ \\  \mathcal{}{S}^h \rightarrow  \mathfrak{u}^{h}_{\,(\boldsymbol{x},\, \huge{t})}=\mathbf{N}_{\,(\boldsymbol{x})} \,\mathbf{d}_{(t)}, { with } {\mathbf{d}_{\mathrm{r}}}_{(t)}=\overline{ \mathfrak{u}}_{(t)}  \end{matrix} </math>
+| style="text-align: center;" | <math>\frac{\mu }{2}\frac{\partial \left(J^{-\frac{2}{3}}\left(-\frac{1}{3}\mathrm{tr}\boldsymbol{C}{\boldsymbol{C}}^{-1}+\boldsymbol{I}\right)\right)}{\partial \boldsymbol{C}}=</math>
+|-
+| style="text-align: center;" | <math> =\frac{\mu }{2}\left(\frac{\partial J^{-\frac{2}{3}}}{\partial \boldsymbol{C}}\left(\boldsymbol{I}-\frac{1}{3}\mathrm{tr}\boldsymbol{C}\boldsymbol{C}^{-1}\right)+J^{-\frac{2}{3}}\frac{\partial \left(-\frac{1}{3} \mathrm{tr}\boldsymbol{C}\boldsymbol{C}^{-1} \right)}{\partial \boldsymbol{C}}\right)</math>
+|-
+| style="text-align: center;" | <math> =\frac{\mu }{2}J^{-\frac{2}{3}}\left(-\frac{1}{3}\boldsymbol{C}^{-1}\otimes \boldsymbol{I}+\frac{\mathrm{tr}\boldsymbol{C}}{9}\boldsymbol{C}^{-1}\otimes \boldsymbol{C}^{-1}-\frac{1}{3}\boldsymbol{I}\otimes \boldsymbol{C}^{-1}-\frac{\mathrm{tr}\boldsymbol{C}}{3}\frac{\partial \boldsymbol{C}^{-1}}{\partial \boldsymbol{C}}   \right) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.11)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.49)
 |}
-In a similar way, velocity <math display="inline">\dot{ \mathfrak{u}}</math> and acceleration <math display="inline">\ddot{ \mathfrak{u}}</math> are approximated as <math display="inline">\dot{ \mathfrak{u}}^h = \mathbf{N}_{\,(\boldsymbol{x})} \,\dot{\mathbf{d}}_{(t)}</math> and <math display="inline">\ddot{ \mathfrak{u}}^h = \mathbf{N}_{\,(\boldsymbol{x})} \,\ddot{\mathbf{d}}_{(t)}</math>, respectively.  With all the previous definitions being formulated, the approximated weak form of the problem can be posed:
+By substituting the results of Equations [[#eq-3.47|3.47]] and [[#eq-3.49|3.49]] in the definition of <math display="inline">\mathrm{C}^{\tau }</math>
-<span id="eq-3.12"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,183: / Line 2,063: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" |
+| style="text-align: center;" | <math>\mathrm{C}^{\tau }=4\left[\left(\boldsymbol{F}\otimes \boldsymbol{F}^T\right):\frac{\partial ^2\Psi \left(\boldsymbol{C}\right)}{\partial \boldsymbol{C}\partial \boldsymbol{C}}:\left(\boldsymbol{F}^T\otimes \boldsymbol{F}\right)\right] </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.12)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.50)
 |}
-Given <math display="inline">\mathbf{f}, \;\overline{ \mathfrak{u}}^{\,i}\,,\;\; \overline{\mathbf{t}}^{\,i}\,,\;\;  \mathfrak{u}^{0}</math> and <math display="inline"> \dot{ \mathfrak{u}}^{0} \;</math> , find <math display="inline"> \mathfrak{u}^h_{(t)} \in \mathcal{}{S}^h\;,\;\; t  \in [ 0, T]\;</math> such that
+the final expressions of the volumetric and deviatoric spatial incremental constitutive tensors are obtained
+<span id="eq-3.51"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,195: / Line 2,076: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\int _{{\mho }} {\mathbf{v}\,^h}^{T} \rho \, \ddot{ \mathfrak{u}}^h d \mho +\int _{{\mho }}  {\mathbf{v}\,^h}^{T} \, \overline{\mu }  \,\dot{ \mathfrak{u}}^h \, d \mho +\int _{{\mho }} \nabla ^{s}  {\mathbf{v}\,^h}^{T} \, \overline{D} \, \nabla ^{s} \dot{ \mathfrak{u}}^h \, d \mho \, +    </math>
+| style="text-align: center;" | <math>\mathrm{C}^{\tau }_{vol}=p  J(\boldsymbol{u})\left(\boldsymbol{I}\otimes \boldsymbol{I} -2\mathrm{I}^{sym}\right)  </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.51)
 |}
+<span id="eq-3.52"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,204: / Line 2,087: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>+ \int _{{\mho }} \nabla ^{s} {\mathbf{v}\,^h}^{T} \,\mathbf{C} \, \nabla ^{s}  \mathfrak{u}^h \, d \mho \;\; = \int _{{\mho }} {\mathbf{v}\,^{h}}^{T} \boldsymbol{f} d \mho +\sum _{i=1}^{n_{ d}} \int _{\Gamma _{\sigma }^{i}} v_{i}^h \overline{t}^{\,i} d \Gamma \,,\;\forall \mathbf{v} \in \mathcal{}{V}   </math>
+| style="text-align: center;" | <math>\mathrm{C}^{\tau }_{dev}= \frac{2}{3}G\mathrm{tr}\left(\bar{\boldsymbol{b}}\right)\mathrm{I}_d-\frac{2}{3}\left(\boldsymbol{\tau }_{dev}\otimes \boldsymbol{I}+\boldsymbol{I}\otimes \boldsymbol{\tau }_{dev}\right)  </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.52)
 |}
+For the sake of clarity, Equations [[#eq-3.36|3.36]] and [[#eq-3.37|3.37]] are used in the push forward operation.
+==3.2 <span id='lb-3.2'></span>Hyperelastic - J₂ plastic law==
-<math display="inline"> \mathfrak{u}^h=\overline{ \mathfrak{u}}^{\,i}\;</math> on <math display="inline">\; \Gamma _{ \mathfrak{u}}^{i}\times [0, T]</math>
+In this section a hyperelastic - <math display="inline">J_2</math> metal plastic law in finite strains regime is presented. In this context, the main hypothesis, on which the elastoplastic constitutive framework is based, is represented by the notion of the multiplicative decomposition of the total deformation gradient <math display="inline">\boldsymbol{F}</math> in an elastic and plastic component of the form
-<math display="inline"> \mathfrak{u}^h_{(\boldsymbol{x}, 0)}= \mathfrak{u}^{0}_{(\boldsymbol{x})}\,</math> and <math display="inline">\;\dot{ \mathfrak{u}}^h_{(\boldsymbol{x}, 0)}={\dot{ \mathfrak{u}}^{0}}_{(\boldsymbol{x})}\;</math> with <math display="inline">\; \boldsymbol{x} \in \mho  </math>
+<span id="eq-3.53"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\boldsymbol{F} = \boldsymbol{F}^e\boldsymbol{F}^p  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.53)
+|}
-===3.2.1 Approximated rotating elastodynamic model===
+This theory, introduced for the first time in <span id='citeF-149'></span><span id='citeF-150'></span>[[#cite-149|[149,150]]], lies on the concept of a local intermediate stress-free configuration defined by the plastic total deformation gradient <math display="inline">\boldsymbol{F}^P</math>, which can be recovered by performing a purely elastic loading from the fully deformed configuration.  As pointed out in <span id='citeF-151'></span>[[#cite-151|[151]]], by formulating the <math display="inline">J_2</math> plastic flow theory based on the concept of elastic-plastic multiplicative decomposition of <math display="inline">\boldsymbol{F}</math>, some features can be observed. The stress-strain relation derives from the ''specific strain energy function'', decoupled into its volumetric and deviatoric parts and the integration algorithm reduces to the radial return mapping in which the elastic predictor is computed through the elastic stress-strain relation. In addition to that, as in the infinitesimal theory, it is possible to linearise the algorithm which allows to define the algorithmic tangent elastoplastic moduli in a closed-form. In the case that the plastic flow does not take place, the solution of finite elasticity is recovered and the procedure presented in Section [[#3.1 Hyperelastic law|3.1]] is valid. In this section, the equations which are used to derive the algorithmic procedure are presented under the hypothesis of isotropic stress response and isochoric plastic flow, i.e., <math display="inline">\mathrm{det}\boldsymbol{F}^P=\mathrm{det}\boldsymbol{C}^P=1</math>.
-Until now, rotation has not been taken into account when modelling the weak form of the elastodynamic problem. Introducing the '''kinematic equations''' obtained in section [[#2.3 Kinematic equations|2.3]], substituting the approximations presented in equation [[#eq-3.11|3.11]] and using the algebraic property <math display="inline">(\mathbf{A B})^{\mathrm{T}}=\mathbf{B}^{\mathrm{T}} \mathbf{A}^{\mathrm{T}}</math>, the above expression can be written in the noninertial frame as
+The first equation to be introduced is the ''specific strain energy function'', from which it is possible to derive the expression of the stress tensor. As previously discussed in Section [[#3.1 Hyperelastic law|3.1]], according to the axioms of ''thermodynamic determinism'', ''material objectivity'', ''material symmetry'' and the aforementioned notion of local stress-free configuration, the following ''specific strain energy function'' <math display="inline">\Psi </math> with uncoupled volumetric (<math display="inline"> U(J^e)) </math> and deviatoric (<math display="inline"> \tilde{\Psi }(\bar{\boldsymbol{C}^e})</math>) part is considered
+<span id="eq-3.54"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,223: / Line 2,119: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{c}^T  \left\{\;\overset{\mathbf{M}}{\overbrace{\left(\int _{{\mho }}^{ } \mathbf{N}^T \rho \;\mathbf{N} \; d\mho   \right)}}\; \ddot{\mathbf{d}} \;  + \; \overset{\mathbf{D}}{\overbrace{\left(\underset{\mathbf{D}_{\,static}}{\underbrace{\int _{{\mho }}^{ } \mathbf{N}^T \overline{\mu }\;\mathbf{N}\;d\mho  \;  + \; \int _{{\mho }}^{ } \mathbf{B}^T \overline{\mathbf{D}}\;\mathbf{B}\;d\mho }} \; + \; \underset{\mathbf{D}_{\,rot}}{\underbrace{2\int _{{\mho }}^{ } \mathbf{N}^T \rho \;\widetilde{\Omega }\;\mathbf{N}\;d\mho }}   \right)}}\; \dot{\mathbf{d}} \;  + \right.  </math>
+| style="text-align: center;" | <math>\Psi = U(J^e)+\tilde{\Psi }(\bar{\boldsymbol{C}^e})=U(J^e)+\frac{1}{2}\mu (\mathrm{tr}(\bar{\boldsymbol{C}^e})-3)  </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.54)
 |}
+with <math display="inline"> \bar{\boldsymbol{C}^e}\equiv J^{-\frac{2}{3}}\boldsymbol{C}^e=J^{-\frac{2}{3}}{\boldsymbol{F}^e}^T\boldsymbol{F}^e</math> being the volume preserving part of the elastic right Cauchy-Green strain tensor. According to Equation [[#eq-3.44|3.44]], the elastic Kirchhoff stress tensor is derived as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,232: / Line 2,131: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>+ \;  \overset{\mathbf{K}}{\overbrace{\left(\; \underset{\mathbf{K}_{\,static}}{\underbrace{ \int _{{\mho }}^{ } \mathbf{B}^T \mathbf{C}\;\mathbf{B} \; d\mho }} \;+ \; \underset{\mathbf{K}_{\,rot}}{\underbrace{\int _{{\mho }}^{ } \mathbf{N}^T \rho \;(\widetilde{\Omega }^2 + \widetilde{\alpha })\;\mathbf{N} \; d\mho  }} \right)}} \; {\mathbf{d}} \;- </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau }=J^e U'(J^e)\boldsymbol{I} + \mu \mathrm{dev}\left(\bar{\boldsymbol{b}^e} \right)=J^e\, p\boldsymbol{I} + \mathrm{dev}\boldsymbol{\tau } </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.55)
 |}
-<span id="eq-3.13"></span>
+where <math display="inline"> \bar{\boldsymbol{b}^e}\equiv J^{-\frac{2}{3}}\boldsymbol{F}^e{\boldsymbol{F}^e}^T</math> is the volume preserving part of <math display="inline">\boldsymbol{b}^e</math>.
+Once defined the stored energy function of Equation [[#eq-3.54|3.54]], it is necessary to introduce the yield condition, which is characteristic of the <math display="inline">J_2</math> plastic theory; the classical Mises-Huber yield condition in terms of <math display="inline"> \boldsymbol{\tau } </math>, graphically represented in Figure [[#img-3.1|3.1]], can be expressed as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,242: / Line 2,145: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left.\; - \;\overset{\mathbf{F}}{\overbrace{\left(\;\underset{\mathbf{F}_{\,static}}{\underbrace{\int _{{\mho }} {\mathbf{N}}^{T} \boldsymbol{f} d \mho +\sum _{i=1}^{n_{ d}} \int _{\Gamma _{\sigma }^{i}} \overline{\mathbf{{N}}}_{i}^T \;\overline{t}^{\,i} d \Gamma }} \;-\; \underset{\mathbf{F}_{\,rot}}{\underbrace{\int _{{\mho }}^{ } \mathbf{N}^T \rho \;(\widetilde{\Omega }^2 + \widetilde{\alpha })\;\mathbf{r}_i \; d\mho }}  \right)}}\right\} = 0  </math>
+| style="text-align: center;" | <math>f(\boldsymbol{\tau },\alpha )={\lVert \boldsymbol{\tau }_{dev}\rVert }-\sqrt{\frac{2}{3}}\left(\sigma _Y+H\alpha \right)\leq 0 , </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.13)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.56)
 |}
-In compact form:
+with <math display="inline"> \sigma _Y </math> being the flow stress, <math display="inline"> H>0 </math> the isotropic hardening and <math display="inline"> \alpha </math> the hardening parameter. The last governing equation, which makes the plastic problem determined, is given by the fundamental form of the corresponding associative flow rule, which can be derived uniquely by satisfying the ''principle of the maximum dissipation'' <span id='citeF-152'></span>[[#cite-152|[152]]]. As shown in <span id='citeF-153'></span><span id='citeF-154'></span>[[#cite-153|[153,154]]], the associative flow rule in strain space reads
+<span id="eq-3.57"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,254: / Line 2,159: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{c}^T\left[\mathbf{M}\,\ddot{\mathbf{{d}}}\;+\;\mathbf{D}\,\dot{\mathbf{{d}}}\;+\;\mathbf{K}\,{\mathbf{{d}}}  - \mathbf{F} \right]= 0 </math>
+| style="text-align: center;" | <math>\left\{    \begin{array}{rcll}\dfrac{\partial }{\partial t}\left(\overline{\boldsymbol{C}}^p\right)^{-1} &= & -\frac{2}{3}\gamma \,  { {\mathrm{tr}}}(\boldsymbol{b}^e)\boldsymbol{F}^{-1}\boldsymbol{n}\boldsymbol{F}^{-T}  \\      \dot{\alpha }&= &\sqrt{\frac{2}{3}}\gamma     \end{array}    \right.  </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.57)
 |}
-<span id="eq-3.14"></span>
+<div id='img-3.1a'></div>
+<div id='img-3.1b'></div>
+<div id='img-3.1'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-J2_model-0_4cm-1cm-14_8cm-0_5cm.png|320px|]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-J2_model-20cm-1cm-0cm-0_5cm.png|260px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 3.1:''' <math>J_2</math> model in the principal stress space (a) and in <math>\pi </math> plane (b)
+|}
+where <math display="inline"> \overline{\boldsymbol{C}}^p </math> is the volume preserving part of the plastic right Cauchy-Green deformation tensor <math display="inline"> \overline{\boldsymbol{C}}^p = {\overline{\boldsymbol{F}}^p}^T\overline{\boldsymbol{F}}^p </math>, <math display="inline"> \gamma </math> is the plastic multiplier, <math display="inline"> \boldsymbol{n} :=  \dfrac{\mathrm{dev}\boldsymbol{\tau }}{\Vert \mathrm{dev}\boldsymbol{\tau }\Vert }</math> is the unit vector of <math display="inline"> {{\mathrm{dev}}}(\boldsymbol{\tau }) </math>. In order to complete the formulation of the model, it is assumed that the parameter <math display="inline"> \alpha </math>, as in the linear theory, is governed by a rate equation <span id='citeF-151'></span>[[#cite-151|[151]]], where <math display="inline"> \gamma </math> is subjected to the standard Kuhn-Tucker loading/unloading condition:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,264: / Line 2,184: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{c}^T\left[\mathbf{M}\,\ddot{\mathbf{{d}}}\;+\;(\mathbf{D}_{\;static} + \mathbf{D}_{\,rot})\,\dot{\mathbf{{d}}}\;+\;(\mathbf{K}_{\,static} + \mathbf{K}_{\,rot})\,{\mathbf{{d}}}  - (\mathbf{F}_{\,static} - \mathbf{F}_{\,rot}) \right]= 0 </math>
+| style="text-align: center;" | <math>\gamma \geq 0 \quad f(\boldsymbol{\tau },\alpha )\leq 0 \quad \gamma f(\boldsymbol{\tau },\alpha )=0 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.14)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.58)
 |}
-Where the subscript ''"static"'' refers to those matrices that are general in the FEM when applied to static structures, whereas those matrices with the subscript ''"rot"'' are particular of this specific case of rotating structure.  Recall that in the above equation, '''d''' stands for the approximate elastic displacement in the rotative frame — <math display="inline">\,\mathbf{u}^h_{(\,\boldsymbol{x},t)} = \mathbf{N}_{(x)}\;\mathbf{d}_{(t)}\,</math>.
+and consistency condition
-The obtained matrices are very common in physical systems. <math display="inline">\mathbf{M}</math> is known as mass matrix, <math display="inline">\mathbf{D}</math> as damping matrix, <math display="inline">\mathbf{K}</math> as stiffness matrix and <math display="inline">\mathbf{F}</math> as force vector. Quite often, <math display="inline">\overline{\mathbf{D}}</math> and <math display="inline">\overline{\boldsymbol{\mu }}</math> are not known. A common approach is to suppose that the system is governed by the '''Rayleigh damping''', which assumes <math display="inline">\overline{\boldsymbol{\mu }} = \overline{\alpha }\rho </math> and <math display="inline">\overline{\mathbf{D}} = \overline{\beta }\mathbf{C}</math>, where <math display="inline">\overline{\alpha }</math> and <math display="inline">\overline{\beta }</math> are assumed to be constant parameters. Thus, the static damping matrix in the noninertial frame can be expressed as:
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\gamma \dot{f}(\boldsymbol{\tau },\alpha )=0 </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.59)
+|}
+The algorithmic procedure to be established in order to solve the plastic problem has to respect the ''material frame indifference'' principle. This can be accomplished by defining the discrete form of the evolution equation (see Equation [[#eq-3.57|3.57]]) in material description.  Accordingly, a time stepping algorithm is conducted by applying a backward Euler difference scheme on Equation [[#eq-3.57|3.57]]:
-<span id="eq-3.15"></span>
+<span id="eq-3.60"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,279: / Line 2,209: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{D}\;_{static}= \overline{\alpha } \,\mathbf{M} + \overline{\beta }\, \mathbf{K}\;_{static} </math>
+| style="text-align: center;" | <math>\left\{    \begin{array}{rcll}(\overline{\boldsymbol{C}}^p_{n+1})^{-1}-(\overline{\boldsymbol{C}}^p_{n})^{-1}&= &-\frac{2}{3}\Delta \gamma \left(\left(\boldsymbol{C}^p_{n+1}\right)^{-1}:\boldsymbol{C}_{n+1}\right)\boldsymbol{F}^{-1}_{n+1}\boldsymbol{n}_{n+1}\boldsymbol{F}^{-T}_{n+1},\\ \alpha _{n+1}-\alpha _n &= &\sqrt{\frac{2}{3}}\Delta \gamma  \end{array}    \right.     </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.15)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.60)
 |}
-It is important to notice that, in the case where the angular velocity <math display="inline">\Omega </math> is '''constant''', the stiffness and damping matrices and the force vector due to rotation can be written as:
+and by operating a push-forward transformation it is possible to recover the spatial form of the discrete spatial evolution equation through some useful and fundamental steps, such as
+<span id="eq-3.61"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,291: / Line 2,222: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}_{\,rot} = \int _{{\mho }}^{ } \mathbf{N}^T \rho \;\widetilde{\Omega }^2\;\mathbf{N} \; d\mho  = -{\Omega }^2 \int _{{\mho }}^{ } \mathbf{N}^T \rho \; \begin{bmatrix}1 & 0 & 0 \\  0 & 1 & 0 \\  0 & 0 & 0  \end{bmatrix} \;\mathbf{N} \; d\mho  </math>
+| style="text-align: center;" | <math>\overline{\boldsymbol{F}}_{n+1}\left(\overline{\boldsymbol{C}}^p_{n+1}\right)^{-1}\overline{\boldsymbol{F}}^T_{n+1}=\overline{\boldsymbol{F}}_{n+1}\left((\overline{\boldsymbol{F}}^p_{n+1})^{T}\overline{\boldsymbol{F}}^p_{n+1}\right)^{-1}\overline{\boldsymbol{F}}^T_{n+1}</math>
+|-
+| style="text-align: center;" | <math> =\overline{\boldsymbol{F}}_{n+1}\left(\overline{\boldsymbol{F}}^p_{n+1}\right)^{-1}\left(\overline{\boldsymbol{F}}^p_{n+1}\right)^{-1}\overline{\boldsymbol{F}}^T_{n+1}=\overline{\boldsymbol{b}}^e_{n+1} </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.61)
 |}
+<span id="eq-3.62"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,300: / Line 2,235: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{D}_{\,rot} = 2\int _{{\mho }}^{ } \mathbf{N}^T \rho \;\widetilde{\Omega }\;\mathbf{N} \; d\mho  = 2 {\Omega } \int _{{\mho }}^{ } \mathbf{N}^T \rho \; \begin{bmatrix}0 & -1 & 0 \\  1 & 0 & 0 \\  0 & 0 & 0  \end{bmatrix} \;\mathbf{N} \; d\mho  </math>
+| style="text-align: center;" | <math>\overline{\boldsymbol{F}}_{n+1}\left(\overline{\boldsymbol{C}}^p_{n}\right)^{-1}\overline{\boldsymbol{F}}^T_{n+1}=\overline{\boldsymbol{f}}_{n+1}\left(\overline{\boldsymbol{F}}_n(\overline{\boldsymbol{C}}^p_{n})^{-1}\overline{\boldsymbol{F}}^T_{n}\right)\overline{\boldsymbol{f}}^T_{n+1}=\overline{\boldsymbol{f}}_{n+1}\overline{\boldsymbol{b}}^e_{n}\overline{\boldsymbol{f}}^T_{n+1} </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.62)
 |}
+<span id="eq-3.63"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,309: / Line 2,246: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}_{\,rot} = \int _{{\mho }}^{ } \mathbf{N}^T \rho \;(\widetilde{\Omega }^2 + \widetilde{\alpha })\;\mathbf{r}_i \; d\mho  = -{\Omega }^2 \int _{{\mho }}^{ } \mathbf{N}^T \rho \; \begin{bmatrix}1 & 0 & 0 \\  0 & 1 & 0 \\  0 & 0 & 0  \end{bmatrix} \;\mathbf{r}_i \; d\mho  </math>
+| style="text-align: center;" | <math>\boldsymbol{C}_{n+1}:\left(\boldsymbol{C}^p_{n+1}\right)^{-1}=\boldsymbol{I}:\boldsymbol{F}_{n+1}\left(\boldsymbol{C}^p_{n+1}\right)^{-1}\boldsymbol{F}_{n+1}^T </math>
+|-
+| style="text-align: center;" | <math> =\boldsymbol{I}:\boldsymbol{F}_{n+1}\left(\left(\boldsymbol{F}^p_{n+1}\right)^T\boldsymbol{F}^p_{n+1}\right)^{-1}\boldsymbol{F}_{n+1}^T </math>
+|-
+| style="text-align: center;" | <math> =\boldsymbol{I}:\boldsymbol{F}_{n+1}\left(\boldsymbol{F}^p_{n+1}\right)^{-1}\left(\boldsymbol{F}^p_{n+1}\right)^{-T}\boldsymbol{F}_{n+1}^T </math>
+|-
+| style="text-align: center;" | <math> = \boldsymbol{I}:\boldsymbol{b}^e_{n+1}=\mathrm{tr}\boldsymbol{b}^e_{n+1} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.16)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.63)
 |}
-In this case, all the above expressions are constant in time. The key fact here is that, for the two-dimensional case, <math display="inline">\mathbf{K}_{\,rot}</math> is proportional to the mass matrix by a factor of <math display="inline">-\Omega ^2</math>. Then:
+where in Equation [[#eq-3.62|3.62]], the relation <math display="inline">\overline{\boldsymbol{F}}_{n+1}=\overline{\boldsymbol{f}}_{n+1}\overline{\boldsymbol{F}}_n</math> and the results of Equation [[#eq-3.61|3.61]] are used. By the use of the expressions earlier derived, the spatial form of Equation [[#eq-3.60|3.60]] is
-<span id="eq-3.17"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,322: / Line 2,264: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K} = \mathbf{K}_{\,static} - \Omega ^2 \mathbf{M} </math>
+| style="text-align: center;" | <math>\left\{    \begin{array}{rcll}\overline{\boldsymbol{b}}^e_{n+1}&= &\overline{\boldsymbol{f}}_{n+1}\overline{\boldsymbol{b}}^e_{n}\overline{\boldsymbol{f}}^T_{n+1}-\frac{2}{3}\Delta \gamma \mathrm{tr}\overline{\boldsymbol{b}}^e_{n+1}\boldsymbol{n}_{n+1},\\ \alpha _{n+1} &= & \alpha _n + \sqrt{\frac{2}{3}}\Delta \gamma  \end{array}    \right. </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.17)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.64)
 |}
-This is a '''softening effect''' induced by the non-inertial character of the frame reference where the equations are posed. This behaviour has already been reviewed in the introductory part of this report (see section [[#1.2.6 Centrifugal softening and stiffening|1.2.6]]). This softening causes a reduction in the natural frequencies of the structure, compared to those of the static case. However, this effect is '''rarely seen''' in actual rotors. In reality, softening is a phenomena eclipsed by the so-called stiffening effect. The problem is that simple rotor models do not account for the latter, and thus the overall effect is of softening <span id='citeF-3'></span>[[#cite-3|[3]]].
+With the evolution equation written in spatial configuration, the yield condition and the definition of the Kirchhoff stress tensor it is possible to define the return mapping algorithm, through which the plastic problem is solved in the time interval <math display="inline">\left[t_n, t_{n+1} \right]</math>.  Let <math display="inline">\boldsymbol{F}_n</math>, <math display="inline">\alpha _n</math>, <math display="inline">\boldsymbol{b}^e_{n}</math> and the configuration <math display="inline">\phi _n</math> be known data at time <math display="inline">t_n</math> and the incremental displacement of the configuration <math display="inline">\phi _n</math>, <math display="inline">\boldsymbol{u}_n\circ \phi _n</math>, at time <math display="inline">t_{n+1}</math> used to defined the incremental deformation gradient <math display="inline">\boldsymbol{f}_{n+1}</math> which can be evaluated as <math display="inline">\boldsymbol{f}_{n+1}=\boldsymbol{1}+\nabla _{x_n}\boldsymbol{u}_n</math>. In order to compute the elasto-plastic response, a trial elastic state is defined where no plastic flow takes place. Through this assumption, the intermediate configuration remains unchanged, i.e.:
-For sure, equation [[#eq-3.14|3.14]] can be posed in the '''inertial''' frame of reference using the rotation matrix and applying the transformation of stresses from equation [[#eq-2.23|2.23]]:
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\left\{    \begin{array}{rcll}\left(\overline{\boldsymbol{C}}^{p^{-1}}_{n+1} \right)^{trial}&= &\overline{\boldsymbol{C}}^{p^{-1}}_{n}, \\ \alpha _{n+1}^{trial} &= & \alpha _n \end{array}    \right. </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.65)
+|}
+which, by a push-forward transformation, corresponds to the spatial form:
+<span id="eq-3.66"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,336: / Line 2,289: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M}_{\;inertial}= \int _{{\mho }}^{ } \mathbf{N}^T \mathbf{{{Q}}_{(\theta )}}\; \rho \;\mathbf{N} \; d\mho  </math>
+| style="text-align: center;" | <math>\left(\overline{\boldsymbol{b}}^{e}_{n+1} \right)^{trial}:=\overline{\boldsymbol{F}}_{n+1}\left(\overline{\boldsymbol{C}}^{p^{-1}}_{n+1} \right)^{trial}\overline{\boldsymbol{F}}^T_{n+1}=\overline{\boldsymbol{f}}_{n+1}\left[\overline{\boldsymbol{F}}_{n}\overline{\boldsymbol{C}}^{p^{-1}}_{n}\overline{\boldsymbol{F}}_{n}^T\right]\overline{\boldsymbol{f}}_{n+1}^T </math>
+|-
+| style="text-align: center;" | <math> =\overline{\boldsymbol{f}}_{n+1}\overline{\boldsymbol{b}}^{e}_{n}\overline{\boldsymbol{f}}_{n+1}^T </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.66)
 |}
+According to Equation [[#eq-3.66|3.66]], the trial state of stress is defined as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,345: / Line 2,303: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{D}_{\;inertial}= \int _{{\mho }}^{ } \mathbf{N}^T \mathbf{{{Q}}_{(\theta )}}\; \overline{\mu }\;\mathbf{N}\;d\mho  \;  + \; \int _{{\mho }}^{ } \mathbf{B}^T \mathbf{{{T}}}_{(\theta )}\; \overline{\mathbf{D}}\; \mathbf{{{T}}}_{(\theta )}^T\;\mathbf{B}\;d\mho  \; + \; 2\int _{{\mho }}^{ } \;\mathbf{N}^T \mathbf{{{Q}}_{(\theta )}}\; \rho \;\widetilde{\Omega }\;\mathbf{N}\;d\mho  </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau }^{trial}_{n+1}=U'(J_{n+1})J_{n+1}\boldsymbol{I}+\mu \mathrm{dev}\left(\overline{\boldsymbol{b}}^{e^{trial}}_{n+1} \right) </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.67)
 |}
+and the discrete governing equations are written as
+<span id="eq-3.68"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,354: / Line 2,316: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}_{\;inertial}= \int _{{\mho }}^{ } \mathbf{B}^T \mathbf{{{T}}}_{(\theta )}\; \mathbf{C}\;\mathbf{{{T}}}_{(\theta )}^T\;\mathbf{B} \; d\mho  \;+ \; \int _{{\mho }}^{ } \mathbf{N}^T \mathbf{{{Q}}_{(\theta )}}\; \rho \;(\widetilde{\Omega }^2 + \widetilde{\alpha })\;\mathbf{N} \; d\mho   </math>
+| style="text-align: center;" | <math>\left\{    \begin{array}{rcll}\overline{\boldsymbol{b}}^{e}_{n+1}&= &\overline{\boldsymbol{b}}^{e^{trial}}_{n+1}-\frac{2}{3}\Delta \gamma \mathrm{tr}\overline{\boldsymbol{b}}^{e}_{n+1}\boldsymbol{n}_{n+1}, \\ \alpha _{n+1} &= & \alpha _n+\sqrt{\frac{2}{3}}\Delta \gamma  \end{array}    \right.  </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.68)
 |}
+<span id="eq-3.69"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,363: / Line 2,327: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}_{\;inertial}= \int _{{\mho }} {\mathbf{N}}^{T}\mathbf{{{Q}}_{(\theta )}}\; \boldsymbol{f} d \mho +\sum _{i=1}^{n_{ d}} \int _{\Gamma _{\sigma }^{i}} \mathbf{N}_{i}^T \;\mathbf{{{T}}}_{(\theta )}\;\overline{t}^{\,i} d \Gamma \;-\; {\int _{{\mho }}^{ } \mathbf{N}^T \mathbf{{{Q}}_{(\theta )}}\;\rho \;(\widetilde{\Omega }^2 + \widetilde{\alpha })\;\mathbf{r}_i \; d\mho } </math>
+| style="text-align: center;" | <math>\left\{    \begin{array}{rcll}\boldsymbol{\tau }_{n+1}&= &U'(J_{n+1})J_{n+1}\boldsymbol{I}+\mu \mathrm{dev}\left(\overline{\boldsymbol{b}}^{e}_{n+1} \right)=U'(J_{n+1})J_{n+1}\boldsymbol{I}+\mathrm{dev}\left(\boldsymbol{\tau }_{n+1} \right), \\ \boldsymbol{n}_{n+1} &= & \dfrac{\mathrm{dev}\left(\boldsymbol{\tau }_{n+1} \right)}{\Vert \mathrm{dev}\left(\boldsymbol{\tau }_{n+1} \right)\Vert } \end{array}    \right.  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.18)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.69)
 |}
-===3.2.2 Block matrix decomposition===
+<span id="eq-3.70"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\Delta \gamma \geq 0, \quad f(\boldsymbol{\tau }_{n+1},\alpha _{n+1})\leq 0, \quad \Delta \gamma f(\boldsymbol{\tau }_{n+1})=0  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.70)
+|}
-The above equations cannot be solved unless boundary values of displacement are specified. Moreover, the involved matrices do not admit inverse, so the only way to tackle the system is to break the presented matrices into '''block matrices'''. Remember that <math display="inline">\mathbf{r}</math> stands for the set of DOFs where displacement is prescribed, whereas <math display="inline">\mathbf{l}</math> holds for the set of DOFs where the displacement field is to be computed. Hence, naming <math display="inline">\mathbf{J}_{rl}</math> the matrix made-up by the elements forming the intersection of the <math display="inline">\mathbf{r}</math> rows and <math display="inline">\mathbf{l}</math> columns of <math display="inline">\mathbf{J}</math>, equation [[#eq-3.14|3.14]] is split into:
+By exploiting the Kuhn-Tucker condition in discrete form of Equation [[#eq-3.70|3.70]], two cases might be encountered. The first one, it is identified with the case the yield condition <math display="inline">f^{trial}_{n+1}\leq 0</math>:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,377: / Line 2,350: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{bmatrix}{\mathbf{c}_\mathbf{l}}^T & {\mathbf{c}_\mathbf{r}}^T  \end{bmatrix} \left(\begin{bmatrix}\mathbf{M}_{\mathbf{ll}}&\mathbf{M}_{\mathbf{lr}} \\  \mathbf{M}_{\mathbf{rl}} & \mathbf{M}_{\mathbf{rr}} \end{bmatrix} \begin{bmatrix}\ddot{\mathbf{d}}_\mathbf{l} \\  \ddot{\mathbf{d}}_\mathbf{r} \end{bmatrix} + \begin{bmatrix}\mathbf{D}_{\mathbf{ll}}&\mathbf{D}_{\mathbf{lr}} \\  \mathbf{D}_{\mathbf{rl}} & \mathbf{D}_{\mathbf{rr}} \end{bmatrix} \begin{bmatrix}\dot{\mathbf{d}}_\mathbf{l} \\  \dot{\mathbf{d}}_\mathbf{r} \end{bmatrix} + \begin{bmatrix}\mathbf{K}_{\mathbf{ll}}&\mathbf{K}_{\mathbf{lr}} \\  \mathbf{K}_{\mathbf{rl}} & \mathbf{K}_{\mathbf{rr}} \end{bmatrix} \begin{bmatrix}{\mathbf{d}}_\mathbf{l} \\  {\mathbf{d}}_\mathbf{r} \end{bmatrix}-\begin{bmatrix}\mathbf{F}_\mathbf{l}\\  \mathbf{F}_\mathbf{r} + \mathbf{R} \end{bmatrix}\right)= 0 </math>
+| style="text-align: center;" | <math>f^{trial}_{n+1}:= f(\boldsymbol{\tau }_{n+1}^{trial},\alpha _{n})=\Vert \mathrm{dev}\boldsymbol{\tau _{n+1}^{trial}}\Vert - \sqrt{\frac{2}{3}}\left(\sigma _Y + H \alpha _n \right) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.19)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.71)
 |}
-Where <math display="inline">\mathbf{R}</math> stands for the elastic reactions at the nodes with prescribed displacement, which are unknown. The above expression can be thought as a system of two equations:
+then, the condition <math display="inline">\Delta \gamma=0</math> is directly satisfied and no plastic flow takes place, leading to a completely elastic response. In the second alternative case, where the yield condition <math display="inline">f^{trial}_{n+1}> 0</math>, it is clear that <math display="inline">\boldsymbol{\tau }_{n+1}^{trial}</math> can not be admitted and, according to Equation [[#eq-3.68|3.68]], <math display="inline">\overline{\boldsymbol{b}}^{e}_{n+1}\neq \overline{\boldsymbol{b}}^{e^{trial}}_{n+1}</math> leading to the condition of <math display="inline">\Delta \gamma{>0}</math>. Thus, in this case the radial return mapping has to be performed. By considering Equation [[#eq-3.68|3.68]] and applying the trace operator to it, it is immediately demonstrated that
+<span id="eq-3.72"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,389: / Line 2,363: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{matrix}\mathbf{c}_\mathbf{l}^T \left( \mathbf{M}_{\mathbf{ll}} \ddot{\mathbf{d}}_\mathbf{l}\;+\;\mathbf{M}_{\mathbf{lr}} \ddot{\mathbf{d}}_\mathbf{r} \;+\;\mathbf{D}_{\mathbf{ll}} \dot{\mathbf{d}}_\mathbf{l}\;+\;\mathbf{D}_{\mathbf{lr}} \dot{\mathbf{d}}_\mathbf{r}  \;+\;\mathbf{K}_{\mathbf{ll}} {\mathbf{d}}_\mathbf{l}\;+\;\mathbf{K}_{\mathbf{lr}} {\mathbf{d}}_\mathbf{r} \;-\;\mathbf{F}_\mathbf{l} \right)= 0\\ \\ \mathbf{c}_\mathbf{r}^T \left( \mathbf{M}_{\mathbf{rl}} \ddot{\mathbf{d}}_\mathbf{l}\;+\;\mathbf{M}_{\mathbf{rr}} \ddot{\mathbf{d}}_\mathbf{r} \;+\;\mathbf{D}_{\mathbf{rl}} \dot{\mathbf{d}}_\mathbf{l}\;+\;\mathbf{D}_{\mathbf{rr}} \dot{\mathbf{d}}_\mathbf{r}  \;+\;\mathbf{K}_{\mathbf{rl}} {\mathbf{d}}_\mathbf{l}\;+\;\mathbf{K}_{\mathbf{rr}} {\mathbf{d}}_\mathbf{r} \;-\;\mathbf{F}_\mathbf{r} \;-\;\mathbf{R} \right)= 0 \end{matrix} </math>
+| style="text-align: center;" | <math>\mathrm{tr}\,\overline{\boldsymbol{b}}^{e}_{n+1}=\mathrm{tr}\,\overline{\boldsymbol{b}}^{e^{trial}}_{n+1}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.20)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.72)
 |}
-The above equations are fulfilled for all <math display="inline">\mathbf{c}^T</math> if and only if the terms multiplying both <math display="inline">\mathbf{c}_\mathbf{l}^T</math> and <math display="inline">\mathbf{c}_\mathbf{r}^T</math> vanish. Solving for the unknown displacements and reactions, and substituting <math display="inline">\ddot{\mathbf{d}}_\mathbf{r}</math>, <math display="inline">\dot{\mathbf{d}}_\mathbf{r}</math> and <math display="inline">{\mathbf{d}}_\mathbf{r}</math> by the corresponding boundary values <math display="inline">\overline{\ddot{\mathbf{u}}}</math>, <math display="inline">\overline{\dot{\mathbf{u}}}</math> and <math display="inline">\overline{{\mathbf{u}}}</math>, respectively, the resulting equations are:
+since <math display="inline">\mathrm{tr}\,\boldsymbol{n}_{n+1}=0</math>. By substituting the expression of Equation [[#eq-3.72|3.72]] in Equation [[#eq-3.68|3.68]] it is found that
-<span id="eq-3.21"></span>
+<span id="eq-3.73"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,402: / Line 2,376: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{matrix}\mathbf{M}_{\mathbf{ll}} \ddot{\mathbf{d}}_\mathbf{l} \;+\;\mathbf{D}_{\mathbf{ll}} \dot{\mathbf{d}}_\mathbf{l} \;+\;\mathbf{K}_{\mathbf{ll}} {\mathbf{d}}_\mathbf{l}=\mathbf{F}_\mathbf{l} - (\;\mathbf{M}_{\mathbf{lr}} \overline{\ddot{\mathbf{u}}} \;+\;\mathbf{D}_{\mathbf{lr}} \overline{\dot{\mathbf{u}}} \;+\;\mathbf{K}_{\mathbf{lr}} \overline{{\mathbf{u}}})\\ \\ \mathbf{R} =  \mathbf{M}_{\mathbf{rl}} \ddot{\mathbf{d}}_\mathbf{l}\;+\;\mathbf{D}_{\mathbf{rl}} \dot{\mathbf{d}}_\mathbf{l}\;+\;\mathbf{K}_{\mathbf{rl}} {\mathbf{d}}_\mathbf{l}\;+\;\mathbf{M}_{\mathbf{rr}} \overline{\ddot{\mathbf{u}}} \;+\;\mathbf{D}_{\mathbf{rr}} \overline{\dot{\mathbf{u}}} \;+\;\mathbf{K}_{\mathbf{rr}} \overline{{\mathbf{u}}}\;-\;\mathbf{F}_\mathbf{r} \end{matrix} </math>
+| style="text-align: center;" | <math>\overline{\boldsymbol{b}}^{e}_{n+1}= \overline{\boldsymbol{b}}^{e^{trial}}_{n+1}-\frac{2}{3}\Delta \gamma \mathrm{tr}\,\overline{\boldsymbol{b}}^{e^{trial}}_{n+1}\boldsymbol{n}_{n+1}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.21)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.73)
 |}
-===3.2.3 Reaction computation===
+which used for the definition of the deviatoric part of <math display="inline">\boldsymbol{\tau }</math> leads to
-Notice that the reactions can only be found once the displacements in the unconstrained degrees of freedom are known. If the equations are posed in the noninertial frame of reference, the obtained value of reaction is the expected from a static structure under the effect of a certain deformation. To account for the total reaction, the '''rigid body''' moment associated to the rotation motion is to be known:
-<span id="eq-3.22"></span>
+<span id="eq-3.74"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,417: / Line 2,389: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{W}_{rb} = \mathbf{J}_\mathbf{z}\; \alpha  </math>
+| style="text-align: center;" | <math>\mathrm{dev}\boldsymbol{\tau }_{n+1}=\mu \mathrm{dev}\overline{\boldsymbol{b}}^{e^{trial}}_{n+1}-\frac{2}{3}\mu \Delta \gamma \mathrm{tr}\left(\overline{\boldsymbol{b}}^{e^{trial}}_{n+1}\right)\boldsymbol{n}_{n+1}</math>
+|-
+| style="text-align: center;" | <math> =\mathrm{dev}\boldsymbol{\tau }_{n+1}^{trial}-\frac{2}{3}\mu \Delta \gamma \mathrm{tr}\left(\overline{\boldsymbol{b}}^{e^{trial}}_{n+1}\right)\boldsymbol{n}_{n+1}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.22)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.74)
 |}
-Where <math display="inline">\mathbf{W}_{rb}</math> stands for the reaction torque associated to the rigid body motion, <math display="inline">\mathbf{J}_\mathbf{z}</math> is the inertia moment around the rotation axis and <math display="inline">\alpha </math> is the angular acceleration. If this reaction torque is added to the torque generated by the elastic reactions in the noninertial frame <math display="inline">\mathbf{R}</math>, the overall torque is obtained. In order to find the value of <math display="inline">\mathbf{W}_{rb}</math>, it will be necessary to compute <math display="inline">\mathbf{J}_\mathbf{z}</math>, which fortunately can be posed in terms of the FEM. Defining the rigid body modes <math display="inline">\mathcal{}{R}</math> in three dimensions:
+By considering the expression <math display="inline">\overline{\mu }=\dfrac{1}{3}\mu \mathrm{tr}\,\overline{\boldsymbol{b}}^{e^{trial}}_{n+1}</math>, Equation [[#eq-3.74|3.74]] can be rewritten as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,429: / Line 2,403: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathcal{}{R} = \begin{bmatrix}1 & 0 & 0 &0  & \Delta z_1 &  -\Delta y_1\\  0 & 1 & 0 & -\Delta z_1 &0  &\Delta x_1 \\  0 & 0&  1& \Delta y_1 & -\Delta x_1 &0 \\  \vdots  &\vdots   & \vdots  &  \vdots  &\vdots   & \vdots \\  1 & 0 & 0 &0  & \Delta z_{n_{pt}} &  -\Delta y_{n_{pt}}\\  0 & 1 & 0 & -\Delta z_{n_{pt}} &0  &\Delta x_{n_{pt}} \\  0 & 0&  1& \Delta y_{n_{pt}} & -\Delta x_{n_{pt}} &0 \\ \end{bmatrix} = \begin{bmatrix}\mathbf{I} & -\Delta \widetilde{\mathbf{r}}\\  \vdots & \vdots  \end{bmatrix} </math>
+| style="text-align: center;" | <math>\left(\Vert \mathrm{dev}\boldsymbol{\tau }_{n+1}\Vert +2\overline{\mu }\Delta \gamma \right)\boldsymbol{n}_{n+1}=\Vert \mathrm{dev}\boldsymbol{\tau }_{n+1}^{trial}\Vert \boldsymbol{n}_{n+1}^{trial} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.23)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.75)
 |}
-Where <math display="inline">\mathcal{}{R}</math> is a matrix with 6 columns, corresponding to the''' rigid body degrees of freedom''': three translations and three rotations. In the above expressions, the first three columns are the translation modes, and can be represented by the identity matrix '''I'''. On the other hand, the last three columns are the rotation modes, and are computed as the '''spin''' of the distance with respect to the rotation axis <math display="inline">-\Delta \widetilde{\mathbf{r}}</math>. The rigid body modes matrix has as many rows as degrees of freedom, which means that the sequence presented above is repeated for each node. In two-dimensional problems, only one rotational and two translational modes exist.
+with <math display="inline">\boldsymbol{n}_{n+1}^{trial}=\dfrac{\mathrm{dev}\boldsymbol{\tau }_{n+1}^{trial}}{\Vert \mathrm{dev}\boldsymbol{\tau }_{n+1}^{trial}\Vert }</math>. From this last equation it is deduced that
-The key fact is that <math display="inline">\mathcal{}{R}</math>  can be used to normalise the mass matrix <math display="inline">\mathbf{M}</math> in such a way that information about the structure is obtained. For a generic two-dimensional case it is found that:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,443: / Line 2,415: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\underset{3\times n_{pt}}{\underbrace{\mathcal{}{R}^T}}\;\underset{n_{pt}\times n_{pt}}{\underbrace{\mathbf{M}}}\;\underset{n_{pt}\times 3}{\underbrace{\mathcal{}{R}}} = \begin{bmatrix}\mathbf{m} & 0 &0\\  0 & \mathbf{m} & 0\\  0& 0 & \mathbf{J}_\mathbf{z} \end{bmatrix} </math>
+| style="text-align: center;" | <math>\Vert \mathrm{dev}\boldsymbol{\tau }_{n+1}\Vert +2\overline{\mu }\Delta \gamma =\mathrm{dev}\boldsymbol{\tau }_{n+1}^{trial} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.24)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.76)
 |}
-Where '''m''' stands for the total mass of the structure, and <math display="inline">\mathbf{J}_\mathbf{z}</math> is the inertia moment around the rotation axis. These properties may seem to be trivial to compute, but determining either the mass or the inertia can be a difficult task if the structure is not defined by a simple geometry. Moreover, when the geometry is simple and those values can be computed analytically, they can be used to asses whether <math display="inline">\mathbf{M}</math> has been computed correctly by comparing the analytic values of mass and inertia with those obtained using the above numerical expression.
+along with
-Once the inertia is known, the torque <math display="inline">\mathbf{W}_{rb}</math> associated to the rigid body motion can be computed. Then, the reaction torque due to elastic reactions <math display="inline">\mathbf{W}_{el}</math> is to be computed by multiplying each term of <math display="inline">\mathbf{R}</math> by the distance with respect to the rotation axis. For a generic case, the '''elastic reaction torque''' is computed in terms of each of the components of <math display="inline">\mathbf{R}</math>:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,457: / Line 2,427: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{w}_{el\;} = \begin{bmatrix}0&-\Delta z\,_1  & \Delta y\,_1\\  \Delta z\,_1 & 0  & -\Delta x\,_1\\  -\Delta y\,_1& \Delta x\,_1 & 0\\ \vdots & \vdots  &  \vdots \\  0&-\Delta z_{{n}_{pt}} & \Delta y_{{n}_{pt}}\\  \Delta z_{{n}_{pt}} & 0  & -\Delta x_{{n}_{pt}}\\  -\Delta y_{{n}_{pt}}& \Delta x_{{n}_{pt}} & 0\\ \end{bmatrix}\begin{pmatrix}{R_x}\,_1\\  {R_y}\,_2\\  {R_z}\,_3 \\ \vdots \\ {R_x}_{{n}_{pt}}\\  {R_y}_{{n}_{pt}}\\  {R_z}_{{n}_{pt}} \end{pmatrix} = \begin{bmatrix}\Delta \widetilde{\mathbf{r}} \\ \vdots  \end{bmatrix} \mathbf{R} </math>
+| style="text-align: center;" | <math>\boldsymbol{n}_{n+1}=\boldsymbol{n}_{n+1}^{trial} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.25)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.77)
 |}
+Thus, it is possible to rewrite the yield condition as
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\Vert \mathrm{dev}\boldsymbol{\tau }_{n+1}\Vert -\sqrt{\frac{2}{3}}\left(\sigma _Y + H\alpha _{n+1} \right)=\Vert \mathrm{dev}\boldsymbol{\tau }_{n+1}^{trial}\Vert{-2}\overline{\mu }\Delta \gamma -\sqrt{\frac{2}{3}}\left(\sigma _Y + H\alpha _{n} \right)</math>
+|-
+| style="text-align: center;" | <math> -\sqrt{\frac{2}{3}}\left(\alpha _{n+1}-\alpha _{n}\right)=f^{trial}_{n+1}-2\overline{\mu }\left(1+\frac{H}{3\overline{\mu }} \right)\Delta \gamma=0 </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.78)
+|}
-The total reaction torque <math display="inline">\mathbf{W}_{el\;}</math> is an <math display="inline">n_{d}</math> vector computed by adding the contributions of <math display="inline">\mathbf{w}_{el\;}</math> for every spatial direction. Finally, the total torque is:
+and, finally, the unknown <math display="inline">\Delta \gamma </math> is determined
-<span id="eq-3.26"></span>
+<span id="eq-3.79"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,472: / Line 2,454: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{T} = \mathbf{W}_{rb} + \mathbf{W}_{el} </math>
+| style="text-align: center;" | <math>\Delta \gamma =\frac{1}{2\overline{\mu }}\frac{f^{trial}_{n+1}}{1+\dfrac{H}{3\overline{\mu }}}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.26)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.79)
 |}
-When the kinematics of the problem are known — which is the case — one of the the most important unknowns to determine are the total forces and moments acting on the structure. The value of <math display="inline">\mathbf{T}</math> gives a hint on how much '''work''' needs to be injected into the blade to ensure rotation at the given angular speed and acceleration.
+which is used for the determination of <math display="inline">\overline{\boldsymbol{b}}^{e}_{n+1}</math> and <math display="inline">\mathrm{dev}\boldsymbol{\tau }_{n+1}</math> (see Equations [[#eq-3.73|3.73]] and [[#eq-3.74|3.74]], respectively).  The procedure defined above is able to guarantee the preservation of the material frame indifference framework and it results in an extension to finite strains of the classical radial return mapping method of infinitesimal plasticity. The governing equations have been derived and the algorithmic procedure to be followed step-by-step is described in Algorithm [[#algorithm-3.1|3.1]].
-==3.3 Local formulation==
+{| style="margin: 1em auto;border: 1px solid darkgray;"
+|-
+|
+:<span style="font-size: 75%;">
+Initial data on material points: <math> \boldsymbol{F}_n </math>, <math> \boldsymbol{b}^e_n</math>
+|-
+| OUTPUT of calculations: <math> \boldsymbol{\tau }_{n+1}</math>,<math> \boldsymbol{b}^e_{n+1}</math>
+|-
+|
+<ol>
+<li>'''UPDATE THE CURRENT CONFIGURATION''' </li>
+:* Compute the the current configuration: <math display="inline"> \boldsymbol{\varphi }_{n+1}=\boldsymbol{\varphi }_n + \boldsymbol{u}_n \circ \boldsymbol{\varphi }_n </math>
+:* Compute the relative deformation gradient: <math display="inline">\boldsymbol{f}_{n+1} = \boldsymbol{1} + \nabla _{\boldsymbol{x}_n} \boldsymbol{u}_n</math>
+:* Compute the total deformation gradient in updated configuration: <math display="inline">\boldsymbol{F}_{n+1} = \boldsymbol{f}_{n+1}\boldsymbol{F}_n</math>
+<li>'''COMPUTE THE ELASTIC PREDICTOR''' </li>
+:* Compute the volume preserving part of <math display="inline">\boldsymbol{f}_{n+1}</math>: <math display="inline"> \overline{\boldsymbol{f}}_{n+1}=\left[\mathrm{det}\boldsymbol{f}_{n+1} \right]^{-\frac{1}{3}}\boldsymbol{f}_{n+1} </math>
+:* Compute the volume preserving part of <math display="inline">{\boldsymbol{b}^e}^{\mathrm{trial}}</math>: <math display="inline">{\overline{\boldsymbol{b}}^e}^{\mathrm{trial}}_{n+1}= \overline{\boldsymbol{f}}_{n+1}{\overline{\boldsymbol{b}}^e}_n {\overline{\boldsymbol{f}}_{n+1}}^T</math>
+:* Compute the deviatoric part of the <math display="inline">\boldsymbol{\tau }</math>: <math display="inline">\mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}=\mu \mathrm{dev}\left[{\overline{\boldsymbol{b}}^e}^{\mathrm{trial}}_{n+1}\right]</math>
+<li>'''CHECK FOR PLASTIC LOADING'''</li>
+:  Evaluate the Mises-Huber yield condition <math display="inline">f^{\mathrm{trial}}_{n+1}:=\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert -\sqrt{\frac{2}{3}}\left(H \alpha _n + \sigma _Y \right)</math>
+{|
+|-
+|
+:a If <math display="inline">f^{\mathrm{trial}_{n+1}}\leq{0}</math>, no plastic loading is observed. Thus, <math display="inline">\mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}} = \mathrm{dev}\boldsymbol{\tau }_{n+1}</math> and <math display="inline"> {\overline{\boldsymbol{b}}^e}^{\mathrm{trial}}_{n+1}={\overline{\boldsymbol{b}}^e}_{n+1} </math>
+:b If <math display="inline">f^{\mathrm{trial}_{n+1}}>0</math>, plastic loading is observed.
+|-
+| Setting <math display="inline"> \overline{\mu }=\dfrac{1}{3}\mu \mathrm{tr}{\overline{\boldsymbol{b}}^e}^{\mathrm{trial}}_{n+1} </math>
+{|
+|-
+|
+::* Compute the plastic multiplier <math display="inline">\Delta \gamma =\dfrac{f^{\mathrm{trial}}_{n+1}/2\overline{\mu }}{1+H/3\overline{\mu }}</math>
+::* Compute the unit tensor <math display="inline"> \boldsymbol{n} = \dfrac{\mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}}{\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert } </math>
+|} Computation of the return mapping
+|-
+| Correct <math display="inline">\mathrm{dev}\boldsymbol{\tau }</math>: <math display="inline">\mathrm{dev}\boldsymbol{\tau }_{n+1}=\mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}-2\overline{\mu }\Delta \gamma \boldsymbol{n}</math>
+|-
+| Correct <math display="inline">\alpha _{n+1}</math>: <math display="inline">\alpha _{n+1} = \alpha _n+\sqrt{\frac{2}{3}}\Delta \gamma  </math>
+|}
+<li>'''EVALUATE THE STRESS IN CURRENT CONFIGURATION'''
+: <math display="inline">\boldsymbol{\tau }_{n+1}=J_{n+1}U'(J_{n+1})+ \mathrm{dev}\boldsymbol{\tau }_{n+1} </math> </li>
+<li>'''UPDATE THE INTERMEDIATE CONFIGURATION'''
+: <math display="inline"> {\overline{\boldsymbol{b}}^e}_{n+1}=\dfrac{\mathrm{dev}\boldsymbol{\tau }_{n+1}}{\mu }+\dfrac{1}{3}\mathrm{tr}\left[{\overline{\boldsymbol{b}}^e}^{\mathrm{trial}}_{n+1}\right]\boldsymbol{1} </math>  </li>
+</ol>
+</span>
+|-
+| style="text-align: center; font-size: 75%;"|
+<span id='algorithm-3.1'></span>'''Algorithm. 3.1''' Return Mapping algorithm.
+|}
-The integrals presented in the previous sections cannot be computed at once unless the geometry of the domain is tremendously simple, which is not the case. The grace of the FEM is that breaks the domain into smaller elements with well defined geometric properties, where the previous integrals are '''approximated''' using Gauss Quadrature, allowing computer implementation . In order to achieve this goal, a new '''local frame '''within a single element <math display="inline">\mho ^e</math> is defined. Let's assume the model consists of <math display="inline">n_{el}</math> elements, and take <math display="inline">e</math> as the variable index for the elements <math display="inline">1\leq e \leq n_{el}</math>.
+Finally, the closed form expression for the algorithmic elasto-plastic moduli <math display="inline"> \boldsymbol{\mathrm{C}^{ep}}</math> is derived through the linearisation of the Kirchhoff stress tensor
-Let's introduce <math display="inline"> \boldsymbol{x}^e \; \in {\overline{\mho }}^{\boldsymbol{\,e}}</math> , the vector of global or '''physical''' coordinates in the local frame, containing the coordinates of a given element <math display="inline">\mathbf{e}</math>. Following this local approach, let's define <math display="inline">\boldsymbol{\xi } = (\xi \;\eta \;\zeta \;)^T \; \in \overline{\mho }_{\boldsymbol{\xi }}\;</math> as the vector of local or element coordinates in the '''parent''' domain. In order to visualise the mapping from the parent domain to the physical domain using isoparametric linear elements, let's consider a simple 1D case. The transformation from one domain to another is represented in the following figure:
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-<div id='img-3.1'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-mapping.png|600px|Representation of the mapping from the physical to the parent domain for a linear 1D element]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 3.1:''' Representation of the mapping from the physical to the parent domain for a linear 1D element
+|-
+| style="text-align: center;" | <math>\boldsymbol{\tau }_{n+1}=J_{n+1}U'(J_{n+1})\boldsymbol{I}+ \mathrm{dev}\boldsymbol{\tau }_{n+1}=J_{n+1}U'(J_{n+1})\boldsymbol{I}+ \mathrm{dev}\boldsymbol{\tau }_{n+1}^{trial}-2\overline{\mu }\Delta \gamma \boldsymbol{n} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.80)
 |}
-The same concept can be applied to 2D and 3D elements. For a generic case, the mapping is constructed using the same shape functions employed in the interpolation of <math display="inline">\mathfrak{u}^{h}</math> (isoparametric elements). Defining <math display="inline">n^e</math> as the number of nodes of the element <math display="inline">e</math>, the mapping is expressed as:
+According to the definition of the tangent modulus in current configuration
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,500: / Line 2,529: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{x}^e_{(\xi )} = \begin{bmatrix}x_1^e & x_2^e & \cdots  & x_{n_e}^e\\     y_1^e & y_2^e & \cdots  & y_{n_e}^e\\     z_1^e & z_2^e & \cdots  & z_{n_e}^e \end{bmatrix} \begin{bmatrix}{N}_1^e\\     {N}_2^e\\     \vdots \\     {N}_{n_e}^e \end{bmatrix} = \mathbf{X}^e {\mathcal{}{N}^e_{(\xi )}}^T </math>
+| style="text-align: center;" | <math>\boldsymbol{\mathrm{C}}^{\tau }:=2\frac{\partial \boldsymbol{\tau }_{n+1}}{\partial \boldsymbol{g}_{n+1}} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.27)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.81)
 |}
-Were <math display="inline"> \mathbf{X}^e</math> stands for the nodal coordinates in the physical domain, and <math display="inline">\mathcal{}{N}^e_{(\xi )} \in \mathbb{R}^{1 \times n^{e}}</math> is the matrix of shape functions for scalar valued fields. In a similar way, one may define the matrix of shape functions for vector valued fields <math display="inline">\mathbf{{N}}^e = [ {N}_1^e \;\mathbf{I} \; \;\; {N}_2^e \;\mathbf{I} \;\;\;\cdots \;\;\;{N}_{n_e}^e \;\mathbf{I} ] \in \mathbb{R}^{n_d \times n_d \; n^{e}} </math> , where <math display="inline">\mathbf{I}</math> is the <math display="inline">n_d\times n_d</math> identity matrix.
+where <math display="inline">\boldsymbol{g}</math> denotes the metric tensor in the current configuration, the expression of <math display="inline">\boldsymbol{\mathrm{C}}^{\tau }</math> in the context of the Hyperelastic-<math display="inline">J_2</math> plastic law reads
-The mapping can be interpreted as a '''change of coordinates''', and thus it is a transformation with an associated Jacobian matrix <math display="inline">\mathbf{J}^e</math>. It allows the differential mapping between physical and parent domain, and vice versa:
+<span id="eq-3.82"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,514: / Line 2,542: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{matrix}\left[\begin{array}{l}{d x} \\ {d y} \\ {d z}\end{array}\right]= \left[\begin{array}{lll}{\frac{\partial x}{\partial \xi }} & {\frac{\partial x}{\partial \eta }} & {\frac{\partial x}{\partial \zeta }} \\ {\frac{\partial y}{\partial \xi }} & {\frac{\partial y}{\partial \eta }} & {\frac{\partial y}{\partial \zeta }} \\ {\frac{\partial z}{\partial \xi }} & {\frac{\partial z}{\partial \eta }} & {\frac{\partial z}{\partial \zeta }}\end{array}\right]\left[\begin{array}{l}{d \xi } \\ {d \eta } \\ {d \zeta }\end{array}\right]\quad \Rightarrow \quad d \boldsymbol{x}=\mathbf{J}^{e} d \boldsymbol{\xi } \; \quad \\ \\  \left[\begin{array}{l}{d \xi } \\ {d \eta } \\ {d \zeta }\end{array}\right]= \left[\begin{array}{lll}{\frac{\partial \xi }{\partial x}} & {\frac{\partial \xi }{\partial y}} & {\frac{\partial \xi }{\partial z}} \\ {\frac{\partial \eta }{\partial x}} & {\frac{\partial \eta }{\partial y}} & {\frac{\partial \eta }{\partial z}} \\ {\frac{\partial \zeta }{\partial x}} & {\frac{\partial \zeta }{\partial y}} & {\frac{\partial \zeta }{\partial z}}\end{array}\right]\left[\begin{array}{l}{d x} \\ {d y} \\ {d z}\end{array}\right]\quad \Rightarrow \quad d \boldsymbol{\xi }={\mathbf{J}^{e}}^{-1} d \boldsymbol{x}  \end{matrix} </math>
+| style="text-align: center;" | <math>\boldsymbol{\mathrm{C}}^{\tau ,ep}=2\frac{\partial \boldsymbol{\tau }_{n+1}^{trial}}{\partial \boldsymbol{g}_{n+1}}-\boldsymbol{n}\otimes 2\overline{\mu }2\frac{\partial \Delta \gamma }{\partial \boldsymbol{g}_{n+1}}-2\overline{\mu }\Delta \gamma 2\frac{\partial \boldsymbol{n}}{\partial \boldsymbol{g}_{n+1}}-2\Delta \gamma \boldsymbol{n}\otimes 2\frac{\partial \overline{\mu }}{\partial \boldsymbol{g}_{n+1}}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.28)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.82)
 |}
+It is deduced that the first term of Equation [[#eq-3.82|3.82]] is given by the sum of Equations [[#eq-3.51|3.51]] and [[#eq-3.52|3.52]] and it reads
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>2\frac{\partial \boldsymbol{\tau }_{n+1}^{trial}}{\partial \boldsymbol{g}_{n+1}}=\boldsymbol{\mathrm{C}}^{\mathrm{trial}} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.83)
+|}
-Taking a closer look to the elemental Jacobian matrix, one founds that it can be expressed as
+In what follows, the derivation of the three last terms is presented. By rewriting Equation [[#eq-3.79|3.79]] as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,528: / Line 2,566: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{J}^{e}=\left[\begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right] \begin{bmatrix}\frac{\partial }{\partial \xi } &  \frac{\partial }{\partial \eta }& \frac{\partial }{\partial \zeta } \end{bmatrix} =\boldsymbol{x}\; \tilde{\boldsymbol{\nabla }}_{\boldsymbol{\xi }}^{T} = (\mathbf{X}^e\; {\mathcal{}{N}^e_{(\xi )}}^T) \tilde{\boldsymbol{\nabla }}_{\boldsymbol{\xi }}^{T} = \mathbf{X}^e\; (\tilde{\boldsymbol{\nabla }}_{\boldsymbol{\xi }}^{T} {\mathcal{}{N}^e_{(\xi )}}^T) = \mathbf{X}^e\;\tilde{\mathbf{B}}_{\xi }^{e^{T}} </math>
+| style="text-align: center;" | <math>2\overline{\mu }\Delta \gamma + \frac{2}{3}H\Delta \gamma =f^{\mathrm{trial}} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.29)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.84)
 |}
-Where <math display="inline">\tilde{\boldsymbol{\nabla }}_{\boldsymbol{\xi }}</math> is the '''gradient operator''' for scalar-valued functions in element coordinates and <math display="inline">\tilde{\mathbf{B}}_{\xi }^{e}</math> is the matrix of the gradient of <math display="inline">\mathcal{}{N}^e</math> for scalar-valued functions in the parent domain. The gradient operator for scalar-valued functions in both parent and physical domains (<math display="inline">\tilde{\boldsymbol{\nabla }}_{\boldsymbol{\xi }}</math> and <math display="inline">\tilde{\boldsymbol{\nabla }}</math>) are linked through the transpose of the inverse of the Jacobian matrix:
+and deriving it with respect to <math display="inline">\boldsymbol{g}_{n+1}</math>, the derivative <math display="inline">\dfrac{\partial \Delta \gamma }{\partial \boldsymbol{g}_{n+1}}</math> is obtained
-<span id="eq-3.30"></span>
+<span id="eq-3.85"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,541: / Line 2,579: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\tilde{\boldsymbol{\nabla }}={\mathbf{J}}^{e-T}\; \tilde{\boldsymbol{\nabla }}_{\boldsymbol{\xi }}\;\;\Rightarrow \;\;\tilde{\mathbf{B}}^{e} = \tilde{\nabla } \mathcal{}{N}^{e} = {\mathbf{J}^{e}}^{-T} \left(\tilde{\nabla }_{\xi } \;\mathcal{}{N}^{e}\right)= {\mathbf{J}^{e}}^{-T}\;\tilde{\mathbf{B}}_{\xi }^{e} </math>
+| style="text-align: center;" | <math>\frac{\partial \Delta \gamma }{\partial \boldsymbol{g}_{n+1}}=\left(\frac{1}{1+\dfrac{H}{3\overline{\mu }}}  \right)\left(\frac{1}{2}\frac{\partial \lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\partial \boldsymbol{g}_{n+1}} - \Delta \gamma \frac{\partial \overline{\mu }}{\partial \boldsymbol{g}_{n+1}}  \right)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.30)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.85)
 |}
-In the above expression, <math display="inline">\tilde{\mathbf{B}}^{e} \in \mathbb{R}^{n_{d} \times n^{e}} </math> is the matrix of gradient of shape functions for scalar valued functions in global coordinates. In a 3D problem:
+The derivative <math display="inline">\dfrac{\partial \overline{\mu }}{\partial \boldsymbol{g}_{n+1}} </math> is solved by firstly rewriting it in reference configuration with the use of Equation [[#eq-3.63|3.63]] as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,553: / Line 2,591: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\tilde{\mathbf{B}}^{e} = \begin{bmatrix}\frac{\partial N_1^e}{\partial x} & \frac{\partial N_2^e}{\partial x} & \cdots  & \frac{\partial N_{n_e}^e}{\partial x}\\ \\ \frac{\partial N_1^e}{\partial y} & \frac{\partial N_2^e}{\partial y} & \cdots  & \frac{\partial N_{n_e}^e}{\partial y}\\ \\ \frac{\partial N_1^e}{\partial z} & \frac{\partial N_2^e}{\partial z} & \cdots  & \frac{\partial N_{n_e}^e}{\partial z} \end{bmatrix} = \left[\begin{array}{cccc}{\tilde{\mathbf{B}}_{1}^{e}} & {\tilde{\mathbf{B}}_{2}^{e}} & {\boldsymbol{\cdots }} & {\tilde{\mathbf{B}}_{n^{e}}^{e}}\end{array}\right] </math>
+| style="text-align: center;" | <math>\frac{\partial \overline{\mu }}{\partial \boldsymbol{C}_{n+1}}=\frac{\partial \left(\frac{1}{3}\mu \left(\boldsymbol{C}^{p^{-1}}_n:\boldsymbol{C}_{n+1} \right)J^{-\frac{2}{3}}_{n+1} \right)}{\partial \boldsymbol{C}_{n+1}}=\frac{1}{3}\mu \left(\frac{\partial J^{-\frac{2}{3}}_{n+1}}{\partial \boldsymbol{C}_{n+1}} + J^{-\frac{2}{3}}_{n+1}\frac{\partial \left(\boldsymbol{C}^{p^{-1}}_n:\boldsymbol{C}_{n+1} \right)}{\partial \boldsymbol{C}_{n+1}}\right)</math>
+|-
+| style="text-align: center;" | <math> =\frac{1}{3}\overline{\mu } \boldsymbol{C}^{-1}_{n+1}+\frac{1}{3}\mu J^{-\frac{2}{3}}\boldsymbol{C}^{p^{-1}}_n </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.31)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.86)
 |}
-However, the displacement field to be computed is a '''vector field''', so a matrix of symmetric gradient of shape functions for vector-valued functions in the physical domain <math display="inline">\mathbf{B}^e</math> is to be defined. For a 3D problem:
+and, then, by performing a push-forward transformation with <math display="inline">\boldsymbol{F}_{n+1}</math>, the derivative in current configuration is obtained
+<span id="eq-3.87"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,565: / Line 2,606: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathbf{B}}^{e} = \begin{bmatrix}\frac{\partial N_1^e}{\partial x} & 0 &  0& \cdots  &\frac{\partial N_{n_e}^e}{\partial x} & 0 &  0  \\ \\  0& \frac{\partial N_1^e}{\partial y} & 0 & \cdots & 0 & \frac{\partial N_{n_e}^e}{\partial y} &  0   \\ \\  0& 0 &  \frac{\partial N_1^e}{\partial z}& \cdots &  0 & 0 &  \frac{\partial N_{n_e}^e}{\partial z}   \\ \\  0& \frac{\partial N_1^e}{\partial z} & \frac{\partial N_1^e}{\partial y} &\cdots  &  0 & \frac{\partial N_{n_e}^e}{\partial z} &  \frac{\partial N_{n_e}^e}{\partial y}   \\ \\  \frac{\partial N_1^e}{\partial z}& 0 & \frac{\partial N_1^e}{\partial x} &\cdots  &  \frac{\partial N_{n_e}^e}{\partial z} & 0 &  \frac{\partial N_{n_e}^e}{\partial x}  \\  \\  \frac{\partial N_1^e}{\partial y}& \frac{\partial N_1^e}{\partial x} & 0 &\cdots  &  \frac{\partial N_{n_e}^e}{\partial y} & \frac{\partial N_{n_e}^e}{\partial x} &  0    \end{bmatrix} = \left[\begin{array}{cccc}{{\mathbf{B}}_{1}^{e}} & {{\mathbf{B}}_{2}^{e}} & {\boldsymbol{\cdots }} & {{\mathbf{B}}_{n^{e}}^{e}}\end{array}\right] </math>
+| style="text-align: center;" | <math>\frac{\partial \overline{\mu }}{\partial \boldsymbol{g}_{n+1}}=\frac{1}{3}\mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}=\frac{1}{3}\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert \boldsymbol{n}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.32)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.87)
 |}
-In order to illustrate this section, which may seem a little bit abstract to the reader, let's return to the '''example''' presented in figure [[#img-3.1|3.1]]. Imagine a 1D problem, where the domain has been split into smaller straight lines (elements). Each element has at least two nodes where the displacements have to be computed. It is not the only possible case: one can compute the displacements at more than two points by increasing the number of nodes of the element. In 1D, an element with two nodes is called linear; if it has three nodes, quadratic; if four, cubic; and so on. These names refer to the '''order''' of the polynomial that has to be used as shape function in each case.
+The derivative <math display="inline">\dfrac{\partial \lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\partial \boldsymbol{g}_{n+1}}</math> is solved by introducing the following equation
-The key is that no matter the length of the element <math display="inline">h^e</math> in the physical domain. When mapped to the parent domain, the element boundaries go from <math display="inline">\xi = -1</math> to <math display="inline">\xi = +1</math>. Is in this parent domain where the shape functions <math display="inline">\mathbf{N}^e</math> are defined: as the element length is normalised, the shape functions can be the same for '''all''' elements.  Recall that in the FEM, a shape function <math display="inline">N^e_a</math> is restricted to have a value of <math display="inline">\mathbf{1}</math> at node <math display="inline">\mathbf{a}</math>, and <math display="inline">\mathbf{0}</math> at every other node. Notice that in order to accomplish the latter, there will be as many shape functions as nodes. With all these properties in mind, the following figures represent shape functions of different order for a 1D element defined in the parent domain:  <div id='img-3.2a'></div>
+<span id="eq-3.88"></span>
-<div id='img-3.2b'></div>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-<div id='img-3.2c'></div>
-<div id='img-3.2'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-linear_shape2.png|600px|]]
+|
-|[[Image:Draft_Samper_987121664-monograph-quadshape.png|600px|]]
+{| style="text-align: left; margin:auto;width: 100%;"
-|- style="text-align: center; font-size: 75%;"
-| (a)
-| (b)
 |-
-| colspan="2"|[[Image:Draft_Samper_987121664-monograph-cubicshape.png|600px|]]
+| style="text-align: center;" | <math>\frac{\partial \lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\partial \boldsymbol{g}_{n+1}}=J^{-\frac{2}{3}}\mathrm{dev}\frac{\partial \lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\partial \overline{\boldsymbol{g}}_{n+1}}  </math>
-|- style="text-align: center; font-size: 75%;"
+|}
-|  colspan="2" | (c)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.88)
-|- style="text-align: center; font-size: 75%;"
-| colspan="2" | '''Figure 3.2:''' 1D shape functions for linear (a), quadratic (b) and cubic (c) isoparametric elements.
 |}
-The same reasoning that has been followed for a 1D element can be developed for 2D and 3D elements. In those cases, many different types of elements can be defined, depending on their geometric properties and number of nodes. The shape functions will then have two or three spatial variables, and are to be defined for each type of element. For instance, in order to complete the previous example, let's consider the case of a linear triangular element:  <div id='img-3.3'></div>
+where <math display="inline">\overline{\boldsymbol{g}}:=J^{-\frac{2}{3}}\boldsymbol{g}</math> and the derivative <math display="inline">\dfrac{\partial \lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\partial \overline{\boldsymbol{g}}_{n+1}}</math>, according to <span id='citeF-153'></span>[[#cite-153|[153]]], is defined as
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-linear_triangular.png|420px|Representation of a triangular linear element in both parent and physical domains]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 3.3:''' Representation of a triangular linear element in both parent and physical domains
+|-
+| style="text-align: center;" | <math>\frac{\partial \lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\partial \overline{\boldsymbol{g}}_{n+1}}=\frac{1}{2\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }(2\mu \mathrm{tr}\boldsymbol{b}^e\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert \boldsymbol{n}+2 J^{\frac{2}{3}}\left(\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert ^2  \boldsymbol{n}^2 \right)</math>
+|-
+| style="text-align: center;" | <math> -\frac{2}{3} J^{-\frac{2}{3}}\mu ^2\left(\boldsymbol{b}^e:\mathrm{dev}\boldsymbol{b}^e \right)\boldsymbol{I} ) </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.89)
 |}
-In the parent domain, the nodes are always listed in '''counter-clockwise''' direction. As can be seen, no matter the shape of the triangle in the physical domain, once it is mapped into the parent space all the elements are geometrically identical. The matrix of shape functions of the above element and its gradient in the parent domain are, then:
+By substituting this last equation in [[#eq-3.88|3.88]], the final expression is obtained
+<span id="eq-3.90"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,606: / Line 2,646: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathcal{}{N}^{e}=\left[\begin{array}{llll}1 {-\xi }  {-\eta } & {\xi } & {\eta }\end{array}\right]\;,\;\;\;\tilde{\boldsymbol{B}}_{\xi }^{e}=\left[\begin{array}{lll}{-1} & {1} & {0} \\ {-1} & {0} & {1}\end{array}\right] </math>
+| style="text-align: center;" | <math>\frac{\partial \lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\partial \boldsymbol{g}_{n+1}}=\overline{\mu }\left(\boldsymbol{n}+\frac{\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\overline{\mu }}\mathrm{dev}\boldsymbol{n}^2 \right)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.33)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.90)
 |}
-===3.3.1 The Gauss Quadrature integration method===
+Having defined Equations [[#eq-3.87|3.87]] and [[#eq-3.90|3.90]], it is possible to write again Equation [[#eq-3.85|3.85]] as
-Although the domain has been divided into smaller elements, the integrals presented earlier are still difficult to compute, as those elements have a wide variety of '''different boundaries'''. If the geometry of the element is sort of complex (nothing unusual in 3D), the integrals would have to be evaluated along those boundaries, which also change from one element to another. The solution to this problem has been presented in the previous section: if the element is mapped into the parent domain, its geometric properties are kind of normalised. The integrals can then be evaluated in the parent domain, and later on be transformed into global coordinates by using the Jacobian matrix (recall equation [[#eq-3.30|3.30]]).
+<span id="eq-3.91"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\frac{\partial \Delta \gamma }{\partial \boldsymbol{g}_{n+1}}=\left(\frac{1}{1+\dfrac{H}{3\overline{\mu }}}  \right)\left(\frac{1}{2}\overline{\mu }\boldsymbol{n}+\frac{\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{2\overline{\mu }}\mathrm{dev}\boldsymbol{n}^2-\frac{\Delta \gamma }{3}\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert \boldsymbol{n}\right)  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.91)
+|}
-Now the question lies in which method use to evaluate those integrals. For sure, the analytic solution is discarded as it would take too much time. From among all the numeric integration methods, one of the most widespread is Gauss Quadrature. It is a fast and simple method, that give great approximations with few operations. Basically, it reduces the integral to a '''summation''' of function values at specific points (Gauss points), multiplied by a certain factor called weight. The great advantage of this method is that as all the elements of the same type are mapped equally in the parent domain, the values of the weights and gauss points will be the same for all elements.
+The last derivative to be solved is
-Imagine that the integral to evaluate <math display="inline">I</math> is of some function <math display="inline">f : \overline{\mho }^{e} \rightarrow \mathbb{R}</math> over the physical domain. First, the integral has to be mapped into the parent domain through a change of variable, accomplished by using the Jacobian matrix of the transformation <math display="inline">\boldsymbol{x}=\boldsymbol{x}(\xi )</math>:
+<span id="eq-3.92"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,624: / Line 2,672: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>I=\int _{\mho ^{e}} f d \mho =\int _{\mho _{\xi }} <code>J</code>^{e}(\xi ) f(\xi ) d \mho _{\xi } </math>
+| style="text-align: center;" | <math>2\frac{\partial \boldsymbol{n}}{\partial \boldsymbol{g}_{n+1}}=2\frac{\partial \left(\dfrac{\mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}}{\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }\right)}{\partial \boldsymbol{g}_{n+1}}=\frac{1}{\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }\left( \frac{\partial \left(\mathrm{dev}\boldsymbol{\tau }_{n+1}^{trial}\right)}{\partial \boldsymbol{g}_{n+1}}-\boldsymbol{n}\otimes 2\frac{\partial \lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\partial \overline{\boldsymbol{g}}_{n+1}} \right)</math>
+|-
+| style="text-align: center;" | <math> =\frac{1}{\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }\left(\boldsymbol{\mathrm{C}}^{\mathrm{trial}}_{dev}-2\overline{\mu }\boldsymbol{n}\otimes \left(\boldsymbol{n}+\frac{\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\overline{\mu }}\mathrm{dev}\boldsymbol{n}^2  \right)\right)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.34)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.92)
 |}
-Where <math display="inline"><code>J</code></math> stands for the determinant of the Jacobian. In 3D, <math display="inline">\mathrm{d} \Omega _{\xi }=\mathrm{d} \xi \;\mathrm{d} \eta \;\mathrm{d} \zeta </math>. As Gaussian rules for integrals in several dimension are constructed by using 1D Gaussian rules on each coordinate separately, hereinafter only the one-dimensional Gauss Quadrature will be considered. The goal of the method is to find a set of Gauss points <math display="inline">\xi _g</math> and weights <math display="inline">w_g</math> such that:
+By substituting the expressions of Equations [[#eq-3.87|3.87]], [[#eq-3.91|3.91]] and [[#eq-3.92|3.92]] in Equation [[#eq-3.82|3.82]], the spatial constitutive tensor is finally obtained
+<span id="eq-3.93"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,636: / Line 2,687: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>I=\int _{\mho _{\xi }} <code>J</code>^{e}(\xi ) f(\xi ) d \mho _{\xi } \approx \sum _{g=1}^{m} w_{g} <code>J</code>^{e}(\xi _g)  f\left(\xi _{g}\right) </math>
+| style="text-align: center;" | <math>\boldsymbol{\mathrm{C}}^{\mathrm{ep}}=\boldsymbol{\mathrm{C}}^{\mathrm{trial}}-\beta _1\boldsymbol{\mathrm{C}}^{\mathrm{trial}}_{dev}-2\overline{\mu }\beta _3\boldsymbol{n}\otimes \boldsymbol{n}-2\overline{\mu }\beta _4[\boldsymbol{n}\otimes \mathrm=\boldsymbol{n}^2]^{sym}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.35)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.93)
 |}
-The above expression give exact results only if <math display="inline"><code>J</code>^{e}(\xi ) f(\xi )</math> is a '''polynomial'''. If not, only approximate results are obtained, whose accuracy improves as more Gauss points are used. Supposing that the integrand <math display="inline">q(\xi ) = \mathbf{J}^{e}(\xi ) f(\xi )</math> is a polynomial of order <math display="inline">p</math>:
+where the coefficients <math display="inline"> \beta _1, \beta _3 </math> and <math display="inline"> \beta _4 </math> are expressed as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,648: / Line 2,699: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>q(\xi )=\sum _{i=0}^{p} \alpha _{i} \xi ^{i}\;\;\Rightarrow \;\;I=\sum _{i=0}^{p} \alpha _{i} \int _{\mho _{\xi }}^{ } \xi ^{i} d \mho _{\xi }=\sum _{i=0}^{p} \alpha _{i}\left(\sum _{g=1}^{m} w_{g} \xi _{g}^{i}\right) </math>
+| style="text-align: center;" | <math>\beta _1=2\overline{\mu }\frac{\Delta \gamma }{\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }, </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.36)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.94)
 |}
-The values of the Gauss points and weights are obtained by '''exactly''' evaluating each monomial <math display="inline">\xi ^{i}</math>, with <math display="inline">i=1,2 \ldots p</math>:
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\beta _2=\frac{2}{3}\left[1 - \frac{1}{1 + \dfrac{H}{3\overline{\mu }}} \right]\frac{\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\overline{\mu }}\Delta \gamma , </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.95)
+|}
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,660: / Line 2,719: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\sum _{g=1}^{m} w_{g} \xi _{g}^{i} = \left.\frac{\xi ^{i+1}}{i+1}\right|_{\Omega _{\xi }} ^{} \; \xrightarrow=  \;\;= \; \left\{\begin{array}{ll}{0} & {{ if } \quad i=1,3,5 \ldots } \\ {\frac{2}{i+1}} & {{ if } \quad i=0,2,4,6 \ldots }\end{array}\right. </math>
+| style="text-align: center;" | <math>\beta _3=\frac{1}{1 + \dfrac{H}{3\overline{\mu }}}-\beta _1+\beta _2, </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3.37)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.96)
 |}
-The previous expressions lead to a '''linear system''' of <math display="inline">p+1</math> equations with <math display="inline">2m</math> unknowns. If <math display="inline">q</math> results to be a polynomial, the exact integration requires from <math display="inline">m \geq \frac{p+1}{2}</math> points. Gauss points and weights will be different depending on the number of points <math display="inline">m</math> used. The values of the Gauss points and weights up to 5 points are presented in the following table:
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
-|+ style="font-size: 75%;" |<span id='table-3.1'></span>'''Table. 3.1''' Gauss points positions and weights <span id='citeF-9'></span>[[#cite-9|[9]]]
 |-
 |
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\beta _4=\left[\frac{1}{1 + \dfrac{H}{3\overline{\mu }}} - \beta _1 \right]\frac{\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert }{\overline{\mu }}, </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.97)
 |}
-=4 FEM programming=
+<math display="inline"> \boldsymbol{\mathrm{C}}^{\mathrm{trial}} </math> and <math display="inline"> \boldsymbol{\mathrm{C}}^{\mathrm{trial}}_{dev} </math> as in Equations [[#eq-3.45|3.45]] and [[#eq-3.52|3.52]], respectively.
-Now that the elastodynamic model for rotating structures has been formulated in terms of the FEM, it is time to tackle its programming. This section covers the different processes involved in the simulation and how the <math display="inline">MATLAB</math> solver works. In this chapter, only the static case is to be addressed, as the methods used to solve the dynamic problem will be reviewed later in the report.
+When the use of a mixed formulation, with mean stress (<math display="inline"> p </math>) and displacements (<math display="inline">\boldsymbol{u}</math>) as primary variables, might be required, the reader has to refer to Equation [[#eq-3.44|3.44]] for the definition of the Kirchhoff stress tensor and to Equation [[#eq-3.93|3.93]] for the definition of the spatial constitutive tensor, keeping in mind that its volumetric part, <math display="inline"> \boldsymbol{\mathrm{C}}{\mathrm{trial}}_{vol} </math>, is obtained as prescribed by Equation [[#eq-3.51|3.51]].
-==4.1 Preprocessing==
+==3.3 Hyperelastic - Mohr-Coulomb plastic law==
-All the finite element simulations, carried out or not by commercial codes, start with the model definition, a critical phase widely known as preprocessing. This first step begins with the definition of the '''geometric domain''' of the problem: in this case, the structure. Nowadays, the most widespread method to define the geometry is by using computer-aided design (CAD) software. Examples of commercial CAD programs are ''CATIA'', ''Rhino'' and ''SolidWorks''.
+The <math display="inline">J_2</math> plastic law, presented in the previous section, is based on the theory which works fine for the modelling of plastic behaviour of metals. As it can be noted, this kind of theory is pressure insensitive since the yield limit does not depend on the mean stress, also graphically represented by Figure [[#img-3.1|3.1]]. For materials, such as soils and other granular materials, whose behaviour is pressure-dependent, the employment of the plastic law, presented in Section [[#lb-3.2|3.2]], might be inappropriate. In this regard, one of the most used plastic model in geotechnical engineering is the Mohr-Coulomb strength theory. This constitutive law is a phenomenological model, based on the fundamental assumption that the macroscopic plastic behaviour is the result of the microscopic mechanism of friction sliding between the single grains which compose the bulk.  This concept is expressed by one of the most important failure criteria proposed by Coulomb in 1776
-Once the geometry is defined, it is exported to what is known as mesh generation software. The final goal of the latter is to create a '''mesh''': a subdivision of the geometry into discrete cells known as elements. As it has been commented in the previous chapter, many different types of elements exist, so its type, length, area and other properties are to be defined. Notice that meshing is not a trivial process as geometries can be quite intricate. In order to accurately capture the input domain geometry, the mesh generation is often aided by computer algorithms. In this step, the domain boundary is also to be defined. Once the mesh has been generated, the software computes what is known as '''connectivity matrix''', which contains all the information about the mesh and defines the relations between elements and nodes (see for instance the numbers printed in figure [[#img-4.1|4.1]]a).
+<span id="eq-3.98"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\tau =c-\sigma _n\, \mathrm{tan}\phi   </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.98)
+|}
-The connectivity matrix <math display="inline">\mathbf{CN }\in \mathbb{R}^{n_{el} \times n^{e}}</math>, also known as element connectivity array, stores all the element definitions. It is an array of integers numbers where the <math display="inline">\mathbf{(e\,,\,a)}</math> cell contains the global number of the <math display="inline">\mathbf{a} — th</math> node of element <math display="inline">\mathbf{e}</math>. It is an essential matrix as it stores all the information regarding the mesh, and it is crucial when transforming from global to local coordinates.
+which states that the plastic flow starts when the state of stress on a specific plane exceeds the shear strength (<math display="inline">\tau </math>) which is a function of the normal stress <math display="inline">\sigma _n</math>, the material constants of cohesion <math display="inline">c</math> and internal friction angle <math display="inline">\phi </math>. By using the Mohr plane representation, where it is possible to visualize the shear stresses as function of the normal stresses, the Coulomb's failure criterion is shown in Figure [[#img-3.2|3.2]].
-Quite often, commercial programs allow both designing and meshing. This is in fact the case of '''GiD''', the pre and postprocessing tool used during this project. However, its designing tools are quite basic, and so it admits the importation of models from other CAD softwares. Nonetheless, those tools are quite useful when editing an imported geometry in order to simplify the meshing procedure. Figure [[#img-4.1|4.1]] illustrates the meshing of a 2D figure with GiD, using triangular and quadrilateral elements. In some cases, the mesh can be very simple (for instance a grid in figure [[#img-4.1|4.1]]a), whereas other geometries require from more complex meshes.
+<div id='img-3.2'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-MohrCoulombMohr_plane.png|444px|Coulomb's failure criterion in Mohr's plane representation ]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding-bottom:10px;"| '''Figure 3.2:''' Coulomb's failure criterion in Mohr's plane representation
+|}
-The next step is to define, for each element, material properties (density, elastic coefficients, heat conductivity, etc.) and loadings (volumetric forces, internal heat sources, etc.). The final part of the preprocessing phase consists in defining boundary conditions, which in our case is to prescribe displacements and tractions along the boundary. Commercial softwares usually include a wide library of material properties and loading scenarios to carry out these last steps.
+According to this stress representation, Equation [[#eq-3.98|3.98]] can be seen as a failure envelope, which can be experimentally determined: the failure occurs when the Mohr's circle is just tangent to the failure envelope.
-<div id='img-4.1a'></div>
+In this Section, a Mohr-Coulomb plastic law for finite strains is presented.  For the derivation of the formulas which are the expressions of the stress return and elasto-plastic moduli, the theory presented by Simo in <span id='citeF-155'></span><span id='citeF-156'></span>[[#cite-155|[155,156]]] is exploited. In these works, the main idea lies on modelling the elastic response by the use of principal stresses, which allows to extend the use of a small strain return mapping in stress space to the finite deformation regime. In Appendix [[#A Plastic flow rule in finite strains regime|A]], the plastic flow rule within the multiplicative plastic framework is presented and the form in terms of Hencky strains is derived.   According to this simplification, the following uncoupled form of the ''specific strain energy function'' is assumed
-<div id='img-4.1b'></div>
-<div id='img-4.1'></div>
+<span id="eq-3.99"></span>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-quad_mesh.png|507px|]]
+|
-|[[Image:Draft_Samper_987121664-monograph-triag_mesh.png|515px|]]
+{| style="text-align: left; margin:auto;width: 100%;"
-|- style="text-align: center; font-size: 75%;"
+|-
-| (a)
+| style="text-align: center;" | <math>\Psi (\boldsymbol{\epsilon }^e_A) = \dfrac{1}{2}\lambda \left[\epsilon ^e_1 +\epsilon ^e_2 +\epsilon ^e_3\right]^2 + \mu \left[\left(\epsilon ^e_1\right)^2+\left(\epsilon ^e_2\right)^2+\left(\epsilon ^e_3\right)^2\right]  </math>
-| (b)
+|}
-|- style="text-align: center; font-size: 75%;"
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.99)
-| colspan="2" | '''Figure 4.1:''' Mesh representation using quadrilateral (a) and triangular (b) elements. In the case of the circle, the mesh has been generated using in-built algorithms.
 |}
-As GiD does not implement this capability, this last part of the preprocessing is carried out with <math display="inline">MATLAB</math>. Once the mesh is created, GiD exports a raw file (''.msh'' extension) including the connectivity matrix, the nodal coordinates, the element typology and boundary information. Those are later recovered and imported to <math display="inline">MATLAB</math>. Material properties are manually assigned to each element, as well as the nodal values of body forces and prescribed boundary displacements and tractions.
+quadratic in the principal Hencky strains <math display="inline"> \boldsymbol{\epsilon }^e </math>, defined as
-==4.2 Solver==
+<span id="eq-3.100"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\boldsymbol{\epsilon }^e = {{\mathrm{ln}}}(\boldsymbol{\lambda })  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.100)
+|}
-Once the formulation of the problem is completed and all the information regarding structure's geometry, material properties, loads and boundary conditions is gathered in a <math display="inline">MATLAB</math> file, it is time to solve the elastostatic equations. In fact, the equations to be solved correspond to the approximated weak form of the problem stated in terms of the FEM (see equation [[#eq-3.21|3.21]]). In this chapter, only the '''static form''' of the problem is to be tackled — that is, <math display="inline">\ddot{\mathbf{d}} = \dot{\mathbf{d}}=0</math> :
+with <math display="inline"> \boldsymbol{\lambda } </math> the eigenvalues of the left Cauchy&#8211;Green deformation tensor <math display="inline"> \boldsymbol{b}^e </math>, which can be calculated according to its spectral decomposition
-<span id="eq-4.1"></span>
+<span id="eq-3.101"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,716: / Line 2,799: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{d}_{\mathrm{l}}=\mathbf{K}_{ \mathrm{ll}}^{-1}\left(\mathbf{F}_{\mathrm{l}}-\mathbf{K}_{\mathrm{lr}}\; \overline{\mathbf{u}}\right) </math>
+| style="text-align: center;" | <math>\boldsymbol{b}^e = \sum \limits _{A=1}^3 \boldsymbol{\lambda }_A^2 \boldsymbol{m}^{A}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.1)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.101)
 |}
-It is a first step in the understanding of the whole phenomena, and the developed code is going to be re-used later on when studying the dynamic case. The code is developed in <math display="inline">MATLAB</math>, with its main working algorithm schematised in the following figure:   <div id='img-4.2'></div>
+where <math display="inline"> \boldsymbol{m}^{A} </math> are the eigenbases associated with <math display="inline"> \boldsymbol{\lambda }_A </math>. If an isotropic elastic response is assumed, by defining the spectral decomposition of the Kirchhoff stress tensor <math display="inline"> \boldsymbol{\tau } </math> as
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+<span id="eq-3.102"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-scheme1.png|600px|MATLAB main code scheme for elastostatic problems]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 4.2:''' <math>MATLAB</math> main code scheme for elastostatic problems
+|-
+| style="text-align: center;" | <math>\boldsymbol{\tau }=\sum \limits _{A=1}^3 \boldsymbol{\tau }_A \boldsymbol{m}^{A}  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.102)
 |}
-As it can be suspected from the previous figure, the main task of the solver is to compute the matrix <math display="inline">\mathbf{K}</math> and vector <math display="inline">\mathbf{F}</math> from equation [[#eq-3.13|3.13]]. Using the local formulation presented in the previous chapter, the computation of the latter can be split into a simpler problem, regarding only '''elemental'''  matrices <math display="inline">\mathbf{K}^e</math> and vectors <math display="inline">\mathbf{F}^e</math>. Once they are obtained for all the elements of the domain, the elemental matrices are joined together into a global stiffness matrix <math display="inline">\mathbf{K}</math> and force vector <math display="inline">\mathbf{F}</math> through a process called assembly. When <math display="inline">\mathbf{K}</math> and <math display="inline">\mathbf{F}</math> are known, equation [[#eq-4.1|4.1]] results in a linear system of algebraic equations, easily handled with native <math display="inline">MATLAB</math> functions. A further explanation regarding elemental matrix computation and assembly can be found in Annex [[#B Elemental matrix computation|B]].
+it can be observed that the eigenbasis <math display="inline"> \boldsymbol{m}^{A} </math> of Equation [[#eq-3.101|3.101]] are the same of those of Equation [[#eq-3.102|3.102]].
-Once the displacement field is known, the final step of the solver consists in finding the reactions and the strain field. Computation of '''elastic reactions''' is straightforward, as they are obtained from equation [[#eq-3.21|3.21]]. On the other hand, the '''strain field''' is to be computed in the local frame, element by element. Strains are defined as the spatial derivative of displacements (see equation [[#eq-A.7|A.7]]), which can be written in terms of the FEM as:
+With the ''specific strain energy function'' of Equation [[#eq-3.99|3.99]] and the aforementioned assumption of isotropy, the stress-strain relation in principal axes takes the form
-<span id="eq-4.2"></span>
+<span id="eq-3.103"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,739: / Line 2,827: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{\varepsilon }=\nabla ^{s} \mathbf{{u}} \;\;\;\Rightarrow \;\;\; \boldsymbol{\varepsilon }^h = \nabla ^{s} \mathbf{{u}}^h = \overset{n_{el}}{\underset{e=1}{\mathbf{A}}} \mathbf{B}^{e} \mathbf{d}^{e}  \;\;\;\Rightarrow \;\;\; \boldsymbol{\sigma }^h = \overset{n_{el}}{\underset{e=1}{\mathbf{A}}}  \mathbf{C}\; \mathbf{B}^{e} \mathbf{d}^{e}  </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau }=\mathrm{a}\boldsymbol{\epsilon }^e  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.2)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.103)
 |}
-Where <math display="inline">\mathbf{A}</math> represents the assembly operator. As <math display="inline">\mathbf{B}^{e}</math> is approximated through Gauss Quadrature, strains and stresses are only computed at the Gauss points <math display="inline">\xi _g</math>.
+with <math display="inline"> \mathrm{a} = K \boldsymbol{I}\otimes \boldsymbol{I} + 2\mu \left[\mathrm{I} - \frac{1}{3}\boldsymbol{I}\otimes \boldsymbol{I} \right]</math> being the Hencky elastic tensor.
-===4.2.1 Efficient assembly===
-In the FEM, computation of global matrices is never done in a single step. Instead, elemental matrices are computed element by element, and then joined together through a process called '''assembly''', constructing this way the global matrix. The intuitive method to perform the assembly is quite simple to program: elemental matrices are computed one by one, and then allocated into a bigger, global matrix using the connectivity array. The connectivity array <math display="inline">\mathbf{CN}</math> links the local numeration of the degrees of freedom with the global one, specifying the row and column of the global matrix where each component of the elemental matrix has to be placed. For a one-dimensional case, the classical assembly of a matrix <math display="inline">\mathbf{M}</math> can be programmed as follows:   <pre>for e = 1 : n_el %Loop over all the elements
+As depicted in Figure [[#img-3.3|3.3]], the Mohr-Coulomb criterion comprises six planes in principal stress space, forming six corners and a common vertex on the tension side of the hydrostatic axis.
-   M_e = Compute_Me(. . .); %Function that computes the elemental matrix
-   for a = 1 : n_e %Loop over the nodes of the element e
-       for b = 1 : n_e
-          A = CN(e,a); B = CN(e,b); %Transformation from local to global
-          M_glo(A,B) = M_glo(A,B)+M_e(a,b); %Allocate M_e into the global matrix
-       end
-   end
-end</pre>
-The previous algorithm has two main drawbacks. The first one is related with the '''memory''' large matrices take up. Each cell of a <math display="inline">MATLAB</math> matrix is defined as ''double'', a type of variable that uses 8 bytes of memory. Using the traditional approach, a 3D system with 30000 nodes would involve matrices with a size bigger than 7.5 gigabytes, which is clearly infeasible. But if one takes closer look to the matrices involved in the FEM, will discover that most of the cells are filled with zeros. Matrices in which most of the elements are zero are known as ''sparse'' matrices.   <div id='img-4.3'></div>
+<div id='img-3.3a'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+<div id='img-3.3b'></div>
+<div id='img-3.3'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-SPARSE.png|390px|Representation of a sparse matrix. Blue squares are non-zero values.]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-mohr_coulomb_model-1cm-0cm-14_5cm-0cm.png|300px|]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-mohr_coulomb_model-16cm-1cm-1cm-1cm.png|300px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 4.3:''' Representation of a sparse matrix. Blue squares are non-zero values.
+| colspan="2" style="padding:10px;"| '''Figure 3.3:''' Mohr Coulomb model in the principal stress space (a) and in <math>\pi </math> plane (b)
 |}
-For sure, storing the zeros of a sparse matrix as double variables is a waste of memory. That is why most of the programming languages have in-built algorithms to store sparse matrices in more efficient ways. The key is to store only the non-zero values and their position in the matrix. Using this approach, systems with a large number of degrees of freedom can be tackled without memory problems.
+If the principal stresses are rearranged as
-The second disadvantage of the classical assembly method is related with its '''computational cost'''. A closer look to the presented algorithm reveals three nested loops. A local matrix has to be computed for each element, and then placed into the global matrix cell by cell. In <math display="inline">MATLAB</math>, this iteration is rather slow. For the static case, where matrices are computed once, it is not a big deal. But when dealing with the dynamic case, global matrices are to be computed repeatedly, increasing considerably the running time of the solver. This problem is solved by using a '''vectorised''' version of the assembly algorithm.
+<span id="eq-3.104"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\tau _1 \geqslant \tau _2 \geqslant \tau _3 ,  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.104)
+|}
-The key of the vectorised algorithm is that computes and assembles all the elemental matrices at once, avoiding iterations <span id='citeF-10'></span>[[#cite-10|[10]]]. It is more difficult to program than the classical algorithm, but its advantages are considerable. For instance, figure [[#img-4.4|4.4]] reveals an outstanding decrease of the computation time for 2D systems. As it can be spotted, although for few degrees of freedom the classical algorithm is quicker, the vectorised algorithm proves to be the most efficient solution as the size of the system grows.
+the stresses are returned to only one of the six faces, the primary yield plane.
-The vectorised assembly algorithm has been adapted for the rotating case from a set of codes given by prof. ''Hernández J.'', which were used in the context of composite materials. An exemplification of the vectorised assembly for a simple mesh can be found in Annex [[#C Example of vectorised assembly|C]].
+Following the reordering of principal stresses as described by Equation [[#eq-3.104|3.104]], the yield function and the plastic potential, in the case of Mohr-Coulomb plasticity, are respectively written as
-<div id='img-4.4'></div>
+<span id="eq-3.105"></span>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-Computation_time_2.png|420px|Comparison of the computation time of F, M and K using the classical and vectorised assembly (logarithmic scale). ]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 4.4:''' Comparison of the computation time of <math>\mathbf{F}</math>, <math>\mathbf{M}</math> and <math>\mathbf{K}</math> using the classical and vectorised assembly (logarithmic scale).
+|-
+| style="text-align: center;" | <math>f(\boldsymbol{\tau })=(\tau _1 - \tau _3)+(\tau _1 + \tau _3)\sin \phi - 2c \cos \phi   </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.105)
 |}
-===4.2.2 Approximated one-dimensional stationary solution===
+<span id="eq-3.106"></span>
-To conclude the ''solver'' section, let's exemplify the explained concepts approximating the solution of the system presented in [[#eq-2.26|2.26]]. It is the simplest case to tackle: a 1D blade rotating at constant speed with no vibration at all. It can be interpreted as a static elastic problem in the rotational frame, where deformations can only occur in the axial direction <math display="inline">\varsigma </math>. The goal is to find the '''approximated solution''' using the FEM, and assess its validity by comparing it with the analytic result obtained in [[#2.5.2 One-dimensional stationary solution|2.5.2]]. Recall the differential equation to solve:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,791: / Line 2,884: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{d^2 \mathbf{u}_\varsigma }{d \varsigma ^2}= -\frac{1}{\textrm{E}}\rho {\Omega }^{\, 2}(\varsigma +{{\mathbf{u}}_\varsigma } ) </math>
+| style="text-align: center;" | <math>g(\boldsymbol{\tau })=(\tau _1 - \tau _3)+(\tau _1 + \tau _3)\sin \psi   </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.3)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.106)
 |}
-Using the FEM formulation, an approximated weak form of the equation can be written:
+where <math display="inline"> \phi </math> is the angle of internal friction, <math display="inline"> c </math> the cohesion and <math display="inline"> \psi </math> the dilation angle. The expressions of Equations [[#eq-3.105|3.105]] and [[#eq-3.106|3.106]] in the principal stress space are represented by planes and, accordingly, they can be rewritten as
+<span id="eq-3.107"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,803: / Line 2,897: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>-\int _{{\mho }} \nabla ^{s} {\mathbf{v}\,^h}^{T} \, \, \textrm{E} \;\;\nabla ^{s}  {\mathbf{u}}_{\varsigma }^h \, d \varsigma + \int _{{\mho }} {\mathbf{v}\,^h}^{T} \;\rho \; \Omega ^{2} \; {\mathbf{u}}_{\varsigma }^h \;\;d \varsigma + \int _{{\mho }} {\mathbf{v}\,^h}^{T} \;\rho \; \Omega ^{2} \; \;\varsigma d \varsigma = 0 </math>
+| style="text-align: center;" | <math>f(\boldsymbol{\tau })=\frac{\partial f}{\partial \boldsymbol{\tau }}\left(\boldsymbol{\tau } - \boldsymbol{\tau _A} \right)=\boldsymbol{a}_1^T\left(\boldsymbol{\tau } - \boldsymbol{\tau _A} \right)=k\tau _1 - \tau _3 - 2c \sqrt{k}=0  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.4)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.107)
 |}
-By replacing the test and trial function with its approximations in terms of the shape functions, the above equation yields to:
+<span id="eq-3.108"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,815: / Line 2,908: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\overset{\mathbf{K}}{\overbrace{\left(\underset{\mathbf{K}_{\hbox{static}}}{\underbrace{ \int _{{\mho }} \mathbf{B}^{T} \, \, \textrm{E} \;\; \mathbf{B}  \;\;d \varsigma }} -  \underset{\mathbf{K}_{\hbox{rot}}}{\underbrace{\int _{{\mho }}\mathbf{N}^{T} \;\rho \; \Omega ^{2} \; \mathbf{N} \;\;d \varsigma }} \right)}}\; \mathbf{d} = \overset{\mathbf{F}\; = \; \mathbf{F}_{\hbox{rot}}}{\overbrace{\int _{{\mho }} {\mathbf{N}}^{T} \;\rho \; \Omega ^{2} \; \;\varsigma \; d \varsigma }} </math>
+| style="text-align: center;" | <math>g(\boldsymbol{\tau })=\frac{\partial g}{\partial \boldsymbol{\tau }}\left(\boldsymbol{\tau } - \boldsymbol{\tau _A} \right)=\boldsymbol{b}_1^T\left(\boldsymbol{\tau } - \boldsymbol{\tau _A} \right)=m\tau _1 - \tau _3  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.5)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.108)
 |}
-Taking into account that displacements are prescribed along the set <math display="inline">\mathbf{r}</math> and unknown in <math display="inline">\mathbf{l}</math>, the above equation can be split into block matrices:
+where <math display="inline">\boldsymbol{a}_1=\left[k\quad 0\quad -1  \right]^T</math> and <math display="inline">\boldsymbol{b}_1=\left[m\quad 0\quad -1  \right]^T</math> are the gradients, and the constants <math display="inline">k</math> and <math display="inline">m</math> are respectively defined as
-<span id="eq-4.6"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,828: / Line 2,920: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{d}_{\mathrm{l}}=\mathbf{K}_{ \mathrm{ll}}^{-1}\left(\mathbf{F}_{\mathrm{l}}-\mathbf{K}_{\mathrm{lr}}\; \overline{\mathbf{u}}_{\varsigma }\right) </math>
+| style="text-align: center;" | <math>k=\frac{1+\mathrm{sin}\phi }{1-\mathrm{sin}\phi } </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.6)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.109)
 |}
-Previous to the solving of the above system of algebraic linear equations, global matrices <math display="inline">\mathbf{K}</math> and <math display="inline">\mathbf{F}</math> are to be known. As it has been discussed, the best choice is to decompose the latter into elemental matrices:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,840: / Line 2,930: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}=\overset{n_{el}}{\underset{e=1}{\mathbf{A}}} \mathbf{K}^{e}\;\;,\;\;\;\;\mathbf{F}=\overset{n_{el}}{\underset{e=1}{\mathbf{A}}} \mathbf{F}^{e} </math>
+| style="text-align: center;" | <math>m=\frac{1+\mathrm{sin}\psi }{1-\mathrm{sin}\psi } </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.7)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.110)
 |}
-Elemental matrices are to be calculated using the approach reviewed in Annex [[#B Elemental matrix computation|B]]. The first step is to define the type of element in the parent domain. For simplicity, let's assume '''linear elements'''. Recall the shape functions in the parent domain for a linear 1D element from figure [[#img-3.2|3.2]](a):
+===3.3.1 The stress regions===
+In the case of the Mohr-Coulomb plastic law, four types of stress returns and constitutive matrices are possible.In Figure [[#img-3.4a|3.4a]] the stress regions are shown: return to a yield plane (with condition of <math display="inline">f(\boldsymbol{\tau }=0)</math>), to the line which corresponds to triaxial compression (<math display="inline">l_1</math>), to the line which corresponds to triaxial tension (<math display="inline">l_2</math>) and to the apex point (point A).  In order to be able to know to which region to apply the return, boundary planes between these regions need to be defined. In the case of a linear yield criterion, boundary planes in the principal stress space are planes (Figure [[#img-3.4b|3.4b]]) and the solution of the problem is found by simply applying geometric arguments.
+<div id='img-3.4a'></div>
+<div id='img-3.4b'></div>
+<div id='img-3.4'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-stress_region-1cm-1cm-12_5cm-1cm.png|420px|]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-stress_region-12_5cm-0_5cm-1cm-1cm.png|420px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding-bottom:10px;padding-top:10px;"| '''Figure 3.4:''' Stress regions (a) and boundary planes (b) in the principal stress space
+|}
+By using the definition of a plane in the principal stress space
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,852: / Line 2,960: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{N}^e_{\xi } = \frac{1}{2}\left[(1-\xi ) \;\;(1+\xi ) \right]\;\; , \;\;\;\;\mathbf{B}^e_{\xi } = \frac{\mathrm{d} \mathbf{N}^e_{\xi } }{\mathrm{d} \varsigma } =  \frac{\mathrm{d} \mathbf{N}^e_{\xi } }{\mathrm{d} \xi } \; \frac{\mathrm{d} \xi }{\mathrm{d}\varsigma }  </math>
+| style="text-align: center;" | <math>p(\boldsymbol{\tau })=\boldsymbol{n}\left(\boldsymbol{\tau } - \boldsymbol{\tau _l} \right) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.8)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.111)
 |}
-To compute <math display="inline">\mathbf{B}^e_{\xi }</math>, the Jacobian of the transformation <math display="inline">\frac{\mathrm{d} \xi }{\mathrm{d}\varsigma }</math> is to be found:
+where <math display="inline">\boldsymbol{n}</math> is the normal to the plane and <math display="inline">\tau _l</math> a stress point laying on the plane, <math display="inline">p_{I-II}</math> and <math display="inline">p_{I-III}</math> are expressed as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,864: / Line 2,972: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{\mathrm{d} \xi }{\mathrm{d}\varsigma } = \left(\frac{\mathrm{d} \mathbf{N}^{e}}{\mathrm{d} \xi } \boldsymbol{\varsigma }^{e}\right)^{-1} = 2\left(-\varsigma _{1}^{e}+\varsigma _{2}^{e}\right)^{-1} = \frac{2}{h^e} </math>
+| style="text-align: center;" | <math>p_{I-II}(\boldsymbol{\tau })=\left(\boldsymbol{r}^p_1 \times \boldsymbol{r}^l_1 \right)^T \left(\boldsymbol{\tau } - \boldsymbol{\tau _A} \right)=0 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.9)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.112)
 |}
-Where <math display="inline">h^e</math> represents the length of the element. Then:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,876: / Line 2,982: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{B}^e_{\xi } = \frac{1}{h^e}\left[-1 \;\;\;\;+1 \right] </math>
+| style="text-align: center;" | <math>p_{I-III}(\boldsymbol{\tau })=\left(\boldsymbol{r}^p_2 \times \boldsymbol{r}^l_2 \right)^T \left(\boldsymbol{\tau } - \boldsymbol{\tau _A} \right)=0 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.10)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.113)
 |}
-With all, the computation of the elemental stiffness matrix is reduced to:
+with
+<span id="eq-3.114"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,888: / Line 2,995: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}^e_{\hbox{static}} =\int _{{\mho }} \mathbf{B}^{e^T} \, \, \textrm{E} \;\; \mathbf{B}^e  \;\;d \varsigma  = \int _{-1}^{1} \frac{1}{h^e}\left[-1 \;\;\;\;+1 \right]^T \textrm{E}\;\;\frac{1}{h^e}\left[-1 \;\;\;\;+1 \right]\;\frac{h^e}{2} d \xi = \frac{\textrm{E}}{h^e}\;\begin{bmatrix}1 & -1\\  -1 & 1 \end{bmatrix} </math>
+| style="text-align: center;" | <math>\boldsymbol{r}^l_1=\left[1\quad 1\quad k \right]^T  </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.114)
 |}
+and
+<span id="eq-3.115"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,897: / Line 3,008: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}^e_{\hbox{rot}} =-\int _{{\mho }} \mathbf{N}^{e^T} \;\rho \; \Omega ^{2} \; \mathbf{N}^e  \;\;d \varsigma  = -\;\rho \; \Omega ^{2} \int _{-1}^{1} \frac{1}{2}\left[ (1-\xi ) \;\;(1+\xi ) \right]^T \frac{1}{2}\left[ (1-\xi ) \;\;(1+\xi )\right]\;\frac{h^e}{2} d \xi = </math>
+| style="text-align: center;" | <math>\boldsymbol{r}^l_2=\left[1\quad k\quad k \right]^T  </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.115)
 |}
+the vectors direction of lines <math display="inline">l_1</math> and <math display="inline">l_2</math>, <math display="inline">\boldsymbol{r}^p_1</math> and <math display="inline">\boldsymbol{r}^p_2</math> the vectors direction of the plastic corrector.
+By using geometric arguments, it is possible to identify the four stress regions without defining the planes <math display="inline">p_{II-IV}</math> and <math display="inline">p_{III-IV}</math>. For instance, by considering the parametric equation of a line in the principal stress space
+<span id="eq-3.116"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,906: / Line 3,023: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>= -\;\rho \; \Omega ^{2} \frac{h^e}{8}\;\int _{-1}^{1}\begin{bmatrix}(1-\xi )^2 & (1-\xi )\;(1+\xi )\\  (1-\xi )\;(1+\xi ) & (1+\xi )^2 \end{bmatrix} d \xi = -\rho \; \Omega ^{2}\frac{h^e}{6} \begin{bmatrix}2 & 1\\  1 & 2 \end{bmatrix} </math>
+| style="text-align: center;" | <math>l : \boldsymbol{\tau }=t \boldsymbol{r}^l + \boldsymbol{\tau _l}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.11)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.116)
 |}
-Recall that for the calculation of the elemental stiffness matrix, integrals are evaluated exactly. This is not the case of the force vector, which is approximated trough Gauss Quadrature. Replacing <math display="inline">\rho \; \Omega ^{2}\;\varsigma </math> by <math display="inline">\mathbf{f}</math>:
+where <math display="inline">t</math> is a parameter with unit of stress and <math display="inline">\boldsymbol{\tau _l}</math> a stress point on the line, the parametric equations of lines <math display="inline">l_1</math> and <math display="inline">l_2</math> are
+<span id="eq-3.117"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,918: / Line 3,036: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}^e =\int _{{\mho }} \mathbf{N}^{e^T}\;\mathbf{f} \;d \varsigma  = \int _{{\mho }} \mathbf{N}^{e^T}\left( \mathbf{N}^{e}\mathbf{f}^e\right)\;d \varsigma  =  \int _{-1}^{1} \mathbf{N}^{e^T}\mathbf{N}^{e} \rho ^e\; \Omega ^{2}\; \varsigma ^e \frac{h^e}{2} d \xi = </math>
+| style="text-align: center;" | <math>l_1 : \boldsymbol{\tau }=t_1 \boldsymbol{r}^l_1 + \boldsymbol{\tau _A}  </math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.117)
 |}
+<span id="eq-3.118"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,927: / Line 3,047: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>=\frac{h^e}{8} \sum _{g=1}^{m} w_g \rho ^e\; \Omega ^{2}\; \begin{bmatrix}(1-\xi _g)^2 & (1-\xi _g)\;(1+\xi _g)\\   (1-\xi _g)\;(1+\xi _g) & (1+\xi _g)^2  \end{bmatrix} \begin{bmatrix}\varsigma _1^e\\   \varsigma _2^e  \end{bmatrix} </math>
+| style="text-align: center;" | <math>l_2 : \boldsymbol{\tau }=t_2 \boldsymbol{r}^l_2 + \boldsymbol{\tau _A}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.12)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.118)
 |}
-As <math display="inline">\mathbf{N}^e</math> is linear in <math display="inline">\xi </math> and <math display="inline">\mathbf{f}</math> is also linear in <math display="inline">\varsigma </math>, three Gauss points (<math display="inline">m=3</math>) will suffice to evaluate the above integral exactly. The values of Gauss points and weights can be found in table [[#table-3.1|3.1]]. Once the elemental stiffness matrix and force vector are known, they are computed for each element of the domain and assembled into global matrix <math display="inline">\mathbf{K}</math> and vector <math display="inline">\mathbf{F}</math>. The last step before finding displacement field is to solve the system of equations stated in [[#eq-4.6|4.6]].
+where parameters <math display="inline">t_1</math> and <math display="inline">t_2</math> are defined in a way that at the apex point <math display="inline">t_1=t_2=0</math>; thus, when the condition <math display="inline">t_1>0</math> and <math display="inline">t_2>0</math> are both satisfied the predictor stress falls in Region IV. Below, in Table [[#table-3.1|3.1]] the conditions, which determine the stress regions and their corresponding return mapping, are listed.
-In order to '''compare''' the results with the exact solution, let's consider the same conditions as in the analytic case. The structure of study will be a blade with constant cross-section and <math display="inline">4\;m</math> radius, rotating at <math display="inline">60\;\hbox{rad/s}</math>. For an aluminium bar (<math display="inline">E = 70\;GPa</math>, <math display="inline">\rho = 2700\; kg/m^3</math>), the results are displayed in figure [[#img-4.5|4.5]] and compared with the exact solution. Recall that computation of stresses once nodal displacements are known is achieved using expression [[#eq-4.2|4.2]].
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-3.1'></span>Table. 3.1 Identification of stress regions
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |   Conditions
+| style="border-left: 2px solid;border-right: 2px solid;" | Region
+| style="border-left: 2px solid;border-right: 2px solid;" | Type of return
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> p_{I-II}\geq{0} </math> and <math display="inline"> p_{I-III}\leq{0} </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | I
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>f(\boldsymbol{\tau }=0)</math>
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> p_{I-II}<0 </math> and <math display="inline"> p_{I-III}<0 </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | II
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>l_1</math>
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> p_{I-II}>0 </math> and <math display="inline"> p_{I-III}>0 </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | III
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>l_2</math>
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> t_1>0 </math> and <math display="inline"> t_2>0 </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | IV
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>apex</math>
-<div id='img-4.5'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-APP1D2.png|600px|Approximated solution of equation [[#eq-2.26|2.26]] using 70 elements, compared with the exact result.  ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 4.5:''' Approximated solution of equation [[#eq-2.26|2.26]] using 70 elements, compared with the exact result.
 |}
-In the FEM, another input parameter to define is the number of elements into which the domain is divided. Theoretically, the approximated solution '''converges''' into the exact one as the number of elements increases. In order to assess this convergence, let's define the error associated to the approximated solution <math display="inline">\mathbf{e}</math> using the '''L2 norm''' of the distance between <math display="inline">\mathbf{u}_\varsigma </math> and <math display="inline">\mathbf{u}^h_\varsigma </math>:
+===3.3.2 Stress update in principal stress space===
+In a general three-dimensional framework the development of the return mapping derivation and implementation in presence of singularities may result tedious. However, if isotropic yield criteria are considered, it is possible to reduce the dimension of the problem from six to three. Further, by taking advantage of considering an isotropic linear yield criterion and a perfect plastic law, for the implementation of the implicit integration scheme in principal stress space, the theory presented in <span id='citeF-157'></span>[[#cite-157|[157]]] is followed. In this work, an efficient return algorithm, based on geometric arguments, is presented for infinitesimal deformation; as previously mentioned, if the ''specific strain energy function'' of Equation [[#eq-3.99|3.99]] is employed, the algorithmic procedure of <span id='citeF-157'></span>[[#cite-157|[157]]] can be extended also to the case of finite strains in a straightforward way, as addressed in Appendix [[#A Plastic flow rule in finite strains regime|A]]. In the following, the stress update formulas and elasto-plastic tangent tensor are presented for each type of return. These expressions are, then, used in Algorithm [[#algorithm-3.2|3.2]] where the algorithmic procedure is described in the framework of a finite strain plastic model.
+====3.3.2.1 Return to a plane====
+In this respect, the evolution equation in finite strains in terms of Hencky strains, derived in Appendix [[#A Plastic flow rule in finite strains regime|A]], is as follows
+<span id="eq-3.119"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,950: / Line 3,094: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\| \mathbf{e}\| _{L_{2}}=\left\|\mathbf{u}_\varsigma -\mathbf{u}^{h}_\varsigma \right\|_{L_{2}}=\left(\int _{\mho }\left(\mathbf{u}_\varsigma -\mathbf{u}^{h}_\varsigma \right)^{2} d \varsigma \right)^{\frac{1}{2}} </math>
+| style="text-align: center;" | <math>\boldsymbol{\epsilon }^e_{n+1} =\boldsymbol{\epsilon }^{e^{trial}}_{n+1}-\Delta \gamma \frac{\partial g_{n+1}}{\partial \boldsymbol{\tau }}   </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.13)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.119)
 |}
-In a similar way, one can define the error <math display="inline">\mathbf{e'}</math> associated to the derivative of <math display="inline">\mathbf{u}</math>, which in this case is linked with stresses. In order to evaluate those integrals, the domain is again split into elements where Gauss Quadrature is applied. As quite often the exact solution is not a polynomial, high order Gauss rules are needed. In general, the logarithm of the error varies linearly with the element size, and its slope depends on the order of the element:
+In addition, in order to obtain the expressions for the return to the yield surface, the following condition is considered
+<span id="eq-3.120"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,962: / Line 3,107: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\| \mathbf{e}\| _{L_{2}}=C h^{\alpha } </math>
+| style="text-align: center;" | <math>\left(\frac{\partial f_{n+1}}{\partial \boldsymbol{\tau }} \right)^T \boldsymbol{\tau }_{n+1}=0  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.14)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.120)
 |}
-Where <math display="inline">\alpha </math> is known as the rate of convergence of the element, while <math display="inline">C</math> is an arbitrary constant and <math display="inline">h</math> the element length (assumed constant). The parameter <math display="inline">\alpha </math> is equal to <math display="inline">p+1</math>, where <math display="inline">p</math> is the order of the used shape function. For the derivative of the error, the slope of convergence is one order lower. Recall that if obtaining the analytic exact solution were impossible, the computation of the error in the way that has been developed would not be practicable. In that case, in order to check the convergence of the method, '''relative error''' should be computed. That is, to compare the error of the function obtained using <math display="inline">n_{el}</math> elements, with a previous function obtained with <math display="inline">n_{el}-k</math> elements.
+which states that in case of perfect plasticity the strain increment must be tangential to the yield surface.
-==4.3 Postprocessing==
+By firstly substituting Equation [[#eq-3.119|3.119]] into Equation [[#eq-3.103|3.103]]
-The last step of every FEM simulation is analysis and evaluation of the solution results, often referred as postprocessing. Once the solution of the problem is computed, it is exported to a postprocess program that interprets, prints and plots the results. Among many other operations, these programs are used to '''draw''' the deformed structural shape, produce colour-coded plots (representing displacements, strains and stresses) and '''animate''' the dynamic behaviour of the system, if that were the case. Equilibrium check (assessing if reactions counteract external forces, for example) is also part of the postprocess stage.
+<span id="eq-3.121"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-In the previous example, results were graphed in a simple plot. This approach is unattainable for multidimensional problems with non-trivial geometry, and thus external programs are to be used. In this project, the postprocessing is carried out by '''GiD''', the same program used for the mesh definition. To illustrate '''GiD''' capabilities, let's consider a fixed-free cantilever beam under the effect of a distributed load <math display="inline">w = -20\;KN/m</math> along its upper side. The beam is shaped as a long rectangle (<math display="inline">0,2 \times 2 m</math>) and meshed using 1000 linear quadrilateral elements. The following figure represents its '''deformed configuration''', and stresses along the beam are represented using colours:
-<div id='img-4.6'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-distribtedload2d.png|480px|Deformed shape (×200) and stress (σₓ) colour map of a 2D beam under the effect of a distributed load on its upper side.  ]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 4.6:''' Deformed shape (<math>\times 200</math>) and stress (<math>\sigma _x</math>) colour map of a 2D beam under the effect of a distributed load on its upper side.
+|-
+| style="text-align: center;" | <math>\boldsymbol{\tau }_{n+1}=\mathrm{a}:\boldsymbol{\epsilon }^{e^{trial}}-\mathrm{a}:\Delta \gamma \frac{\partial g_{n+1}}{\partial \boldsymbol{\tau }}   </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.121)
 |}
-The colour legend has not been displayed as it is not from interest in this example. Remember that although displacement is continuous along the domain, its derivatives (stress and strains) are only computed at Gauss points. Modern postprocessing programs have in-built tools that blur the latter in order to hide discontinuities. The engineer has to be aware that misuse of blurring tools can induce notable errors.
-To both asses the accuracy and validate the 2D solver, FEM results can be contrasted with the '''analytic solution''' obtained from beam theory. In particular, vertical displacement along the mean line of the beam is the variable that is being compared in figure [[#img-4.7|4.7]]. Analytic results are those of a beam with squared cross section, whose deflection can be expressed as:
+and, then, Equation [[#eq-3.121|3.121]] in Equation [[#eq-3.120|3.120]], the following expression is obtained
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,991: / Line 3,134: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>y=\frac{-w x^{2}}{24 E I}\left(x^{2}-4 L x+6 L^{2}\right) </math>
+| style="text-align: center;" | <math>\left(\frac{\partial f_{n+1}}{\partial \boldsymbol{\tau }} \right)^T \left(\mathrm{a}:\boldsymbol{\epsilon }^{e^{trial}}-\mathrm{a}:\Delta \gamma \frac{\partial g_{n+1}}{\partial \boldsymbol{\tau }} \right)=0 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4.15)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.122)
 |}
-<div id='img-4.7'></div>
+After rearranging the terms of the last equation, the expression of <math display="inline">\Delta \gamma </math> is found
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-2dstaticbeamanalytic2.png|600px|Comparison of the approximated and theoretical vertical displacement of a beam under the effect of a distributed load.  ]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 4.7:''' Comparison of the approximated and theoretical vertical displacement of a beam under the effect of a distributed load.
+|-
+| style="text-align: center;" | <math>\Delta \gamma =\frac{\left(\dfrac{\partial f_{n+1}}{\partial \boldsymbol{\tau }} \right)^T : \mathrm{a}:\boldsymbol{\epsilon }^{e^{trial}}}{\left(\dfrac{\partial f_{n+1}}{\partial \boldsymbol{\tau }} \right)^T : \mathrm{a}:\dfrac{\partial g_{n+1}}{\partial \boldsymbol{\tau }}} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.123)
 |}
-=5 Dynamic response=
+By substituting the expression of <math display="inline">\Delta \gamma </math> in Equation [[#eq-3.121|3.121]]
-So far, the problem regarding static deformations in the noninertial frame is fully settled. However, although it can be used to determine initial conditions, the developed model is unable to account for '''time-dependant variables''' such as velocity or acceleration in the rotating frame. In other words: it cannot capture the phenomena of vibration. In order to study the dynamic response of the system, new approaches are to be considered. Hence, this chapter will review two main methodologies to tackle time-dependant problems: modal decomposition analysis and numerical time integration. A brief description regarding the singular value decomposition (SVD) will be presented at the end of this chapter, presenting a promising method to handle nonlinear dynamic systems.
+<span id="eq-3.124"></span>
-==5.1 Modal decomposition analysis==
-Modal analysis is one of the most widespread techniques to tackle structural vibration problems, for both discrete and continuous systems. The key of this method is that it studies the problem in the '''frequency domain''', rather than in the temporal one.  To start with, recall the equation describing a rotating structure in the noninertial frame of reference:
-<span id="eq-5.1"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,018: / Line 3,159: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M}_{\mathbf{ll}} \;\ddot{\mathbf{d}}_\mathbf{l} \;+\;\mathbf{D}_{\mathbf{ll}}\; \dot{\mathbf{d}}_\mathbf{l} \;+\;\mathbf{K}_{\mathbf{ll}}\; {\mathbf{d}}_\mathbf{l}=\mathbf{F}_\mathbf{l} - (\;\mathbf{M}_{\mathbf{lr}}\; \overline{\ddot{\mathbf{u}}} \;+\;\mathbf{D}_{\mathbf{lr}} \;\overline{\dot{\mathbf{u}}} \;+\;\mathbf{K}_{\mathbf{lr}}\; \overline{{\mathbf{u}}}) </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau }_{n+1}=\mathrm{a}:\left(\boldsymbol{\epsilon }^{e^{trial}} -  \dfrac{\left(\dfrac{\partial f_{n+1}}{\partial \boldsymbol{\tau }} \right)^T : \mathrm{a}:\boldsymbol{\epsilon }^{e^{trial}}}{\left(\dfrac{\partial f_{n+1}}{\partial \boldsymbol{\tau }} \right)^T : \mathrm{a}:\dfrac{\partial g_{n+1}}{\partial \boldsymbol{\tau }}} \frac{\partial g_{n+1}}{\partial \boldsymbol{\tau }}     \right)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.1)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.124)
 |}
-Where <math display="inline">\mathbf{K}</math>, <math display="inline">\mathbf{D}</math> and <math display="inline">\mathbf{F}</math> are the sum of two contributions: a first one typical of static conditions, and a particular second one accounting for the effect of rotation. For the sake of notational simplicity, on now on let's assume the following redefinitions:
+it can be defined the elasto-plastic fourth order constitutive tensor <math display="inline">\mathrm{a}^{ep}</math>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,030: / Line 3,171: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{matrix}\mathbf{M}_{\mathbf{ll}}\rightarrow \mathbf{M}\;\;,\;\;\;\;\;\;\mathbf{K}_{\mathbf{ll}}\rightarrow \mathbf{K}\;\;,\;\;\;\;\;\;\mathbf{D}_{\mathbf{ll}}\;\rightarrow \mathbf{D}\;\;,\;\;\;\;\;\;\mathbf{F}_\mathbf{l} - (\;\mathbf{M}_{\mathbf{lr}} \;\overline{\ddot{\mathbf{u}}} \;+\;\mathbf{D}_{\mathbf{lr}}\; \overline{\dot{\mathbf{u}}} \;+\;\mathbf{K}_{\mathbf{lr}}\; \overline{{\mathbf{u}}})\rightarrow \mathbf{F}\\ \\ \mathbf{d}_{\mathbf{l}}\rightarrow \mathbf{d}\;\;,\;\;\;\;\;\;\mathbf{d}^0_{\mathbf{l}}\rightarrow \mathbf{d}^0\;\;,\;\;\;\;\;\;\dot{\mathbf{d}}^0_\mathbf{l}\rightarrow \dot{\mathbf{d}}^0 \end{matrix} </math>
+| style="text-align: center;" | <math>\mathrm{a}^{ep}=\mathrm{a}-\frac{\left(\mathrm{a}:\dfrac{\partial g_{n+1}}{\partial \boldsymbol{\tau }} \right)\otimes \left(\mathrm{a}: \dfrac{\partial f_{n+1}}{\partial \boldsymbol{\tau }}\right)}{\left(\dfrac{\partial f_{n+1}}{\partial \boldsymbol{\tau }}\right)^T :\mathrm{a}:\dfrac{\partial g_{n+1}}{\partial \boldsymbol{\tau }}} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.2)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.125)
 |}
-The advantage of the presented reformulation is that allows equation [[#eq-5.1|5.1]] to be expressed as:
+Moreover, according to Equation [[#eq-3.124|3.124]], it is found that the plastic corrector of <math display="inline">\boldsymbol{\tau }</math> reads
-<span id="eq-5.3"></span>
+<span id="eq-3.126"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,043: / Line 3,184: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M} \;\ddot{\mathbf{d}} \;+\;\mathbf{D} \;\dot{\mathbf{d}} \;+\;\mathbf{K}\; {\mathbf{d}}=\mathbf{F}  </math>
+| style="text-align: center;" | <math>\mathrm{a}:\left(\frac{\dfrac{\partial g_{n+1}}{\partial \boldsymbol{\tau }}\left(\dfrac{\partial f_{n+1}}{\partial \boldsymbol{\tau }} \right)^T : \mathrm{a}:\boldsymbol{\epsilon }^{e^{trial}}}{\left(\dfrac{\partial f_{n+1}}{\partial \boldsymbol{\tau }} \right)^T:\mathrm{a}:\dfrac{\partial g_{n+1}}{\partial \boldsymbol{\tau }}}\right)=\frac{f^{trial}\mathrm{a}:\dfrac{\partial g_{n+1}}{\partial \boldsymbol{\tau }}}{\left(\dfrac{\partial f_{n+1}}{\partial \boldsymbol{\tau }} \right)^T:\mathrm{a}:\dfrac{\partial g_{n+1}}{\partial \boldsymbol{\tau }}}=f^{trial}\boldsymbol{r}^p  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.3)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.126)
 |}
-With <math display="inline">\mathbf{d} :[0, \textrm{T}] \rightarrow \mathbb{R}^{n}</math> and subject to initial conditions <math display="inline">\mathbf{d}_{(0)} = \mathbf{d}^0</math> and <math display="inline">\dot{\mathbf{d}}_{(0)} =\dot{\mathbf{d}}^0</math>. The previous expression can be thought as a second order ordinary linear differential equation. From calculus, it is known that the solution of this kind of ODE is the sum of a particular <math display="inline">\mathbf{d}^\mathbf{p}</math> and homogeneous <math display="inline">\mathbf{d}^\mathbf{z}</math> solution:
+with <math display="inline">\boldsymbol{r}^p</math> representing the direction of the plastic corrector in principal space and <math display="inline">f^{trial}</math> defined as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,055: / Line 3,196: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathbf{d}}={\mathbf{d}}^\mathbf{z}+{\mathbf{d}}^\mathbf{p} </math>
+| style="text-align: center;" | <math>f^{trial}=\left(\frac{\partial f_{n+1}}{\partial \boldsymbol{\tau }} \right)^T : \mathrm{a}:\boldsymbol{\epsilon }^{e^{trial}}=\left(\frac{\partial f_{n+1}}{\partial \boldsymbol{\tau }} \right)^T:\boldsymbol{\tau }^{trial} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.4)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.127)
 |}
-The homogeneous solution of the system is found by making <math display="inline">\mathbf{F} = 0</math>, whereas the particular solution is to be computed for the specific case of study. One of the main drawbacks of the modal analysis is that, as it works in the frequency domain, forcing functions need to be '''harmonic''': <math display="inline">\mathbf{F} = \overline{\textrm{F}}\, \sin (\,\textrm{w t}\,)</math>.  This is not the case of the system of study, where forces due to rotation are either constant or increasing in time, and external forces can adopt any arbitrary shape. One way of tackling this problem is to decompose <math display="inline">\mathbf{F}</math> into the sum of harmonic functions through the '''Fourier transform'''. As the system of study is linear, its response can be expressed as the sum of the responses due to each of the obtained harmonic forcing functions.
+====3.3.2.2 Return to a line====
-If <math display="inline">\mathbf{F}</math> is difficult to pose in terms of harmonic functions, another approach consists in splitting <math display="inline">\mathbf{F}</math> into the sum of <math display="inline">n</math> '''impulses'''. The response of the system to an impulse is identical to the free response with certain initial conditions <span id='citeF-11'></span>[[#cite-11|[11]]]. Again, the system is studied under the effect of each of those individual impulses, and taking advantage of the linearity of the system, the solution is expressed as the sum of the latter individual responses. As modal study of structures under a generic forcing function <math display="inline">\mathbf{F}</math> can easily become complicated, the present section will only cover the homogeneous solution of equation  [[#eq-5.3|5.3]].
+If it is found that the return has to be performed to a line, the parametric equation which defines a line in the principal stress space has to be considered (see Equation [[#eq-3.116|3.116]]). By observing Equation [[#eq-3.116|3.116]], the direction vector of the line <math display="inline">\boldsymbol{r}^l</math> is given by the cross product of the perpendicular vector of the two adjacent planes
-===5.1.1 Undamped free vibrations===
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>t\propto \boldsymbol{a}_1\times \boldsymbol{a}_2 </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.128)
+|}
-In order to introduce the basics of the modal analysis, let's consider the simplest case to handle: a system where both <math display="inline">\mathbf{F}</math> and <math display="inline">\mathbf{D}</math> are zero, so damping and external forces play no role whatsoever. With this assumption, equation [[#eq-5.3|5.3]] reduces to:
+Similarly, the direction of the plastic potential line <math display="inline">\boldsymbol{r}_g^l</math> is defined by
-<span id="eq-5.5"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,074: / Line 3,222: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M} \; \ddot{\mathbf{d}}  \;+\;\mathbf{K}\; {\mathbf{d}}=0 </math>
+| style="text-align: center;" | <math>\boldsymbol{r}_g^l\propto \boldsymbol{b}_1\times \boldsymbol{b}_2 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.5)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.129)
 |}
-The modal analysis starts supposing a solution with the following form:
+In the return mapping to a line the plastic strain increment must be perpendicular to the potential line
-<span id="eq-5.6"></span>
+<span id="eq-3.130"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,087: / Line 3,235: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathbf{d}}= \boldsymbol{\Phi } \;\; \mathbf{q}_{(t)} = \boldsymbol{\Phi } \; \left(\mathbf{A} \cos (\boldsymbol{\omega }\textrm{ t})+\mathbf{B} \sin (\boldsymbol{\omega }\textrm{ t}) \right)\;\;\;,\;\;\;\;\;\boldsymbol{\Phi } \in \mathbb{R}^{n} </math>
+| style="text-align: center;" | <math>\left(\Delta \epsilon ^p \right)\boldsymbol{r}_g^l=0  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.6)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.130)
 |}
-Where <math display="inline">\textrm{'''q'''}_{(t)}</math> is a known solution for homogeneous second order differential equations. Since <math display="inline"> \ddot{\mathbf{q}} = -\boldsymbol{\omega }^2 \; {\mathbf{q}}</math> , the above equation can be written as:
+By considering Equation [[#eq-3.121|3.121]], the condition of Equation [[#eq-3.130|3.130]] can be expressed as
+<span id="eq-3.131"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,099: / Line 3,248: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left(\mathbf{K}  -\boldsymbol{\omega }^2  \; \mathbf{M} \right)\boldsymbol{\Phi } \;\; \textrm{q}_{(t)} = 0 \;\;\;\;\mapsto \;\;\;\;\left(\mathbf{K}  -\boldsymbol{\omega }^2  \; \mathbf{M} \right)\boldsymbol{\Phi } = 0 </math>
+| style="text-align: center;" | <math>\left[\left(\boldsymbol{\tau }^{trial}- \boldsymbol{\tau }_{n+1}\right):\mathrm{a}^{-1}  \right]\boldsymbol{r}_g^l=0  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.7)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.131)
 |}
-The preceding equation is known as the ''eigenvalue problem''. The trivial solution <math display="inline">\boldsymbol{\Phi } = 0</math> is not from interest, whereas the non-trivial solution is given by a set of eigenvalues <math display="inline">\boldsymbol{\lambda } = \boldsymbol{\omega }^2  </math>, where <math display="inline">\boldsymbol{\omega }</math> represents the diagonal matrix of natural or resonant vibration frequencies. For a system with <math display="inline">\mathbf{n}</math> degrees of freedom, <math display="inline">\mathbf{n}</math> eigenvalues exist that satisfy the above expression. Being <math display="inline">j</math> an index that goes from <math display="inline">1</math> to <math display="inline">\mathbf{n}</math>, the eigenvalue problem can be interpreted as a system of equations of the form
+Since <math display="inline">\boldsymbol{\tau }_{n+1}</math> has to belong to the line, the expression of Equation [[#eq-3.116|3.116]] is substituted in Equation [[#eq-3.131|3.131]]
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,111: / Line 3,260: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left(\mathbf{K}  - {\lambda }_j  \; \mathbf{M} \right)\boldsymbol{\Phi }_j = 0 </math>
+| style="text-align: center;" | <math>\left[\left(\boldsymbol{\tau }^{trial}- \left(t\,\boldsymbol{r}^l + \boldsymbol{\tau }_A  \right)\right):\mathrm{a}^{-1}  \right]\boldsymbol{r}_g^l=0 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.8)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.132)
 |}
-Where <math display="inline">\boldsymbol{\Phi }_j</math> is referred as the ''eigenvector'' associated to the eigenvalue <math display="inline">{\lambda }_j </math>. Notice that eigenvectors do never define absolute values of the different independent coordinates of the system, but inform about the phase and amplitude '''relations''' between degrees of freedom. In structural problems, eigenvectors are called natural modes or mode shapes, and can be though as possible deformed configurations oscillating at a given frequency stated by its corresponding eigenvalue. In matrix form, eigenvalues and eigenvectors are expressed as:
+By rearranging the terms of the last equation, the expression for t is obtained
+<span id="eq-3.133"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,123: / Line 3,273: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{\lambda } = \begin{pmatrix}\omega _1\,^2 & 0 &  \cdots & 0\\   0 & \omega _2\,^2 & \cdots & 0\\   \vdots  & \vdots  & \ddots  & \vdots \\   0& 0 & \cdots & \omega _n\,^2   \end{pmatrix} \;\;\;,\;\;\;\;\;\; \boldsymbol{\Phi } = \begin{bmatrix}\boldsymbol{\Phi }_1 & \boldsymbol{\Phi }_2& \cdots  & \boldsymbol{\Phi }_n  \end{bmatrix}=\begin{pmatrix}\Phi _{11} & \Phi _{12}  &  \cdots & \Phi _{1n} \\   \Phi _{21}  & \Phi _{22}  & \cdots & \Phi _{2n} \\   \vdots  & \vdots  & \ddots  & \vdots \\   \Phi _{n1} & \Phi _{n2}  & \cdots & \Phi _{nn}   \end{pmatrix} </math>
+| style="text-align: center;" | <math>t=\frac{\left(\boldsymbol{r}_g^l \right)^T\mathrm{a}^{-1}\left(\boldsymbol{\tau }^{trial}- \boldsymbol{\tau }_A  \right)}{\left(\boldsymbol{r}_g^l \right)^T\mathrm{a}^{-1}\boldsymbol{r}^l}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.9)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.133)
 |}
-The main advantage of this method is that natural frequencies and mode shapes are relatively easy to obtain. <math display="inline">MATLAB</math>, for instance, has in-built functions that compute a set of <math display="inline">\mathbf{m}\leq \mathbf{n}</math> eigenvectors with the smallest associated frequency. The key of this approach is that there is no need to compute all the <math display="inline">\mathbf{n}</math> eigenvalues and eigenvectors, and instead the solution of the system can be '''approximated''' using a small number of them. In structural analysis, shape modes associated to low frequencies are the most dangerous, and the ones from interest when studying vibration. Recall that <math display="inline">\boldsymbol{\Phi }</math> and <math display="inline">\boldsymbol{\lambda }</math> only depend on the matrices <math display="inline">\mathbf{K}</math> and <math display="inline">\mathbf{M}</math>, which can be computed using the static FEM program previously presented. For instance, mode shapes associated to the four lowest natural frequencies of the cantilever beam defined in [[#img-4.6|4.6]] are displayed in the following figure:
+In the Mohr-Coulomb plastic law a plane is delimited by two lines expressed by Equations [[#eq-3.117|3.117]] and [[#eq-3.118|3.118]] and the corresponding potential directions are
-<div id='img-5.1'></div>
+<span id="eq-3.134"></span>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-modes2D2.png|600px|Shape modes associated to the first four lower natural frequencies of a 2D fixed-free beam]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 5.1:''' Shape modes associated to the first four lower natural frequencies of a 2D fixed-free beam
+|-
+| style="text-align: center;" | <math>\boldsymbol{r}_{g,1}^l=\left[1 \quad 1\quad m \right]^T  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.134)
 |}
-An important property of eigenvectors is that they are orthogonal with respect to <math display="inline">\mathbf{K}</math> and <math display="inline">\mathbf{M}</math>:
-<span id="eq-5.10"></span>
+<span id="eq-3.135"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,145: / Line 3,297: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{\Phi }_{i}^{T} \mathbf{M} \;\boldsymbol{\Phi }_{j}=0 \;, \quad \boldsymbol{\Phi }_{i}^{T} \mathbf{K} \; \boldsymbol{\Phi }_{j}=0 \quad i \neq j </math>
+| style="text-align: center;" | <math>\boldsymbol{r}_{g,2}^l=\left[1 \quad m\quad m \right]^T  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.10)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.135)
 |}
-The previous equation can only be applied if the matrices involved in the system are symmetric and positive semidefinite, and implies that the matrix of natural modes <math display="inline">\boldsymbol{\Phi }</math> diagonalizes both <math display="inline">\mathbf{K}</math> and <math display="inline">\mathbf{M}</math>. It is a common approach to normalise <math display="inline">\boldsymbol{\Phi }</math> such that <math display="inline">\boldsymbol{\Phi }_{i}^{T} \mathbf{M} \boldsymbol{\Phi }_{i} = \mathbf{I}</math>, which results into <math display="inline">\boldsymbol{\Phi }_{i}^{T} \mathbf{K}\, \boldsymbol{\Phi }_{i} = \boldsymbol{\omega }^2</math>. Introducing the generalised coordinates <math display="inline">\mathbf{q} \in \mathbb{R}^{n}</math> such that <math display="inline">\mathbf{d} = \boldsymbol{\Phi }\;\mathbf{q}</math>, and pre-multiplying by <math display="inline">\boldsymbol{\Phi }_{i}^{T}</math> equation [[#eq-5.5|5.5]]:
+With the definitions of Equations [[#eq-3.114|3.114]], [[#eq-3.115|3.115]], [[#eq-3.134|3.134]] and [[#eq-3.135|3.135]] in hand, it is possible to evaluate <math display="inline">t_1</math> and <math display="inline">t_2</math> with the use of Equation [[#eq-3.133|3.133]] and the updated stress <math display="inline">\boldsymbol{\tau }_{n+1}</math> laying either on line <math display="inline">l_1</math> or <math display="inline">l_2</math>, depending on the type of return.
+In the definition of the elastoplastic fourth-order constitutive tensor <math display="inline">\mathrm{a}^{ep}_l</math> the following observation are made. Firstly, in the case of return to a line the updated stress lays on the line and the stress increment has the same direction of <math display="inline">\boldsymbol{r}^l</math>, thus, the elastic strain increment must have the direction
-<span id="eq-5.11"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,158: / Line 3,311: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\overset{\mathbf{I}}{\overbrace{\boldsymbol{\Phi }^{T}\mathbf{ M}\; \boldsymbol{\Phi }}}\; \; \ddot{\mathbf{q}}+\overset{\boldsymbol{\omega }^2}{\overbrace{\boldsymbol{\Phi }^{T} \mathbf{K} \; \boldsymbol{\Phi }}} \;\;  \mathbf{q}=0 \; \; \; \; \Rightarrow \; \; \; \;  \ddot{\mathbf{q}}+\boldsymbol{\omega }^2 \; \mathbf{q}=0 </math>
+| style="text-align: center;" | <math>\boldsymbol{r}^e=\mathrm{a}^{-1}\boldsymbol{r}^l </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.11)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.136)
 |}
-The above expression is, in fact, a system of uncoupled differential equations subject to the following initial conditions:
+This means that <math display="inline">\mathrm{a}^{ep}_l</math> has to be singular with respect to the strain directions associated with both the yield planes that define the line, <math display="inline">\boldsymbol{b}_1</math> and <math display="inline">\boldsymbol{b}_2</math> and any linear combination of the two
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,170: / Line 3,323: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{q}_{(0)}=\mathbf{q}^{0}=\boldsymbol{\Phi }^{T} \mathbf{M}\; \mathbf{d}^{0}\;\;,\;\;\;\; \dot{\mathbf{q}}_{(0)}=\dot{\mathbf{q}}^{0}=\boldsymbol{\Phi }^{T} \mathbf{M}\;  \dot{\mathbf{d}}^{0} </math>
+| style="text-align: center;" | <math>\mathrm{a}^{ep}_l\left(\gamma _1\boldsymbol{b}_1 + \gamma _2\boldsymbol{b}_2\right)=0 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.12)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.137)
 |}
-Recalling the definition of <math display="inline">\mathbf{q}</math> stated in equation [[#eq-5.6|5.6]], the exact solution of the system can be expressed in uncoupled form as:
+where <math display="inline">\gamma _1</math> and <math display="inline">\gamma _2</math> are plastic multipliers. After these considerations the following system of equations is defined:
-<span id="eq-5.13"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,183: / Line 3,335: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{d}=\sum _{i=1}^{n} \boldsymbol{\Phi }_{i}\left(\mathbf{q}_{i}^{0} \cos \omega _{i} \textrm{ t}+\frac{\dot{\mathbf{q}}_{i}^{0}}{\omega _{i}} \sin \omega _{i} \textrm{ t}\right) </math>
+| style="text-align: center;" | <math>\left\{    \begin{array}{rcll}\mathrm{a}^{ep}_l \boldsymbol{r}^e &= & \boldsymbol{r}^l  \\      \mathrm{a}^{ep}_l\boldsymbol{b}_1&= & 0 \\       \mathrm{a}^{ep}_l\boldsymbol{b}_2&= & 0 \\    \end{array}    \right. </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.13)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.138)
 |}
-In order to postprocess the results, displacements of every degree of freedom are to be computed at different values of time and joined together into a matrix. The time between two consecutive samples is called '''time-step''', and in order to capture the vibration phenomena it has to be less than half the period of the highest frequency of interest. Once displacements are computed, they can be animated through '''GID'''.
+and the solution of it leads to the expression of <math display="inline">\mathrm{a}^{ep}_l</math>
-<span id="eq-5.14"></span>
+<span id="eq-3.139"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,196: / Line 3,348: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{d} = \begin{bmatrix}\mathbf{d}_{(\textrm{t}_1)} & \mathbf{d}_{(\textrm{t}_2)}  & \cdots  & \mathbf{d}_\textrm{(T)} \end{bmatrix} = \begin{pmatrix}{\textrm{d}_1}_{(\textrm{t}_1)}&  {\textrm{d}_1}_{(\textrm{t}_2)} & \cdots  &  {\textrm{d}_1}_{(\textrm{T})}\\  {\textrm{d}_2}_{(\textrm{t}_1)}& {\textrm{d}_2}_{(\textrm{t}_2)} & \cdots  & {\textrm{d}_2}_{(\textrm{T})} \\  \vdots &  \vdots & \ddots  &\vdots \\  {\textrm{d}_n}_{(\textrm{t}_1)} & {\textrm{d}_n}_{(\textrm{t}_2)} &\cdots   & {\textrm{d}_{\mathit{n}}}_{(\textrm{T})} \end{pmatrix} </math>
+| style="text-align: center;" | <math>\mathrm{a}^{ep}_l=\frac{\boldsymbol{r}^l\left(\boldsymbol{r}^l_g \right)^T}{\left(\boldsymbol{r}^l \right)^T\mathrm{a}^{-1}\boldsymbol{r}^l_g}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.14)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.139)
 |}
-As it has been pointed before, sometimes it is enough with an approximation of the solution, obtained using only <math display="inline">\mathbf{m}<\mathbf{n}</math> terms. A good strategy to asses which shape modes are the most relevant for a given structure is to use the '''normalised initial amplitude''' associated to each mode as guide:
+The expression of Equation [[#eq-3.139|3.139]] has only elements related to the principal stresses; in order to consider the shear stiffness, the elasto-plastic constitutive tensor in principal space is modified as follows
+<span id="eq-3.140"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,208: / Line 3,361: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\overline{\mathbf{q}}^{\,0}_i = \left|\frac{\mathbf{q}^{\,0}_i}{\left\|\mathbf{q}^{\,0}_i \right\|}  \right| </math>
+| style="text-align: center;" | <math>\widehat{\mathrm{a}}^{ep}_l=\mathrm{a}^{ep}_l + \mathrm{G}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.15)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.140)
 |}
-For the presented 2D beam, normalised initial amplitudes are plotted in a bar graph in figure [[#img-5.2|5.2]] for the six shape modes with lower frequency. It can be seen that the most excited mode shape is the one corresponding to pure bending (<math display="inline">\boldsymbol{\Phi }_{1}</math>), which makes sense since the beam is under the effect of a distributed load on its upper side. Notice that in order to find this out, '''initial conditions''' need to be defined. Different modes can dominate the same structure under distinct initial conditions.  <div id='img-5.2'></div>
+where <math display="inline">\mathrm{G}</math> reads
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-normal2.png|300px|Normalised initial amplitude versus mode shape number for a 2D fixed-free beam under the effect of a distributed load]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 5.2:''' Normalised initial amplitude versus mode shape number for a 2D fixed-free beam under the effect of a distributed load
+|-
+| style="text-align: center;" | <math>\mathrm{G}=\frac{E}{2\left(1 + \nu \right)}\quad \begin{bmatrix}\mathrm{0}_{3\times{3}} & \mathrm{0}_{3\times{3}} \\ \mathrm{0}_{3\times{3}} & \mathrm{1}_{3\times{3}}  \end{bmatrix} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.141)
 |}
-===5.1.2 Damped free vibrations===
+====3.3.2.3 Return to a point====
-Modal analysis can handle damped systems only if some requirements regarding the damping matrix are met: <math display="inline">\mathbf{D}</math> needs to be either diagonal or a linear combination of <math display="inline">\mathbf{M}</math> and <math display="inline">\mathbf{K}</math>. The last assumption is known as Rayleigh damping, and has already been presented in equation [[#eq-3.15|3.15]]. The equation describing a system under Rayleigh damping is written as:
+In the case of return to the apex, no calculation is needed since <math display="inline">\boldsymbol{\tau }=\boldsymbol{\tau }_A</math>, defined as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,230: / Line 3,387: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M} \; \ddot{\mathbf{d}}\;+\;\mathbf{D}\; \dot{\mathbf{d}}  \;+\;\mathbf{K}\; {\mathbf{d}}=0 \;\;\; \Rightarrow \;\;\; \mathbf{M} \; \ddot{\mathbf{d}}\;+\;\left(\overline{\alpha }\;\mathbf{M}+\overline{\beta }\;\mathbf{K}\right)\; \dot{\mathbf{d}}  \;+\;\mathbf{K}\; {\mathbf{d}}=0  </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau }_A=\frac{2c\sqrt{k}}{k-1}\left[1\quad 1\quad 1  \right]^T </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.16)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.142)
 |}
-Introducing the change of variable <math display="inline">\mathbf{d} = \boldsymbol{\Phi }\;\mathbf{q}</math>, and pre-multiplying by <math display="inline">\boldsymbol{\Phi }_{i}^{T}</math>:
+With respect to the definition of the elasto-plastic fourth order constitutive tensor <math display="inline">\mathrm{a}^{ep}_{apex}</math>, this tensor has to be singular with respect to any direction in the principal stress space, i.e.
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,242: / Line 3,399: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\overset{\mathbf{I}}{\overbrace{\boldsymbol{\Phi }^{T} \mathbf{M} \boldsymbol{\Phi }}} \, \ddot{\mathbf{q}} +  (\overline{\alpha }\;\overset{\mathbf{I}}{\overbrace{\boldsymbol{\Phi }^{T} \mathbf{M} \boldsymbol{\Phi }}} + \overline{\beta }\;\overset{\boldsymbol{\omega }^2}{\overbrace{\boldsymbol{\Phi }^{T} \mathbf{K} \; \boldsymbol{\Phi }}}  )\, \dot{\mathbf{q}} + \overset{\boldsymbol{\omega }^2}{\overbrace{\boldsymbol{\Phi }^{T} \mathbf{K} \; \boldsymbol{\Phi }}}\,\mathbf{q} \;\;\; \Rightarrow \;\;\; \ddot{\mathbf{q}}+\left(\overline{\alpha }+\overline{\beta }\,\boldsymbol{\omega }^{2}\right)\dot{\mathbf{q}}+\boldsymbol{\omega }^{2} \mathbf{q}=\mathbf{0} </math>
+| style="text-align: center;" | <math>\mathrm{a}^{ep}_{apex}=\mathrm{0} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.17)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.143)
 |}
-Which is a system of uncoupled differential equations. Defining the damping ratio associated to the <math display="inline">i—th</math> eigenvalue <math display="inline">\overline{\xi }_{i}</math> and the damped oscillation frequency <math display="inline">\overline{\omega _{i}}</math> :
+and the final expression which consider the shear stiffness, as well, reads
+<span id="eq-3.144"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,254: / Line 3,412: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\overline{\xi }_{i} = \frac{1}{2 \omega _{i}}\left(\overline{\alpha }+\overline{\beta }\, \omega _{i}^{2}\right)\;\;\; , \;\;\;\;\;\;\overline{\omega }_{i}= \omega _{i} \sqrt{1-\overline{\xi }_{i}^{2}} </math>
+| style="text-align: center;" | <math>\widehat{\mathrm{a}}^{ep}_{apex}=\mathrm{G}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.18)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.144)
 |}
-The system of equations can be decoupled into:
+{| style="margin: 1em auto;border: 1px solid darkgray;"
+|-
+|
+:<span style="font-size: 75%;">
+Initial data on material points: <math> \boldsymbol{F}_n </math>, <math> \boldsymbol{b}^e_n</math>
+|-
+| OUTPUT of calculations: <math> \boldsymbol{\tau }_{n+1}</math>,<math> \boldsymbol{b}^e_{n+1}</math>
+|-
+|
+<ol>
+<li>'''UPDATE THE CURRENT CONFIGURATION''' </li>
+:* Compute the the current configuration: <math display="inline"> \boldsymbol{\varphi }_{n+1}=\boldsymbol{\varphi }_n + \boldsymbol{u}_n \circ \boldsymbol{\varphi }_n </math>
+:* Compute the relative deformation gradient: <math display="inline">\boldsymbol{f}_{n+1} = \boldsymbol{1} + \nabla _{\boldsymbol{x}_n} \boldsymbol{u}_n</math>
+:* Compute the total deformation gradient in updated configuration: <math display="inline">\boldsymbol{F}_{n+1} = \boldsymbol{f}_{n+1}\boldsymbol{F}_n</math>
+<li>'''COMPUTE THE ELASTIC PREDICTOR'''  </li>
+:* Compute the trial elastic left Cauchy-Green tensor of <math display="inline">{\boldsymbol{b}^e}^{\mathrm{trial}}= \boldsymbol{f}_{n+1}{\boldsymbol{b}^e}_n {\boldsymbol{f}_{n+1}}^T</math>
+:* Compute the spectral decomposition of <math display="inline">{\boldsymbol{b}^e}^{\mathrm{trial}}= \sum \limits _{A=1}^3 \lambda _A^2 \boldsymbol{m}^{A}</math>
+:* Compute the logarithmic principal stretches <math display="inline">\boldsymbol{\epsilon }^{e, \mathrm{trial}} = {{\mathrm{ln}}}(\lambda _A)</math>
+:* Compute the principal trial Kirchhoff stresses <math display="inline">\boldsymbol{\tau }^{\mathrm{trial}}=\mathrm{a}\boldsymbol{\epsilon }^{e, \mathrm{trial}}</math>
+<li>'''CHECK FOR PLASTIC LOADING'''
+</li>
+:  Evaluate the Mohr-Coulomb yield condition <math display="inline">f(\boldsymbol{\tau }^{\mathrm{trial}})=(\tau _1 - \tau _3)+(\tau _1 + \tau _3)\sin \phi - 2c \cos \phi </math>
+:a If <math display="inline">f^{\mathrm{trial}}_{n+1}\leq{0}</math>, no plastic loading is observed. Thus, <math display="inline">\boldsymbol{\tau }_{n+1} = \boldsymbol{\tau }^{\mathrm{trial}}</math> and <math display="inline"> {\boldsymbol{b}^e}_{n+1} = {\boldsymbol{b}^e}^{\mathrm{trial}}_{n+1}</math>
+:b If <math display="inline">f^{\mathrm{trial}}_{n+1}>0</math>, plastic loading is observed.
+::* Define the type of return (see Section [[#3.3.1 The stress regions|3.3.1]])
+::* Applied the corresponding return stress and compute the elasto-plastic tangent moduli (see Section [[#3.3.2 Stress update in principal stress space|3.3.2]])
+::* Correct <math display="inline">\boldsymbol{\tau }</math>: <math display="inline">\boldsymbol{\tau }_{n+1}=\boldsymbol{\tau }^{\mathrm{trial}}-\Delta \gamma \mathrm{a}\frac{\delta g(\boldsymbol{\tau }^{\mathrm{trial}})}{\delta \boldsymbol{\tau }}</math>
+::* Correct <math display="inline">\boldsymbol{\epsilon }^{e}</math>: <math display="inline">\boldsymbol{\epsilon }^{e}=\mathrm{a}^{-1}\boldsymbol{\tau }_{n+1}</math>
+<li>'''EVALUATE THE STRESS IN CURRENT CONFIGURATION'''
+</li>
+:  Transform <math display="inline">\boldsymbol{\tau }_{n+1}</math> and elasto-plastic tangent moduli back to the original coordinate system
+{|
+<li>'''UPDATE THE INTERMEDIATE CONFIGURATION'''
+</li>
+|}
+:  Update <math display="inline"> {\boldsymbol{b}^e}_{n+1}=\sum \limits _{A=1}^3 exp\left[2 \boldsymbol{\epsilon }^{e}_A \right]\boldsymbol{m}^{A} </math>
+</ol>
+</span>
+|-
+| style="text-align: center; font-size: 75%;"|
+<span id='algorithm-3.2'></span>'''Algorithm. 3.2''' Return Mapping algorithm.
+|}
+===3.3.3 The consistent elasto-plastic tangent moduli===
+As highlighted in <span id='citeF-155'></span><span id='citeF-156'></span>[[#cite-155|[155,156]]], the exact closed-form linearisation of the return mapping algorithm produces a modified elasto-plastic moduli, referred to as the consistent algorithmic moduli, which compared to the classical elasto-plastic moduli, presented in the earlier sections, is able to restore the quadratic rate of convergence exhibited by Newton-like iterative methods.
+Since the consistent tangent operator represents the instantaneous variation of the stress tensor <math display="inline">\boldsymbol{\tau }</math> (Equation [[#eq-3.102|3.102]]) with respect to the strain tensor <math display="inline">\boldsymbol{\epsilon }</math> (Equation [[#eq-3.100|3.100]]) and the updated expression of the stress tensor is a function dependent on the trial deformation state, the consistent tangent moduli has the following form
+<span id="eq-3.145"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,266: / Line 3,480: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\ddot{\mathbf{q}}_{i}+2 \,\overline{\xi }_{i} \,\omega _{i}\, \dot{\mathbf{q}}_{i}+\omega _{i}^{2}\, \mathbf{q}_{i}=0\;\;\;,\;\;\;\;i=1,2 \ldots n </math>
+| style="text-align: center;" | <math>\mathrm{C}^{cep}=\sum \limits _{A=1}^3\sum \limits _{B=1}^3\frac{\partial \tau _A}{\partial \epsilon _B^{trial}}\boldsymbol{m}^{(A, trial)}\otimes \frac{\partial \epsilon _B^{trial}}{\partial g}+\sum \limits _{A=1}^3 2\tau _A \frac{\partial \boldsymbol{m}^{(A)}}{\partial g}</math>
+|-
+| style="text-align: center;" | <math> =\sum \limits _{A=1}^3\sum \limits _{B=1}^3 \mathrm{a}^{ep}_{AB}\boldsymbol{m}^{(A, trial)}\otimes \boldsymbol{m}^{(B, trial)}+\sum \limits _{A=1}^3 2\tau _A\mathrm{c}^{A, trial} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.19)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.145)
 |}
-Up to this point, two different situations may occur depending on the value of <math display="inline">\overline{\xi }_{i}</math>. If <math display="inline">0 \leq \overline{\xi }_{i}<1</math>, then <math display="inline">\overline{\omega }_{i}</math> is real valued and the exact expression for the generalised coordinates is:
+where the relation
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,278: / Line 3,494: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{q}_{i}=e^{-\overline{\xi }_{i} \omega _{i} \textrm{t}}\left(\mathbf{q}_{i}^{0} \cos \overline{\omega }_{i} \textrm{t}+\frac{\dot{\mathbf{q}}_{i}^{0}+\overline{\xi }_{i} \omega _{i}\, \mathbf{q}_{i}^{0}}{\overline{\omega }_{i}} \sin \overline{\omega }_{i} \textrm{t}\right) </math>
+| style="text-align: center;" | <math>2\frac{\partial \boldsymbol{\epsilon }^{trial}_A}{\partial \boldsymbol{g}}=2\boldsymbol{F}\frac{\partial \boldsymbol{\epsilon }^{trial}_A}{\partial \boldsymbol{C}}\boldsymbol{F}^T=\boldsymbol{m}^{(A, trial)} \quad for\, A=1,2,3 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.20)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.146)
 |}
-This response is known as '''subcritical oscillation''', and can be interpreted as an undamped free vibration whose amplitude decreases exponentially in time. On the other hand, if <math display="inline">\overline{\xi }_{i}>1</math>, damped oscillation frequencies are no longer real valued and generalised coordinates are expressed as:
+is used.
+The expression of the consistent tangent moduli is given by the sum of two terms. In the first term it can be found the elasto-plastic tangent moduli <math display="inline">\mathrm{a}^{ep}</math> dependent on the specific plastic model and the structure of return mapping algorithm. In the case of a Mohr-Coulomb plastic model, the definition, depending on the type of return, can be found in Equations [[#eq-3.126|3.126]], [[#eq-3.140|3.140]] or [[#eq-3.144|3.144]]. On the other hand, the tensor product <math display="inline">\boldsymbol{m}^{(A, trial)}</math> and the moduli <math display="inline">\mathrm{c}^{A, trial}</math> are independent on the plastic model; with regards to <math display="inline">\mathrm{c}^{A, trial}</math>, this tensor is dependent only on the ''specific strain energy function'' and it reflects the changing orientation of the spectral direction of <math display="inline">\boldsymbol{\tau }</math>. The closed-form of <math display="inline">\mathrm{c}^{A, trial}</math> is derived in Appendix [[#B Derivatives of the rank-one matrices principal direction|B]] and in the following the final spatial form is presented
+<span id="eq-3.147"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,290: / Line 3,509: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{q}_{i}=\mathbf{q}_{i}^0 \;e^{-\omega _{i} \left(\overline{\xi }_{i} + \sqrt{\overline{\xi }_{i}^{\,2}-1}\,\right)\,\textrm{t}} </math>
+| style="text-align: center;" | <math>\mathrm{c}^{A, trial}  := \frac{1}{D_A}\left[\mathrm{I}_b - \boldsymbol{b}\otimes \boldsymbol{b} + I_3 \lambda _A^{-2}\left(\boldsymbol{I}\otimes \boldsymbol{I} - \mathrm{I} \right) \right]</math>
+|-
+| style="text-align: center;" | <math>  + \frac{1}{D_A}\left[\lambda _A^{2}\left(\boldsymbol{b}\otimes \boldsymbol{m}^{(A)}+\boldsymbol{m}^{(A)}\otimes \boldsymbol{b}  \right)-\frac{1}{2}{D'}_A\lambda _A \boldsymbol{m}^{(A)}\otimes \boldsymbol{m}^{(A)}  \right]</math>
+|-
+| style="text-align: center;" | <math>  - \frac{1}{D_A}\left[I_3 \lambda _A^{-2}\left(\boldsymbol{I}\otimes \boldsymbol{m}^{(A)}+\boldsymbol{m}^{(A)}\otimes \boldsymbol{I}  \right)\right] </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.21)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.147)
 |}
-This is a non-oscillating response known as '''overcritical motion'''. In both cases, physical displacements can be computed in the same way as in the undamped case:
+where <math display="inline">\left(\mathrm{I}_b\right)=\frac{1}{2}\left(b_{ac}b_{bd}+b_{ad}b_{bc} \right)</math>, <math display="inline">D_A=2\lambda _A^4-I_1\lambda _A^2+I_3\lambda _A^{-2}</math>, <math display="inline">{D'}_A=8\lambda _A^3-2 I_1-2 I_3\lambda _A^{-3}</math>, <math display="inline">I_1=\mathrm{tr}\boldsymbol{C}</math>, <math display="inline">I_3=J^2</math> and <math display="inline">\lambda _A</math> are the principal stretches defined as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,302: / Line 3,525: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{d}=\sum _{i=1}^{n} \boldsymbol{\Phi }_{i} \; \mathbf{q}_{i} </math>
+| style="text-align: center;" | <math>\lambda _A = \sqrt{\lambda _A^2} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.22)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.148)
 |}
-===5.1.3 Modal analysis of rotating structures===
+with <math display="inline">\lambda _A^2</math> are the eigenvalues of <math display="inline">\boldsymbol{b}^{e^{trial}}</math>, according to Equation [[#eq-3.101|3.101]]. In Equation [[#eq-3.147|3.147]] the superscript <math display="inline">trial</math> has been omitted for the tensor <math display="inline">\mathrm{I}_b</math>, <math display="inline">\boldsymbol{b}</math> and <math display="inline">\boldsymbol{m}^{(A)}</math>.
-The presented method of modal decomposition can be applied to the case of study of rotating structures, but subject to some limitations. The approach that has been developed previously is only valid if the matrices involved in the system are '''constant'''. Recall from equation [[#eq-3.13|3.13]] that for rotation problems, the stiffness and damping matrices are the addition of a static component, which is constant, and a rotating one that depends on angular velocity and acceleration. The latter varies with time if the angular acceleration is different from zero. Thus, the only possibility for the system to be tackled through modal analysis is to rotate under '''constant angular velocity''' <math display="inline">\Omega </math>. In that case, the eigenvalue problem is written as:
+When the use of a mixed formulation, with mean stress (<math display="inline"> p </math>) and displacements (<math display="inline">\boldsymbol{u}</math>) as primary variables, might be required, the reader has to consider the volumetric part of the Kirchhoff stress tensor defined as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,316: / Line 3,539: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left(\mathbf{K}_{\,\hbox{static}} + \mathbf{K}_{\,\hbox{rot}} -\boldsymbol{\omega }^2  \; \mathbf{M} \right)\boldsymbol{\Phi } = 0 </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau }_{vol}=J p\boldsymbol{I} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.23)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3.149)
 |}
-And if the problem is either 1D or 2D:
+and the volumetric part of the spatial constitutive tensor expressed by Equation [[#eq-3.51|3.51]].
+=4 Irreducible formulation=
+The simulation of granular flow problems, which involve large deformation and complex history-dependent constitutive laws, is of paramount importance in several industrial and engineering processes. Particular attention has to be paid to the choice of a suitable numerical technique such that reliable results can be obtained.  In Chapter [[#2 Particle Methods|2]] a review of several numerical techniques is presented in order to individuate the most suited methods for the numerical analysis of granular flows under both quasi-static and inertial regime. For the achievement of such purpose, it is found that the Material Point Method (MPM) and the Galerkin Meshfree Method (GMM) might be two good candidates, as previously shown.   In this Chapter, firstly, an irreducible formulation is presented. The displacement-based formulation, defined in a Update Lagrangian framework under finite strain regime, is implemented in both the MPM and GMM strategy. Afterwards, these two numerical strategies, already presented in Chapter [[#2 Particle Methods|2]], are verified against classical benchmarks in solid and geo-mechanics. The aim is to assess their validity in the simulation of cohesive-frictional materials, both in static and dynamic regimes and in problems dealing with large deformations.  The vast majority of MPM techniques in the literature is based on some sort of explicit time integration. The techniques proposed in the current work, on the contrary, are based on implicit approaches which can also be easily adapted to the simulation of static cases. Although both methods are able to give a good prediction, it is observed that, under very large deformation of the medium, GMM lacks in robustness due to its meshfree nature, which makes the definition of the meshless shape functions more difficult and expensive than in MPM. On the other hand, the mesh-based MPM demonstrates to be more robust and reliable for extremely large deformation cases.
+==4.1 Governing equations==
+Let us consider the body <math display="inline">  \mathcal{B} </math> which occupies a region <math display="inline"> \Omega </math> of the three-dimensional Euclidean space <math display="inline"> \mathcal{E} </math> with a regular boundary <math display="inline"> \partial \Omega </math> in its reference configuration. A deformation of <math display="inline">  \mathcal{B} </math> is defined by a one-to-one mapping
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,328: / Line 3,559: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left(\mathbf{K}_{\,\hbox{static}} - \Omega ^2\mathbf{M} \; -\boldsymbol{\omega }^2  \; \mathbf{M} \right)\boldsymbol{\Phi } = 0 \;\;\;\Rightarrow \;\;\;\mathbf{K}_{\,\hbox{static}} - (\Omega ^2 \; +\boldsymbol{\omega }^2)  \; \mathbf{M}   = 0 </math>
+| style="text-align: center;" | <math>\varphi :\Omega \rightarrow \mathcal{E} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.24)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.1)
 |}
-The above expression unfolds the '''softening effect''' predicted in [[#eq-3.17|3.17]], which produces a decrease in the natural frequencies as angular velocity increases. The relationship between natural frequency of the static and rotating case is:
+that maps each point ''p'' of the body <math display="inline">  \mathcal{B} </math> into a spatial point <math display="inline">\boldsymbol{x}</math>
-<span id="eq-5.25"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,341: / Line 3,571: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{\omega }_{\,\hbox{rot}} = \sqrt{\boldsymbol{\omega }_{\,\hbox{static}}^2 - \Omega ^2} </math>
+| style="text-align: center;" | <math>\boldsymbol{x} = \varphi \left(\boldsymbol{p}\right) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.25)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.2)
 |}
-According to the above equation, lower frequencies are the most affected by the softening effect. This behaviour can be represented through a Campbell diagram (see figure [[#img-5.3|5.3]]), where natural frequencies of a rotating structure are plotted as a function of the angular speed. This softening effect, however, is rarely seen in real scenarios: it is a consequence of the '''limitations''' of the model, which is not able to account for a counteracting stiffening effect. The decrease of the natural frequencies emerges in fact due to the non-inertial character of the frame of study.
+which represents the location of  ''p'' in the deformed configuration of <math display="inline">\mathcal{B}</math>. The region of <math display="inline">\mathcal{E}</math> occupied by <math display="inline">\mathcal{B}</math> in its deformed configuration is denoted as <math display="inline">\varphi \left(\Omega \right)</math>.
-Supposing that the 2D beam studied in figure [[#img-5.1|5.1]] undergoes a rotation at constant angular velocity, then the natural frequencies linked with the presented mode shapes would decrease as the spinning speed grows in the way it is displayed in the following graph:  <div id='img-5.3'></div>
+The problem is governed by mass and linear momentum balance equations <span id="eq-4.3"></span>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+<span id="eq-4.3.a"></span>
+<span id="eq-4.3.b"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-campbell.png|360px|Campbell diagram representing the natural frequencies of a 2D beam rotating at constant speed]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 5.3:''' Campbell diagram representing the natural frequencies of a 2D beam rotating at constant speed
+|-
+| style="text-align: center;" | <math>\frac{D\rho }{Dt}+\rho \nabla \cdot \boldsymbol{v} = 0 \qquad \textrm{in}  \quad \varphi (\Omega ) </math>
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.3.a)
+|-
+| style="text-align: center;" | <math> \rho \boldsymbol{a} - \nabla \cdot \boldsymbol{\sigma } = \rho \boldsymbol{b} \qquad \textrm{in}  \quad \varphi (\Omega )  </math>
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.3.b)
+|}
+|}
+where <math display="inline"> \rho </math> is the mass density, <math display="inline"> \boldsymbol{a} </math> is the acceleration, <math display="inline"> \boldsymbol{v} </math> is the velocity, <math display="inline">\boldsymbol{\sigma }</math> is the symmetric Cauchy stress tensor and <math display="inline"> \boldsymbol{b} </math> is the body force.  Acceleration and velocity are, by definition, the material derivatives of the velocity, <math display="inline"> \boldsymbol{v} </math>, and the displacement, <math display="inline"> \boldsymbol{u} </math>, respectively.  For a compressible material the conservation of mass is satisfied by
+<span id="eq-4.4"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\rho = \frac{\rho _0}{det(\boldsymbol{F})}  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.4)
 |}
-===5.1.4 Limitations===
+where <math display="inline">\rho _0</math> is the density in the undeformed configuration and <math display="inline">det(\boldsymbol{F})</math> is the determinant of the total deformation gradient <math display="inline"> \boldsymbol{F} := d\boldsymbol{x}/d\boldsymbol{X} </math> with <math display="inline">\mathbf{x}</math> and <math display="inline">\mathbf{X}</math> representing the current and initial position, respectively. Equation [[#eq-4.4|4.4]] holds at any point, and in particular at the sampling points where the equation is written, e.g. the material points.  Thermal effects are not considered in the present work, so the energy balance is considered implicitly fulfiled.
-Until now, only stationary scenarios have been considered as modal analysis is not capable of accounting the effect of angular acceleration. Another drawback of this method is related with the requirement that <math display="inline">\mathbf{D}</math> needs to be a linear combination of <math display="inline">\mathbf{K}</math> and <math display="inline">\mathbf{M}</math>. This is not the case of rotating structures, as under the effect of rotation, the damping matrix adopts the form:
+The balance equations are solved numerically in a three-dimensional region <math display="inline"> \Omega \subseteq \mathcal{R}^3 </math>, in the time range <math display="inline">t \in [0,T] </math>, given the following boundary conditions on the Dirichlet (<math display="inline">\varphi (\partial \Omega _D)</math>) and Neumann boundaries (<math display="inline">\varphi (\partial \Omega _N)</math>), respectively <span id="eq-4.5"></span>
+<span id="eq-4.5.a"></span>
+<span id="eq-4.5.b"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,365: / Line 3,619: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{D }= {\mathbf{D}_{\,\hbox{static}}} + {\mathbf{D}_{\,\hbox{rot}}} = \overline{\alpha }\;\mathbf{M}+\overline{\beta }\;{\mathbf{K}_{\,\hbox{static}}} + {\mathbf{D}_{\,\hbox{rot}}} </math>
+| style="text-align: center;" | <math>\boldsymbol{u} = \boldsymbol{\overline{u}} \quad  \textrm{on}  \quad \varphi (\partial \Omega _D) </math>
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.5.a)
+|-
+| style="text-align: center;" | <math> \boldsymbol{\sigma } \cdot \boldsymbol{n} = \boldsymbol{\overline{t}} \quad  \textrm{on}  \quad \varphi (\partial \Omega _N)  </math>
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.5.b)
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.26)
 |}
-Where <math display="inline">{\mathbf{D}_{\,\hbox{rot}}}</math>, that accounts for the Coriolis effect, cannot be expressed in terms of <math display="inline">\mathbf{K}</math> nor <math display="inline">\mathbf{M}</math> and neither is a symmetric matrix. Therefore, modal analysis in rotating structures becomes impracticable unless '''Coriolis effect''' is neglected. Nonetheless, it is a powerful tool that allows extraction of important information about the system, such as natural frequencies and mode shapes. Moreover, modal decomposition leads to exact results if all the <math display="inline">\mathbf{n}</math> terms of the summation are taken into consideration. This exact solution will help assessing the validity of the approximated results obtained through the numerical methods presented in the next section.
+where <math display="inline"> \boldsymbol{n} </math> is the unit outward normal.
-==5.2 Numerical integration==
+In order to fully define the Boundary Value Problem a stress-strain relation, like those ones defined in Chapter [[#3 Constitutive Models|3]], is needed.
-In order to overcome the limitations modal analysis is subject to, numerical integration methods arose. The key is to solve the elastodynamic differential equation by integrating over the time domain. As these integrals do not have exact close solution, they are approximated trough a set of numerical methods and algorithms. In particular, the implicit '''Newmark''' <math display="inline">\boldsymbol{\beta }</math>'''-method''' <span id='citeF-12'></span>[[#cite-12|[12]]] is the one covered in this report. To start with, the time interval is divided into <math display="inline">\textrm{N}</math> time steps:
+==4.2 Weak form==
+In Section [[#4.1 Governing equations|4.1]] the strong form of the problem has been defined. In this section, the  weak form is derived, following the formulation explained in <span id='citeF-148'></span>[[#cite-148|[148]]], a displacement-based finite element procedure.
+Let the displacement space <math display="inline">\mathcal{V} \in [\mathbf{\mathbf{H}}^{1} (\mathcal{B})]^{d}</math> be the space of vector functions whose components and their first derivatives are square-integrable, the integral form of the problem is
+<span id="eq-4.6"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,381: / Line 3,643: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>[0, \textrm{T}]=\left[\textrm{t}^{0}, \textrm{t}^{1}\right]\cup \left[\textrm{t}^{1}, \textrm{t}^{2}\right]\ldots \cup \left[\textrm{t}^{n}, \textrm{t}^{n+1}\right]\ldots \left[\textrm{t}^{N-1},\textrm{t}^{N}\right] </math>
+| style="text-align: center;" | <math>\int _{\varphi (\Omega )} (\nabla \cdot \boldsymbol{\sigma })\cdot  \boldsymbol{w} dv + \int _{\varphi (\Omega )} \rho \left(\boldsymbol{b}-\boldsymbol{a}\right)\cdot \boldsymbol{w} dv -\int _{\varphi (\partial \Omega _N)}\left(\boldsymbol{\sigma } \cdot \boldsymbol{n}- \boldsymbol{\overline{t}} \right) \cdot \boldsymbol{w} da=0, </math>
+|-
+| style="text-align: center;" | <math>\quad \forall \boldsymbol{w}\in \mathcal{V}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.27)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.6)
 |}
-Where <math display="inline">\textrm{t}^n</math> is a generic time value. Following this description, equation [[#eq-5.3|5.3]] is discretised:
+where <math display="inline">\boldsymbol{w}</math> is an arbitrary test function, such that <math display="inline">\boldsymbol{w} = \left\lbrace \boldsymbol{w}\in \mathcal{V}\mid \boldsymbol{w} = \boldsymbol{0} \, \textrm{on}  \, \varphi (\partial \Omega _D) \right\rbrace </math>, <math display="inline"> dv </math> is the differential volume and <math display="inline"> da </math> the differential boundary surface. By integrating by parts, applying the divergence theorem and considering the symmetry of the stress tensor, the following expression is obtained
-<span id="eq-5.28"></span>
+<span id="eq-4.7"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,394: / Line 3,658: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M} \;\mathbf{a}^{n+1} \;+\;\mathbf{D}^{n+1}  \;\mathbf{v}^{n+1} \;+\;\mathbf{K}^{n+1}\; {\mathbf{d}}^{n+1}=\mathbf{F}^{n+1} \;\;\;\;\;\hbox{where}\;\;\;\;\;\mathbf{a}=\ddot{\mathbf{d}}\;\;,\;\;\;\mathbf{v}=\dot{\mathbf{d}} </math>
+| style="text-align: center;" | <math>\int _{\varphi (\Omega )} \boldsymbol{\sigma }: (\nabla ^S\boldsymbol{w}) dv - \int _{\varphi (\Omega )} \rho \left(\boldsymbol{b}-\boldsymbol{a}\right)\cdot \boldsymbol{w} dv - \int _{\varphi (\partial \Omega _N)}\boldsymbol{\overline{t}} \cdot \boldsymbol{w} da=0  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.28)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.7)
 |}
-Recall that <math display="inline">\mathbf{D}</math> and <math display="inline">\mathbf{K}</math> are no longer constant, as angular acceleration is now taken into consideration.  Newmark <math display="inline">\beta </math>-methods are a family of integration schemes in which displacements and velocities at time <math display="inline">\textrm{t}^{n+1}</math> are uploaded through the following formulae:
+Under the assumption that the stress tensor is a function of the current strain only
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,406: / Line 3,670: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf\hbox{Initial conditions}\;\;\Rightarrow \;\;\mathbf{a}^{0}= \mathbf{M}^{-1}\, \left(\mathbf{F}^{\,0}  - \mathbf{D}^{\,0}\,\mathbf{v}^0 - \mathbf{K}^{\,0}\,\mathbf{d}^0 \right) </math>
+| style="text-align: center;" | <math>\boldsymbol{\sigma } = \boldsymbol{\sigma }(\boldsymbol{\epsilon }) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.29)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.8)
 |}
-<span id="eq-5.30"></span>
+the problem is reduced to find a kinematically admissible field <math display="inline"> \boldsymbol{u} </math> that satisfies
+<span id="eq-4.9"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,417: / Line 3,683: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{d}^{n+1}=\tilde{\mathbf{d}}^{n}+\beta \,\Delta \textrm{t}^{2} \;\mathbf{a}^{n+1} \;\;,\;\;\;\hbox{with}\;\;\;\;\;\tilde{\mathbf{d}}^{n} =\mathbf{d}^{n}+\Delta \textrm{t}\; \mathbf{v}^{n}+\frac{\Delta \textrm{t}^{2}}{2}(1-2 \beta ) \;\mathbf{a}^{n} </math>
+| style="text-align: center;" | <math>G(\boldsymbol{u},\boldsymbol{w})=0\quad \forall \boldsymbol{w}\in \mathcal{V}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.30)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.9)
 |}
-<span id="eq-5.31"></span>
+where <math display="inline"> G </math> is the virtual work functional defined as
+<span id="eq-4.10"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,428: / Line 3,696: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{v}^{n+1}=\tilde{\mathbf{v}}^{n}+\gamma \,\Delta \textrm{t}\; \mathbf{a}^{n+1} \;\;,\;\;\;\hbox{with}\;\;\;\;\;\tilde{\mathbf{v}}^{n}=\mathbf{v}^{n}+(1-\gamma ) \Delta \textrm{t} \;\mathbf{a}^{n} </math>
+| style="text-align: center;" | <math>G(\boldsymbol{u},\boldsymbol{w})=\int _{\varphi (\Omega )} \boldsymbol{\sigma }: (\nabla ^S\boldsymbol{w}) dv - \int _{\varphi (\Omega )} \rho \left(\boldsymbol{b}-\boldsymbol{a}\right)\cdot \boldsymbol{w} dv  - \int _{\varphi (\partial \Omega _N)}\boldsymbol{\overline{t}} \cdot \boldsymbol{w} da  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.31)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.10)
 |}
-Where <math display="inline">\Delta \textrm{t} = \textrm{t}^{n+1}-\textrm{t}^{n}</math> is the time step  and <math display="inline">\beta </math> and <math display="inline">\gamma </math> are numerical parameters. The parameter <math display="inline">\gamma </math> introduces damping to the numerical method, controlling what is called '''artificial viscosity''' <span id='citeF-12'></span>[[#cite-12|[12]]]. It is used to reduce noise in the solution. If <math display="inline">\gamma = \frac{1}{2}</math>, no damping is added, whereas if <math display="inline">\gamma > \frac{1}{2}</math> the artificial viscosity added to the system is proportional to <math display="inline">\gamma -\frac{1}{2}</math>. On the other hand, the <math display="inline">\beta </math> parameter defines how much implicit or explicit the method is. Continuing with the formulation of the numerical scheme, if equation [[#eq-5.31|5.31]] is introduced into [[#eq-5.28|5.28]]:
+==4.3 Linearisation of the spatial weak formulation==
-<span id="eq-5.32"></span>
+In this work the boundary value problems (BVP) is characterized by both geometrical and material non-linearity. When a non-linear BVP is considered, the discretisation of the weak form results in a system of non-linear equations; for the solution of such a system, a linearisation is, therefore, needed.  The most used and known technique is the Newton-Raphson's iterative procedure, which makes use of directional derivatives to linearise the non-linear equations. The virtual work functional of Equation [[#eq-4.10|4.10]] is linearised with respect to the unknown <math display="inline"> \boldsymbol{u} </math>, using an arbitrary argument <math display="inline"> \boldsymbol{u}^* </math>, which is chosen to be the last known equilibrium configuration. The linearised problem is to find <math display="inline"> \delta \boldsymbol{u} </math> such that
+<span id="eq-4.11"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,441: / Line 3,711: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>(\mathbf{M}+\gamma \,\Delta \textrm{t} \;\mathbf{D}^{n+1})\; \mathbf{a}^{n+1}+\mathbf{K}^{n+1}\; \mathbf{d}^{n+1} =\mathbf{F}^{n+1}-\mathbf{D}^{n+1} \;\tilde{\mathbf{v}}^{n} </math>
+| style="text-align: center;" | <math>L(\delta \boldsymbol{u}, \boldsymbol{w})\simeq G(\boldsymbol{u}^*,\boldsymbol{w})+DG(\boldsymbol{u}^*,\boldsymbol{w})[\delta \boldsymbol{u}]=0,\quad \forall \boldsymbol{w}\in \mathcal{V}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.32)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.11)
 |}
-Acceleration at <math display="inline">\textrm{t}^{n+1}</math> is obtained from equation [[#eq-5.30|5.30]]:
+where <math display="inline"> L </math> is the linearised virtual work function and
-<span id="eq-5.33"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,454: / Line 3,723: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{a}^{n+1}=\frac{1}{\beta \, \Delta \textrm{t}^{2}}\left(\mathbf{d}^{n+1}-\tilde{\mathbf{d}}^{n}\right) </math>
+| style="text-align: center;" | <math>DG(\boldsymbol{u}^*,\boldsymbol{w})[\delta \boldsymbol{u}]=\dfrac{d}{d\gamma }\bigg|_{\gamma=0}G(\boldsymbol{u}^*+\mathfrak{\gamma }\delta \boldsymbol{u},\boldsymbol{w}) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.33)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.12)
 |}
-Then, expression [[#eq-5.32|5.32]] transforms into:
+is the directional derivative of <math display="inline"> G </math> at <math display="inline"> \boldsymbol{u}^* </math> in the direction of <math display="inline"> \delta \boldsymbol{u} </math>, given by
-<span id="eq-5.34"></span>
+<span id="eq-4.13"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,467: / Line 3,736: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>(\tilde{\mathbf{M}}^{n+1}+\mathbf{K}^{n+1})\; \mathbf{d}^{n+1}=\mathbf{F}^{n+1}-\mathbf{D}^{n+1} \;\tilde{\mathbf{v}}^{n}+\tilde{\mathbf{M}}^{n+1}\; \tilde{\mathbf{d}}^{n} \;\;,\;\;\;\hbox{with}\;\;\;\;\;\tilde{\mathbf{M}}^{n+1} =\frac{\mathbf{M}+\gamma \Delta \textrm{t}\; \mathbf{D}^{n+1}}{\beta \, \Delta \textrm{t}^{2}} </math>
+| style="text-align: center;" | <math>DG(\boldsymbol{u}^*,\boldsymbol{w})[\delta \boldsymbol{u}]  =\dfrac{d}{d\gamma }\bigg|_{\gamma=0}\int _{\varphi (\Omega )} \left[\boldsymbol{\sigma }(\boldsymbol{\epsilon }(\gamma )):(\nabla ^S\boldsymbol{w}) - \rho \left(\boldsymbol{b}-\boldsymbol{a}\right)\cdot \boldsymbol{w} \right]dv+ </math>
+|-
+| style="text-align: center;" | <math> - \dfrac{d}{d\gamma }\bigg|_{\gamma=0} \int _{\varphi (\partial \Omega _N)}\boldsymbol{\overline{t}} \cdot \boldsymbol{w}da   </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.34)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.13)
 |}
-The algorithm for the implicit case (<math display="inline">\beta >0</math>) can be summarised as:
+Under the assumption of conservative external loads, only the terms related to the internal and inertial forces are dependent on the deformation. Using the following definitions
-<ol>
-<li>Compute <math display="inline">\tilde{\mathbf{d}}^{n}</math> and <math display="inline">\tilde{\mathbf{v}}^{n}</math> through equations [[#eq-5.30|5.30]] and [[#eq-5.31|5.31]].    </li>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-<li>Obtain <math display="inline">{\mathbf{d}}^{n+1}</math> solving the system of equations stated in [[#eq-5.34|5.34]].    </li>
-<li>Update accelerations <math display="inline">{\mathbf{a}}^{n+1}</math> and velocities <math display="inline">{\mathbf{v}}^{n+1}</math>   using expressions [[#eq-5.33|5.33]] and [[#eq-5.31|5.31]].  </li>
-</ol>
-<div id='img-5.4'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-scheme2.png|600px|Newmark β-method algorithm schematisation for the case of rotating structures. ]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 5.4:''' Newmark <math>\beta </math>-method algorithm schematisation for the case of rotating structures.
+|-
+| style="text-align: center;" | <math>\boldsymbol{\epsilon }(\gamma )=\nabla ^S(\boldsymbol{u}^*+\gamma \delta \boldsymbol{u}) = \boldsymbol{\epsilon }^* + \gamma \nabla ^S(\delta \boldsymbol{u}) </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.14)
 |}
-This scheme is found to be '''unconditionally stable''' for <math display="inline">\beta \geq \frac{\gamma }{2} \geq \frac{1}{4}</math>. It is found that <math display="inline">\beta =\frac{1}{4}</math> and <math display="inline">\gamma =\frac{1}{2}</math> yield to reasonable convergence rates. If those particular values of numeric parameters are chosen, then the scheme is known as constant average acceleration method. Recall that if <math display="inline">\;\mathbf{F}^{n+1} = f({\mathbf{d}}^{n+1}, \,\textrm{t}^{n+1})</math>, then [[#eq-5.34|5.34]] would be a nonlinear system of equations, which is to be tackled through numerical schemes such as Newton-Raphson. Fortunately, this is not the case of study and displacements are simply computed as:
+where <math display="inline">\boldsymbol{\epsilon }^*=\nabla ^S(\boldsymbol{u}^*)</math>  is the strain field at <math display="inline"> \boldsymbol{u}^* </math> and <math display="inline">\boldsymbol{u}(\gamma )= \boldsymbol{u}^* + \gamma \delta \boldsymbol{u}</math>,  the directional derivative <math display="inline"> DG(\boldsymbol{u}^*,\boldsymbol{w})[\delta \boldsymbol{u}] </math> reduces to
+<span id="eq-4.15"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,497: / Line 3,763: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{d}^{n+1}=\left(\tilde{\mathbf{M}}^{n+1}+\mathbf{K}^{n+1}\right)^{-1}\;\left(\mathbf{F}^{n+1}-\mathbf{D}^{n+1} \;\tilde{\mathbf{v}}^{n}+\tilde{\mathbf{M}}\; \tilde{\mathbf{d}}^{n}\right) </math>
+| style="text-align: center;" | <math>DG(\boldsymbol{u}^*,\boldsymbol{w})[\delta \boldsymbol{u}]=\dfrac{d}{d\gamma }\bigg|_{\gamma=0}\left(\int _{\varphi (\Omega )} \left[\boldsymbol{\sigma }(\boldsymbol{\epsilon }(\gamma )):(\nabla ^S\boldsymbol{w})+\rho \boldsymbol{a}(\boldsymbol{u}(\gamma )) \cdot \boldsymbol{w} \right]dv \right)</math>
+|-
+| style="text-align: center;" | <math>= DG^{static}(\boldsymbol{u}^*,\boldsymbol{w})[\delta \boldsymbol{u}] + DG^{dynamic}(\boldsymbol{u}^*,\boldsymbol{w})[\delta \boldsymbol{u}]  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.35)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.15)
 |}
-===5.2.1 Timestep selection===
+which can be split in a static and dynamic contribution.
-The presented method is a practical tool that allows the solving of complex nonlinear systems. However, convergence of the solution is only granted for sufficient small values of <math display="inline">\Delta \textrm{t}</math>. Its value is directly linked with the highest frequency involved in the problem. According to the '''Nyquist-Shannon''' theorem, sampling frequency <math display="inline">f_s</math> needs to be at least twice the maximum frequency of the signal. In the case of study, two different phenomena are to be taken into account: '''vibration''' in the noninertial frame and '''rotation''' in the inertial one.
-The period of the rotation motion, supposing constant angular acceleration <math display="inline">\alpha </math> and initial velocity <math display="inline">v_0</math>, can be posed as:
+Under the assumption of finite strains and adopting an Updated Lagrangian kinematic framework,  the expression of the directional derivative (Equation [[#eq-4.15|4.15]]) should be derived in spatial form. A common way to do that consists in linearising the material weak form and in doing a  push-forward operation to recover the spatial form <span id='citeF-148'></span>[[#cite-148|[148]]]. Therefore, the linearisation of the weak form derived with respect to the initial configuration reads:
+<span id="eq-4.16"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,513: / Line 3,780: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\hbox{period} = \frac{1}{\alpha } \left(  \sqrt{v_0 + 4\,\pi \,\alpha \,(k+1)} - \sqrt{v_0 + 4\,\pi \,\alpha \,k}\,\right)\;\;,\;\;\;\; k = \hbox{floor}\left[\frac{1}{2\pi } (v_0 +\frac{1}{2} \alpha \,\textrm{t}^2)\right] </math>
+| style="text-align: center;" | <math>DG(\boldsymbol{u}^*,\boldsymbol{w})[\delta \boldsymbol{u}]  =\int _{\Omega }\nabla _X\delta \boldsymbol{u}\boldsymbol{S}\cdot \nabla _X\boldsymbol{w} dV </math>
+|-
+| style="text-align: center;" | <math> + \int _{\Omega }\left[(\boldsymbol{F}^T\nabla ^{S}_x\boldsymbol{w}\boldsymbol{F}):\mathbb{C}(\boldsymbol{F}^T\nabla ^{S}_{x}\delta \boldsymbol{u}\boldsymbol{F})\right]dV </math>
+|-
+| style="text-align: center;" | <math>+\int _{\Omega }\rho _0\frac{d\boldsymbol{a}}{d\boldsymbol{u}} \cdot \boldsymbol=\delta\boldsymbol{u}  dV  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.36)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.16)
 |}
-As rotations undergoing angular acceleration for a long period of time are not usual, let's consider a rotation at very high constant spin speed, say <math display="inline">\Omega = 15000 \;\hbox{rpm}</math>. In consequence, the  sampling frequency should be greater than <math display="inline">500\; Hz</math>, and the timestep smaller than 2 <math display="inline">m s</math>. To put these numbers in context, recall natural frequencies from figure [[#img-5.1|5.1]]: the third mode presented a frequency of <math display="inline">637\; Hz</math>, implying a sampling frequency greater than <math display="inline">1275\; Hz</math>.
+where <math display="inline"> \nabla _X </math> and <math display="inline"> \nabla _x </math> are the material and spatial gradient  operator, respectively, <math display="inline"> \boldsymbol{S} </math> is the Second Piola Kirchhoff stress tensor, <math display="inline"> \mathbb{C} </math> is the fourth order incremental constitutive tensor and <math display="inline"> dV </math> is the differential volume element in the underformed configuration. The linearisation of the weak form with respect to the current configuration can be derived by pushing-forward the linearisation of Equation [[#eq-4.16|4.16]]. The first term can be directly written in terms of the Kirchhoff stress <math display="inline"> \boldsymbol{\tau } = \boldsymbol{F}\boldsymbol{S}\boldsymbol{F}^T </math> as
-In general, vibration frequencies are higher than the rotation velocity, and thus the selection of <math display="inline">\Delta \textrm{t}</math> will not be influenced by the velocity of the rotation motion. The criteria to follow when picking a sampling frequency is to ensure that it is at least twice the highest vibration frequency to capture.
+<span id="eq-4.17"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\nabla _X\delta \boldsymbol{u}\boldsymbol{S}\cdot \nabla _X\boldsymbol{w} = \nabla _X\delta \boldsymbol{u}\boldsymbol{F}^{-1}\boldsymbol{\tau }\boldsymbol{F}^{-T}\cdot \nabla _X\boldsymbol{w}  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.17)
+|}
-===5.2.2 Validation===
+and using this standard identity <math display="inline">\nabla _x a = \nabla _X a\boldsymbol{F}^{-1}</math>, Equation [[#eq-4.17|4.17]] can be written as
-In order to asses whether the numerical scheme is performing properly, numerical results will be compared with those obtained with modal analysis. As the latter is restricted to free undamped static structures, the case of study will involve the free vibration of the 2D cantilever rectangular beam presented in [[#img-4.7|4.7]]. Imagine the beam  under the effect of a distributed load along its upper side, when all of the sudden the force '''vanishes''' and the structure starts to oscillate from its deformed configuration around its equilibrium position.
+<span id="eq-4.18"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-Although highly visual, animations are not a proper way of comparing quantitatively the two responses. Instead, the plot of dynamic displacements at different degrees of freedom will be compared for both modal and numerical cases. In fact, the nodes whose vertical displacement will be graphed are those located at the centre and tip of the '''mean line''' of the beam. The modal solution will be obtained taking into account the six first mode shapes with the lowest frequency, whereas the numerical scheme will be based on the constant average acceleration method (<math display="inline">\gamma =\frac{1}{2}</math> and <math display="inline">\beta =\frac{1}{4}</math>). The selected timestep depends of the highest natural frequency to capture, which in this case is linked with the pure bending mode. Hence, <math display="inline">\Delta \textrm{t} = 1 \, ms</math>. The results are compared in the following figure, where dots represent the approximated numerical solution: <div id='img-5.5'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-modalnum1.png|420px|Comparison of modal and numerical methods to solve the free undamped vibration of a 2D fixed-free beam. ]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 5.5:''' Comparison of modal and numerical methods to solve the free undamped vibration of a 2D fixed-free beam.
+|-
+| style="text-align: center;" | <math>\nabla _X\delta \boldsymbol{u}\boldsymbol{S}\cdot \nabla _X\boldsymbol{w} = \nabla _x\delta \boldsymbol{u}\boldsymbol{\tau }\cdot \nabla _x\boldsymbol{w}  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.18)
 |}
-As can be spotted, amplitude in the tip is higher than in the centre. Furthermore, the frequency of vibration is that of the bending motion (<math display="inline">\sim 41\, Hz</math>), as was expected from figure [[#img-5.2|5.2]]. The main problem of comparing both solutions is that numerical methods, unlike modal analysis, cannot '''filter''' high frequency modes. Thus, the numeric solution is always affected by high frequency components and is not capable of representing the bending motion in isolation. In consequence, modal and numeric results will only be coincident if all the mode shapes are taken into consideration and the time step is sufficiently small. The following figure compares horizontal displacements at the tip using 600 different eigenvalues, and a numeric timestep of <math display="inline">0.1\mu s</math>: <div id='img-5.6'></div>
+The second  integral of Equation [[#eq-4.16|4.16]] can be re-written as:
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-horizontal_dis_1_2.png|420px|Modal and numeric (red crosses) methods comparison, this time with higher accuracy.]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 5.6:''' Modal and numeric (red crosses) methods comparison, this time with higher accuracy.
+|-
+| style="text-align: center;" | <math>\int _{\Omega }\nabla ^S_{x}\boldsymbol{w}:\widehat{\mathbb{C}}\left[\nabla ^S_{x}\delta \boldsymbol{u}\right]dV </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.19)
 |}
-===5.2.3 Limitations===
+adopting the transformation of the fourth order incremental constitutive tensor <math display="inline"> \mathbb{C} </math> in Voigt notation <span id='citeF-148'></span>[[#cite-148|[148]]]:
-Numerical methods allow complex and even nonlinear systems to be tackled, obtaining results with a proper degree of accuracy. Even so, one of their main drawbacks is linked with the requirement of really small time steps in order to capture structural vibrations. This implies high computation time and memory usage. Recall that in order to animate the results, displacements need to be computed for each timestep. For instance, the displacements of a 3D system with 30000 nodes studied during a timespan of <math display="inline"> 10s</math> with <math display="inline">\Delta \textrm{t} = 0.1\,ms</math> would take up 67 gigabytes of memory. In reality, postprocessing tools also require from other inputs, increasing even more the memory consumption. This makes numeric simulations of complex structures only feasible for institutions with high computational resources.
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\widehat{\mathbb{C}}_{iklm}=F_{iA}F_{lC}F_{mD}F_{kB}\mathbb{C}_{ABCD} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.20)
+|}
-Another disadvantage of numerical schemes with respect to modal analysis is that no mode shapes nor natural frequencies are obtained. In fact, what is computed through numerical methods is the temporal evolution of the individual displacements at each degree of freedom. The whole phenomena of vibration is difficult to capture by only looking at individual graphs, so postprocessing tools use this information to create an animation of the dynamic response. But animations are not an outright solution, as no information regarding the predominating modes of the structure and its frequency are obtained.
+where lowercase indexes are referred to the incremental constitutive tensor relative to the Kirchhoff stress, while uppercase indexes to the incremental constitutive tensor relative to the Second Piola Kirchhoff stress.
-When very different frequencies play an important role in the system, the latter has to be studied at different '''time scales''' as there will be motions with very long period coexisting with other ones which elapse very fast. The study of both motions at the same time based on animations is very tricky, and requires from patience and experience. In order to overcome this limitation, the next section reviews a promising method that captures the predominating modes and frequencies of a structure using only the raw data obtained from the numerical approach.
+With these transformations, the linearisation of the static contribution at the current configuration is
-==5.3 Singular value decomposition==
+<span id="eq-4.21"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>DG^{static}(\boldsymbol{u}^*,\boldsymbol{w})[\delta \boldsymbol{u}] = \int _{\Omega }\nabla _x\delta \boldsymbol{u}\boldsymbol{\tau }\cdot \nabla _x\boldsymbol{w} + \nabla ^S_{x}\boldsymbol{w}:\widehat{\mathbb{C}} [\nabla ^S_{x}\delta \boldsymbol{u}]dV  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.21)
+|}
-The singular value decomposition, commonly referred as SVD, is one of the most versatile tools of linear algebra. It allows the factorisation of a rectangular matrix <math display="inline">\mathbf{A}</math> of the form:
+Considering the definition of the determinant of the deformation gradient:
-<span id="eq-5.37"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,560: / Line 3,861: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{A} = \mathbf{U} \,\boldsymbol{\Sigma }\, \mathbf{V}^T \;,\;\;\;\hbox{or} \;\;\;\;\textrm{a}_{\textrm{ij}}=\sum _{k=1}^{n} \textrm{u}_{\textrm{ik}} \;\sigma _{k} \;\textrm{v}_{\textrm{jk}} </math>
+| style="text-align: center;" | <math>det(\boldsymbol{F}) = J = \dfrac{dv}{dV} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.37)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.22)
 |}
-Where:
+the following relations hold
-* <math display="inline">\mathbf{A}</math> is a <math display="inline">\textrm{m}\times \textrm{n}</math> matrix.
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-* <math display="inline">\mathbf{U}</math> is a <math display="inline">\textrm{m}\times \textrm{m}</math> orthonormal matrix known as the matrix of left side vectors (LSV).
+|-
-* <math display="inline">\mathbf{V}</math> is a <math display="inline">\textrm{n}\times \textrm{n}</math> orthonormal matrix known as the matrix of right side vectors (RSV).
+|
-* <math display="inline">\boldsymbol{\Sigma }</math> is a <math display="inline">\textrm{m}\times \textrm{n}</math> diagonal matrix known as the matrix of singular values.
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\boldsymbol{\sigma } = \dfrac{1}{J}\boldsymbol{\tau } </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.23)
+|}
-The diagonal entries of <math display="inline">\boldsymbol{\Sigma }</math> are referred as singular values <math display="inline">\sigma _{i}</math>, fulfilling that <math display="inline">\sigma _{i}>0</math>. Quite often, the matrix of singular values is sorted in ascending order, so that <math display="inline">\sigma _{i+1} \leq \sigma _{i}</math>. If <math display="inline">\mathbf{A}</math> is a real matrix, then both <math display="inline">\mathbf{U}</math> and <math display="inline">\mathbf{S}</math> are orthonormal: <math display="inline">\mathbf{U}^T \, \mathbf{U} = \mathbf{V}^T \, \mathbf{V} = \mathbf{I} </math>. An intuitive approach to SVD is to think of <math display="inline">\mathbf{A}</math> as a linear transformation. Then, the singular value decomposition can be though as a composition of three geometrical transformations: a rotation <math display="inline">\mathbf{U}</math>, a scaling or stretching <math display="inline">\boldsymbol{\Sigma }</math> and another rotation <math display="inline">\mathbf{V}^T</math>.
+<span id="eq-4.24"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\overline{\widehat{\mathbb{C}}} = \dfrac{1}{J}\widehat{\mathbb{C}}  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.24)
+|}
-The SDV can be interpreted as a generalisation of the eigendecomposition, which is restricted to positive semidefinite matrices. In fact, the SVD can be split into  a pair of eigenproblems:
+where <math display="inline"> \boldsymbol{\sigma } </math> and <math display="inline"> \boldsymbol{\tau } </math> are the Cauchy and Kirchhoff stress tensor, respectively, and <math display="inline"> \overline{\widehat{\mathbb{C}}} </math> is the incremental constitutive tensor relative to the Cauchy stress. Equation [[#eq-4.16|4.16]] can now be re-written in the current configuration as
+<span id="eq-4.25"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,581: / Line 3,897: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{matrix}\mathbf{A}\;\mathbf{A}^T = \mathbf{U} \,\boldsymbol{\Sigma }\, \overset{\mathbf{I}}{\overbrace{\mathbf{V}^T \, \mathbf{V}}}\,\boldsymbol{\Sigma }^T\,\mathbf{U}^T \;\;\;\Rightarrow \;\;\;\mathbf{A}\;\mathbf{A}^T \mathbf{U} = \mathbf{U}\,\boldsymbol{\Sigma }^2 \\  \mathbf{A}^T\mathbf{A} = \mathbf{V}\boldsymbol{\Sigma }^T\,\overset{\mathbf{I}}{\overbrace{\mathbf{U}^T \, \mathbf{U}}} \,\boldsymbol{\Sigma }\,\mathbf{V}^T\;\;\;\Rightarrow \;\;\;\mathbf{A}^T\mathbf{A}\,\mathbf{V} =\mathbf{V}\,\boldsymbol{\Sigma }^2 \end{matrix} </math>
+| style="text-align: center;" | <math>DG(\boldsymbol{u}^*,\boldsymbol{w})[\delta \boldsymbol{u}] = \int _{\varphi (\Omega )} \left(\nabla _x\delta \boldsymbol{u}\boldsymbol{\sigma }\cdot \nabla _x\boldsymbol{w} + \nabla ^S_{x}\boldsymbol{w}:\overline{\widehat{\mathbb{C}}} [\nabla ^S_{x}\delta \boldsymbol{u}]+\rho \frac{d\boldsymbol{a}}{d\boldsymbol{u}} \cdot \boldsymbol=\delta\boldsymbol{u}\right)dv  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.38)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.25)
 |}
-Hence, <math display="inline">\mathbf{U}</math> can be thought as the set of orthonormal eigenvectors of <math display="inline">\mathbf{A}\;\mathbf{A}^T</math> and <math display="inline">\mathbf{U}</math>  as the set of orthonormal eigenvectors of <math display="inline">\mathbf{A}^T\,\,\mathbf{A}</math>. The singular values of <math display="inline">\mathbf{A}</math> are in fact the square root of the eigenvalues of both <math display="inline">\mathbf{A}\;\mathbf{A}^T</math> and <math display="inline">\mathbf{A}^T\,\,\mathbf{A}</math>. This property brings to light the capacity of the SVD to discover some of the same kind of information as with the '''modal analysis'''. In fact, the simplest numerical scheme to tackle the SVD problem consists in solving one of the above eigenproblems, obtaining either <math display="inline">\mathbf{U}</math> or <math display="inline">\mathbf{V}</math> as eigenvectors and <math display="inline">\boldsymbol{\Sigma }^2</math> as eigenvalues. Then, the remaining matrix of vectors is obtained using equation [[#eq-5.37|5.37]].
+Equation [[#eq-4.25|4.25]] represents the linearisation of the spatial weak formulation, also known as the Updated Lagrangian formulation, since the deformation state <math display="inline"> \boldsymbol{u}^* </math> is continuously updated during the non-linear incremental solution procedure, e.g. the Newton Raphson's method.
-===5.3.1 Truncated SVD===
+==4.4 Spatial Discretisation==
-The SVD is an interdisciplinary  technique used in diverse fields such as signal processing, pattern recognition, data compression, weather prediction and social-media algorithms among many others. A popular application is related with '''dimensionality reduction''': a process through which a given matrix <math display="inline">\mathbf{A}</math> is approximated by a lower rank matrix <math display="inline">\mathbf{A}^{(r)}</math>. The SVD provides the best k-rank approximation to <math display="inline">\mathbf{A}</math> of the form:
+Let <math display="inline">\mathcal{V}_{h}</math>  be a finite element space to approximate <math display="inline">\mathcal{V}</math>.  The problem is now finding <math display="inline">\mathbf{u}_{h} \in \mathcal{V}_{h}</math>  such that
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,597: / Line 3,913: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{A} \approx  \mathbf{A}^{(r)}=\sum _{i=1}^{r} \mathbf{U}_{i}\, \sigma _{i}\, \mathbf{V}_{i} </math>
+| style="text-align: center;" | <math>DG(\boldsymbol{u}^*_h,\boldsymbol{w}_{h})[\delta \boldsymbol{u}_{h}]= -G(\boldsymbol{u}^*_h,\boldsymbol{w}_{h}),\quad \forall \boldsymbol{w}_{h}\in \mathcal{V}_{h} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.39)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.26)
 |}
-The key is to perform a SVD picking only the <math display="inline">\textrm{r}</math>-largest values of <math display="inline">\sigma _i</math>. If <math display="inline">r<n</math>, then the dimension of the matrices involved in the SVD problem are reduced: <math display="inline">\mathbf{U}</math> becomes <math display="inline">\textrm{m}\times \textrm{r}</math>, <math display="inline">\mathbf{V}^T</math> is now <math display="inline">\textrm{r}\times \textrm{m}</math> and <math display="inline">\boldsymbol{\Sigma }</math> is reduced into a <math display="inline">\textrm{r}\times \textrm{r}</math> matrix. The truncated version of the SVD is tackled as an optimization of the Frobenius norm of the difference between the original matrix and its approximation. Numerical schemes use a given value of tolerance to limit the truncation error.
+or using Equation [[#eq-4.25|4.25]]
-Even considering the truncated model, computing the SVD can be extremely time-consuming for  large-scale problems. One of the most used techniques to speed up the algorithm relies on block matrix decomposition. The factorisation no longer processes the whole matrix, but performs successive SVDs over smaller sub-matrices. Moreover, in this way the quality of the approximation can be controlled blockwise. These kind of method is known as '''partitioned''' truncated SVD.
+<span id="eq-4.27"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\int _{\varphi (\Omega )}\left\lbrace \nabla _x\delta \boldsymbol{u}_{h}\boldsymbol{\sigma }\cdot \nabla _x\boldsymbol{w}_{h}  +  \nabla ^S\boldsymbol{w}_{h}:\overline{\widehat{\mathbb{C}}} [\nabla ^S\delta \boldsymbol{u}_{h}]+\rho \frac{d\boldsymbol{a}_{h}}{d\boldsymbol{u}_{h}} \cdot \boldsymbol{w}_{h}[\delta \boldsymbol{u}_{h}]\right\rbrace  dv =  </math>
+|-
+| style="text-align: center;" | <math> -\left(\int _{\varphi (\Omega )} \boldsymbol{\sigma }: (\nabla ^S\boldsymbol{w}_{h}) dv - \int _{\varphi (\Omega )} \rho \left(\boldsymbol{b}-\boldsymbol{a}_{h}\right)\cdot \boldsymbol{w}_{h} dv - \int _{\varphi (\partial \Omega _N)}\boldsymbol{\overline{t}} \cdot \boldsymbol{w}_{h} da \right)  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.27)
+|}
-To improve even more the performance of the algorithm, '''randomised''' techniques <span id='citeF-10'></span>[[#cite-10|[10]]] are used to compute an approximation to the range of <math display="inline">\mathbf{A}</math>. These techniques are of special interest when the data comes from discretization of continuum systems. An intuitive approach is to estimate the range of <math display="inline">\mathbf{A}</math> by picking a random collection of vectors of the matrix and examining the subspace formed by the action of <math display="inline">\mathbf{A}</math> on each of them. For sure, randomised SVD algorithms are not that simple and require a deeper knowledge on linear algebra.
+Let us assume to discretise the continuum body <math display="inline"> \mathcal{B} </math> by a set of <math display="inline"> n_p </math> material points and to assign a finite volume of the body <math display="inline"> \Omega _p </math> to each of those material points.  Thus, the geometrical representation (<math display="inline">\mathcal{B}_h</math>) of <math display="inline"> \mathcal{B} </math> reads
-===5.3.2 SVD in structural analysis===
+<span id="eq-4.28"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\mathcal{B}\approx \mathcal{B}_h=\bigcup _{p = 1}^{n_p}\Omega _p  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.28)
+|}
-In the case of study, which is structural analysis, it is from interest to work in the basis defined by the mass matrix <math display="inline">\mathbf{M}</math> instead of the euclidean space. The goal is to re-express <math display="inline">\mathbf{A}</math> in such a way that redundancies are filtered out and actual dominant displacement fluctuation patterns are revealed <span id='citeF-13'></span>[[#cite-13|[13]]]. In order to develop this normalisation, <math display="inline">\mathbf{M}</math> needs to be a symmetric positive definite matrix such that admits the Cholesky factorisation:
+and with this approximation the integrals of the weak form can be written as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,617: / Line 3,953: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M} =  \overline{\mathbf{M}}^T \;\overline{\mathbf{M}}  </math>
+| style="text-align: center;" | <math>\int _{\mathcal{B}}(...)dV\approx \int _{\mathcal{B}^h}(...)dV=\bigcup _{p = 1}^{n_p}\int _{\Omega _p}(...)d\Omega _p </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.40)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.29)
 |}
-Where <math display="inline">\overline{\mathbf{M}}</math> is an upper triangular matrix. Then, the truncated SVD is not performed to the matrix <math display="inline">\mathbf{A}</math> but to the normalised matrix <math display="inline">\overline{\mathbf{A}} = \overline{\mathbf{M}}\;\mathbf{A}  </math> :
+For the computation of the linearised system of equations, an integration is necessary over the volume occupied by each material point <math display="inline"> \Omega _p </math>. By using the spatial discretisation defined in Equation [[#eq-4.28|4.28]], the linearised system of equations (see Equation [[#eq-4.27|4.27]]) is rewritten as
+<span id="eq-4.30"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,629: / Line 3,966: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\overline{\mathbf{A}} = \overline{\mathbf{U}} \,\boldsymbol{\Sigma }\, \mathbf{V}^T \;\;\;\Rightarrow \;\;\;\overline{\mathbf{A}}^{\,(r)}\approx \sum _{i=1}^{r} \overline{\mathbf{U}}_{i}\, \sigma _{i}\, \mathbf{V}_{i} </math>
+| style="text-align: center;" | <math>\bigcup _{p = 1}^{n_p}\int _{\Omega _p}\left(\left\lbrace \nabla _x\delta \boldsymbol{u}_{h}\boldsymbol{\sigma }\cdot \nabla _x\boldsymbol{w}_{h}  +  \nabla ^S\boldsymbol{w}_{h}:\overline{\widehat{\mathbb{C}}} [\nabla ^S\delta \boldsymbol{u}_{h}]+\rho \frac{d\boldsymbol{a}_{h}}{d\boldsymbol{u}_{h}} \cdot \boldsymbol{w}_{h}[\delta \boldsymbol{u}_{h}]\right\rbrace \right)d\Omega _p </math>
+|-
+| style="text-align: center;" | <math> = - \bigcup _{p = 1}^{n_p}\left(\int _{\Omega _p}\left(\boldsymbol{\sigma }: (\nabla ^S\boldsymbol{w}_{h})-\rho \left(\boldsymbol{b}-\boldsymbol{a}_{h} \right)\right)d\Omega _p- \int _{\partial{\Omega _N}_p}\boldsymbol{\overline{t}} \cdot \boldsymbol{w}_{h} da_p\right)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.41)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.30)
 |}
-Finally, the right side vectors are obtained from <math display="inline">\overline{\mathbf{U}}</math> :
+and by exploiting the finite element approximation with particle integration the final discretised form is obtained
+<span id="eq-4.31"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,641: / Line 3,981: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{U} =  \overline{\mathbf{M}}^{\,-1}\; \,\overline{\mathbf{U}} </math>
+| style="text-align: center;" | <math>\bigcup _{p = 1}^{n_p}\sum ^{n}_{I=1}\sum ^{n}_{K=1}\boldsymbol{w}_{I}^T\left( \left(\nabla _x N_I \right)^T\boldsymbol{\sigma }\left(\nabla _x N_K \right)\mathbf{I} + \mathbf{B}_{I}^T\mathbf{D}\mathbf{B}_{K}+\frac{N_I\rho N_K}{\beta \Delta t^2 } \mathbf{I}\right)V_p \delta \boldsymbol{u}_K </math>
+|-
+| style="text-align: center;" | <math> = - \bigcup _{p = 1}^{n_p}\sum ^{n}_{I=1}\boldsymbol{w}_{I}^T\left(\mathbf{B}_{I}\boldsymbol{\sigma }- \rho \boldsymbol{b}N_I+\sum ^{n}_{K=1}N_I\rho N_K\boldsymbol{a}_K \right)V_p - \bigcup _{l = 1}^{n_l}\sum ^{n_m}_{I=1}\boldsymbol{w}_{I}^TN_I\boldsymbol{\overline{t}} A_l  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.42)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.31)
 |}
-Following this approach, the SVD can be posed in terms of structural analysis. The matrix <math display="inline">\mathbf{A}</math> is nothing else but the matrix of displacements <math display="inline">\mathbf{d}</math> defined in equation [[#eq-5.14|5.14]]. Each column contains the displacements of the unconstrained degrees of freedom at a given value of time, whereas each row accounts for the temporal evolution of the displacement at a single degree of freedom. Hence, defining <math display="inline">\textrm{n}_{\,\mathbf{l}}</math> as the number of unconstrained degrees of freedom and <math display="inline">\textrm{n}_{\,\Delta \textrm{t}}</math> as the number of time-steps into which the time domain is split, <math display="inline">\mathbf{A}</math> is a <math display="inline">\textrm{n}_{\,\mathbf{l}}\times \textrm{n}_{\,\Delta \textrm{t}}</math> matrix.
+where <math display="inline">I  </math> and <math display="inline"> K </math> are the indexes of the finite element's nodes, <math display="inline"> \nabla _x N_I </math> is the spatial gradient of the shape function evaluated at node <math display="inline">I </math>, <math display="inline"> \mathbf{D} </math> is the matrix form of the incremental constitutive tensor <math display="inline"> \overline{\widehat{\mathbb{C}}} </math>, <math display="inline"> V_p </math> is the volume relative to a single material point, <math display="inline"> A_l </math> is the surface  and <math display="inline"> \mathbf{B}_{I} </math> is the deformation matrix relative to node <math display="inline">I</math>, expressed here for a 2D problem as:
-From the structural analysis perspective, <math display="inline">\mathbf{U}</math> is a <math display="inline">\textrm{n}_{\,\mathbf{l}}\times \textrm{r}</math> matrix containing in each column a certain combination of nodal displacement patterns which are predominating in the system as time passes by. These can be thought, in analogy with modal decomposition, as the principal modes of the structure. In reality, left side vectors are not modal mode shapes but a certain combination of the sadistically reigning ones. Hence, on now on the vectors <math display="inline">\mathbf{U}_i</math> will be referred as '''singular modes'''.
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\mathbf{B}_{I} = \begin{bmatrix}{\displaystyle \frac{\partial N_{I}}{\partial x}} & 0 \\       0             & {\displaystyle \frac{\partial N_{I}}{\partial y}}\\      {\displaystyle \frac{\partial N_{I}}{\partial y}}  & {\displaystyle \frac{\partial N_{I}}{\partial x}}   \end{bmatrix} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.32)
+|}
-On the other hand, <math display="inline">\mathbf{V}^T</math> is a <math display="inline">\textrm{r}\times \textrm{n}_{\,\Delta \textrm{t}}</math> matrix each row of which contains a statistically predominant temporal evolution of the whole structure. In contrast with modal analysis, each <math display="inline">\mathbf{V}_i</math> describes a vibration motion which is not necessarily harmonic. The vectors <math display="inline">\mathbf{V}_i</math> will referred as '''singular oscillations'''.
+The left hand side  of Equation [[#eq-4.31|4.31]] is given by three addends multiplied by the increment of the unknowns. The first one is commonly known as the ''geometric'' stiffness matrix
-The grace of this method is that each <math display="inline">\mathbf{V}_i</math> is linked with <math display="inline">\mathbf{U}_i</math>, allowing a certain intuition on how each of the singular modes evolves in time. Through an adequate analysis, one can find out the predominating frequencies of each singular oscillation (RSV), and consequently the vibration frequencies <math display="inline">\overline{\boldsymbol{f}}_i</math> associated to each singular mode (LSV). Moreover, singular values <math display="inline">\sigma _i</math> give a hint on the intensity of each singular mode, allowing the truncated study of the system accounting only for the most relevant ones. The individual motions can then be animated using '''GID''', obtaining a more clear sight of the phenomena than the one obtained with numerical integration.
+<span id="eq-4.33"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\mathbf{K}^{G}_{IK} = \left(\nabla _x N_I \right)^T\boldsymbol{\sigma }\left(\nabla _x N_K \right)\mathbf{I}V_p  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.33)
+|}
-As it has been pointed out, singular oscillations are no longer a pure harmonic and thus may contain the contribution of more than one natural frequency. In order to extract information regarding predominant frequencies, two numerical tools can be used. The most intuitive approach is based on looking for the '''local maxima''' of <math display="inline">\mathbf{V}_i</math>, and approximating the natural frequency as the average of the time intervals between successive maximums. The main drawback of the latter method is that if more than two frequencies are involved in the temporal description of the motion, only the highest one can be extracted from <math display="inline">\mathbf{V}_i</math>. Furthermore, the induced error on the measure of the latter is no longer negligible.
+while the second term is known as the ''material'' stiffness matrix
-A more feasible approach consists in applying the '''discrete Fourier transform''' on <math display="inline">\mathbf{V}_i</math> using in-built <math display="inline">MATLAB</math> functions. By doing so, the singular oscillation is mapped into the frequency domain. Predominating frequencies of each singular mode can be found by looking for the peaks on the power spectral density estimation constructed from the Fourier transform. Recall that the obtained vectors <math display="inline">\mathbf{U}_i</math> and <math display="inline">\mathbf{V}_i</math> have been scaled — in space and temporal coordinates, respectively — during the SVD, and thus information regarding the sampling frequency used during the numeric integration is needed to recover the actual frequencies involved in the vibration motion.
+<span id="eq-4.34"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\mathbf{K}^{M}_{IK} = \mathbf{B}_{I}^T\mathbf{D}\mathbf{B}_{K}V_p  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.34)
+|}
-For sure, the development of a numerical algorithm capable of handling the truncated SVD problem in a short amount of time is far out of the scope of this report. Instead, a <math display="inline">MATLAB</math> tool developed by prof. ''Hernández J.'' will be used to perform a partitioned randomised SVD. This software limits the number of terms of the truncated approximation <math display="inline">\mathbf{A}^{(r)}</math> by using a value of tolerance introduced by the user.
+and their sum represents the static contribution to the tangent stiffness matrix
-===5.3.3 Validation===
+<span id="eq-4.35"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-In order to determine whether the singular value decomposition is a suitable technique for structural analysis, its performance will be assessed by comparing the singular modes and predominant frequencies with the results obtained through modal analysis. Again, the case of study involves a '''2D cantilever beam''' in a free undamped oscillation, as covered in [[#img-5.5|5.5]]. Let's consider the dynamic displacements obtained from modal decomposition as the '''input''' data matrix for the SVD. The goal is to check whether this method is able to recover the information regarding natural modes and frequencies properly.
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\mathbf{K}^{static}_{IK} = \mathbf{K}^{G}_{IK} + \mathbf{K}^{M}_{IK}  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.35)
+|}
-In this particular case, the modal response has been approximated using only the six natural modes with the lowest frequency. The sampling frequency used to discretise equation [[#eq-5.13|5.13]] is <math display="inline">500000 \,Hz</math>. On the other hand, the truncated SVD will only account for the six more relevant singular modes. To help comparison with modal analysis, one may define the '''normalised singular values''' as
+The ''dynamic'' component is given by
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,671: / Line 4,046: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\overline{\sigma }_i = \frac{{\sigma }_i}{\sum _{i =1}^{r}{\sigma }_i} </math>
+| style="text-align: center;" | <math>\mathbf{K}^{dynamic}_{IK} = \frac{N_I\rho N_K}{\beta \Delta t^2 } \mathbf{I} V_p </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5.43)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.36)
 |}
-With this in mind, the SVD and modal results are compared in the following table:
+Finally the tangent stiffness matrix is given by
-{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
-|+ style="font-size: 75%;" |<span id='table-5.1'></span>'''Table. 5.1''' Frequencies and intensities comparison: SVD vs. modal analysis
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
 |
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\mathbf{K}^{tan}_{IK} = \mathbf{K}^{static}_{IK} + \mathbf{K}^{dynamic}_{IK} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.37)
 |}
-The above table reveals an outstanding degree of accuracy in the frequency determination using the SVD. The singular modes associated to the four lower predominant vibration frequencies are displayed in the following figure: <div id='img-5.7'></div>
+and represents the submatrix relative to one node of the discretisation with dimension <math display="inline"> \left[n_{dof} \times n_{dof} \right]</math>, where <math display="inline"> n_{dof} </math> is the number of degrees of freedom of a single node. This matrix can be considered as the Jacobian matrix of the right hand side of Equation [[#eq-4.31|4.31]], i.e., the residual <math display="inline"> \mathbf{R_I} </math>.  Equation [[#eq-4.31|4.31]] can be rewritten in compact form as
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+<span id="eq-4.38"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-modessvd.png|600px|Singular modes associated to the first four lower vibration frequencies. ]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 5.7:''' Singular modes associated to the first four lower vibration frequencies.
+|-
+| style="text-align: center;" | <math>\mathbf{K}^{tan}_{IK}   \delta \boldsymbol{u}_K  = - \mathbf{R}_I.  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.38)
 |}
-Comparing the above illustration with natural modes (figure [[#img-5.1|5.1]]), one finds out that they are recovered almost exactly. The only exception is found in the third natural mode, related with the axial deformation. In this case, neither the predominant frequency nor the singular mode match exactly with the modal results. The reason behind this discordance is that the axial mode is merely activated in a bending scenario. In consequence, being the SVD a statistical method, less reliability is to be expected when recovering the axial mode.
+==4.5 Numerical verification==
-With all, the singular value decomposition has been proven to be a good approach in recovering relevant information from a given matrix of data. In structural analysis, it is a promising tool for obtaining the '''predominant motions''' when modal analysis is not practicable. The next chapter will cover the application of the SVD to a set of rotating scenarios.
+In this section three benchmark tests are considered for the comparison of the MPM and GMM formulations. Firstly, the static analysis of a 2D cantilever beam subjected to its self-weight is analysed and a mesh convergence study is performed. Secondly, the rolling of a rigid disk on inclined plane is studied.  Finally, a cohesive-soil column collapse is analysed. All the numerical experiments have been performed on a PC with one Intel(R) Core(TM) i7-4790 CPU at 3.60GHz.
-=6 Analysis of rotating beams using the SVD=
+===4.5.1 2D cantilever beam. Static analysis===
-Once the methodology for handling rotating structures in non-stationary regime is fully developed, it is time to tackle a few scenarios involving rotating machinery. The simulations will be carried taking advantage of the Newman-<math display="inline">\beta </math> algorithm presented in the previous section, and then the SVD will used to disclose the dominant modes and frequencies of the dynamic response. The results will be '''compared''' with those obtained through modal analysis, which is restricted to non-rotating vibrations. Hence, a hint on the effects centrifugal, tangential and Coriolis forces have on rotating structures will be brought to light in this chapter.
+The static analysis of a 2D cantilever beam subjected to its self-weight under the assumption of plain strain is presented. The cantilever beam has a length <math display="inline">l =8 m</math> and a square cross section of unit side (<math display="inline">b=h=1m</math>) (Figure [[#img-4.1|4.1]]). The beam is modelled with a hyperelastic material (presented in Section [[#3.1 Hyperelastic law|3.1]]): the density is <math display="inline">\rho = 1000kg/m^3</math>, the Young's modulus is <math display="inline">E = 90MPa</math> and the Poisson's ratio is <math display="inline">\nu = 0</math>. The results obtained with the MPM and GMM algorithms are compared with a standard FEM code using the same UL formulation.
-Postprocessing of results will be carried out by '''GiD''', which uses the generic spatial basis <math display="inline">\left\{\textrm{xyz}\right\}^T</math>. The simulations performed in the present chapter will consider only variables in the local frame of reference <math display="inline">\left\{\varsigma \,\varrho \,\varpi  \right\}^T</math>, so although some figures incorporate a set of <math display="inline">\textrm{xyz}</math> axes, keep in mind that they represent in fact the corotational frame.
+<div id='img-4.1'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
-==6.1 Two-dimensional cantilever beam==
-The first case of study continues with the analysis of the rectangular 2D '''cantilever beam''' studied in the previous section. The beam is <math display="inline">0.2 \times 2\,m</math> and modelled using 1000 quadrilateral elements. The main difference is that the beam now undergoes a '''rotation''' perpendicular to the plane, with its axis located at the centre of the fixed end of the beam. The beam is made from 6061 T6 aluminium alloy (<math display="inline">E = 70 \,GPa</math>, <math display="inline">\nu = 0.3</math>), and the considered density is <math display="inline">\rho =\; 2700 kg/m^2</math>.  <div id='img-6.1'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-cantilever.png|600px|Rectangular cantilever beam subject to rotation.]]
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-cantilever_static_ilaria.png|480px|Static 2D cantilever analysis: geometry]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 6.1:''' Rectangular cantilever beam subject to rotation.
+| colspan="1" style="padding:10px;" | '''Figure 4.1:''' Static 2D cantilever analysis: geometry
 |}
+A mesh convergence study is carried out adopting five different mesh sizes, <math display="inline">h</math> = 0.5, 0.25, 0.125, 0.0625 and 0.01<math display="inline">m</math>, respectively.  Quadrilateral elements are used in FEM, MPM and GMM with four integration points per cell (in the case of MPM and GMM the integration points coincide with the material points). In GMM, the mesh is only initially used for the creation of the material points and then deleted. Regarding the spatial search and the evaluation of the shape functions in GMM, a search radius <math display="inline">R = \sqrt{2h^2}</math>, dilation parameters <math display="inline">R_{eff} = R/2</math> and <math display="inline">\gamma = 1.8</math> are adopted in GMM-MLS and GMM-LME, respectively. Under the assumption of linear regime, the vertical deflection at point A of the free edge can be evaluated analytically according to Timoshenko <span id='citeF-158'></span>[[#cite-158|[158]]] as:
-The loading scenario is quite similar as the presented in [[#img-4.6|4.6]]: the beam is under the effect of a distributed <math display="inline">\varrho </math> load on its upper side (-20 kN/m), while it is '''rotating''' at a certain angular velocity. Initial displacements under these conditions will be computed using the static FEM software. The dynamic case will account for a '''sudden''' vanishing of the distributed load and the consequent vibration of the beam. This oscillating behaviour will be analysed under the effect of different angular speeds and accelerations. No internal '''damping''' will be considered in this section. For comparison purposes, the first eight natural frequencies and modes of the presented structure can be found in the report attachments (figure [[#D.1.1 Natural modes and frequencies|D.1.1]]).
+<span id="eq-4.39"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\delta = -\left(\dfrac{\rho g (b h l) l^3}{8EI} + \dfrac{\rho g l^2}{2 G A_s}\right)= -0.67806m  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.39)
+|}
-===6.1.1 Constant angular velocity===
+where <math display="inline"> g </math> is the gravity acceleration, <math display="inline">I = \dfrac{b h^3}{12}</math> the inertia of the beam section and <math display="inline">A_s = \dfrac{5}{6}A</math> the reduced cross section area due to the shear effect. However, the solution is computed under the assumption of non-linearity and, as benchmark solution, the deflection evaluated through the finest mesh is considered. This value is <math display="inline">\delta = -0.67433 m</math> and is equally reached by all the methods.
-The first analysis will involve constant angular speed. The goal is to keep track of the vibration frequencies at different angular velocities, and compare the results with the '''softening effect''' predicted by modal analysis. The imposed angular speed cannot be indiscriminately high, as deformations would no longer be considered small. Using the 1D solver developed in [[#4.2.2 Approximated one-dimensional stationary solution|4.2.2]], it has been found that elastic displacements reach a 1% of the length of the beam when <math display="inline">\Omega = 4000</math> rpm. Hence, this value will not be exceeded in this analysis.
+Figures [[#img-4.2|4.2]] and [[#img-4.3|4.3]] compare the solutions obtained with an Updated Lagrangian FEM, MPM, GMM-MLS and GMM-LME code, respectively, in terms of vertical displacement and Cauchy stress along the horizontal direction. One can observe that the results are in good agreement for all the methods.
-It is important to select an adequate timespan and timestep, as numerical results are sensitive on these parameters. The '''timestep''' is chosen depending on the maximum frequency to capture, in this case <math display="inline">f_{\hbox{max}}\approx 2500\, Hz</math>, corresponding to the eighth natural frequency. Hence, <math display="inline">\Delta \,\textrm{t} < 0.2\,ms</math>. In order to gain accuracy, the used timestep will be <math display="inline">\Delta \,\textrm{t} = 5\,\mu s</math>. To keep computation time low, the '''timespan''' will be no longer than <math display="inline">0.2\,s</math>. With these numeric parameters, and using the constant average acceleration method, the mean time of computation is around 25 minutes. Smaller time-steps or greater time-spans would be too much time-consuming.  '''
+<div id='img-4.2a'></div>
+<div id='img-4.2b'></div>
-====6.1.1.1 Campbell diagram===='''
+<div id='img-4.2c'></div>
+<div id='img-4.2d'></div>
-The dynamic analysis is to be developed for three different angular velocities: 50, 100 and 200 rad/s. The SVD is truncated to the '''eight''' higher singular values. The left and right side vectors obtained for each case are displayed in [[#D.1.2 Constant angular velocity|D.1.2]]. To ease visualisation of the RSVs, each one is plotted for a certain fraction of the total timespan. The dominant frequencies are found looking for the peaks of the '''discrete Fourier transform''' of each RSV. The obtained frequencies are presented in the following figure, and compared with the softening effect predicted by modal decomposition.  <div id='img-6.2'></div>
+<div id='img-4.2'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-campbellnew.png|588px|Campbell diagram: comparison between modal prediction and SVD results.]]
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-static_dispy_fem.png|270px|FEM code]]
+|[[Image:Draft_Samper_987121664-monograph-static_dispy_mpm.png|270px|MPM code]]
+|rowspan="4" style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-static_dispy_legend.png|100px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a) FEM code
+| (b) MPM code
+|-
+|style="padding-top:10px;"|[[Image:Draft_Samper_987121664-monograph-static_dispy_gmm_mls.png|270px|GMM-MLS code]]
+|style="padding-top:10px;"|[[Image:Draft_Samper_987121664-monograph-static_dispy_gmm_lme.png|270px|GMM-LME code]]
+|- style="text-align: center; font-size: 75%;"
+| (c) GMM-MLS code
+| (d) GMM-LME code
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 6.2:''' Campbell diagram: comparison between modal prediction and SVD results.
+| colspan="3" style="padding:10px;"| '''Figure 4.2:''' Static cantilever. Displacement along y-direction
 |}
-As can be spotted, in most cases the dominant frequency obtained using SVD matches almost exactly with the '''predicted''' natural frequency. In other cases (modes 5, 6 and 8), the values are not coincident but very similar (<math display="inline">\pm 1 Hz</math>). In all likelihood, these differences can be attributed to the '''probabilistic''' behaviour of the SVD, as the numeric sample has a timespan of only <math display="inline">0.2\,s</math>.  Even so, in all cases the vibration frequencies predicted by the SVD decrease with the spin speed, matching with the expected softening effect. Thereby, it can be deduced that neither Coriolis nor centrifugal forces induce a '''significant''' change in the natural frequencies of rotating beams. In the same way, by comparing the LSVs from [[#D.1.2 Constant angular velocity|D.1.2]] with the natural modes of the beam, it can be stated that mode shapes do not either change with the angular velocity.
-Recall that the obtained results shown in [[#D.1.2 Constant angular velocity|D.1.2]] are just '''predominant''' patterns of the dynamic response. Thus, the RSVs representing reigning vibrations are no longer pure harmonics, but mostly the result of a certain combination of natural frequencies. In most cases, they match almost perfectly to a sine or cosine function, and hence the predominant frequency is easy to determine. That is not the case of the RSVs associated to axial vibrations (natural modes 2 and 7 from figure [[#D.1.1 Natural modes and frequencies|D.1.1]]), which are the result of more than one predominant frequency. For instance, the pure axial motion has a <math display="inline">0</math> Hz frequency associated to it, caused by the offset in displacements the centrifugal force induces. To visualise the frequencies involved in the vibration phenomena, figure [[#img-D.11|D.11]] displays the '''power spectral density''' estimation of the sum of the right side vectors.
+<div id='img-4.3a'></div>
+<div id='img-4.3b'></div>
-'''
+<div id='img-4.3c'></div>
+<div id='img-4.3d'></div>
-====6.1.1.2 Singular values===='''
+<div id='img-4.3'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
-Although Coriolis and centrifugal acceleration induce little changes in natural frequencies, they do change considerably the '''intensity''' in which natural modes show up. Considering the normalised singular values as the indicator of intensity, figure [[#img-D.8|D.8]] shows how axial modes (2 and 7) gain intensity as rotation speed increases, whereas flexural modes weaken. This effect is caused by the centrifugal force, the responsible of axial excitation, which increases proportionally to <math display="inline">\Omega ^2</math>. The numeric values of predominant frequencies and normalised singular values are presented in table [[#table-D.2|D.2]].
-'''
-====6.1.1.3 Reactions===='''
-Another interesting variable to keep track of is the '''reaction torque''', which at last instance is the one determining the required torque to spin at the desired speed. For the presented scenario and studied angular speeds, the reaction torque is found to oscillate along with the beam, as displayed in figure [[#img-D.9|D.9]].
-As the motion is not under the effect of angular acceleration, no rigid body reactions are present. Thus, figure [[#img-D.9|D.9]] only accounts for the elastic reactions described in [[#eq-3.21|3.21]]. The torque is mainly induced by the effect of transverse reactions, as local horizontal reactions due to centrifugal forces produce no net torque around the specified rotation axis. Hence, the required torque is closely related with the bending modes, and thus, with its natural frequency, which decreases with <math display="inline">\Omega </math>. In this case, reactions are not generated by a certain applied force but as a consequence of the '''inertia forces''' induced by the high accelerations present in the vibration phenomena.
-At some intervals, the required torque becomes negative: half of the time, the beam is bending in the direction of rotation, tending to accelerate the structure. Hence, the overall work is zero. This is an '''artificial scenario''': the angular speed of actual rotors is never constant, whereas the torque injected by the driving device is in fact uniform. In real cases, the vibration of the beam accelerates and decelerates the rotation motion in a small amount (as stated in [[#eq-1.2|1.2]]). Moreover, real rotors work within viscous fluids, which act as dampers of the vibration motion.
-'''
-====6.1.1.4 Displacements===='''
-Regarding dynamic results, displacements at the tip of the beam will be reviewed for different rotation velocities. Displacements in the local '''perpendicular''' direction are induced mostly by the pure flexural mode, and hence its vibration motion is easy to capture as it is associated to a low frequency. The amplitude of these displacements does not increase significantly with <math display="inline">\Omega </math>, but it is the decrease in frequency the most remarkable effect rotation velocity introduces. Figure [[#img-D.10|D.10]] presents the numeric results and their lower rank approximation computed using the truncated SVD with eight singular values.
-On the other hand, '''axial''' displacements are highly dependant on <math display="inline">\Omega </math>, as the centrifugal force that excites axial modes increases with the square of the angular velocity. In fact, the amplitude increases by a factor of 16 when comparing figure [[#img-D.12|D.12]] (a) with figure [[#img-D.12|D.12]] (c).  Moreover, the local horizontal displacements are linked to a relatively high frequency, and thus the truncated SVD is not capable to recreate the numerical result exactly. This issue is easily overcome by decreasing the value of tolerance of the SVD.
-'''
-====6.1.1.5 Stresses===='''
-In structural analysis, one of the most usual parameters used to determine whether the structure is going to fail or not is the '''security factor'''. In this example, computation of stresses is quite straightforward, as no internal damping is considered. Stresses and strains are obtained directly from the numerical scheme, but unlike displacements, they are not passed into the SVD for dominant patterns recognition. When dealing with stresses, their distribution is not so important as it is the peak value, from which depends the security factor. Figure [[#img-6.3|6.3]] displays the peak values of <math display="inline">\sigma _{\varsigma \varsigma }</math>, <math display="inline">\sigma _{\varrho \varrho }</math> and <math display="inline">\tau _{\varsigma \varrho }</math> as time passes by for the most favourable case (<math display="inline">\Omega = 50</math> rad/s).
-From figure [[#img-6.3|6.3]], it is found that maximum stresses oscillate at twice the natural frequency of the flexural mode. That is caused by the fact that maximum stresses take place at maximum deformed configurations — highest strains — which occur two times per oscillation. For the used aluminium, the yield strength is around 276 MPa <span id='citeF-14'></span>[[#cite-14|[14]]], and so the security factors are 50, 13, and 4 for 0, 50 and 100 rad/s, respectively.  <div id='img-6.3'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-stresses_50.png|510px|Temporal evolution of the maximum stresses (absolute value) for Ω = 50 rad/s.]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-static_stressx_fem.png|270px|FEM code]]
+|[[Image:Draft_Samper_987121664-monograph-static_stressx_mpm.png|270px|MPM code]]
+|rowspan="4" style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-static_stressx_legend.png|135px|]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 6.3:''' Temporal evolution of the maximum stresses (absolute value) for <math>\Omega </math> = 50 rad/s.
+| (a) FEM code
+| (b) MPM code
+|-
+|style="padding-top:10px;"|[[Image:Draft_Samper_987121664-monograph-static_stressx_gmm_mls.png|270px|GMM-MLS code]]
+|style="padding-top:10px;"|[[Image:Draft_Samper_987121664-monograph-static_stressx_gmm_lme.png|270px|GMM-LME code]]
+|- style="text-align: center; font-size: 75%;"
+| (c) GMM-MLS code
+| (d) GMM-LME code
+|- style="text-align: center; font-size: 75%;"
+| colspan="3" style="padding:10px;"| '''Figure 4.3:''' Static cantilever. Cauchy stress along x-direction
 |}
-'''
-====6.1.1.6 Results animation===='''
-To close down this section, the outputs from the dynamic analysis will be animated through <math display="inline">\mathbf{GiD}</math>. However, postprocessing is a heavy task for computers with humble computation capabilities, as results are encoded in ASCII. For that reason, only the case corresponding to <math display="inline">\Omega = 50</math> rad/s will be dynamically animated. To simplify the animation, the deformed configuration will be displayed in the rotating frame, and only stresses in the local horizontal direction will be represented through a colour map.
-During the bending motion, one expects to found a symmetric distribution of stresses with respect to the beam's axis: one side under traction and the other under compression. Figure [[#img-D.13|D.13]] shows how this symmetry is '''broken''' in rotating scenarios, as the centrifugal force introduces an axial displacement proportional to the distance from the rotation axis that counteracts the compressing stresses.
+A convergence study is performed to analyse the accuracy of MPM and GMM in comparison with the UL-FEM. The error is evaluated as
-===6.1.2 Constant angular acceleration===
-The second and last analysis regarding the rectangular cantilever beam accounts for the effect of a constant acceleration. Recall that angular acceleration was neglected in modal analysis, as variable rotation speed is not compatible with its formulation. Hence, this analysis aims to unfold the effect angular acceleration has on rotating structures. For the sake of simplicity, only two simulations will be carried out, corresponding to <math display="inline">\alpha = 5</math> and <math display="inline">50\, \hbox{rad/s}^2</math>. In both cases, the initial angular velocity will be <math display="inline">\Omega _0 = 50\; \hbox{rad/s}</math>.
-The numerical parameters will be the exact same as in the case of constant spin speed. Thus, the timespan will not be large enough for the angular acceleration to introduce significant changes in angular velocity — in the case of higher acceleration, <math display="inline">\Delta \Omega = 10 \;\hbox{rad/s}</math>. For this reason, no Campbell diagram will be represented, as the softening effect induced by <math display="inline">\alpha </math> will be relatively small and difficult to measure. Remember that the rotation stiffness matrix is proportional to:
+<span id="eq-4.40"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,789: / Line 4,164: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{pmatrix}-\Omega ^2 &  -\alpha & 0\\  \alpha & -\Omega ^2 & 0\\  0 & 0 & 0  \end{pmatrix} </math>
+| style="text-align: center;" | <math>error =\left|\dfrac{\delta{-}u_{num}}{\delta } \right|  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (6.1)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.40)
 |}
-Thus, as <math display="inline">\Omega ^2 \gg  \alpha </math>, no significant effect is expected. The first difference with the previous case is the '''tangential''' force introduced by <math display="inline">\alpha </math>, acting perpendicular to the beam. In consequence, the flexural mode is highly activated when angular acceleration is present. In fact, the flexural motion is so predominant that the SVD is no longer capable of extracting neither the natural frequency nor the mode of the axial vibration. Figure  [[#img-D.16|D.16]] reveals how the second predominant mode from SVD now corresponds to a '''combination''' of both axial and perpendicular displacements. Intensity of axial modes (three and seven from figure [[#D.1.1 Natural modes and frequencies|D.1.1]]) decreases with angular acceleration, as shown in  figure [[#img-D.18|D.18]], that compares normalised singular values of different natural modes for different values of <math display="inline">\alpha </math>.
+where <math display="inline">u_{num}</math> is the numerical solution measure at point A (see Figure [[#img-4.1|4.1]]). Figure  [[#img-4.4|4.4]] depicts the error evolution in function of the inverse of the mesh size <math display="inline">h</math>. It is demonstrated that all the methods have a quadratic rate of convergence. In particular, the UL-FEM, MPM and GMM-MLS error curves coincide. Regarding the error, evaluated with the GMM-LME algorithm, the quadratic rate is maintained, but the curve is shifted a bit upwards, which makes this technique less accurate than GMM-MLS in the benchmark case studied.
-Notice that the truncated SVD is not capable of recovering the seventh natural mode when <math display="inline">\alpha = 50\; \hbox{rad/s}^2</math>. In fact, in this case the seventh and eight LSVs are the same and correspond to the eight natural mode <math display="inline">\Phi _8</math>. Predominant frequencies and singular values obtained are summarised in table [[#table-D.3|D.3]].   '''
+<div id='img-4.4'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-convergence_study.png|360px|Static cantilever. Convergence analysis]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding-bottom:10px;"| '''Figure 4.4:''' Static cantilever. Convergence analysis
+|}
-====6.1.2.1 Displacements===='''
+===4.5.2 Rolling of a rigid disk on an inclined plane===
-Again, displacements at the tip will be considered representative of the vibration phenomena. Displacements in the perpendicular direction oscillate at the first natural frequency, but this time the vibration is not around zero but to a new equilibrium position which depends on the magnitude of the applied tangential force. The higher the angular acceleration, the more deformed the configuration at equilibrium will be. This behaviour can be spotted in figure [[#img-D.19|D.19]].
+The second benchmark test is a rigid disk rolling without slipping on an inclined plane. The geometry of the problem is depicted in Figure [[#img-4.5|4.5]]. The disk is made of a hyperelastic material (presented in Section [[#3.1 Hyperelastic law|3.1]]): the density is <math display="inline">\rho = 7800 kg/m^3</math>, the Young's modulus is <math display="inline">E = 200 MPa</math> and the Poisson's ratio is <math display="inline">\nu = 0.3</math>.
-On the other hand, axial displacements are relatively smaller (one order of magnitude below perpendicular ones). They oscillate at two different frequencies, corresponding to the flexural and axial natural modes. It is worth pointing that, as the angular velocity increases with time, so does the centrifugal force. For that reason, the mean value of local horizontal displacements '''increases''' with time, as captured by the second RSV in [[#img-D.17|D.17]]. The rate of this rise is proportional to the angular acceleration. This effect can be identified in figure [[#img-6.4|6.4]], where horizontal displacements are represented. The increase in length of the beam <math display="inline">\Delta \varsigma </math> during the analysed timespan are <math display="inline"> 0.11</math> and <math display="inline"> 0.01</math> mm for <math display="inline">\alpha = 50</math> and <math display="inline">5\; \hbox{rad/s}^2</math>, respectively. If the angular acceleration (for instance <math display="inline">\alpha = 50\; \hbox{rad/s}^2</math>) persisted in time, that would result in an increase of half millimetre in the beam's length every second.  <div id='img-6.4'></div>
+<div id='img-4.5'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-xdisp_acc_1.png|600px|Local horizontal displacements (m) at the tip of the beam for different values of angular acceleration.  ]]
+|style="padding:10px;"|[[File:Draft_Samper_987121664_5803_rolling_geometry.png]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 6.4:''' Local horizontal displacements (m) at the tip of the beam for different values of angular acceleration.
+| colspan="1" style="padding-bottom:10px;"| '''Figure 4.5:''' Rolling disk. Geometry
 |}
-'''
-====6.1.2.2 Reactions===='''
+This test is chosen for an objective assessment of the robustness of the MPM and GMM algorithm. The rolling on the plane implies a contact between the nodes belonging to the inclined plane and the nodes belonging to the disk. In a UL-FEM code a contact algorithm would be necessary to set this boundary condition. On the contrary, by using either MPM or GMM, the contact is implicitly caught. The analytical acceleration (<math display="inline">a</math>) can be computed imposing the equilibrium of momentum at the contact point <math display="inline">P</math>
-As in the previous case, the obtained reactions are those the driving device must inject into the structure to ensure the desired acceleration. In this case of constant <math display="inline">\alpha </math>, the elastic reaction torque oscillates at the bending frequency, but with a mean value different from zero as the equilibrium position of the beam is no longer the horizontal configuration. Figure [[#img-D.20|D.20]] displays how the mean value of elastic reaction torque increases with <math display="inline">\alpha </math>.
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>a = \dfrac{2}{3} g sin\theta  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.41)
+|}
-It is worth pointing that, in the case of variable angular speed, a rigid body motion must be taken into account and added to the previous one, as stated in equation [[#3.2.3 Reaction computation|3.2.3]]. In this case, the rigid body torque is constant:
+where <math display="inline">g</math> is gravity and <math display="inline">\theta </math> the angle of the inclined plane. Integrating over time the acceleration, velocity and displacement projected on the x-axis can be obtained as a function of time
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,822: / Line 4,210: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{W}_{rb} = \mathbf{J}_\mathbf{z}\; \alpha = \overset{ kg \; m^2 }{\overbrace{14436}}\; \alpha \;\; \xrightarrow= \;\;\mathbf{W}_{rb} = 721,8 \; kN\cdot m </math>
+| style="text-align: center;" | <math>v(t) = a(t)  \hbox{cos}\theta \; t </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (6.2)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.42)
 |}
-This constant value is to be added to the elastic torque from [[#img-D.20|D.20]], which is significantly smaller. Recall that this is not a real scenario, as the kinematics of the motion are often the unknown rather than the input parameter. The required power the driving engine must provide is given by the product of the total reaction torque (rigid body and elastic) by the instantaneous angular velocity <math display="inline">\Omega </math>.
+<span id="eq-4.43"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>u(t) = v(t)  \hbox{cos}\theta \; t + \dfrac{1}{2} a(t) \hbox{cos}\theta \; t^2  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.43)
+|}
-'''
+For the study of this test case, the analytical solution of Equation  [[#eq-4.43|4.43]] is used for the assessment of the absolute error obtained with MPM, GMM-MLS and GMM-LME, evaluated as
-====6.1.2.3 Stresses===='''
+<span id="eq-4.44"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-In order to validate structural reliability, one needs to focus on the study of stresses, which determine the security factor. Again, as no internal damping is considered, computation of stresses is done as in the static scenario. For simplicity, only peaks values will be considered in this section. As expected, stresses increase with <math display="inline">\alpha </math>, although they are most sensitive to a change in angular velocity. Figure  [[#img-6.5|6.5]] reveals how stresses in the local horizontal direction are the higher ones, and oscillate in the flexural natural frequency. The peaks in stresses correspond to the maximum deformed configurations.  <div id='img-6.5'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-max_stress_acc1.png|480px|Temporal evolution of the maximum stresses (absolute value) for Ω₀ = 50 rad/s and α = 50 rad/s². ]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 6.5:''' Temporal evolution of the maximum stresses (absolute value) for <math>\Omega _0 </math> = 50 rad/s and <math>\alpha </math> = 50 rad/s<math>^2</math>.
+|-
+| style="text-align: center;" | <math>error =\sqrt{\sum _{t_i}\left(u(t_i)-u_{num}(t_i)\right)^2 }  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4.44)
 |}
-It is worth pointing that the mean value of <math display="inline">\sigma _{\varsigma \varsigma }</math> increases with time, consequence of the increasing centrifugal force that is acting on the structure. Although only absolute values are represented in figure [[#img-6.5|6.5]], looking into the dynamic results (figure [[#img-D.21|D.21]]) one finds out that traction stresses are the most relevant contribution. As seen in the previous case, centrifugal force tends to pull the beam decreasing the effect of compression stresses.
+where <math display="inline">t_i</math> is the time where the numerical result is calculated. As a mesh-based and a meshless techniques are compared in a dynamic test, for a more objective comparison, the error is analysed along with the total computational time, needed to finalize the simulation.
-'''
+A triangular mesh with mesh size <math display="inline">h=0.01m</math> is used for MPM and GMM simulations. In both techniques the same initial distribution of material points is used, which counts for three initial particles for cell.  Regarding the GMM-MLS the approximants are constructed by adopting a search radius <math display="inline">R = 1.5\sqrt{2h^2}</math> and a dilation parameter <math display="inline">R_{eff} = \sqrt{2h^2}</math>. In GMM-LME the basis functions are evaluated using a search radius <math display="inline">R = \sqrt{2h^2}</math> and three values of dilation parameter <math display="inline">\gamma = 0.8, 1.8, 2.8</math>. All the numerical tests are repeated for three different time steps with <math display="inline">\Delta t_1 = 2 \Delta t_2 = 4\; \Delta t_3</math>.
-====6.1.2.4 Results animation===='''
+Table [[#table-4.1|4.1]] shows the results of the analysis, in terms of errors and computational times, performed through MPM, GMM-MLS and GMM-LME.
-To ease visualisation of results, dynamic displacements and stress distribution will be animated in the rotation frame trough '''GiD'''. Only the case of higher angular acceleration (<math display="inline">\alpha = 50</math> rad/s<math display="inline">^2</math>) is presented in figure [[#img-D.21|D.21]]. The animation captures one oscillation of the beam, with a timespan of 25 ms. As commented, higher stresses are found in the most deformed configurations of the beam.
-==6.2 Three-dimensional cantilever beam==
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-4.1'></span>Table. 4.1 Rolling disk. Absolute errors and computational times.
-For the next case of study, a new geometry will be defined. In order to keep the problem simple, a cantilever beam will again be the structure to study. In this case, however, the beam is modelled in three dimensions using  3200 hexahedral elements. The beam is <math display="inline">2\times 0.25 \times 0.25\,m</math>,  with a thickness of <math display="inline">0.05\, m</math>. Again, this study considers that the structure undergoes a rotation in the vertical -<math display="inline">z</math>- axis. The material from which the beam is made of is aluminium, with a density <math display="inline">\rho = 2700\; Kg/m^3</math>, a Young's modulus <math display="inline">E = 70 \;GPa</math> and a Poisson's coefficient <math display="inline">\nu = 0.3</math>.  <div id='img-6.6'></div>
+|-style="font-size: 85%;"
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+|
-|-
+| colspan='2' | <math>\Delta t_1</math>
-|[[Image:Draft_Samper_987121664-monograph-3dbeam.png|420px|3D cantilever beam subject to rotation around the vertical axis. ]]
+| colspan='2' | <math>\Delta t_2</math>
-|- style="text-align: center; font-size: 75%;"
+| colspan='2' | <math>\Delta t_3</math>
-| colspan="1" | '''Figure 6.6:''' 3D cantilever beam subject to rotation around the vertical axis.
+|-style="font-size: 85%;"
+|
+| <math>error [m]</math>
+| <math>t_{comp} [s]</math>
+| <math>error [m]</math>
+| <math>t_{comp} [s]</math>
+| <math>error [m]</math>
+| <math>t_{comp} [s]</math>
+|-style="font-size: 85%;"
+|  MPM
+| 2.34
+| 232.72
+| 1.27
+| 472.92
+| 0.91
+| 894.91
+|-style="font-size: 85%;"
+| GMM-MLS
+| 0.84
+| 264.78
+| 0.26
+| 517
+| 0.20
+| 981.82
+|-style="font-size: 85%;"
+| GMM-LME <math display="inline">\gamma = 0.8</math>
+| 1.37
+| 234.32
+| 0.07
+| 460.73
+| 0.06
+| 971.29
+|-style="font-size: 85%;"
+| GMM-LME <math display="inline">\gamma = 1.8</math>
+| 0.9
+| 237.22
+| 0.23
+| 460.50
+| 0.07
+| 1005.21
+|-style="font-size: 85%;"
+| GMM-LME <math display="inline">\gamma = 2.8</math>
+| 0.52
+| 232.42
+| 0.07
+| 466.63
+| 0.06
+| 1038.70
 |}
-The loading scenario is the following: the beam is rotating at a given angular speed, subjected to <math display="inline">\alpha = 10</math>  rad/s<math display="inline">^2</math>, when the angular acceleration vanishes instantaneously and so does the tangential force. This section will analyse the vibrations this '''sudden stop''' in acceleration induces, for different values of angular speed <math display="inline">\Omega </math>. Again, no source of damping will be considered. In contrast with the last analysis, this time the weight of the beam will be taken into consideration, and supposed constant and negative in the <math display="inline">z</math> axis. For comparison purposes, the first eight natural frequencies and modes of the presented structure can be found in [[#D.2.1 Natural modes and frequencies|D.2.1]].
-===6.2.1 Constant angular velocity===
+Regarding the absolute errors, it can be observed that, for a given computational cost, GMM is generally more accurate than MPM, because of the use of smooth basis functions which provide a better approximation of the unknown variables. In particular GMM-LME presents smaller errors in comparison to GMM-MLS. In all the three cases considered (with <math display="inline">\gamma = 0.8, 1.8, 2.8</math>) the errors converge to a unique value at the same computational time, while in the case of GMM-MLS, the advantage of using higher order elements is lost for the smallest delta time. Regarding MPM, it is established that to achieve the same order of accuracy of GMM a higher computational time must be expected, due to either a finer discretisation in space or in time. However, it is worth highlighting that GMM is much more time consuming than MPM, showing an increment of computational time of <math display="inline">10%</math> in the case of GMM-MLS and from <math display="inline">8.5%</math> up to <math display="inline">16%</math> in the case of GMM-LME.
-For the sake of simplicity, the presented geometry will only be studied for the case of constant angular velocity. In fact, the dynamic behaviour will be analysed for three values of <math display="inline">\Omega </math>: 25, 50 and 100 rad/s. The goal is to keep track of the vibration frequencies using the truncated SVD as <math display="inline">\Omega </math> increases. For the numerical computations, the used timestep will be <math display="inline">\Delta \,\textrm{t} = 25\,\mu s</math>. To keep computation time low, the timespan has been selected in order to capture a full oscillation of first natural mode, so <math display="inline">\textrm{T}_f = 17\, ms</math>. For this case, the truncated SVD will compute the eight highest singular values and their associated LSVs and RSVs.   '''
+In this example, some essential conclusions can be drawn. In Section [[#4.5.1 2D cantilever beam. Static analysis|4.5.1]] it was observed that in a static case the rate of convergence is the same for all the methods under analysis, but the accuracy of GMM-MLS and MPM is better than the GMM-LME. On the contrary, in a dynamic case with a contact problem, the result is overturned. In fact a better behaviour is noted if LME approximants are employed.
-====6.2.1.1 Campbell diagram===='''
+In Figures [[#img-4.6a|4.6a]], [[#img-4.6b|4.6b]] and [[#img-4.6c|4.6c]] the distribution of the module of velocity field within the disk is shown for the test case solved by means of the MPM, GMM-MLS and GMM-LME, respectively. As expected, the minimum velocity is localized in the region of the disk close to the contact point with the inclined plane; while maximum velocity is observed on the opposite part of the disk.
-One of the main applications of the truncated SVD is to recover predominant frequencies of the dynamic vibration motion. The goal is to recover these frequencies for different values of <math display="inline">\Omega </math> applying the discrete Fourier transform to each RSV. Recall that in the 2D case, predominant frequencies were compared with the softening effect predicted by modal analysis (see equation [[#eq-5.25|5.25]] ). However, this approach is not possible in three dimensions as the rotation stiffness matrix is no longer proportional to the mass matrix. The eigenvalue problem admits no further simplification than:
+<div id='img-4.6a'></div>
+<div id='img-4.6b'></div>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+<div id='img-4.6c'></div>
+<div id='img-4.6'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
 |-
-|
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-rolling_mpm.png|420px|]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
+| (a)
 |-
-| style="text-align: center;" | <math>\left(\mathbf{K}_{\,\hbox{static}} + \mathbf{K}_{\,\hbox{rot}} -\boldsymbol{\omega }^2  \; \mathbf{M} \right)\boldsymbol{\Phi } = 0 </math>
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-rolling_mls.png|420px|]]
-|}
+|- style="text-align: center; font-size: 75%;"
-| style="width: 5px;text-align: right;white-space: nowrap;" | (6.3)
+| (b)
+|-
+| style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-rolling_lme.png|420px|]]
+|- style="text-align: center; font-size: 75%;"
+| (c)
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 4.6:''' Rolling disk. Absolute velocity field in test case solved with MPM (a), GMM-MLS (b) and GMM-LME with <math>\gamma=1.8</math> (c)
 |}
-The only way to keep track of the natural frequencies as angular velocity increases is by solving the previous equation for different values of <math display="inline">{\mathbf{K}_{\hbox{rot}}}\,(\Omega )</math>, obtaining '''numerically''' the desired Campbell diagram (<math display="inline">\omega \;vs.\;\Omega </math>). By doing so, it is found that in most cases natural frequencies decrease in the same way as they did in 2D. The biggest '''difference''' is that in three dimensions, natural frequencies linked with modes acting in the <math display="inline">z</math> direction — the rotation axis — decrease much slower and in a different pattern.
+===4.5.3 Cohesive soil column collapse===
-This effect is displayed in a Campbell digram in [[#img-D.29|D.29]], where modal prediction and SVD results are compared. It is worth pointing that, in still conditions, natural modes and frequencies appear in '''pairs''', as the beam is symmetric with respect to the <math display="inline">\varsigma \varrho </math> and <math display="inline">\varsigma z</math> planes. However, as rotation velocity increases, vibration frequencies of the modes acting on <math display="inline">\varrho </math> and <math display="inline">z</math> diverge, as their decrease rate is not the same (see <math display="inline">\Phi _7</math> vs <math display="inline">\Phi _8</math> in  [[#img-D.29|D.29]]).
+The third example is the simulation of a soil column collapse. The column is modelled with a cohesive-frictional material, defined by a cohesion <math display="inline">c = 5kPa</math>, a friction angle <math display="inline">\phi = 25^{\circ }</math>, an elastic bulk modulus <math display="inline">K = 1.5MPa</math> and a density <math display="inline">\rho = 1850 kg/m^3</math>. In the current work the Mohr-Coulomb plastic law in finite strains with implicit integration scheme in principal stress space, presented in Section [[#3.3 Hyperelastic - Mohr-Coulomb plastic law|3.3]], is employed.
-Recall that one of the advantages of the SVD with respect to modal analysis is that it accounts for the effect of Coriolis, centrifugal and tangential forces. On the other hand, its main drawback is that as it is a '''probabilistic method''', great timespans and small timesteps are needed to get a reliable value of predominant frequency. Nonetheless, the results displayed in   [[#img-D.29|D.29]] agree (<math display="inline">\pm 1 Hz</math>) with the predicted softening effect. It is found that the more intensively a mode is activated, the greater the accuracy in which the SVD can extract its predominant frequency. The RSVs from which the predominant frequency has been obtained are plotted for different values of <math display="inline">\Omega </math> in [[#img-D.24|D.24]], [[#img-D.26|D.26]]  and [[#img-D.28|D.28]].
+This test has been chosen for the assessment of the robustness of MPM and GMM when the body undergoes really large deformation. The results are compared with the work of <span id='citeF-56'></span>[[#cite-56|[56]]], where a Smooth Particle Hydrodynamics method (SPH) is applied to geotechnical problems.
-Another limitation of the SVD is that it is not able to isolate natural modes. The LSVs the SVD computes are just predominant patterns, which quite often correspond to the combination of two or more natural modes. Figures  [[#img-D.23|D.23]], [[#img-D.25|D.25]] and [[#img-D.27|D.27]] — which represent the predominant modes of the beam for different values of <math display="inline">\Omega </math> — evince for instance how the first predominant mode <math display="inline">\mathbf{U}_1</math> is in reality the combination of two natural modes: axial in <math display="inline">\varsigma </math> (<math display="inline">\boldsymbol{\Phi }_6</math>) and bending in <math display="inline">z</math> (<math display="inline">\boldsymbol{\Phi }_2</math>). The higher <math display="inline">\Omega </math> is, the higher contribution the axial mode has, as centrifugal forces increase with <math display="inline">\Omega ^2</math> whereas weight is constant.  '''
+The initial geometry and the boundary conditions are described by Figure [[#img-4.7|4.7]].
-====6.2.1.2 Displacements===='''
+<div id='img-4.7'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-dam_break_geom.png|360px|Granular column collapse. Geometry]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding-bottom:10px;"| '''Figure 4.7:''' Granular column collapse. Geometry
+|}
-Once more, displacements at the tip — upper side, at half the thickness — will be considered representative of the vibration phenomena. This time there is not a predominant direction in which displacements are considerably greater than in the others. Displacements in the <math display="inline">\varrho </math> axis are those with the lowest frequency and linked with the natural mode <math display="inline">\boldsymbol{\Phi }_1</math> (see [[#D.2.1 Natural modes and frequencies|D.2.1]]), as the beam was under the effect of a '''transverse force''' which vanished at <math display="inline">t=0</math>, oscillating thus around the equilibrium position <math display="inline">\varrho=0</math> (see figure [[#img-D.30|D.30]]).
+Quadrilateral elements with an initial distribution of four material points per cell are used in the simulations. Two different mesh sizes are considered: <math display="inline">h_1 = 0.05\;m</math> (Mesh1) and <math display="inline">h_2 = 0.025\;m</math> (Mesh2). In GMM the basis functions are evaluated using an initial search radius <math display="inline">R = \sqrt{2h^2}</math>, a dilation parameter <math display="inline">R_{eff} = 0.5 \sqrt{2h^2}</math> and <math display="inline">\gamma = 1.8</math>, in GMM-MLS and GMM-LME, respectively.  In this particular case, the procedure for the evaluation of the basis functions in MLS and LME technique has been modified to avoid the creation of a non-convex hull of nodes which might lead to an incorrect set of approximants. This is required because the column is subjected to extremely large deformations. While in the previous examples a constant radius was used for the definition of the cloud of nodes surrounding a material point, in the current example a variable radius is adopted to guarantee a minimum number of nodes in each connectivity. In the case of LME, as a Newton iterative procedure is used for the evaluation of the shape functions, a measure of the goodness of the solution is represented by the condition number <math display="inline">k(A)</math> of the Hessian matrix <math display="inline">A</math>, defined in <span id='citeF-145'></span>[[#cite-145|[145]]]. If <math display="inline">k(A)</math> exceeds a user-defined tolerance, the LME algorithm is repeated considering the old connectivity plus an additional node, chosen as the next node closer to the material point. In the case of MLS, it has been sufficient to impose a minimum number of six nodes in each cloud of nodes. In Figure [[#img-4.8|4.8]] a comparison of the column deformation at different representative time instants is shown. The SPH model taken from <span id='citeF-56'></span>[[#cite-56|[56]]] predicts a higher final run-out of the column collapse, while the final configurations at time <math display="inline">2.0s</math> of MPM, GMM-LME and GMM-MLS are almost coincident using Mesh1 and Mesh2.  It is worth highlighting that GMM-MLS and MPM results of Figure [[#img-4.8|4.8]] are always in good agreement. However, this is not the case if the evolution of the equivalent plastic strains is observed (see Figures [[#img-4.9|4.9]] and [[#img-4.10|4.10]]). In the case of GMM-LME, an improvement of the results is noted by using the finer mesh (Mesh2) in terms of displacements (Figure [[#img-4.8b|4.8b]]) and equivalent plastic strains distribution (Figure [[#img-4.11b|4.11b]]). Regarding MPM, it is proved that a good approximation can be obtained using both meshes.
-The structure undergoes as well a bending in the '''vertical''' direction <math display="inline">z</math>, but with a much higher frequency and a considerably smaller amplitude of oscillation. This effect is produced because, unlike in the <math display="inline">\varrho </math> direction, the distributed force in <math display="inline">z</math> — the weight — does not vanishes but remains constant as time passes by. Hence, vibrations in the vertical axis are not caused by the bending mode <math display="inline">\boldsymbol{\Phi }_2</math> but are the result of higher frequency modes. Even though vibration amplitudes are relatively small in <math display="inline">z</math>, displacements on this axis have a considerable offset induced by the weight, as can be spotted in figure [[#img-D.31|D.31]].
+<div id='img-4.8a'></div>
+<div id='img-4.8b'></div>
+<div id='img-4.8'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-config_mesh1b.png|194px|Mesh 1]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-config_mesh2b.png|193px|Mesh 2]]
+|rowspan="2" style="padding-right:10px;"| [[Image:Draft_Samper_987121664-monograph-legend_config_b.png|137px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a) Mesh 1
+| (b) Mesh 2
+|- style="text-align: center; font-size: 75%;"
+| colspan="3" style="padding:10px;"| '''Figure 4.8:''' Soil column collapse. Configurations of the column at different representative time instants.
+|}
-A similar offset is found in the axial displacements, induced by the constant centrifugal force. As in the vertical direction, displacements in <math display="inline">\varsigma </math> are subject to high frequency vibration, but to small oscillation amplitudes. One of the vibration frequencies corresponds to <math display="inline">\omega _1</math>, as bending in <math display="inline">\varrho </math> also causes a small displacement in <math display="inline">\varsigma </math>. For sure, there is as well a component linked with the axial mode <math display="inline">\boldsymbol{\Phi }_6</math>, but its amplitude is considerably smaller. These three effects — offset, <math display="inline">\omega _1</math> and <math display="inline">\omega _6</math> — can be easily identified In the following figure:
-<div id='img-6.7'></div>
+<div id='img-4.9a'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+<div id='img-4.9b'></div>
+<div id='img-4.9'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-xdisp3d.png|510px|Temporal evolution of ς displacements at the tip of a cantilever beam under constant angular velocity.]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-MPM_eq_plastic-0-0-5cm-0-0-0-5cm-0.png|200px|Mesh 1]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-MPM_eq_plastic-5cm-0-0-0-5cm-0-0-0.png|200px|Mesh 2]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 6.7:''' Temporal evolution of <math>\varsigma </math> displacements at the tip of a cantilever beam under constant angular velocity.
+| (a) Mesh 1
+| (b) Mesh 2
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 4.9:''' Soil column collapse. Distribution of equivalent plastic strains for different representative time instants in MPM results.
 |}
-With no surprise, all displacements '''increase''' (in absolute value) with <math display="inline">\Omega </math>, being the ones in <math display="inline">\varsigma </math> the more susceptible to changes in angular velocity, as the centrifugal force that acts in the axial direction is proportional to <math display="inline">\Omega ^2</math>.
+<div id='img-4.10a'></div>
+<div id='img-4.10b'></div>
+<div id='img-4.10'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-GMM_MLS_eq_plastic-0-0-5cm-0-0-0-5cm-0.png|200px|Mesh 1]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-GMM_MLS_eq_plastic-5cm-0-0-0-5cm-0-0-0.png|200px|Mesh 2]]
+|- style="text-align: center; font-size: 75%;"
+| (a) Mesh 1
+| (b) Mesh 2
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 4.10:''' Soil column collapse. Distribution of equivalent plastic strains for different representative time instants in GMM-MLS results.
+|}
-'''
+<div id='img-4.11a'></div>
+<div id='img-4.11b'></div>
+<div id='img-4.11'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-GMM_LME_eq_plastic-0-0-5cm-0-0-0-5cm-0.png|200px|Mesh 1]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-GMM_LME_eq_plastic-5cm-0-0-0-5cm-0-0-0.png|200px|Mesh 2]]
+|- style="text-align: center; font-size: 75%;"
+| (a) Mesh 1
+| (b) Mesh 2
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 4.11:''' Soil column collapse. Distribution of equivalent plastic strains for different representative time instants in GMM-LME results.
+|}
-====6.2.1.3 Reactions===='''
+In this example, the capability of handling history-dependent materials, such as cohesive-frictional materials, is verified for both methods. It is noted that, in MPM large deformations can be naturally tracked without modifying the algorithm and accurate results are obtained also using the coarser mesh. In GMM, despite the remarkable features highlighted in the previous benchmark tests, when the continuum undergoes extremely large deformations, special care should be taken in the definition of the MLS and LME approximants. In this regard a lack of robustness of the GMM algorithm is observed due to the impossibility of guaranteeing a correct evaluation of the shape functions during the whole deformation process without an ''ad hoc'' modification of the procedure for the definition of the connectivity. Thus, for the solution of this example a correction of the algorithm has been performed and verified to work properly, albeit an increase in the computational time is registered. The establishment of a more general procedure is left for future work.
-From a dynamic point of view, it is always interesting to keep track of the elastic reactions. Remember that like in the previous case, the beam is considered to rotate at a given angular speed, set by the driving device (recall the assumptions made in [[#2.1 Problem formulation and hypotheses|2.1]]). This implies that no matter how great the deformations are, the device will always inject the necessary torque to ensure the desired kinematic conditions. Although it is an artificial scenario, it allows '''uncoupling''' of rigid body and elastic equations, simplifying considerably the dynamic study.
+==4.6 Discussion==
-In this particular case, the elastic reaction torque in <math display="inline">z</math> oscillates with the frequency linked with the transverse motion, the most predominant one. The latter result is quite intuitive: the more deformed the configuration is, the more torque is required. When the deformation is in the direction of rotation, the reaction torque becomes negative. This oscillation in the torque is not caused by the effect of any real force, but because of the inertia induced by accelerations. On the other hand, the biggest component of the torque is actually constant, and produced by the centrifugal force. Thereby, the mean value of the elastic reaction torque in the rotation axis increases with <math display="inline">\Omega </math>, as graphed in figure [[#img-D.32|D.32]].  '''
+In this Chapter, two particle methods: a Material Point Method and a Galerkin Meshless Method are tested and compared to assess their capabilities in solving large displacement and large deformation problems. A variational displacement-based formulation, based on an Updated Lagrangian description, is presented and its derivation is described in detail.
-====6.2.1.4 Stresses===='''
+A comparison of MPM and GMM is performed through three benchmark tests and the methods are assessed in terms of accuracy, computational time and robustness. The first example is a static cantilever beam. A convergence analysis is performed and all the techniques have a quadratic convergence rate (compared to a FEM code). Secondly, the dynamic test of a rolling disk on an inclined plane is considered. The robustness of MPM and GMM in dealing with contact between two rigid bodies is tested and an analysis in terms of computational time and error is performed. It is found that GMM, in dynamic cases, has a higher accuracy than MPM, despite a higher computational cost. This is because in MPM linear basis functions are considered, while in GMM smooth basis functions are computed allowing to obtain a superior approximation of the unknown variables. As a last example, a cohesive soil column collapse is analysed. In this case, it is assessed the robustness of both methods when the continuum undergoes extremely large deformation. Firstly, it is demonstrated that MPM and GMM can be easily coupled with local plastic laws. Furthermore, it is noted that MPM leads to more accurate results and the algorithm does not need to be modified in a large deformation case. On the contrary, in GMM, a modification of the algorithm has to be considered to avoid the formation of non-convex hull of nodes when the connectivity is defined. Nonetheless, in spite of this modification, a discrepancy in the results is noted, by using either the MLS or the LME technique.
-Once more, validation of the structure relies in the study of stresses. In order to simplify the analysis, only the maximum absolute value of stress has been dynamically tracked.  Without considering any source of internal damping, these values are found to oscillate at twice the frequency of the flexural motion. Recall that stresses and strains adopt its higher value at maximum deformed configuration, which takes place twice per period. With all, it has been found that like in the 2D case, the higher stresses are those in the direction of the centrifugal force — that is, <math display="inline">\sigma _{\varsigma \varsigma }</math>. Maximum stresses for <math display="inline">\Omega = 50</math> rad/s are represented in the following figure:  <div id='img-6.8'></div>
+In conclusion, the standard version of MPM represents a good choice to handle problems involving history-dependent materials and large deformations. Regarding GMM, the accuracy of the solution strictly depends on the chosen basis functions. If large deformation of the continuum is not taken into account, this method could be preferred to MPM due to its remarkable feature in getting accurate results at a limited computational time. However, under finite strains regime, independently on the material to model, the construction of a connectivity in the meshless method becomes more complex and, at least to the authors’ knowledge, a general methodology is still missing to properly define a correct connectivity under any deformation condition. Thus, despite the promising features of this approach, an improvement in the robustness of the GMM algorithm is needed to obtain more accurate and reliable solutions in large deformation and failure problems, leaving the MPM, currently, the most suited numerical strategy for the analysis of granular flows.
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-maxstress3d.png|540px|Temporal evolution of the maximum stresses (absolute value) for Ω = 50 rad/s. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 6.8:''' Temporal evolution of the maximum stresses (absolute value) for <math>\Omega </math> = 50 rad/s.
-|}
-In 3D cases where six different components of stress play a significant role, its worth considering the equivalent stress. Using the '''Von Mises''' yield criterion for beams and considering a 6061 T6 alloy, it is found that the security factor of the structure is <math display="inline">14.5</math>.
-'''
+=5 Mixed formulation=
-====6.2.1.5 Results animation===='''
+In the field of granular flow modelling, there might be some cases where the granular matter undergoes undrained conditions. This corresponds to a situation where the bulk modulus K is much higher than the shear modulus G, and shearing deformations will be more important than dilation or compression in the overall response of the body. In this case, the behaviour of the body is usually approximated by assuming it to be incompressible. In the field of Finite Element Analysis (FEA) it is well established that results may suffer from volumetric locking issues, which can be detrimental for the solution itself when an irreducible formulation is employed.  In this Chapter, the numerical strategy of a stabilized mixed formulation for the solution of non-linear solid mechanics problems in nearly-incompressible conditions is presented. The proposed mixed formulation, with displacement and pressure as primary variables, is implemented in the implicit MPM strategy, whose algorithm has been previously described in Chapter [[#2 Particle Methods|2]]. The mixed formulation is tested through classical benchmarks in solid mechanics where a hypereleastic Neo-Hookean and a J2-plastic laws are employed. Further, the stabilized mixed formulation is compared with a displacement-based formulation, described in Chapter [[#4 Irreducible formulation|4]] to demonstrate how the proposed approach gets better results in terms of accuracy, not only when incompressible materials are simulated, but also in the case of compressible ones.
-The dynamic results have been postprocessed and animated using '''GiD'''. The deformed configuration and the stress map for different timesteps are presented in figure [[#img-D.33|D.33]]. For the sake of simplicity, only <math display="inline">\sigma _{\varsigma \varsigma }</math> stresses are coloured. Regarding displacements, it is relatively difficult to keep track of the bending motion in <math display="inline">\varrho </math>, as axial displacements are considerably high and '''eclipse''' the flexural vibration.
+==5.1 Introduction==
-=7 Study of an aeronautics-related case=
+The solution of solid mechanics problems in large displacement and large deformation regime, dealing with incompressible or nearly incompressible materials, is a topic of paramount importance in the computational mechanics community since many engineering problems present such conditions. It is well known that overly stiff numerical solutions appear when Poisson's ratio <math display="inline"> \nu </math> tends to 0.5 or when plastic flow is constrained by the volume conservation condition. In these cases, a standard Galerkin displacement-based formulation (''u'' formulation) fails <span id='citeF-159'></span><span id='citeF-146'></span>[[#cite-159|[159,146]]] due to the inability to evaluate the correct strain field. In the literature, many possible solutions can be found. For instance, Simo and Rifai introduced the ''Mixed Enhanced Element'' for small deformation problems <span id='citeF-160'></span>[[#cite-160|[160]]]. This is a special three-field mixed finite element method in which the space of discrete strains is augmented with local functions. It is worth mentioning that also the class of B-bar methods <span id='citeF-161'></span>[[#cite-161|[161]]] and the classical incompatible modes formulation <span id='citeF-162'></span>[[#cite-162|[162]]] fall under this theory. For general purposes, some variants of this procedure are analysed in <span id='citeF-163'></span>[[#cite-163|[163]]]. Alternative procedures suitable for geometrically non-linear regimes, are given by the F-BAR method <span id='citeF-164'></span>[[#cite-164|[164]]], a technique based on the concept of multiplicative deviatoric/volumetric split in conjunction with the replacement of the compatible deformation gradient field, the non-linear B-bar method <span id='citeF-165'></span>[[#cite-165|[165]]] and the family of enhanced elements <span id='citeF-166'></span>[[#cite-166|[166]]], which represents an extension to the non-linear regime of the procedures exposed in <span id='citeF-161'></span>[[#cite-161|[161]]] and <span id='citeF-162'></span>[[#cite-162|[162]]], respectively. Though the good performance of all the aforementioned methods, none of such techniques is, however, suitable for application on simplicial meshes <span id='citeF-167'></span><span id='citeF-146'></span><span id='citeF-168'></span>[[#cite-167|[167,146,168]]]. In this regards, among the successful strategies for the fulfilment of the incompressibility constraint, it is worth mentioning the group of the ''Mixed Variational Methods''. Different researchers worked on mixed finite element formulations with displacement and mean stress as primary variables <span id='citeF-169'></span><span id='citeF-170'></span><span id='citeF-171'></span><span id='citeF-172'></span><span id='citeF-173'></span>[[#cite-169|[169,170,171,172,173]]]; Cervera and coworkers, for instance, proposed a strain/displacement mixed formulation in the context of compressible and incompressible plasticity <span id='citeF-174'></span><span id='citeF-175'></span>[[#cite-174|[174,175]]]; Simo et al. introduced a non linear version of a three-field Hu-Washizu Variational principle, where displacement, pressure and the Jacobian of the deformation gradient are independent field variables <span id='citeF-176'></span>[[#cite-176|[176]]]. The use of ''Mixed Variational Methods'' and the difficulties encountered when applying them with different elements have been largely discussed in the 1970s. In <span id='citeF-177'></span><span id='citeF-178'></span><span id='citeF-179'></span><span id='citeF-180'></span>[[#cite-177|[177,178,179,180]]] the need to satisfy the stability condition, the so-called ''inf-sup condition'', is demonstrated and the instability and ineffectiveness of elements with equal-order interpolations for all the primary variables are proved. This has motivated the development of a series of stabilization techniques, which allow the employment of low order Galerkin finite elements in computational fluid dynamics and solid mechanics problems <span id='citeF-181'></span><span id='citeF-182'></span><span id='citeF-183'></span><span id='citeF-184'></span><span id='citeF-185'></span><span id='citeF-186'></span><span id='citeF-187'></span><span id='citeF-188'></span>[[#cite-181|[181,182,183,184,185,186,187,188]]].
-So far, the presented chapters encompassed a relatively abstract subject, and has only been put in practice to test the capabilities of the SVD in beam analysis. The proposed cases where artificial scenarios that could be easily studied by following the hypotheses posed in [[#2.1 Problem formulation and hypotheses|2.1]]. The aim of this chapter is to modify the developed numerical tool in order to tackle a case of aeronautical interest, which is the starting of an '''helicopter rotor'''.
+The treatment of the incompressibility constraint is relatively new in the context of the Material Point Method (MPM). Most MPM formulations deal with compressible materials, avoiding the issues arising from the imposition of the incompressibility constraint. However, some procedures for the treatment of locking issues can be found in the literature. For instance, in <span id='citeF-189'></span>[[#cite-189|[189]]] an approach for the solution of kinematic (shearing and volumetric) locking is proposed. The authors identified the employment of linear shape functions in conjunction with a regular, rectangular grid, as cause of the locking. The mixed formulation, employed in such work, is derived from the definition of a three-field Hu-Washizu potential, with stress, strain and displacement considered as primary variables. In <span id='citeF-190'></span>[[#cite-190|[190]]] the formulation presented makes use of the Chorin's projection <span id='citeF-191'></span>[[#cite-191|[191]]], a popular fractional step formulation solved implicitly for fluid mechanics problems and in <span id='citeF-192'></span>[[#cite-192|[192]]] a similar strategy, based on a splitting operator technique for solving the momentum equation, is proposed for the treatment of the incompressibility constraint.
-==7.1 Case of study==
+In this Chapter, the computational strategy proposed in <span id='citeF-139'></span>[[#cite-139|[139]]] for the solution of solid mechanics problems characterized by plastic incompressibility in large displacement and large deformation regime, is described in detail and applied to some representative test examples. A mixed ''u-p'' formulation, where the displacement and mean stress are considered as primary variables, is implemented within the framework of the implicit MPM strategy, developed in the ''Kratos Multiphysics'' open-source platform <span id='citeF-29'></span><span id='citeF-30'></span>[[#cite-29|[29,30]]]. A monolithic solution strategy, which allows not to impose "spurious" pressure boundary conditions on the Neumann boundary, as done in <span id='citeF-190'></span><span id='citeF-192'></span>[[#cite-190|[190,192]]], is used. In the current work, only simplicial elements are considered and a stabilization technique is adopted for the satisfaction of the ''inf-sup condition''. The stabilization, based on the Polynomial Pressure Projection (PPP), presented in <span id='citeF-193'></span>[[#cite-193|[193]]], is chosen for its ease of implementation and good performance demonstrated in previous works <span id='citeF-194'></span><span id='citeF-195'></span>[[#cite-194|[194,195]]]. The proposed approach is validated through a series of benchmark examples, where an elastic Neo-Hookean and a J2 plastic material are employed. Further, for each test, the results obtained through a displacement-based (''u'') and the stabilized mixed (''u-p'') formulation are compared.
-The main goal of the presented chapter is the study of the dynamic behaviour of an helicopter's rotor during the starting of the engines. That is, the blades will start from idle configuration and will end rotating at its '''design''' angular velocity. Helicopter dynamics is a wide and complex subject combining both structural analysis and aerodynamics. As the understanding of the whole phenomena is not the goal of the present report, the following hypotheses will be assumed:
+In what follows, the ''u-p'' formulation is derived in matrix form. Afterwards, the numerical examples are illustrated and the results are discussed.
-* Blades will be modelled as two-dimensional structures. Three-dimensional effects (structural and aerodynamic) will be neglected, as well as gravity forces.
+==5.2 The mixed formulation==
-* Rotation axis is fixed and perpendicular to the plane of the blade.
-* The fluid will not be simulated but rather aerodynamic forces will be modelled.
-* The rotor will produce zero lift, and thus, no induced velocity along the span of the beam will be generated. Parasite drag is the only aerodynamic force acting on the rotor.
-* The helicopter is performing hovering flight with no wind speed.
-* The blade is fixed-free: no articulated parts are considered. Hence, the mechanical twist angle will always be zero.
-The rotor is made of three equally distributed blades, whose geometry is defined in figure [[#img-7.1|7.1]]. The radius of the blade is <math display="inline">R = 5\, m</math>, whereas the chord is <math display="inline">0.2</math> and <math display="inline">0.4\,m</math> at the root and the tip, respectively. The maximum chord is <math display="inline">0.55\,m</math> and its located at the 8% of the span. The total surface of the blade is <math display="inline">A_{\,\hbox{blade}} = 2.335\, m^2</math>. The blade has been modelled using a CAD software (''SolidWorks'') and then exported to '''GiD''' to proceed with the mesh generation. Its geometry is defined in figure [[#img-7.1|7.1]].
+In this section the mixed (''u-p'') formulation is briefly introduced and derived in matrix form.
-<div id='img-7.1'></div>
+===5.2.1 Governing equations in strong form===
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-blade.png|600px|Geometry and of a 2D blade.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 7.1:''' Geometry and of a 2D blade.
-|}
-In order to mesh the above geometry using only quadrilateral elements, the blade needs to be modified. By eliminating curves and roundings, the blade is discretised using a structured mesh made of 1340 elements:
+Let us consider the body <math display="inline">  \mathcal{B} </math> which occupies a region <math display="inline"> \Omega </math> of the three-dimensional Euclidean space <math display="inline"> \mathcal{E} </math> with a regular boundary <math display="inline"> \partial \Omega </math> in its reference configuration. A deformation of <math display="inline">  \mathcal{B} </math> is defined by a one-to-one mapping
-<div id='img-7.2'></div>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-new_blade.png|600px|Structured mesh of a simplified 2D blade.]]
+|
-|- style="text-align: center; font-size: 75%;"
+{| style="text-align: left; margin:auto;width: 100%;"
-| colspan="1" | '''Figure 7.2:''' Structured mesh of a simplified 2D blade.
+|-
+| style="text-align: center;" | <math>\varphi :\Omega \rightarrow \mathcal{E} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.1)
 |}
-In three dimensions, the blades can be thought as an extrusion of a ''NACA 0012'' airfoil, whose cross section <math display="inline">A_{\,\hbox{airfoil}} = 0.02081\, m^2</math> has been computed integrating the profile shape <span id='citeF-15'></span>[[#cite-15|[15]]]. To approximate the superficial density of the blade, it will be considered that its total mass is one fifth of the mass of a solid aluminium blade:
+that maps each point ''p'' of the body <math display="inline">  \mathcal{B} </math> into a spatial point <math display="inline">\mathbf{x}</math>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,972: / Line 4,456: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>m = \frac{1}{5}\,\rho \,(A_{\,\hbox{airfoil}}\;\cdot \;R)\approx 56.19 Kg \;\; \Rightarrow \;\;\rho _{s} = \frac{m}{A_{\,\hbox{blade}}} \approx 24\,  \frac{Kg}{m^2}  </math>
+| style="text-align: center;" | <math>\mathbf{x} = \varphi \left(\boldsymbol{p}\right) </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.1)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.2)
 |}
-Regarding the '''driving device''', it will be supposed that the helicopter has an alternative engine able to inject power to the rotor. As engines cannot produce power when the angular velocity is zero, an electric motor will be used to generate a torque that starts the rotation motion. Remember that the power <math display="inline">\textrm{P}</math> of a rotating object is given by:
+which represents the location of  ''p'' in the deformed configuration of <math display="inline">\mathcal{B}</math>. The region of <math display="inline">\mathcal{E}</math> occupied by <math display="inline">\mathcal{B}</math> in its deformed configuration is denoted as <math display="inline">\varphi \left(\Omega \right)</math>.
-<span id="eq-7.2"></span>
+The boundary value problem of finite elastostatics consists in finding a displacement field <math display="inline"> \boldsymbol{u} : \varphi \left(\Omega \right)\rightarrow \mathcal{E}</math> such that the equilibrium equations and the kinematic conditions are satisfied
+<span id="eq-5.3"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,985: / Line 4,471: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\textrm{P} = \Omega \; \textrm{T} </math>
+| style="text-align: center;" | <math>\left\{\begin{array}{rcll}-\nabla \cdot \boldsymbol{\sigma } &= &\boldsymbol{b}  &\textrm{in}  \quad \varphi \left(\Omega \right)\\         \boldsymbol{\sigma }\cdot \boldsymbol{n} &= &\boldsymbol{\overline{t}} &\textrm{on}  \quad \varphi (\partial \Omega _N) \\         \boldsymbol{u} &= &\boldsymbol{\overline{u}}  &\textrm{on}  \quad \varphi (\partial \Omega _D)         \end{array}\right.  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.2)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.3)
 |}
-Where <math display="inline">\textrm{T}</math> is the torque produced by the device. To avoid sudden jumps in the variables, neither the torque nor the power will be injected instantaneously. Instead, they will vary from an initial to a final value describing a '''sinusoidal''' shape:
+where <math display="inline"> \boldsymbol{\sigma } </math> is the Cauchy stress tensor, <math display="inline"> \boldsymbol{b} </math> denotes the body forces and <math display="inline"> \varphi (\partial \Omega _N) </math> and <math display="inline"> \varphi (\partial \Omega _D) </math> the boundaries of <math display="inline">\varphi \left(\Omega \right)</math>, where both the normal tension (<math display="inline">\boldsymbol{\overline{t}} </math>) (being <math display="inline">\boldsymbol{n} </math> the outer normal) and the displacements (<math display="inline">\boldsymbol{\overline{u}} </math>) are prescribed.
+As described in <span id='citeF-159'></span>[[#cite-159|[159]]], the mixed formulation can be obtained expressing the system of Equations [[#eq-5.3|(5.3)]] in function of two primary variables: the displacement <math display="inline"> \boldsymbol{u} </math> and the mean stress <math display="inline"> p </math> by splitting the stress tensor in its volumetric and deviatoric part <math display="inline"> \boldsymbol{\sigma }^{\mathrm{dev}} </math>. Thus, the system can be rewritten as
+<span id="eq-5.4"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 2,997: / Line 4,486: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{matrix}\textrm{T}= \textrm{T}_0 + \textrm{T}_\hbox{nom} \,\sin \left(\frac{\pi }{2\,\textrm{t}_{a}}\,\textrm{t} \right) \qquad 0\leq t< \textrm{t}_{a}\,,\qquad \qquad  \textrm{T} = \textrm{T}_\hbox{nom}   \qquad  \textrm{t}_{a}\leq t< \textrm{t}_{b} \\ \\ \textrm{P} = \frac{1}{2}\left( \textrm{P}_0 + \textrm{P}_\hbox{nom}\right)+ \frac{1}{2}\left(\textrm{P}_\hbox{nom} -\textrm{P}_0\right)  \,\sin \left(\frac{\pi }{\textrm{t}_c-\textrm{t}_b}\,\textrm{t} + \pi \left(\frac{1}{2}-\frac{\textrm{t}_c}{\textrm{t}_c-\textrm{t}_b} \right)\right)\qquad \textrm{t}_b\leq t< \textrm{t}_{c}\\ \\ \textrm{P}=\textrm{P}_{\hbox{nom}} \qquad \qquad \textrm{t}_{c}\leq t \end{matrix} </math>
+| style="text-align: center;" | <math>\left\{\begin{array}{rcll}-\nabla \cdot \left(\boldsymbol{\sigma }^{\mathrm{dev}} + p \boldsymbol{I}\right)&= &\boldsymbol{b}  &\textrm{in}  \quad \varphi \left(\Omega \right)\\         p - \left(\frac{1}{3} \boldsymbol{I}:\boldsymbol{\sigma }\right)&= &0 &\textrm{in} \quad \varphi \left(\Omega \right)\\         \left(\boldsymbol{\sigma }^{\mathrm{dev}} + p \boldsymbol{I}\right)\cdot \boldsymbol{n} &= &\boldsymbol{\overline{t}} &\textrm{on}  \quad \varphi (\partial \Omega _N) \\         \boldsymbol{u} &= &\boldsymbol{\overline{u}}  &\textrm{on}  \quad \varphi (\partial \Omega _D)         \end{array}\right.  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.3)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.4)
 |}
-<math display="inline">\textrm{W}_0</math> and <math display="inline">\textrm{T}_\hbox{nom}</math> are the initial and nominal values, respectively, of torque generated by the electric motor. The motor reaches the nominal value of torque at <math display="inline">\textrm{t}_{a}</math>, and maintains it constant until <math display="inline">\textrm{t}_{b}</math>, when the engine starts operating. To avoid discontinuities in the injected power, the following statement has to be met:
+being <math display="inline">\boldsymbol{I}</math> the second order identity tensor. We can observe that if <math display="inline"> \boldsymbol{u} </math> is a solution of Equation [[#eq-5.3|(5.3)]], then <math display="inline"> \left(\boldsymbol{u}, p\right)</math>, satisfying also <math display="inline"> p - \left(\frac{1}{3} \boldsymbol{I}:\boldsymbol{\sigma }\right)= 0 </math>, is a solution of Equation [[#eq-5.4|(5.4)]].
+===5.2.2 Weak form and linearisation of the weak form in spatial form===
+According to the standard FEM procedure, the weak form of Equation [[#eq-5.4|(5.4)]] is obtained by employing the Galerkin method and is written in spatial configuration, adopting an Updated Lagrangian framework.
+For sake of clarity the weak form Equation [[#eq-5.3|5.3]], previously derived in Chapter [[#4 Irreducible formulation|4]], is provided below.
+<span id="eq-5.5"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,009: / Line 4,505: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\textrm{P}_0 = \Omega _{(\textrm{t}_{b})}\;\textrm{T}_{\hbox{nom}} </math>
+| style="text-align: center;" | <math>G(\boldsymbol{u},\boldsymbol{w})=\int _{\varphi (\Omega )} \boldsymbol{\sigma }: \left[\nabla ^{s}\boldsymbol{w}\right]\, dv - \int _{\varphi (\Omega )} \boldsymbol{b} \cdot  \boldsymbol{w}\, dv - \int _{\varphi (\partial \Omega _N)}\boldsymbol{\overline{t}} \cdot  \boldsymbol{w}\, da=0, \quad \forall \boldsymbol{w}\in \mathcal{V}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.4)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.5)
 |}
-The engine starts to operate at <math display="inline">\textrm{t}_{b}</math> and reaches its design value <math display="inline">\textrm{P}_\hbox{nom}</math> at <math display="inline">\textrm{t}_{c}</math>, maintaining this value constant for further values of time until normal flight conditions are reached (see figure  [[#img-7.3|7.3]]). No automatic control that regulates the power as a function of the angular velocity is taken into consideration.
+using the notation <math display="inline">\mathbf{A}^s = \frac{1}{2}\left(\mathbf{A} + \mathbf{A}^T\right)</math>.
-==7.2 Rigid-elastic coupling==
+With regard to the mixed formulation, linear interpolation finite elements both for displacement and pressure (''u-p'') are considered. The weak form of the balance of the linear momentum (Equation [[#eq-5.5|(5.5)]]) can be rewritten as
-Unlike in previous chapters, kinematics of the rotating motion are no longer known. This implies breaking the first hypothesis defined in [[#2.1 Problem formulation and hypotheses|2.1]], and thus, the developed model is no longer applicable. In previous cases (see [[#6 Analysis of rotating beams using the SVD|6]]), angular velocity and acceleration were known at every time, being the reaction forces the unknowns. In this case, what is known is the power provided by the engine, whereas the unknowns are the angular variables <math display="inline">\theta </math>, <math display="inline">\Omega </math> and <math display="inline">\alpha </math>. This is a major change in the problem formulation, as rotation and vibration motions become '''coupled'''. From solid mechanics, it is found that the relationship between the torque and the angular velocity of a rigid body is given by:
+<span id="eq-5.6"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,025: / Line 4,520: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{W}_a = \frac{\mathrm{d} \left( \mathbf{J}_a \,\boldsymbol{\Omega }\right)}{\mathrm{d} \textrm{t}}+\mathbf{AG}\times \textrm{m} \,\mathbf{a}_a </math>
+| style="text-align: center;" | <math>G(\boldsymbol{u}, p,\boldsymbol{w})=\int _{\varphi (\Omega )} \left(\boldsymbol{\sigma }^{\mathrm{dev}} + p\boldsymbol{I}\right): \left[\nabla ^{s}\boldsymbol{w}\right]\, dv - \int _{\varphi (\Omega )} \boldsymbol{b} \cdot  \boldsymbol{w}\, dv - </math>
+|-
+| style="text-align: center;" | <math>\int _{\varphi (\partial \Omega _N)}\boldsymbol{\overline{t}} \cdot  \boldsymbol{w}\, da = 0, \quad \forall \boldsymbol{w} \in \mathcal{V}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.5)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.6)
 |}
-Where <math display="inline">a</math> is a generic point where the torque is to be evaluated, <math display="inline">\mathbf{J}_a</math> is the inertia tensor of the solid with respect to <math display="inline">a</math>, <math display="inline">\boldsymbol{\Omega }</math> is the vector of angular velocities, <math display="inline">\mathbf{a}_a</math> the linear acceleration of the point <math display="inline">a</math> and <math display="inline">\mathbf{AG}</math> the vector pointing from <math display="inline">a</math>  to the centre of mass of the solid. If the inertia matrix is posed in such a way that is '''constant''' in time, the above expression transforms into Euler's equations for rigid bodies (see [[#eq-1.2|1.2]]). Luckily enough, with the assumed hypotheses, the torque equation can be expressed as stated in [[#eq-3.22|3.22]]:
+where the Cauchy stress tensor <math display="inline">\boldsymbol{\sigma }</math> is decomposed in its deviatoric and volumetric component, denoted as <math display="inline">\boldsymbol{\sigma }^{\mathrm{dev}} </math> and <math display="inline">p </math>, respectively. The weak form of the pressure continuity equation is obtained by performing a <math display="inline"> L_2</math> inner product of the second equation of [[#eq-5.4|(5.4)]] with an arbitrary test function <math display="inline">q\in \mathcal{Q}</math>, where <math display="inline">\mathcal{Q}</math> is the space of virtual pressure. Finally the weak form of the pressure continuity equation is expressed as
+<span id="eq-5.7"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,037: / Line 4,535: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathrm{W}_z =  \mathrm{J}_z \;\alpha  </math>
+| style="text-align: center;" | <math>G(\boldsymbol{u}, p,q)=\int _{\varphi (\Omega )}q\left[\left(\frac{1}{3} \boldsymbol{I}:\boldsymbol{\sigma }\right)- p\right]dv=0,\quad \forall q\in \mathcal{Q}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.6)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.7)
 |}
-And taking into account that <math display="inline">\mathrm{W}_z </math> results from the contribution of the torque generated by the engine and the elastic reaction torque induced by the structure:
+In this work a Newton-Raphson's iterative procedure is employed for the solution of problems characterized by material and geometrical non-linearities. The non-linear weak forms of Equations [[#eq-5.6|(5.6)]] and [[#eq-5.7|(5.7)]] have to be linearised through an expansion in Taylor's series, evaluated at the last known equilibrium configuration <math display="inline">\boldsymbol{u}^*</math> and <math display="inline">p^*</math>.
+In this way the solution system of linearised equations can be derived and expressed in matrix form as
-<span id="eq-7.7"></span>
+<span id="eq-5.8"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,050: / Line 4,550: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathrm{T} - \mathrm{W}_{el}  =  \mathrm{J}_z \;\alpha  </math>
+| style="text-align: center;" | <math>\begin{bmatrix}^m\mathbf{K}^{\mathrm{tan}} & \mathbf{B} \\ \mathbf{B}^* & -\mathbf{M} \end{bmatrix}   \begin{bmatrix}\delta \boldsymbol{u}\\ \delta \boldsymbol{p} \end{bmatrix}   = - \begin{bmatrix}\boldsymbol{R}_{\boldsymbol{u}}\\ \boldsymbol{R}_p \end{bmatrix}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.7)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.8)
 |}
-And recall how the elastic reaction torque was computed:
+where <math display="inline"> \mathbf{R}_{\boldsymbol{u}} = G(\boldsymbol{u}, p,\boldsymbol{w}) </math> and <math display="inline"> \boldsymbol{R}_p = G(\boldsymbol{u}, p,q) </math> are the components of the residual vector, <math display="inline"> \delta \boldsymbol{u} </math> and <math display="inline"> \delta \boldsymbol{p} </math> are the vector of unknown displacements and unknown mean stresses, respectively. The components of the matrix on the left hand side (lhs) of Equation [[#eq-5.8|(5.8)]] are given by the tangent stiffness matrix <math display="inline"> ^m\mathbf{K}^{\mathrm{tan}} = D_{\boldsymbol{u}}G(\boldsymbol{u}, p,\boldsymbol{w}) </math>, which can be seen as the sum of the material stiffness matrix
+<span id="eq-5.9"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,062: / Line 4,563: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathrm{W}_{el}}_{(i)} = \sum _{j=1}^{n_{pt}}{\mathbf{w}_{el}}_{\,(i \cdot j)} = \sum _{j=1}^{n_{pt}}\left( \begin{bmatrix}\Delta \widetilde{\mathbf{r}} \\ \vdots  \end{bmatrix} \mathbf{R} \right)_{\,(i \cdot j)} </math>
+| style="text-align: center;" | <math>^m\mathbf{K}^{M} := \int _{\varphi (\Omega )}\left[\nabla ^{s}\boldsymbol{w}\right]\left(\mathbf{D}^{\mathrm{dev}} + p (\boldsymbol{I}\otimes \boldsymbol{I} - 2\mathbb{I})\right)\left[\nabla ^{s}\delta \boldsymbol{u}\right]dv  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.8)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.9)
 |}
-Accounting only for the torque in the rotation axis, and including equation [[#eq-3.21|3.21]] equation [[#eq-7.2|7.2]], we arrive at:
+being <math display="inline">\mathbb{I}</math> the fourth order identity tensor, and the geometric stiffness matrix
-<span id="eq-7.9"></span>
+<span id="eq-5.10"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,075: / Line 4,576: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{matrix}\mathbf{M}_{\mathbf{ll}} \ddot{\mathbf{d}}_\mathbf{l} \;+\;\mathbf{D}_{\mathbf{ll}} \dot{\mathbf{d}}_\mathbf{l} \;+\;\mathbf{K}_{\mathbf{ll}} {\mathbf{d}}_\mathbf{l}=\mathbf{F}_\mathbf{l} - (\;\mathbf{M}_{\mathbf{lr}} \overline{\ddot{\mathbf{u}}} \;+\;\mathbf{D}_{\mathbf{lr}} \overline{\dot{\mathbf{u}}} \;+\;\mathbf{K}_{\mathbf{lr}} \overline{{\mathbf{u}}})\\ \\ \mathrm{J}_z \;\alpha = \frac{\textrm{P}}{\Omega } -  \sum _{j=1}^{n_{pt}}\left( \begin{bmatrix}\Delta \widetilde{\mathbf{r}} \\ \vdots  \end{bmatrix} \left[\mathbf{M}_{\mathbf{rl}} \ddot{\mathbf{d}}_\mathbf{l}\;+\;\mathbf{D}_{\mathbf{rl}} \dot{\mathbf{d}}_\mathbf{l}\;+\;\mathbf{K}_{\mathbf{rl}} {\mathbf{d}}_\mathbf{l}\;+\;\mathbf{M}_{\mathbf{rr}} \overline{\ddot{\mathbf{u}}} \;+\;\mathbf{D}_{\mathbf{rr}} \overline{\dot{\mathbf{u}}} \;+\;\mathbf{K}_{\mathbf{rr}} \overline{{\mathbf{u}}}\;-\;\mathbf{F}_\mathbf{r} \right]\right)_{\,(3 \cdot j)} \end{matrix} </math>
+| style="text-align: center;" | <math>^m\mathbf{K}^{G} := \int _{\varphi (\Omega )}\left[\nabla \boldsymbol{w}\right]\left(\boldsymbol{\sigma }^{\mathrm{dev}} + p\boldsymbol{I}\right)\left[\nabla \delta \boldsymbol{u}\right]dv  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.9)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.10)
 |}
-The main obstacle in the resolution of the above system is that all <math display="inline">\mathbf{K}</math>, <math display="inline">\mathbf{D}</math> and <math display="inline">\mathbf{F}</math> depend on both <math display="inline">\Omega </math> and <math display="inline">\alpha </math>. Moreover, the way in which the previous matrices are defined (see [[#eq-3.13|3.13]]) make the equations highly '''nonlinear''' in <math display="inline">\Omega </math> and <math display="inline">\alpha </math>. Hence, a proper numerical method has to be proposed in order to tackle the presented system of non-linear differential equations.
+Furthermore, <math display="inline"> \mathbf{M} = D_{p}G(\boldsymbol{u}, p,q) </math> is
-==7.3 Numeric integration==
+<span id="eq-5.11"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\mathbf{M} = \int _{\varphi (\Omega )} q\, \delta p \, dv  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.11)
+|}
-In the context of coupled rigid-elastic dynamics, the unknowns are the vectors <math display="inline">\mathbf{d}</math>, <math display="inline">\dot{\mathbf{d}}</math> and <math display="inline">\ddot{\mathbf{d}}</math> regarding the vibration motion, and the scalars <math display="inline">{\theta }</math>, <math display="inline">\dot{{\theta }}</math> and <math display="inline">\ddot{{\theta }}</math> regarding the rotation motion. Using the presented '''Newmark''' <math display="inline">\boldsymbol{\beta }</math>'''-method''', the previous system of differential equation can be transformed into a system of algebraic equations. The transformation this method proposes has already been reviewed in [[#5.2 Numerical integration|5.2]] for the elastic variables, and the only thing left is to apply it to the angular variables:
+and the mixed terms <math display="inline"> \mathbf{B} = D_{p}G(\boldsymbol{u}, p,\boldsymbol{w}) </math> and <math display="inline"> \mathbf{B^*} = D_{\boldsymbol{u}}G(\boldsymbol{u}, p,q) </math>, are defined, respectively, as
-<span id="eq-7.10"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,092: / Line 4,601: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\Omega }^{n+1}={\Omega }^{n}+(1-\gamma ) \Delta \textrm{t} \;\alpha ^{n} + \gamma \,\Delta \textrm{t}\; {\alpha }^{n+1}   </math>
+| style="text-align: center;" | <math>\mathbf{B} = \int _{\varphi (\Omega )} \left(\nabla \cdot \boldsymbol{w}\right)\, \delta p\, dv </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.10)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.12)
 |}
-Hence, what is left is a system of non-linear '''algebraic''' equations with the unknowns <math display="inline">\alpha ^{n+1}</math> and <math display="inline">{\mathbf{d}}^{n+1}</math>. This problem could be tackled through a non-linear solving algorithm such as Newton-Raphson, widely used in computational mechanics. It is an iterative procedure based on the solving of a sequence of linear models <span id='citeF-12'></span>[[#cite-12|[12]]]. The Newton method can in fact be applied to arbitrary systems of algebraic equations with <math display="inline">n</math> unknowns. However, its implementation is not an easy task and requires a great understanding of algebra. Moreover, as it is an iterative algorithm, it requires the different matrices to be computed more than once at each timestep, increasing considerably the computation time.
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\mathbf{B^*} = \int _{\varphi (\Omega )} D_{\boldsymbol{u}}\left(\frac{1}{3} \boldsymbol{I}:\boldsymbol{\sigma }\right)\left(\nabla \cdot \delta \boldsymbol{u}\right)\, q\, dv </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.13)
+|}
-A simpler approach is based on the '''explicit''' central difference scheme, which computes accelerations at <math display="inline">\textrm{t}^n</math> and uses this value to update velocities and displacements at <math display="inline">\textrm{t}^{n+1}</math>. Although very easy to program, explicit methods require from very '''little''' timesteps to ensure conditional stability. Thus, computation time will arise with respect to the previous case of known kinematics. The general algorithm can be summarised as:
+where <math display="inline">D_{\boldsymbol{u}}\left(\frac{1}{3} \boldsymbol{I}:\boldsymbol{\sigma }\right)</math> can be derived once determined the volumetric stress as function of the strain field.
-<ol>
+One can observe that <math display="inline">^m\mathbf{K}^{M}</math> and <math display="inline">^m\mathbf{K}^{G}</math> are distinguished from <math display="inline"> \mathbf{K}^{M} </math> and <math display="inline"> \mathbf{K}^{G} </math>, defined for the irreducible formulation (Equations [[#eq-4.34|(4.34)]] and [[#eq-4.33|(4.33)]]). In the mixed case, the deviatoric part of <math display="inline"> \mathbf{D} </math> and <math display="inline"> \boldsymbol{\sigma } </math> is separated by the volumetric one and an evaluation of the latter is done, not using the material response of the constitutive law, but the interpolation of the nodal pressure field on the material points, i.e., the integration points.
-<li>Obtain <math display="inline">{\mathbf{d}}^{n+1}</math> solving the system of equations stated in [[#eq-5.34|5.34]] (using <math display="inline">\Omega ^n</math> and <math display="inline">\alpha ^n</math>) .    </li>
+===5.2.3 The stabilized mixed formulation===
-<li>Once displacements at <math display="inline">\textrm{t}^{n+1}</math> are known, compute the elastic reaction torque <math display="inline">\mathrm{W}_{el}^{n+1}</math> from [[#eq-7.9|7.9]].   </li>
-<li>Find <math display="inline">\alpha ^{n+1}</math> from equation [[#eq-7.7|7.7]] and update <math display="inline">\Omega ^{n+1}</math> using expression [[#eq-7.10|7.10]].   </li>
-</ol>
+For the treatment of the incompressibility constraint, the Polynomial Pressure Projection (PPP), introduced by Dohrmann and Bochev <span id='citeF-193'></span>[[#cite-193|[193]]], is used.  This stabilization procedure is obtained by modifying the mixed variational equation by using a <math display="inline"> L^2 </math> polynomial pressure projection. If <math display="inline"> k </math> is the order of the continuous polynomial shape functions used to approximate <math display="inline"> p </math>, the pressure projection is performed into a polynomial space with order of <math display="inline"> k-1 </math>. As in the current work linear shape functions are used for the pressure, the <math display="inline"> L^2 </math> polynomial pressure projection is made in a discontinuous space and, consequently, it can be performed at the element level as
-The initial value of angular accelerations is found by solving the non-linear algebraic equation that involves initial conditions <math display="inline">\mathbf{d}_0</math>, <math display="inline">\dot{\mathbf{d}}_0</math>, <math display="inline">\ddot{\mathbf{d}}_0</math> and <math display="inline">\Omega _0</math>. As <math display="inline">\alpha _0</math> is the only unknown, the equation is solved by using a simple '''fixed-point iteration''', using as initial guess <math display="inline">\alpha _0 = 0</math>. It has been found that four iterations suffice to obtain a converged solution.
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\int _{\varphi (\Omega )} \tilde{q}\,  \left(p-\tilde{p}\right)\, dv=0,\quad \forall \tilde{q}\in \mathcal{Q}^0 </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.14)
+|}
-==7.4 Aerodynamic forces==
+being <math display="inline"> \tilde{p} </math> the best approximation of <math display="inline"> p </math> in <math display="inline"> (\mathcal{Q}^0) </math> and <math display="inline">\tilde{q}\in \mathcal{Q}^0</math> an arbitrary test function, where <math display="inline">\mathcal{Q}^0</math> is the space of polynomial functions with zero degree in each coordinate direction.  Unlike other stabilization techniques, the pressure stabilization is accomplished without the use of the residual of the momentum equation; thus, the calculation of higher-order derivatives and the specification of a mesh-dependent stabilization parameter are avoided. Moreover, it is demonstrated that symmetry of the mixed formulation is retained.
-Performing a simulation that accounts for fluid-structure interaction is far out of the scope of this section. Instead, the aerodynamic forces acting on the blade will be '''modelled''' using simplified aerodynamic relations. As the rotor is modelled in two dimensions, no forces acting perpendicular to the plane will be taken into consideration. Following the hypotheses presented in [[#7.1 Case of study|7.1]], the only force acting on the rotor plane is the parasite drag.
+In the case of simplicial elements, as in the current work, the stabilization of the unstable mixed formulation requires only the addition of the bilinear form
-This aerodynamic resistance force depends on many parameters, such as the angle of attack, the surface roughness of the blade, the Mach number and the Reynolds number — a non-dimensional parameter that measures the strength of inertial forces of the fluid with respect to viscous ones. In order simplify things up, the effect of the latter variables will be encapsulated in a non-dimensional '''drag coefficient''', <math display="inline">c_D</math>, which will be considered known. Otherwise, this analysis would be impracticable. The drag differential (<math display="inline">\mathrm{d} \textrm{D}</math>) acting on a surface differential (<math display="inline">\mathrm{d} \textrm{A}</math>) of the blade is given by:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,122: / Line 4,643: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{\mathrm{d} \textrm{D}}{\mathrm{d} \textrm{A}} = \frac{1}{2}\,\rho _{\,\hbox{air}}\,c_D\,\textrm{v}^2 </math>
+| style="text-align: center;" | <math>\int _{\varphi (\Omega )^e} \left(q-\tilde{q}\right)\dfrac{\alpha }{G}\left(p-\tilde{p}\right)\, dv=0 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.11)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.15)
 |}
+to Equation [[#eq-5.7|(5.7)]], where <math display="inline"> \alpha </math> is a parameter to be selected for stability and <math display="inline"> G </math> the shear modulus. The weak form of the pressure continuity equation (Equation [[#eq-5.7|(5.7)]]) can be rewritten as
+<span id="eq-5.16"></span>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>G(\boldsymbol{u}, p,q)=\int _{\varphi (\Omega )}q\left[\left(\frac{1}{3} \boldsymbol{I}:\boldsymbol{\sigma }\right)- p\right]- \dfrac{\alpha }{G}\left[q\, p - \tilde{q}\, \tilde{p} \right]dv=0  </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.16)
+|}
-Where <math display="inline">\textrm{v}</math> stands for the relative velocity of the surface differential with respect the fluid. As the blades are rotating in a plane, their solid rigid velocity is proportional to <math display="inline">\Omega \,\mathbf{r}_i</math>, where <math display="inline">\mathbf{r}_i</math> stands for the initial position. Taking into account the velocity induced by the elastic vibrations, and the change in radial position due to elastic displacements:
+and the matrix system (Equation [[#eq-5.8|(5.8)]]) becomes
-<span id="eq-7.12"></span>
+<span id="eq-5.17"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,137: / Line 4,669: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{v}^2 = (\widetilde{\Omega }\,(\mathbf{r}_i+\mathbf{u})+\dot{\mathbf{u}})^2 = \left[\dot{\mathbf{u}}^2 +2\,(\widetilde{\Omega }\,\,\mathbf{u})\, \dot{\mathbf{u}} \,+2\,(\widetilde{\Omega }\,\mathbf{r}_i)\, \,\dot{\mathbf{u}} \, + (\widetilde{\Omega }\,\mathbf{u})^2 +  2\,(\widetilde{\Omega }\, \mathbf{r}_i)\,(\widetilde{\Omega }\, \mathbf{u})\, + (\widetilde{\Omega }\,\mathbf{r}_i)^{2} \right] </math>
+| style="text-align: center;" | <math>\begin{bmatrix}^m\mathbf{K}^{\mathrm{tan}} & \mathbf{B} \\ \mathbf{B}^* & -\mathbf{M}-\mathbf{M}^{\mathrm{stab}} \end{bmatrix}   \begin{bmatrix}\delta \boldsymbol{u}\\ \delta \boldsymbol{p} \end{bmatrix}   = - \begin{bmatrix}\boldsymbol{R}_{\boldsymbol{u}}\\ \boldsymbol{R}_p + \boldsymbol{R}_p^{\mathrm{stab}} \end{bmatrix}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.12)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.17)
 |}
-As the geometry is 2D, the drag force can be modelled in terms of the FEM as a '''body force''', proportional to the area of each element. In three-dimensional problems this approach would not be correct, as aerodynamic forces only act on the surface and hence are to be modelled as boundary tractions. With this in mind, the term <math display="inline">\mathrm{d} \textrm{D}</math> is to be inserted as a body force <math display="inline">\boldsymbol{f}</math> in equation [[#eq-3.13|3.13]]. As a result, the following expression is obtained:
+where
+<span id="eq-5.18"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,149: / Line 4,682: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{f}_D = \,\int _{{\mho }}^{ }\mathbf{N}^T\,\frac{1}{2}\,\rho _{\,\hbox{air}}\,c_D\,\mathbf{v}^2 \;\textrm{d}\mho  </math>
+| style="text-align: center;" | <math>\mathbf{M}^{\mathrm{stab}} = \int _{\varphi (\Omega )} \dfrac{\alpha }{G}\left(q\, \delta p - \tilde{q}\, \delta \tilde{p}\right)\, dv  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.13)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.18)
 |}
-The problem is that from <math display="inline">\boldsymbol{f}_D</math> will not only emerge components proportional to <math display="inline">\dot{\mathbf{u}}</math> and <math display="inline">\mathbf{u}</math>, but also non-linear terms such as <math display="inline">\dot{\mathbf{u}}^2</math>, <math display="inline">\mathbf{u}^2</math> and <math display="inline">\dot{\mathbf{u}}\,\mathbf{u}</math>. The responsible for these nonlinearities to appear is the term <math display="inline">\mathbf{V}^2</math> defined in [[#eq-7.12|7.12]]. To make the problem more attainable, it will be supposed that <math display="inline">\widetilde{\Omega }\,\mathbf{r}_i \gg \dot{\mathbf{u}}</math> and <math display="inline">\mathbf{r}_i \gg \mathbf{u}</math>. Moreover, air density and drag coefficient will be considered constant along the domain. These hypotheses allow <math display="inline">\boldsymbol{f}_D</math> to be simplified as:
+and
+<span id="eq-5.19"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,161: / Line 4,695: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{f}_D = \frac{1}{2}\,\rho _{\,\hbox{air}}\,c_D\,\int _{{\mho }}^{ }\mathbf{N}^T\,\widetilde{\Omega ^2}\,\mathbf{r}_i^{\,2}\;\textrm{d}\mho  </math>
+| style="text-align: center;" | <math>R_p^{\mathrm{stab}} = \int _{\varphi (\Omega )} \dfrac{\alpha }{G}\left[q\, p - \tilde{q}\, \tilde{p} \right]\, dv  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.14)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.19)
 |}
-The above integral is decomposed into elemental matrices as reviewed in [[#B Elemental matrix computation|B]], and computed using Gauss Quadrature. The major drawback of this simplified scenario is that drag forces no longer account for the effect of vibration, and thus air does not damp oscillating motions. In order to overcome this limitation, internal Rayleigh '''damping''' will be taken into consideration (recall equation [[#eq-3.15|3.15]]). This cushioning will induce a new sources of stresses as expected from equation [[#eq-2.15|2.15]]:
+==5.3 The MPM algorithm in the framework of a mixed formulation==
+If a mixed (''u-p'') formulation is used in the framework of the MPM, it is important to highlight that some changes have to be considered in the initialization and convective phase of standard algorithm, described in Section [[#2.4.1 MPM Algorithm|2.4.1]]. In the initialization phase, initial nodal pressure values <math display="inline">p_{I}^n </math>, related to the previous time <math display="inline">t^n</math>, have to be evaluated, in addition to the mass, velocity and acceleration ones, using the following expression:
+<span id="eq-5.20"></span>
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 |-
@@ Line 3,173: / Line 4,710: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{\sigma }= \mathbf{C}\;\boldsymbol{\varepsilon } + \overline{\mathbf{D}}\;\dot{\boldsymbol{\varepsilon }} \;\rightarrow \; \mathbf{C}\; \nabla ^{s} \,\mathbf{N} \;\mathbf{d} + \overline{\beta }\,\mathbf{C}\; \nabla ^{s}\,\mathbf{N} \;\dot{\mathbf{d}} = \mathbf{C} \; \mathbf{B}\; \mathbf{d} + \overline{\beta }\,\mathbf{C} \;\mathbf{B} \;\dot{\mathbf{d}}  </math>
+| style="text-align: center;" | <math>p_{I}^n = \frac{\sum _p N_I m_p p_{p}^n}{\sum _p N_I m_p}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.15)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.20)
 |}
-To close down this section, let's make an estimation of how much '''power''' is required to compensate the parasite drag when operating at nominal speed, say 400 rpm. Considering a constant chord <math display="inline">c</math>, the power the aerodynamic resistance of each blade induces is:
+where <math display="inline">N_I</math> is the shape function of node <math display="inline">I</math> evaluated at the position of the <math display="inline">p-th</math> material point, and <math display="inline">m_p</math> and <math display="inline">p_{p}^n</math> are the mass and the pressure of the material point, respectively. The nodal pressure evaluated in Equation [[#eq-5.20|(5.20)]] is used in the predictor step of the Newmark scheme. Once the solution is iteratively computed using the linearised system of Equations [[#eq-5.17|(5.17)]], the convective phase is performed, as explained in detail in Section [[#2.4.1 MPM Algorithm|2.4.1]]. The pressure on the material points is updated in addition to the material point displacement, velocity and acceleration, through an interpolation of the converged nodal pressure values <math display="inline">p_{I}^{n+1}</math> on the material point position
-<span id="eq-7.16"></span>
+<span id="eq-5.21"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,186: / Line 4,723: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\textrm{P}_{\hbox{drag}} = \Omega \int _{ S}^{ } \varsigma \, \hbox{d}\,\hbox{D} = \Omega \int _{0}^{R} \frac{1}{2}\,\rho _{\,\hbox{air}}\,c_D\,\mathbf{v}^2\, c\,\varsigma \,\hbox{d}\varsigma = \frac{1}{2}\,\rho _{\,\hbox{air}}\,c_D\,\Omega ^3\,c\int _{0}^{R}\varsigma ^3\,\hbox{d}\varsigma = \frac{1}{8}\,\rho _{\,\hbox{air}}\,c_D\,\Omega ^3\,c\,R^4 </math>
+| style="text-align: center;" | <math>p_{p}^{n+1} = \sum _I N_I p_{I}^{n+1}.  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.16)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5.21)
 |}
-Considering flight at sea level (<math display="inline">\rho _{\,\hbox{air}} = 1.225\, Kg/m^3</math>), a mean chord <math display="inline">c = 0.45\,m</math> and a drag coefficient <math display="inline">c_D = 0.008</math>, each blade requires from <math display="inline">25.32\,kW</math> to surpass aerodynamic resistance. Nevertheless, the engine will need to inject more power to counteract the elastic reaction torque.
+The Algorithm [[#algorithm-2.1|2.1]], previously provided in Section [[#2.4.1 MPM Algorithm|2.4.1]], is below presented for a static case, together with the modifications aforementioned.
-==7.5 Rotor starting simulation==
-Once the FEM software is updated with the developed models from the previous sections, it is time to test it out simulating the starting of a rotor blade. Imagine that the studied helicopter has the following properties:
+{| style="margin: 1em auto;border: 1px solid darkgray;"
+|-
+|
+:
+(we will use <math>(\bullet )^n = (\bullet )(t_{n}) </math>), Material DATA: E, <math> \nu </math>, <math> \rho </math>
+|-
+| Initial data on material points: <math> m_p </math>, <math> \mathbf{x}_p^n</math>, <math>p_{p}^n</math>, <math>\mathbf{F}_p^n = \displaystyle{ \sum _I\frac{\partial N_I}{\partial \mathbf{x}_I^{0}}}\cdot \mathbf{x}_I^{n}</math> and <math> \Delta \mathbf{F}_p =\displaystyle{ \sum _I\frac{\partial N_I}{\partial \mathbf{x}_I^{n}}}\cdot \mathbf{x}_I^{n+1} </math>
+|-
+| Initial data on nodes: '''NONE - everything is discarded in the initialization phase'''
+|-
+| OUTPUT of calculations: <math> \Delta \mathbf{u}^{n+1}_I, \boldsymbol{\sigma }^{n+1}_p </math>
+|-
+|
+<ol>
+<li>'''INITIALIZATION PHASE''' </li>
+:* Clear nodal info and recover undeformed grid configuration
+:* Calculation of initial nodal conditions.
+::(a) for p = 1:<math display="inline"> N_p </math>
+:::* Calculation of nodal data
+::::* <math display="inline"> l_I^n = \sum _p N_I \, m_p p_p^n </math>
+::::* <math display="inline"> m_I^n = \sum _p N_I m_p </math>
+::(b) for I = 1:<math display="inline"> N_I </math>
+:::* <math display="inline"> \widetilde{p}_{I}^n = \dfrac{l_I^n}{m_I^n}</math>
-* '''Number of blades''': 3
+:* Newmark method: PREDICTOR. Evaluation of <math display="inline">^{it+1}\Delta \mathbf{u}_I^{n+1} , ^{it+1}p_I^{n+1}</math> by using Equations <math display="inline">^{it+1}\Delta \boldsymbol{u}_{I}^{n+1}= 0.0</math> and <math display="inline">^{it+1}p_I^{n+1}=\widetilde{p}_{I}^n</math>
-* '''Mean chord''': 0.45
-* '''Nominal speed''': 400 rpm
-* '''Motor nominal torque''': 180 kN <math display="inline">\cdot </math> m
-* '''Engine nominal power''': 580 kW
-* '''Nominal torque given at''': <math display="inline">t_a</math> = 25 ms
-* '''Engine starting time''': <math display="inline">t_b</math> = 50 ms
-* '''Nominal power given at''': <math display="inline">t_c</math> = 120 ms
-From basic helicopter theory, it is found that for hovering flight, a simple approximation of the power required to compensate the weight is:
+<li>'''UL-FEM PHASE''' </li>
+:* for p = 1:<math display="inline"> N_p </math>
+::(a) Evaluation of local residual (<math display="inline">rhs</math>) (RHS of Equation [[#eq-5.17|5.17]])
+::(b) Evaluation of local Jacobian matrix of residual (<math display="inline">lhs</math>) (LHS of Equation [[#eq-5.17|5.17]])
+::(c) Assemble rhs and lhs to the global vector <math display="inline">RHS</math> and global matrix <math display="inline">LHS</math> (see the matricial system of Equation [[#eq-5.17|5.17]]))
+:* Solving system <math display="inline"> (^{it+1}\delta \boldsymbol{u}_I^{n+1}, ^{it+1}\delta p_I^{n+1}) </math>
+:* Newmark method: CORRECTOR by using the Equations <math display="inline">^{it+1}\Delta \boldsymbol{u}_{I}^{n+1} =^{it}\Delta \boldsymbol{u}_{I}^{n+1} +^{it+1}\delta \boldsymbol{u}_I^{n+1}</math>(Equation [[#eq-2.13|2.13]]) and <math display="inline">^{it+1}p_I^{n+1} = ^{it+1}\delta p_I^{n+1}</math>
+:* Check convergence
+::(a) NOT converged: go to Step 2
+::(b) Converged: go to Step 3
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+<li>'''CONVECTIVE PHASE''' </li>
+:* Update the solution on the material points by means of an interpolation of nodal information by using the Equations <math display="inline">\Delta \boldsymbol{u}_{p}^{n+1} = \sum _{n = 1}^{n_n} N_I \Delta \boldsymbol{u}_{I}^{n+1}</math> (Equation [[#eq-2.14|2.14]]) and <math display="inline">p_{p}^{n+1} = \sum _I N_I p_{I}^{n+1}</math> (Equation [[#eq-5.21|5.21]])
+:* Save the stress <math display="inline"> \boldsymbol{\sigma }^{n+1}_p </math>, strain <math display="inline"> \boldsymbol{\epsilon }^{n+1}_p </math>, pressure <math display="inline">p^{n+1}_p</math> and total deformation gradient <math display="inline"> \mathbf{F}^{n+1}_p </math> on material points (the latter by&nbsp;<math display="inline">\mathbf{F}_p^{n+1}=\Delta \mathbf{F}_p \cdot \mathbf{F}_p^n</math>)
+</ol>
 |-
-|
+| style="text-align: center; font-size: 75%;"|
-{| style="text-align: left; margin:auto;width: 100%;"
+<span id='algorithm-5.1'></span>'''Algorithm. 5.1''' MPM algorithm in the framework of a mixed formulation.
+|}
+==5.4 Numerical Examples==
+In this section, two numerical examples are presented for the validation of the mixed formulation. Firstly, the well-known benchmark test of a Cook's elastic membrane is considered and a mesh convergence study is performed. The stability of the mixed formulation is assessed in a quasi-incompressible elastic case. Secondly, a plane strain tension test of a J2-plastic plate in compressible and incompressible state is analysed. In this example, the performance of the irreducible ''u'' and the mixed ''u-p'' formulations are compared in the case of incompressible plastic flow. The results obtained with the ''u'' and ''u-p'' formulations are compared and used to demonstrate that a mixed MPM formulation can provide more accurate and reliable results, not only under the assumption of elastic and plastic incompressibility, but even in compressible situations.
+In this work, a stabilization parameter (<math display="inline">\alpha </math>) with value of 1 has been used. The direct solver SuperLU is employed for the solution of the system of linearised equations, both in the case of ''u'' and ''u-p'' formulations.
+===5.4.1 Cook's membrane problem===
+As a first numerical example, we consider the well known Cook's membrane test, proposed for the first time by Cook <span id='citeF-196'></span>[[#cite-196|[196]]]. This test is often used as a benchmark to check the element formulation under compressible and incompressible conditions. In the literature, the Cook's membrane is commonly tested in infinitesimal deformation assumption and material linearity <span id='citeF-171'></span>[[#cite-171|[171]]], geometric non-linearity and material linearity <span id='citeF-197'></span>[[#cite-197|[197]]] and, finally, in geometric and material non-linearities <span id='citeF-155'></span><span id='citeF-164'></span><span id='citeF-170'></span><span id='citeF-194'></span>[[#cite-155|[155,164,170,194]]]. The geometry and material properties of the problem are shown in Figure [[#img-5.1|5.1]]. A clamped trapezoidal plate, subjected to a distributed shear load, whose resultant force is <math display="inline">P = 1N </math>, applied along the right side, is analysed. The static case is solved studying the response of a compressible and a quasi-incompressible Neo-Hookean material, whose formulation is presented in Section [[#3.1 Hyperelastic law|3.1]]. The convergence study is performed using six structured triangular meshes each of which uses an initial value of one material point per element.
+<div id='img-5.1'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
 |-
-| style="text-align: center;" | <math>\textrm{P}_{\,\hbox{lift}} = \frac{(m\,g)^{\frac{3}{2}}}{\sqrt{2\;\rho _{air}\;\pi \;R^2}} </math>
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-geometry_cook.png|480px|Cook's membrane. Geometry, material properties and boundary conditions]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding-bottom:10px;"| '''Figure 5.1:''' Cook's membrane. Geometry, material properties and boundary conditions
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7.17)
+Since the formulations under study are based on the assumption of finite deformation and material non-linearity, the results relative to a very fine mesh (256 elements per side) of a FEM analsys is considered as reference solution in the compressible case, while the result of <span id='citeF-194'></span>[[#cite-194|[194]]] is the benchmark solution for the quasi-incompressible case. The reference solution of vertical displacement at point A (Figure [[#img-5.1|5.1]]) is found to be 0.323m, in the compressible case, and 0.275m in the quasi-incompressible cases, respectively.   The results of ''u'' and ''u-p'' formulations, with and without stabilization term (UP No Stab and UP Stab) are summarized in Table [[#table-5.1|(5.1)]] for both the compressible and nearly incompressible cases. The same results can be observed graphically in Figures [[#img-5.2|5.2]] and [[#img-5.3|5.3]].
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-5.1'></span>Table. 5.1 Cook's membrane. Compressible case: vertical displacement at point A obtained with the U, UP formulation without and with stabilization
+|- style="border-top: 2px solid;font-size: 85%;"
+| colspan='1' style="border-left: 2px solid;border-right: 2px solid;border-right: 2px solid;" | Elements per side
+| colspan='3' style="border-right: 2px solid;border-left: 2px solid;" | Compressible case
+| colspan='3' style="border-right: 2px solid;border-left: 2px solid;" | Quasi-incompressible case
+|- style="border-top: 2px solid;font-size: 85%;"
+| colspan='1' style="border-left: 2px solid;border-right: 2px solid;border-right: 2px solid;" |
+| style="border-left: 2px solid;" | U
+| UP No Stab
+| style="border-right: 2px solid;" | UP Stab
+| style="border-left: 2px solid;" | U
+| UP No Stab
+| style="border-right: 2px solid;" | UP Stab
+|- style="border-top: 2px solid;font-size: 85%;"
+| colspan='1' style="border-left: 2px solid;border-right: 2px solid;border-right: 2px solid;" | 2
+| style="border-left: 2px solid;" | 0.089
+| 0.1013
+| style="border-right: 2px solid;" | 0.1172
+| style="border-left: 2px solid;" | 0.0723
+| 0.0788
+| style="border-right: 2px solid;" | 0.1277
+|- style="border-top: 2px solid;font-size: 85%;"
+| colspan='1' style="border-left: 2px solid;border-right: 2px solid;border-right: 2px solid;" | 4
+| style="border-left: 2px solid;" | 0.1415
+| 0.1718
+| style="border-right: 2px solid;" | 0.1953
+| style="border-left: 2px solid;" | 0.0736
+| 0.1157
+| style="border-right: 2px solid;" | 0.1932
+|- style="border-top: 2px solid;font-size: 85%;"
+| colspan='1' style="border-left: 2px solid;border-right: 2px solid;border-right: 2px solid;" | 8
+| style="border-left: 2px solid;" | 0.2183
+| 0.2511
+| style="border-right: 2px solid;" | 0.2669
+| style="border-left: 2px solid;" | 0.0742
+| 0.1821
+| style="border-right: 2px solid;" | 0.2424
+|- style="border-top: 2px solid;font-size: 85%;"
+| colspan='1' style="border-left: 2px solid;border-right: 2px solid;border-right: 2px solid;" | 16
+| style="border-left: 2px solid;" | 0.2771
+| 0.2952
+| style="border-right: 2px solid;" | 0.3025
+| style="border-left: 2px solid;" | 0.075
+| 0.2356
+| style="border-right: 2px solid;" | 0.2648
+|- style="border-top: 2px solid;font-size: 85%;"
+| colspan='1' style="border-left: 2px solid;border-right: 2px solid;border-right: 2px solid;" | 32
+| style="border-left: 2px solid;" | 0.30386
+| 0.3119
+| style="border-right: 2px solid;" | 0.315
+| style="border-left: 2px solid;" | 0.0775
+| 0.2606
+| style="border-right: 2px solid;" | 0.2725
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| colspan='1' style="border-left: 2px solid;border-right: 2px solid;border-right: 2px solid;" | 64
+| style="border-left: 2px solid;" | 0.3133
+| 0.3176
+| style="border-right: 2px solid;" | 0.319
+| style="border-left: 2px solid;" | 0.0862
+| 0.2702
+| style="border-right: 2px solid;" | 0.275
 |}
-Hence, considering an helicopter with a mass <math display="inline">m = 2800 \;Kg</math>, the power required to ensure hovering conditions at sea level is <math display="inline">\textrm{P}_{\,\hbox{lift}} =\; 328.18 \,kW</math>. Subtracting this value to the nominal power, it is found that the '''available power''' to counteract the elastic reaction torque is around <math display="inline">84\; kW</math> per blade, which is higher than the parasite power induced by aerodynamic drag obtained in equation [[#eq-7.16|7.16]].
-To be conservative, it will be supposed that the engine is not operating at nominal power but at its 60% (<math display="inline">50 \; kW </math> per blade). Regarding the electric motor, it will be considered that at the beginning of the rotation motion the blades are undeformed, and hence all the torque — 60 kN <math display="inline">\cdot </math> m per blade — is used to '''accelerate''' the rotor. For sure, this development is an oversimplification of the real scenario, but still is good enough to provide a glimpse on how the blade dynamically behaves.
+The ''u'' formulation is less accurate than the ''u-p'' formulation both for the UP No Stab and UP Stab cases, not only for the nearly incompressible condition, as expected, but also for the compressible one. However, the discrepancy is clearly visible in the quasi-incompressible problem (Figure [[#img-5.3|5.3]]), where the capability of the ''u'' formulation to predict the displacement field is compromised due to volumetric locking.
-As the numerical integration requires from very small timesteps (<math display="inline">\Delta \textrm{t} = 5 \mu \,s</math>), the timespan is limited to <math display="inline">5</math> s. To reduce computation time, stresses will only be computed every 20 timesteps. Dynamic displacements will also be saved at the same rate to reduce memory consumption. The integration is going to be performed using the constant average acceleration scheme. The considered structural Rayleigh damping is given by <math display="inline">\overline{\alpha } = 0.25</math> and <math display="inline">\overline{\beta } = 5\,\cdot \,10^{-6}</math>. With these inputs, the simulation takes up to three hours to complete.
+<div id='img-5.2'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-vertical_disp_0_33_new2.png|420px|Cook's membrane. Compressible case: vertical displacement at point A]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding-bottom:10px;"| '''Figure 5.2:''' Cook's membrane. Compressible case: vertical displacement at point A
+|}
-The full results of the simulations are presented in the appendix (section [[#D.3 Starting of a rotor blade|D.3]]). The temporal evolution of the presented variables is '''closely linked''' with the way torque is injected into the blade, as angular accelerations are proportional to it. Different input torque will result in a completely different dynamic behaviour of the blade. For the studied case, torque and power are displayed in the following figure: <div id='img-7.3'></div>
+<div id='img-5.3'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-torqpow.png|510px|Injected power and torque as a function of time.]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-vertical_disp_0_4995_new.png|420px|Cook's membrane. Quasi-incompressible case: vertical displacement at point A ]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 7.3:''' Injected power and torque as a function of time.
+| colspan="1" style="padding-bottom:10px;"| '''Figure 5.3:''' Cook's membrane. Quasi-incompressible case: vertical displacement at point A
 |}
-Considering displacements at the tip of the blade as representative from the vibration motion, it is found that the blade is not undergoing continuous oscillation around and equilibrium configuration. Although it is true that oscillations are governing at the very beginning, displacements in <math display="inline">\varrho </math> tend to reach a non vibrating '''steady state'''. This effect was not sighted in previous simulations, where vibrations persisted in time. In all likelihood, the reduction of high frequency vibrations is induced by the structural damping acting on the blade. In fact, the SVD has been able to recover predominant '''damped oscillations''' — see RSVs 3 and 4 in figure [[#img-D.37|D.37]] — that would otherwise be impossible to spot.
+Regarding the mixed approaches, from Figure [[#img-5.3|5.3]] it is possible to infer that even not using a stabilization term the solution is not affected by volumetric locking. However, through the stabilized ''u-p'' formulation it is also possible to prevent pressure oscillation issues in the mean stress field, as can be observed in Figure [[#img-5.4|5.4]], where the pressure values of Figure [[#img-5.4a|5.4a]] are all out of the threshold defined by the solution of Figure [[#img-5.4b|5.4b]].
-Much the same can be said about local axial displacements, but this time their tendency is not towards equilibrium but into a raise in time. This same behaviour has been spotted in earlier simulations, as the centrifugal force inducing axial displacements '''increases''' with <math display="inline">\Omega ^2</math>. Both displacements in <math display="inline">\varsigma </math> and angular speed are compared in figure [[#img-D.34|D.34]].
+<div id='img-5.4a'></div>
+<div id='img-5.4b'></div>
+<div id='img-5.4'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-CookPressureUPNoStab0_5.png|310px|''u-p'' without stabilization]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-CookPressureUPStab0_5.png|310px|''u-p'' with stabilization]]
+|- style="text-align: center; font-size: 75%;"
+| (a) ''u-p'' without stabilization
+| (b) ''u-p'' with stabilization
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 5.4:''' Cook's membrane. Quasi-incompressible case: Pressure counter fill. The mixed formulation without any stabilization (a) fails to predict the pressure field, while it is correctly evaluated using the PPP stabilization (b). Black contour colour should be intended as out of range.
+|}
-Regarding the forces acting on the blade, there is a considerable ''peak'' at the beginning of the rotation motion, when the motor is injecting torque to the standstill rotor. During these first instants, all forces, accelerations and even stresses undergo oscillations. As time passes by and the engine achieves its nominal conditions, both reaction forces and accelerations '''stabilise''' into a considerably lower value. The only exception has to do with the torque induced by aerodynamic drag, that increases with the angular speed. In figure [[#img-D.35|D.35]], angular acceleration is graphed along with the reaction torque, showing how acceleration increases when reactions decrease. <div id='img-7.4'></div>
+===5.4.2 2D tension test===
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+As second numerical example, a plane strain tension problem is considered to test the mixed formulation in an elasto-plastic regime. A 2D plate, clamped at the bottom of the specimen, is subjected to a prescribed vertical displacement on the upper side. Both geometry and material properties are taken from <span id='citeF-173'></span>[[#cite-173|[173]]] and are depicted in Figure [[#img-5.5|5.5]]. The plate is made by a hyperelastic perfectly-plastic material which is simulated using a J2 plastic law, whose formulation is presented in Section [[#lb-3.2|3.2]]. An unstructured triangular background mesh with a mesh size of 0.001m and  an initial distribution of 12 material points per cell, which is found to give the optimal trade-off between accuracy of the results and computational cost in both the compressible and incompressible cases, are adopted.
+<div id='img-5.5'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-stressblade.png|540px|Deformed configuration (×1000) and σ<sub>ςς</sub> (KPa) of the blade for the moment of maximum stress. ]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_geometry.png|330px|Tension test. Geometry, material properties and boundary conditions]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 7.4:''' Deformed configuration (<math>\times 1000</math>) and <math>\sigma _{\varsigma \varsigma }</math> (KPa) of the blade for the moment of maximum stress.
+| colspan="1" style="padding:10px;"| '''Figure 5.5:''' Tension test. Geometry, material properties and boundary conditions
 |}
-When reviewing maximum stresses, it is found that the peaks correspond to the maximum deformed configuration. Again, this value reaches a steady condition as time passes by. What it has been found when carefully checking the results is that, after a few seconds, stresses tend to raise again. This effect is caused by the increase of aerodynamic drag, that induces forces proportional to <math display="inline">\Omega ^2</math>. This same condition is found in transversal displacements, hinting that there is a certain value of angular speed at which stresses and deformations are '''minimum'''. This effect can be spotted in the following figure, where these minimums are represented with a red cross:
+The results of the compressible case are shown in Figures [[#img-5.6|5.6]], [[#img-5.7|5.7]], [[#img-5.8|5.8]] and [[#img-5.9|5.9]], where the displacement along x and y-direction, the equivalent plastic strains and the vertical Cauchy stresses are shown. Volumetric locking is not affecting the numerical results, as the plate is working under compressible conditions. However, the ''u-p'' formulation is more accurate than the ''u'' one, not only in the evaluation of the stress field, but also of the displacement field. Moreover, the goodness of the solution can be appreciated looking at Figure [[#img-5.8b|5.8b]]: the equivalent plastic strains are distinctly distributed along a cross shape, while the result of Figure [[#img-5.8a|5.8a]] revokes the same shape, but without the same order of precision. In conclusion, even if a compressible material is simulated, the results obtained with the ''u-p'' formulation present a higher order of accuracy, by using the same mesh size and the same number of material points per element.
-<div id='img-7.5'></div>
+The results of the incompressible case are shown in Figures [[#img-5.10|5.10]], [[#img-5.11|5.11]], [[#img-5.12|5.12]] and [[#img-5.13|5.13]]. In this case, the ''u'' formulation fails in the simulation of the tension test. As expected, the displacement and stress fields are affected by volumetric locking and the plastic deformations are incorrectly localized. On the other hand, Figures [[#img-5.10b|5.10b]], [[#img-5.11b|5.11b]], [[#img-5.12b|5.12b]] and [[#img-5.13b|5.13b]] show that the ''u-p'' formulation is able to evaluate correctly the displacement and stress field under incompressible conditions. The results are similar to those depicted in Figures [[#img-5.6b|5.6b]], [[#img-5.7b|5.7b]], [[#img-5.8b|5.8b]] and [[#img-5.9b|5.9b]]: the cross-shape distribution of the equivalent plastic strains and stresses are recovered. Furthermore, Figures [[#img-5.14a|5.14a]], [[#img-5.14b|5.14b]], [[#img-5.14c|5.14c]] and [[#img-5.14d|5.14d]] show a comparison in the nearly-incompressible case between the reference solution obtained with the formulation proposed in <span id='citeF-173'></span>[[#cite-173|[173]]] and the results obtained with the MPM ''u-p'' formulation presented in the current work. We can observe that there is a good agreement both in the distribution of equivalent plastic strains and pressure fields and in their values range.  Finally, the stress - displacement curve, evaluated with the mixed formulation, is shown in Figure [[#img-5.15|5.15]]. The results for the compressible and incompressible cases are in good agreement. Both correctly predict the elastic regime and the inception of the plastic flow when the yield stress is reached.
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+Since a mixed formulation with displacement and pressure as primary variables is adopted, strains are not linearly distributed within the element, but these coincide with a constant function. It is worth highlighting that through this numerical procedure while it is possible to avoid the volumetric locking, the problems related with strain localization are still present. This means that the width of the shear bands still depends on the size of the elements.  This problem can be solved by regularization of the element size as proposed, e.g. in <span id='citeF-198'></span><span id='citeF-174'></span><span id='citeF-175'></span>[[#cite-198|[198,174,175]]] or <span id='citeF-156'></span>[[#cite-156|[156]]], where the formulations consider the strain field as primary variable and, therefore, its linear distribution can be evaluated, which allows to accurately predict strain localization with mesh independence.
+<div id='img-5.6a'></div>
+<div id='img-5.6b'></div>
+<div id='img-5.6'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-stressandy.png|468px|Temporal evolution of ϱ displacements at the tip of the beam and maximum σ<sub>ςς</sub>. ]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_u_compr_dispx.png|282px|''u'' formulation]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_up_compr_dispx.png|282px|''u-p'' formulation]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 7.5:''' Temporal evolution of <math>\varrho </math> displacements at the tip of the beam and maximum <math>\sigma _{\varsigma \varsigma }</math>.
+| (a) ''u'' formulation
+| (b) ''u-p'' formulation
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 5.6:''' Tension test. Compressible case: horizontal displacement
 |}
-It is worth pointing that in the simulated timespan, the angular velocity of the rotor has not reached its nominal value, but further timespans are far too restrictive in terms of computation time. The rotor would eventually surpass the nominal speed, as no '''control mechanism''' has been considered. To overcome this issue, injected power needs to be reduced when approaching the desired conditions.
+<div id='img-5.7a'></div>
+<div id='img-5.7b'></div>
+<div id='img-5.7'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_u_compr_dispy.png|282px|''u'' formulation]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_up_compr_dispy.png|282px|''u-p'' formulation]]
+|- style="text-align: center; font-size: 75%;"
+| (a) ''u'' formulation
+| (b) ''u-p'' formulation
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 5.7:''' Tension test. Compressible case: vertical displacement
+|}
-=8 Closure=
+<div id='img-5.8a'></div>
+<div id='img-5.8b'></div>
+<div id='img-5.8'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_u_compr_eq_strain.png|282px|''u'' formulation]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_up_compr_eq_strain.png|282px|''u-p'' formulation]]
+|- style="text-align: center; font-size: 75%;"
+| (a) ''u'' formulation
+| (b) ''u-p'' formulation
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 5.8:''' Tension test. Compressible case: equivalent plastic strain
+|}
-Rotordynamics has been a field of great interest for centuries, being credited for the huge improvement in rotating machinery since the industrial revolution. Modern rotating components such as turbines, rotor blades and engines are still being designed using classical models developed centuries ago. What really created a game-changing shift in rotordynamics was the emergence of '''numerical methods''' in the earlier sixties. Implementation of the FEM gave rise to a completely new approach for structural analysis, allowing complex geometries to be solved in very short amounts of time.
+<div id='img-5.9a'></div>
+<div id='img-5.9b'></div>
+<div id='img-5.9'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_u_compr_stressy2.png|282px|''u'' formulation]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_up_compr_stressy.png|282px|''u-p'' formulation]]
+|- style="text-align: center; font-size: 75%;"
+| (a) ''u'' formulation
+| (b) ''u-p'' formulation
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 5.9:''' Tension test. Compressible case: Cauchy stress along loading axis.  Black contour colour should be intended as out of range.
+|}
-Nowadays, the use of numerical software in structural analysis is a quite common solution. Despite of the great improvement in numerical capabilities over the last years, the FEM still requires from great processing power to perform complex simulations. Hence, it is not surprising that the actual tendency goes in the direction of more efficient methods instead of more accurate models. In this context, the present report has reviewed some of the available '''numerical tools''' to tackle the equations describing the vibration behaviour of a rotatory blade.
+<div id='img-5.10a'></div>
+<div id='img-5.10b'></div>
+<div id='img-5.10'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_u_incompr_dispx.png|282px|''u'' formulation]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_up_incompr_dispx.png|282px|''u-p'' formulation]]
+|- style="text-align: center; font-size: 75%;"
+| (a) ''u'' formulation
+| (b) ''u-p'' formulation
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 5.10:''' Tension test. Incompressible case: horizontal displacement
+|}
-This thesis started with the elastic modelling of a simplified rotor blade undergoing one great rotation. The model supposes small strains and neglects the gyroscopic effect. When the obtained equations are posed in terms of the FEM, a '''softening effect''' is unfolded. The latter effect causes natural frequencies to decrease with rotation speed. However, real rotors do not exhibit this behaviour, but instead are affected by an opposite stiffening effect. Hence, the simplified model is not capable to capture the complete vibration phenomena, but even so it is a great tool that offers a glimpse on how rotordynamics solvers work.
+<div id='img-5.11a'></div>
+<div id='img-5.11b'></div>
+<div id='img-5.11'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_u_compr_dispy.png|282px|''u'' formulation]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_up_compr_dispy.png|282px|''u-p'' formulation]]
+|- style="text-align: center; font-size: 75%;"
+| (a) ''u'' formulation
+| (b) ''u-p'' formulation
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 5.11:''' Tension test. Incompressible case: vertical displacement
+|}
-For the static case, the developed FEM tool has proven to provide '''correct results''', as it has been validated with analytic expressions.  Regarding the dynamic scenario, the present work has focused on the numeric methods itself rather than on the description of the physical model. The goal of this thesis is not to bring to light new discoveries in the field of rotordynamics, but to give a glimpse on how numeric schemes work and in which way could they be more performing. The problem with structural analysis is that the governing equations rapidly become '''nonlinear''', being numerical integration the only possible path to handle them.
+<div id='img-5.12a'></div>
+<div id='img-5.12b'></div>
+<div id='img-5.12'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_u_compr_eq_strain.png|282px|''u'' formulation]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_up_compr_eq_strain.png|282px|''u-p'' formulation]]
+|- style="text-align: center; font-size: 75%;"
+| (a) ''u'' formulation
+| (b) ''u-p'' formulation
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 5.12:''' Tension test. Incompressible case: equivalent plastic strain
+|}
-For that purpose, a numerical scheme based on the Newmark-<math display="inline">\beta </math> method has been developed to study the dynamic response of rotating structures. It has been found that the rotation condition increases considerably the computational costs of the simulations, as matrices linked with rotating effects need to be computed and assembled at each timestep. In order to reduce simulation times, a more '''efficient assembly''' method has been used instead of the traditional approach, leading to an increase in performance by a factor of 15.
+<div id='img-5.13a'></div>
+<div id='img-5.13b'></div>
+<div id='img-5.13'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_u_incompr_stressy2.png|282px|''u'' formulation]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-tension_test_up_incompr_stressy.png|282px|''u-p'' formulation]]
+|- style="text-align: center; font-size: 75%;"
+| (a) ''u'' formulation
+| (b) ''u-p'' formulation
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 5.13:''' Tension test. Incompressible case: Cauchy stress along loading axis.  Black contour colour should be intended as out of range.
+|}
-On the opposite side, when the physics describing the dynamic behaviour of a system are simple, '''modal decomposition analysis''' can be used to study its vibration behaviour. Modal analysis is a widespread tool that unfolds relevant information about the system with low computation costs. In structural analysis, modal decomposition is used to recover natural frequencies and mode shapes. The biggest drawback of this method is its inherent limitation to work only with simple dynamic systems.
+<div id='img-5.14a'></div>
+<div id='img-5.14b'></div>
+<div id='img-5.14c'></div>
+<div id='img-5.14d'></div>
+<div id='img-5.14'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 85%;"
+|-
+|style="padding-top:10px;"|[[Image:Draft_Samper_987121664-monograph-comparison_tension_test_cervera--0_5cm-9_0cm-18cm-0_0cm.png|252px|Equivalent plastic strain]]
+|style="padding-top:10px;"|[[Image:Draft_Samper_987121664-monograph-comparison_tension_test_cervera-17_0cm-9_0cm-0cm-0_0cm.png|282px|Pressure]]
+|- style="text-align: center; font-size: 75%;"
+| (a) Equivalent plastic strain
+| (b) Pressure
+|-
+|style="padding-bottom:10px;"|[[Image:Draft_Samper_987121664-monograph-comparison_tension_test_cervera--0_5cm-0_0cm-18cm-9_0cm.png|252px|Equivalent plastic strain]]
+|style="padding-bottom:10px;"|[[Image:Draft_Samper_987121664-monograph-comparison_tension_test_cervera-17_0cm-0_0cm-0cm-9_0cm.png|282px|Pressure]]
+|- style="text-align: center; font-size: 75%;"
+| (c) Equivalent plastic strain
+| (d) Pressure
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 5.14:''' Tension test. Incompressible case: results evaluated at a total imposed vertical displacement of 0.0001m. a) and b) Results in terms of equivalent plastic strain and pressure using a T1/P1 ''u-p'' formulation, taken from <span id='citeF-173'></span>[[#cite-173|[173]]]. c) and d) Results in terms of equivalent plastic strain and pressure evaluated with the MPM ''u-p'' formulation presented in the current study.
+|}
-Being aware of this situation, the truncated SVD has been found to be an alternative method that combines both numerical integration and modal decomposition to recover predominant patterns of the dynamic behaviour of the structure. If applied to structural analysis, it has been proven to recover predominant modes and frequencies of the system with '''great accuracy'''. The studied cases implied 2D and 3D rotating cantilever beams, although it could be extended to any arbitrary geometry. In order to recover oscillation frequencies from predominant patters, the developed method uses the '''discrete Fourier transform''' to catch the higher components in the frequency domain.  One has, however, to keep in mind that the SVD is a probabilistic method, and thus gains reliability as the size of the sample increases.
+<div id='img-5.15'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-stress_disp.png|360px|Tension test. Stress-Displacement curve. Comparison between the compressible case (red curve) and the incompressible curve (green curve).]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 5.15:''' Tension test. Stress-Displacement curve. Comparison between the compressible case (red curve) and the incompressible curve (green curve).
+|}
-The reader could come to think that the SVD entails no real advantage, as the input data matrix is still computed using numeric integration. Nothing further from the truth: the proposed method allows a '''deeper understanding''' of the vibration phenomena as it breaks up the whole problem into simpler components, easier to interpret and to postprocess. The postprocessing becomes thereby much simpler and faster, as the complete behaviour of the structure can be decomposed into a linear combination of '''predominant''' modes and frequencies. Hence, the problem numerical integration has about encountering multiple time and space scales in the postprocessing step is overcome. This make results analysis an easier and quicker task for the engineer, reducing both the time and computation resources consumed.
+==5.5 Discussion==
-When testing the SVD in rotating beams, it has been found that the method is able to keep track of the softening effect predicted by the model. This capacity of capturing the reduction in vibration frequencies as angular velocity increases would be unthinkable through traditional approaches.  Moreover, it has been spotted that the dynamic displacements can be approximated using only a few singular values, allowing an extensive reduction in postprocessing requirements.
+In this Chapter, a stabilized mixed  ''u-p'' formulation is presented in its strong and weak forms. The formulation is implemented within the framework of the implicit MPM strategy, able to solve problems which involve large displacements and large deformations. The irreducible ''u'' and mixed ''u-p'' formulations are tested and compared through a series of benchmark examples. Firstly, the Cook's membrane problem, a bending dominated test, is investigated. Two cases, a compressible and a nearly-incompressible one, are solved through the ''u'' and ''u-p'' formulations. From the results, it is demonstrated that the ''u-p'' formulation always gives the best performance in term of convergence. In the quasi-incompressible case, the volumetric locking issue is overcome and pressure oscillations are avoided if a stabilization term is added to the mixed finite element formulation. In the second example, a J2 plastic plate, subjected to uniform tension on one side and fixed to the other side, is under study and both formulations are tested under an isochoric plastic flow condition. By comparing the displacement-based and mixed formulation it is shown that, even in this case, better results are obtained through the ''u-p'' procedure. Indeed, a higher definition of displacement, equivalent plastic strains and vertical Cauchy stress fields is observed. Despite volumetric locking issue is fixed in the case of the ''u-p'' formulation, further problems, such as, mesh independence and strain localization, are not addressed in the current work and they would represent interesting topics for a future research.
-In order to locate the developed work in the context of '''aerospace''' engineering, the built software has been adapted to work together with a rigid body model. This has allowed the transient study of the starting of an helicopter's rotor, giving plausible results and proving the actual capabilities of the method. When thinking about the potential capabilities of the SVD in structural analysis, it is too soon to reach a conclusion. The work is still in a very '''early state''', and much investigation needs to be carried in this subject. This report has only accounted for a very simple rotor model, so nothing can be said about the validity of the results as there is no way they could be compared with empirical data. Until the proposed method is further developed, the present thesis will remain as a theoretical but plausible approach for real rotordynamics studies.
+In conclusion, it is demonstrated that the ''u-p'' formulation can evaluate more accurate results in terms of displacement and stress fields, not only under near-incompressible state, avoiding the typical drawback of volumetric locking, but even under compressible conditions.
-=A Theoretical frame=
+=6 Validation=
-==A.1 Introduction to elasticity==
+In Chapter [[#2 Particle Methods|2]] the Material Point Method and its algorithm have been presented. Under the assumption of finite strains an irreducible (see Chapter [[#4 Irreducible formulation|4]]) and a <math display="inline">\boldsymbol{u}-p</math> mixed formulation (see Chapter [[#5 Mixed formulation|5]]) are described and verified by using the constitutive laws, whose algorithms are shown in detail in Chapter [[#3 Constitutive Models|3]].  In the current Chapter, the MPM numerical strategy, implemented within the ''Kratos Multiphysics'' framework, is employed for the solution of typical problems concerning granular flows, involving large displacement and large deformation of the continuum under study.
-===A.1.1 The elastic problem===
+Firstly, the typical granular column collapse is considered. For the validation of such a case, the comprehensive experimental work of Lube and co-workers <span id='citeF-199'></span>[[#cite-199|[199]]] is used as a reference. In the second part of the current Chapter, a second example is taken into account: the rigid strip footing test, a typical test in geomechanics for the assessment of the bearing capacity of a soil to an imposed displacement or force. In the examples considered, experimental results will be used for validation, otherwise, solutions of other studies, available in the literature, will be used as a reference.
-Fundamentals of Theory of Elasticity date from the seventeenth century, back when Robert Hooke posed the law of elasticity, which states that the stretching of a solid is proportional to the force applied to it.  This law was latter generalised to three dimensional bodies by Cauchy, introducing six components of stress <math display="inline">\sigma </math>, linearly related to six components of strain <math display="inline">\varepsilon </math>. Cauchy was the first to introduce the notion of '''stress''' at a point, given by the tractions per unit area across all plane elements trough the point.
-This linear behaviour of deformable bodies is governed by 15 coupled '''partial differential equations''', first derived by Navier. Even today, no closed form of the solution exists, even for simple structures. Elasticians have used energy principles to tackle this problem, which have become a fundamental tool to study elastic bodies, including those problems involving vibrations (see [[#A.2 Classical analysis methods|A.2]]). The Navier-Cauchy equations are <span id='citeF-1'></span>[[#cite-1|[1]]]:
+In this section, the granular collapse on a horizontal plane is considered as a test case for validation. This test has been chosen because, despite the apparent simplicity of the experiment, the description and prediction of the collapse is still a challenge from an experimental, numerical and theoretical point of view <span id='citeF-200'></span>[[#cite-200|[200]]]. It is a perfect example to test the numerical technique in case of large displacement and large strain. Indeed, inertial granular flows are characterized by unsteady motion, a large variation of the free surface with time and propagation toward the free surface of internal interface separating the static and flowing regions. During the last decades, this test has been object of numerical study of many research groups by using techniques based either on discrete or continuum mechanics. Regarding the use of DEM, Staron and Hinch <span id='citeF-201'></span>[[#cite-201|[201]]] showed that their results have good agreement with the experimental results in terms of run-out distance, but they did not provide a physical argument able to explain the relation between the initial aspect ratio and the final run-out. Moreover, they focused on the final deposition profiles without paying attention to the influence of material properties and the collapse mechanism. In Lacaze et al. <span id='citeF-202'></span>[[#cite-202|[202]]], DEM simulations are performed providing good results; they focused on both flow behaviour and final deposition of the collapse. In Kumar <span id='citeF-203'></span>[[#cite-203|[203]]] DEM simulations are carried out with an analysis on the initial grain properties, which is demonstrated that can influence the structure of internal flow and the kinematic of failure mechanism. The use of DEM, in the context of micro-mechanical analysis, is really helpful in providing some insights. However, when extending to upper scales, the method suffers from extremely demanding computational cost, which is detrimental for its usage to practical application in the engineering and industrial framework. Due to this aspect, a high interest of the computational community is sparked in solving granular flow problems with techniques based on continuum mechanics, which have the advantage to reduce tremendously the computational cost.   Some attempts have been carried out by using an ALE FEM <span id='citeF-204'></span>[[#cite-204|[204]]] or SPH <span id='citeF-205'></span><span id='citeF-206'></span>[[#cite-205|[205,206]]]. However, as pointed out in Chapter [[#2 Particle Methods|2]], these methods suffer from some issues which do not make them particularly suitable for the modelling of granular flows. The granular column collapse has been also modelled with the MPM <span id='citeF-118'></span><span id='citeF-207'></span>[[#cite-118|[118,207]]]. In <span id='citeF-118'></span>[[#cite-118|[118]]] a validation is performed by employing a Mohr-Coulomb plastic law, while in <span id='citeF-207'></span>[[#cite-207|[207]]] an analysis of different constitutive laws implemented in a MPM code is carried out and interesting insights are provided on the choice of a suitable constitutive model regarding the modelling of granular material flows.
-''' Equilibrium equations :'''
+In this section, the results of the comprehensive experimental work of Lube and co-worker <span id='citeF-199'></span>[[#cite-199|[199]]] are used as a reference for the validation of the MPM code implemented in the ''Kratos Multiphysics'' framework. In their works, the authors observed that the final run-out, the final maximum height and the corresponding time, indicated with <math display="inline"> d_{\infty } </math>, <math display="inline"> h_{\infty } </math>   and <math display="inline"> t_{\infty } </math>, respectively, to be consistent with the reference works, are found to be independent on the different grains and roughness of the lower boundary and only the initial geometry, the initial aspect ratio, could affect those results.  The experimental test consists in a granular column inside a channel, wide enough in order to avoid the wall influence, at one side sustained by a fixed wall and on the other side by a moving wall. At the beginning, the column is at rest and the experiment starts when the moving wall is removed and the granular material is free to collapse.
-<span id="eq-A.1"></span>
+In this validation work, granular columns of different aspect ratios, <math display="inline"> a = \frac{h_i}{d_i}= 1.2, 3, 5, 7</math>, are considered. It is well known that the first failure surface, which generates after the opening of the moving wall, is very similar in all the sample independently of the initial geometry <span id='citeF-199'></span><span id='citeF-207'></span>[[#cite-199|[199,207]]]. Hence, the amount of mass which starts moving from the static zone increases with the initial aspect ratios and, consequently, different flow regimes might take place depending on the initial geometry of the column. For taller aspect ratios (<math display="inline">a > 2.8</math>), the flow regime is mainly dominated by the inertia (''Regime I''); in the case of low aspect ratios, the flow behaviour is more dominated by the friction and energy dissipation (''Regime II'') which takes place at the bottom layer, at the interface between static and dynamic zone and in the moving mass, as well. With regards to the kinematic of the granular column collapse, it has been experimentally observed that in ''Regime I'' three transient stages can be distinguished: a constant acceleration phase at <math display="inline">0.75g</math>, a constant velocity and a final deceleration stage. The duration of the second stage decreases with <math display="inline">a</math> and for aspect ratios lower than 1.5 does not appear. Thus, in ''Regime II'' only two stages of initial acceleration and final deceleration take place.
+In this test case, the Mohr-Coulomb plastic law, presented in Section [[#3.3 Hyperelastic - Mohr-Coulomb plastic law|3.3]], is employed. It is well known that one of the limitations of this constitutive model lies on the inability to express the dissipation due to friction between grains during the transition from static to flowing regime and vice-versa. As done in <span id='citeF-118'></span>[[#cite-118|[118]]], also in this work of validation a numerical dissipation is added to the matricial formulation to be solved, presented in Chapter [[#4 Irreducible formulation|4]]. It is found that the Rayleigh damping alpha coefficient with a value of 1.5 can replicate with good accuracy the reference solution, as shown in the following paragraphs.
+The power-laws deduced in the experimental study <span id='citeF-199'></span>[[#cite-199|[199]]] are used for the prediction of <math display="inline"> d_{\infty } </math>, <math display="inline"> h_{\infty } </math> and <math display="inline"> t_{\infty } </math> as
+<span id="eq-6.1"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,296: / Line 5,086: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{align}{\frac{\partial \sigma _{x x}}{\partial x}+\frac{\partial \sigma _{y x}}{\partial y}+\frac{\partial \sigma _{z x}}{\partial z}+b_{x}=0} \\ \\  {\frac{\partial \sigma _{x y}}{\partial x}+\frac{\partial \sigma _{y y}}{\partial y}+\frac{\partial \sigma _{z y}}{\partial z}+b_{y}=0} \\ \\  {\frac{\partial \sigma _{x z}}{\partial x}+\frac{\partial \sigma _{y z}}{\partial y}+\frac{\partial \sigma _{z z}}{\partial z}+b_{z}=0}  \end{align} </math>
+| style="text-align: center;" | <math>d_{\infty } = \begin{cases}d_i(1+1.6 a), & \mbox{ if } a < \mbox{1.8} \\ d_i(1+2.2 a^{\frac{2}{3}}), & \mbox{ if } a > \mbox{2.8} \end{cases}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.1)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (6.1)
 |}
-'''Strain displacement relations: '''
+<span id="eq-6.2"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,308: / Line 5,097: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{align}\varepsilon _{x x} &=\frac{\partial u}{\partial x}, \quad \varepsilon _{y z}=\frac{1}{2}\left(\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}\right)=\varepsilon _{z y} \\ \\ \varepsilon _{y y} &=\frac{\partial v}{\partial y}, \quad \varepsilon _{z x}=\frac{1}{2}\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\right)=\varepsilon _{x z} \\ \\  \varepsilon _{z z} &=\frac{\partial w}{\partial z}, \quad \varepsilon _{x y}=\frac{1}{2}\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)=\varepsilon _{y x} \end{align} </math>
+| style="text-align: center;" | <math>h_{\infty } = \begin{cases}h_i, & \mbox{ if } a \leq \mbox{ 1.15} \\ d_i a^{\frac{2}{5}}, & \mbox{ if } a > \mbox{1.15} \end{cases}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.2)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (6.2)
 |}
-==Compatibility relations : ==
+<span id="eq-6.3"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,320: / Line 5,108: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{align}\frac{\partial ^{2} \varepsilon _{x x}}{\partial y^{2}}+\frac{\partial ^{2} \varepsilon _{y y}}{\partial x^{2}}=2 \frac{\partial ^{2} \varepsilon _{x y}}{\partial x \partial y}, \quad \frac{\partial ^{2} \varepsilon _{x x}}{\partial y \partial z}=\frac{\partial }{\partial x}\left(-\frac{\partial \varepsilon _{y z}}{\partial x}+\frac{\partial \varepsilon _{z x}}{\partial y}+\frac{\partial \varepsilon _{x y}}{\partial z}\right)\\ \\  \frac{\partial ^{2} \varepsilon _{y y}}{\partial z^{2}}+\frac{\partial ^{2} \varepsilon _{z z}}{\partial y^{2}}=2 \frac{\partial ^{2} \varepsilon _{y z}}{\partial y \partial z}, \quad \frac{\partial ^{2} \varepsilon _{y y}}{\partial z \partial x}=\frac{\partial }{\partial y}\left(\frac{\partial \varepsilon _{y z}}{\partial x}-\frac{\partial \varepsilon _{z x}}{\partial y}+\frac{\partial \varepsilon _{x y}}{\partial z}\right)\\ \\  \frac{\partial ^{2} \varepsilon _{z z}}{\partial x^{2}}+\frac{\partial ^{2} \varepsilon _{x x}}{\partial z^{2}}=2 \frac{\partial ^{2} \varepsilon _{z x}}{\partial z \partial x}, \quad \frac{\partial ^{2} \varepsilon _{z z}}{\partial x \partial y}=\frac{\partial }{\partial z}\left(\frac{\partial \varepsilon _{y z}}{\partial x}+\frac{\partial \varepsilon _{z x}}{\partial y}-\frac{\partial \varepsilon _{x y}}{\partial z}\right) \end{align} </math>
+| style="text-align: center;" | <math>t_{\infty } = \begin{cases}c \sqrt{\frac{h}{g}} , & \mbox{ if } a \leq \mbox{ 1.15} \\ 3.3 \sqrt{\frac{h}{g}}, & \mbox{ if } a > \mbox{1.15} \end{cases}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.3)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (6.3)
 |}
-Equilibrium equations are obtained considering a general state of stress at an arbitrary point in the deformable body through an infinitesimal approach (see figure below). Note that shear stresses are represented with a <math display="inline">\tau </math>. If one approximates the spatial variations of the stresses in terms of first-order Taylor series, the resulting equilibrium equations are the ones posed in equation [[#eq-A.1|A.1]]. The strain-displacement equations link both strain and displacement field, defining the second as the derivative of the first. However, this is not enough, as another condition needs to be met: the body is still '''continuous''' after the deformation. The latter is ensured by the compatibility relations, which involve second derivatives.
+In this validation study three different mesh discretisations are employed with a cell size of <math display="inline">0.0065m, 0.005m</math> and <math display="inline">0.0025m</math>, defined as Mesh 1, Mesh 2, Mesh 3, respectively, and in what follows, the effect of mesh refinement on the kinematic of granular column collapse is analysed. A number of initial material points per element is set to 9 in all the analyses and the material properties are listed in Table table:  granular column material properties.
-<div id='img-A.1'></div>
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+|+ style="font-size: 75%;" |<span id='table-6.1'></span>Table 6.1. Granular column collapse: material properties
+|- style="border-top: 1px solid;font-size: 85%;"
+|  style="border: 2px solid;" | Young Modulus
+| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | Poisson ratio
+| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | Density
+| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | Internal friction angle
+| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | Dilatancy angle
+| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | Cohesion
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.84e6 Pa
+| style="border-left: 2px solid;border-right: 2px solid;" | 0.3
+| style="border-left: 2px solid;border-right: 2px solid;" | 2600 kg/mc
+| style="border-left: 2px solid;border-right: 2px solid;" | 31 deg
+| style="border-left: 2px solid;border-right: 2px solid;" | 1 deg
+| style="border-left: 2px solid;border-right: 2px solid;" | 0 Pa
+|}
+The numerical results are summarized in Tables [[#table-6.2|6.2]], [[#6.1 Granular collapse on a horizontal plane|6.1]] and [[#6.1 Granular collapse on a horizontal plane|6.1]], and the results, evaluated according to the power laws of Equations [[#eq-6.1|6.1]], [[#eq-6.2|6.2]] and [[#eq-6.3|6.3]], in Table [[#6.1 Granular collapse on a horizontal plane|6.1]]. It is found that, in general, by using more fine mesh the values of <math display="inline"> d_{\infty } </math>, <math display="inline"> h_{\infty } </math> and <math display="inline"> t_{\infty } </math> approach the empirical values of Table [[#6.1 Granular collapse on a horizontal plane|6.1]].
+{| class="floating_tableSCP" style="width: 100%; text-align: left; margin: 1em auto;min-width:30%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-stress_state.png|372px|General stress state for a 3D deformable body  <span id='citeF-16'></span>[[#cite-16|[16]]] ]]
+|
-|- style="text-align: center; font-size: 75%;"
+{|  class="floating_tableSCP wikitable" style="width: 50%; text-align: left; margin: 1em auto;min-width:30%;"
-| colspan="1" | '''Figure A.1:''' General stress state for a 3D deformable body  <span id='citeF-16'></span>[[#cite-16|[16]]]
+|+ style="font-size: 75%;" |<span id='table-6.2'></span>Table 6.2. Granular column collapse: numerical results Mesh 1
+|- style="border-top: 2px solid;font-size: 85%;"
+| colspan='4' style="text-align: center;border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | Mesh 1
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  a
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> d_{\infty } </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> h_{\infty } </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> t_{\infty } </math>
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  1.2
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.332
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.106
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.47
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  3
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.566
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.149
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.64
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  5
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.709
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.162
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.71
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  7
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.811
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.191
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.82
+|}
+|
+{|  class="floating_tableSCP wikitable" style="width: 50%; text-align: left; margin: 1em auto;min-width:30%;"
+|+ style="font-size: 75%;" |<span id='table-6.2'></span>Table 6.3. Granular column collapse: numerical results Mesh 2
+|- style="border-top: 2px solid;"
+| colspan='4' style="text-align: center;border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;font-size: 85%;" | Mesh 2
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  a
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> d_{\infty } </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> h_{\infty } </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> t_{\infty } </math>
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  1.2
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.33
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.104
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.47
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  3
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.55
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.145
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.6
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  5
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.699
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.165
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.71
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  7
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.795
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.196
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.83
+|}
 |}
-===A.1.2 Generalised Hooke's law===
+{| class="floating_tableSCP" style="width: 100%; text-align: left; margin: 1em auto;min-width:30%;"
+|-
+|
+{|  class="floating_tableSCP wikitable" style="width: 50%; text-align: left; margin: 1em auto;min-width:30%;"
+|+ style="font-size: 75%;" |<span id='table-6.2'></span>Table 6.4. Granular column collapse: numerical results Mesh 3
+|- style="border-top: 2px solid;"
+| colspan='4' style="text-align: center;border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;font-size: 85%;" | Mesh 3
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  a
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> d_{\infty } </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> h_{\infty } </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> t_{\infty } </math>
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  1.2
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.31
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.102
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.4
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  3
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.531
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.144
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.59
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  5
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.664
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.162
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.71
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  7
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.766
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.17
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.87
+|}
+|
+{|  class="floating_tableSCP wikitable" style="width: 50%; text-align: left; margin: 1em auto;min-width:30%;"
+|+ style="font-size: 75%;" |<span id='table-6.2'></span>Table 6.5. Granular column collapse: empirical results
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |   a
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> d_{\infty } </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> h_{\infty } </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> t_{\infty } </math>
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  1.2
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.264
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.108
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.35
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  3
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.505
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.14
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.55
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  5
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.673
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.172
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.71
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  7
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.819
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.197
+| style="border-left: 2px solid;border-right: 2px solid;" |  0.838
+|}
+|}
-Robert Hooke was one of the first scientists to notice elastic behaviour of materials. He studied springs under traction and compression, and tried to pose an equation linking both the applied force and the elongation of the spring. He discovered that, for '''small deformations''' <math display="inline">x</math>, the required force was <math display="inline">F = kx</math>, where <math display="inline">k</math> is a constant characteristic of the spring. This law was later generalised to describe elastic objects in general, posing that the strain <math display="inline">\varepsilon </math> (that is, nondimensional deformation) is proportional to the stress <math display="inline">\sigma </math> (that is, force per unit area) applied to it.
-Remembering the equations from the previous section, one can realise that general stresses and strains have multiple independent components, and thus the proportionally factor is no longer a single value, but a '''tensor'''. The tensorial form of Hooke's law introduces a concept that might not be intuitive: some elastic bodies deform in one direction when subjected to a force with a different direction. In such cases, proportionality will hold: if the directions are not changed, the deformation will increase proportional to the applied force. For that cases, the formulation in three dimensions has the following vectorial form:
+In order to carry out a comparison between the experimental and numerical results with regards to the kinematic of the failure mechanism, in Figures [[#img-6.1|6.1]] and [[#img-6.2|6.2]] experimental and numerical normalized distance-time data of the flow front for the test with aspect ratios 3, 5 and 7 are depicted. The normalize distance is evaluate as <math display="inline">d/d_\infty </math>, while the normalized time has the following expression
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,346: / Line 5,273: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left[\begin{array}{c}{F_{1}} \\ {F_{2}} \\ {F_{3}}\end{array}\right]=\left[\begin{array}{ccc}{k_{11}} & {k_{12}} & {k_{13}} \\ {k_{21}} & {k_{22}} & {k_{23}} \\ {k_{31}} & {k_{32}} & {k_{33}}\end{array}\right]\left[\begin{array}{c}{x_{1}} \\ {x_{2}} \\ {x_{3}}\end{array}\right] </math>
+| style="text-align: center;" | <math>T \equiv \frac{t}{t_\infty }= \frac{t}{3.3\sqrt{h_i/g}} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.4)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (6.4)
 |}
-When moving from a discrete spring to '''continuous media''', the formulation changes a little as the strain and stress state around a point cannot longer be described by a vector as all pushing, pulling and shearing effects have to be taken into account. This complexity is captured using a second order tensor, and in the tensorial formulation, forces and displacements are replaced by stresses and strains, respectively. In a Cartesian base, stresses and strains are represented with a first order tensor each:
+where it is used the empirical law <math display="inline">t_\infty=3.3\sqrt{h_i/g}</math> for high aspect ratios, obtained in <span id='citeF-199'></span>[[#cite-199|[199]]], which states that the time for the column to spread depends only on the initial height <math display="inline">h_i</math>.
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+In Figure [[#img-6.1|6.1]] the results obtained with the finer mesh (Mesh 3) are depicted. It is found that the sample with the aspect ratio of 3 is the numerical test which fully matches the yellow region, within which all the experimental normalized distance-time data collapse. The higher the aspect ratio, the higher the mismatch between numerical and experimental results. By observing Figures [[#img-6.2a|6.2a]], [[#img-6.2b|6.2b]] and [[#img-6.2c|6.2c]], it is found that the numerical results in terms of kinematic of the moving front do not show a sensitivity to the different spatial discretisations. Moreover, the typical three stages of the failure mechanism: the initial constant acceleration, the constant velocity and the final deceleration step, are recognized for all aspect ratios. If in Figure [[#img-6.2a|6.2a]] it is shown that the numerical results fall in the yellow zone, for higher values of aspect ratio (see Figures [[#img-6.2b|6.2b]] and [[#img-6.2c|6.2c]]), the curves slightly deviate from the yellow zone. It is deduced that the velocity, in the second stage, is higher than what experimentally observed. This implies that in the numerical simulations the amount of energy dissipated is underestimated, even if a damping term is added to the matricial system to solve.
+<div id='img-6.1'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
-|
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-normalized_dist_time_mesh3_text-0_4cm-0_6cm-0_7cm-1_8cm.png|500px|Granular column collapse: comparison between normalized distance-time data for different aspect ratios with Mesh 3 and experimental results <span id='citeF-199'></span>[[#cite-199|[199]]].]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 6.1:''' Granular column collapse: comparison between normalized distance-time data for different aspect ratios with Mesh 3 and experimental results <span id='citeF-199'></span>[[#cite-199|[199]]].
+|}
+<div id='img-6.2a'></div>
+<div id='img-6.2b'></div>
+<div id='img-6.2c'></div>
+<div id='img-6.2'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{\varepsilon }=\left[\begin{array}{lll}{\varepsilon _{xx}} & {\varepsilon _{xy}} & {\varepsilon _{xz}} \\ {\varepsilon _{yx}} & {\varepsilon _{yy}} & {\varepsilon _{yz}} \\ {\varepsilon _{zx}} & {\varepsilon _{zy}} & {\varepsilon _{zz}}\end{array}\right]; \quad \boldsymbol{\sigma }=\left[\begin{array}{lll}{\sigma _{xx}} & {\tau _{xy}} & {\tau _{xz}} \\ {\tau _{yx}} & {\sigma _{yy}} & {\tau _{yz}} \\ {\tau _{zx}} & {\tau _{zy}} & {\sigma _{zz}}\end{array}\right] </math>
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-normalized_dist_time_a3_text-0_7cm-0_6cm-0_0cm-1_2cm.png|482px|Case a = 3]]
+|- style="text-align: center; font-size: 75%;"
+| (a) Case a = 3
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-normalized_dist_time_a5_text-0_5cm-0_6cm-0_7cm-1_8cm.png|482px|Case a = 5]]
+|- style="text-align: center; font-size: 75%;"
+| (b) Case a = 5
+|-
+| style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-normalized_dist_time_a7_text-0_5cm-0_6cm-0_7cm-1_8cm.png|482px|Case a = 7]]
+|- style="text-align: center; font-size: 75%;"
+|   (c) Case a = 7
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 6.2:''' Granular column collapse: comparison between normalized distance-time data of the flow front for different aspect ratios a.
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.5)
+In addition, for the test case with <math display="inline">a = 7</math> the configurations of the granular column at different times of the analysis are compared with the experimental ones in Figure [[#img-6.4|6.4]]. From the comparison it is found that the numerical results have a good match with the experimental one, except for the maximum height of the column at the position <math display="inline">x=0m</math> in all the time instants considered, most probably due to the adoption of a constitutive law unable to predict the initial failure surface and, thus, the correct flowing behaviour.
+The results of Figure [[#img-6.4|6.4]], to some extent, can be considered accurate, providing a good description not only of the final configuration, but also of the dynamic flowing behaviour at different times. Nevertheless, as pointed out in Fern <span id='citeF-207'></span>[[#cite-207|[207]]], numerical damping aims to mitigate numerical oscillations by reducing the out-of-balance force and is, hence, reducing the dynamic effects. The use of numerical damping to reduce the run-out distance is rather a modification of the dynamic problem than a proper energy dissipating mechanism.  In <span id='citeF-207'></span>[[#cite-207|[207]]] it is confirmed that the initial geometry of the column plays an important role in the dynamics of the failure mechanism, since it establishes the initial potential energy available in the system. From the modelling point of view of such test case an accurate representation of the real triggering mechanism and the flow behaviour it is on the constitutive law which it is found to have two main roles. It defines the first failure surface, thus, the amount of potential energy to be transformed in kinetic energy and to be dissipated, and the way energy is dissipated in the system of the moving mass.   As a last observation, in <span id='citeF-207'></span>[[#cite-207|[207]]] they recognized the initial density as important variable to be considered in the model, able to influence the constitutive model enhancing the mechanical response. It influences the dilatancy characteristics and consequently the failure angle. The enhancement of the angle of failure by density influences, in turn, the volume of the mobilised mass, its potential energy and, in some cases, the dissipation of that energy. They noticed that the initial density affects more the results in low aspect ratios column and this could explain the difference between ''Regime I'' and ''Regime II'' from a physical point of view. The observations made in the work of Fern and Soga give a critical view of some constitutive models, for instance, the Mohr-Coulomb plastic law adopted in the current work, which are not able to predict the real energy dissipation that should have taken place during the failure mechanism (observed in the current work mainly for higher aspect ratios) and the evolution of density before reaching the critical state.
+<div id='img-6.3a'></div>
+<div id='img-6.3b'></div>
+<div id='img-6.3c'></div>
+<div id='img-6.3d'></div>
+<div id='img-6.3e'></div>
+<div id='img-6.3f'></div>
+<div id='img-6.3g'></div>
+<div id='img-6.3h'></div>
+<div id='img-6.3'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granular_column_a7_t1.png|228px|t1]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granular_column_a7_t2.png|228px|t2]]
+|- style="text-align: center; font-size: 75%;"
+| (a) t1
+| (b) t2
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granular_column_a7_t3.png|228px|t3]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granular_column_a7_t4.png|228px|t4]]
+|- style="text-align: center; font-size: 75%;"
+| (c) t3
+| (d) t4
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granular_column_a7_t5.png|228px|t5]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granular_column_a7_t6.png|228px|t6]]
+|- style="text-align: center; font-size: 75%;"
+| (e) t5
+| (f) t6
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granular_column_a7_t7.png|228px|t7]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granular_column_a7_t8.png|228px|t8]]
+|- style="text-align: center; font-size: 75%;"
+| (g) t7
+| (h) t8
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 6.3:''' Granular column collapse: Contour fill of the numerical results and the experimental shape of the column collapse <span id='citeF-199'></span>[[#cite-199|[199]]] in different time instants.
 |}
-Stress and strain tensors vary from a point to another inside continuum materials: <math display="inline">\mathbf{\varepsilon }</math> gives information about the displacements in the neighbourhood of the point, while <math display="inline">\mathbf{\sigma }</math> stands for the forces per unit area that neighbouring parcels exert on each other. The tensor linking <math display="inline">\mathbf{\varepsilon }</math> and <math display="inline">\mathbf{\sigma }</math> is known as the elasticity tensor <math display="inline">\mathbf{C}</math>, represented by a matrix of 81 real numbers <math display="inline">C_{i j k l}</math>:
-<span id="eq-A.6"></span>
+<div id='img-6.4a'></div>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+<div id='img-6.4b'></div>
+<div id='img-6.4c'></div>
+<div id='img-6.4'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
 |-
-|
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-granular_column_a7_t9.png|298px|t9]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
+| (i) t9
 |-
-| style="text-align: center;" | <math>\sigma _{i j}=\sum _{k=1}^{3} \sum _{l=1}^{3} C_{i j k l} \varepsilon _{k l} </math>
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-granular_column_a7_t10.png|298px|t10]]
+|- style="text-align: center; font-size: 75%;"
+| (j) t10
+|-
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-granular_column_a7_t11.png|298px|t11]]
+|- style="text-align: center; font-size: 75%;"
+| (k) t11
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 6.4:''' Granular column collapse: Contour fill of the numerical results and the experimental shape of the column collapse <span id='citeF-199'></span>[[#cite-199|[199]]] in different time instants.
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.6)
+==6.2 Plain strain rigid footing on undrained soil==
+The second test of validation is a plain strain rigid strip footing for the evaluation of the bearing capacity of the soil in undrained conditions, underneath the foundation. The soil is modelled as a purely cohesive weightless elastic-perfectly plastic Mohr-Coulomb material with associative flow rule, which is presented in Section [[#3.3 Hyperelastic - Mohr-Coulomb plastic law|3.3]]. The geometry, the boundary conditions and material properties are represented in Figure [[#img-6.5|6.5]], where for symmetry only half of the domain is considered.
+<div id='img-6.5'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-geometry2.png|500px|Rigid strip footing. Geometry, material properties, boundary conditions and initial material points density. 12 material points (MP) per element are used in the vicinity of the footing while only 4 are used in the rest of the domain.]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding-bottom:10px;"| '''Figure 6.5:''' Rigid strip footing. Geometry, material properties, boundary conditions and initial material points density. 12 material points (MP) per element are used in the vicinity of the footing while only 4 are used in the rest of the domain.
 |}
-Due to inherent symmetries of the model, the elasticity tensor <math display="inline">\mathbf{C}</math> can be reduced to 21 independent coefficients, while <math display="inline">\mathbf{\varepsilon }</math> and <math display="inline">\mathbf{\sigma }</math> are found to be symmetrical matrices (<math display="inline">\sigma _{i j} = \sigma _{j i}</math> and <math display="inline">\varepsilon _{k l} = \varepsilon _{l k}</math> ), having thus only 6 independent values. In this context, the '''Voigt notation''' is introduced: strain and stress tensors are substituted by vectors including only independent coefficients.  For the case of shear strains, they are substituted by '''engineering strains''' <math display="inline">\gamma </math> in Voigt notation (<math display="inline">2 \varepsilon _{k l} = \gamma _{k l}</math> ) :
-<span id="eq-A.7"></span>
+In the geomechanics community this is a classical benchmark for the validation of the constitutive law and of the numerical method adopted for its simulation. In the literature, the rigid strip footing has been studied by many authors. In <span id='citeF-208'></span>[[#cite-208|[208]]] Nazem and coworkers solved this example in three different kinematics frameworks: a Total Lagrangian (TL), an Updated Lagrangian (UL) and an Arbitrary Lagrangian Eulerian (ALE) Finite Element Methods. They show that for large deformations an ALE method is more suitable than UL and TL strategies, avoiding mesh distortion with a remeshing technique. Even if the remeshing could smear a stress concentration and compromise the strain localization, they found that the load-displacement curve is comparable with the numerical solutions available in the literature. In <span id='citeF-209'></span>[[#cite-209|[209]]] the technique of <span id='citeF-208'></span>[[#cite-208|[208]]] is generalized to the case of higher order elements. The same  test example has been also used to prove that the MPM represents an ideal numerical approach since it naturally tracks large deformations without the need of remeshing procedures.  For instance, in <span id='citeF-118'></span>[[#cite-118|[118]]] this example is successfully solved exploiting the capability of MPM to track large deformation and large displacement of the solid. However, the work of <span id='citeF-118'></span>[[#cite-118|[118]]] is limited to the infinitesimal strain assumption.
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+For the validation of this test case, the stabilized mixed formulation, valid under geometric and material non-linearities, presented in Chapter [[#5 Mixed formulation|5]], is employed, generalizing the approach used in <span id='citeF-118'></span>[[#cite-118|[118]]].  The simulation is performed using displacement control with steps of incremental vertical displacement <math display="inline">\Delta \boldsymbol{u} = -0.001m</math>. The total displacement has been imposed in 2000 time steps which corresponds to twice the foundation width <math display="inline"> B </math>. The discretisation of the computational domain is performed through a unstructured triangular background mesh with a mesh size of 0.05m. At the interface between the foundation and the soil, where the largest deformations take place, a higher initial number of material points per element is used for a better resolution of the results (Figure [[#img-6.5|6.5]]).
+In Figures [[#img-6.6|6.6]], [[#img-6.7|6.7]] and [[#img-6.8|6.8]] the displacement and stress fields obtained with the ''u'' and ''u-p'' formulations are compared. As expected, the latter shows to be more reliable and accurate than the first one. It can be noted that the final deformation is accurately described and an improvement is registered if the final deformation is compared with the numerical results of <span id='citeF-208'></span>[[#cite-208|[208]]] and <span id='citeF-209'></span>[[#cite-209|[209]]] which are more similar to the final configuration obtained through the displacement-based formulation. The need for a mixed formulation is evident when evaluating the vertical stress field. In Figure [[#img-6.8a|6.8a]] the displacement-based formulation fails to evaluate a reliable stress response, as the magnitude of the vertical Cauchy stress is out of the expected range in the area where the foundation buries itself. On the other hand, the mixed formulation is able to evaluate a continuous stress field and using such result it is possible to evaluate the normalised load-displacement response of the foundation, which is used for the validation of the current example.
+<div id='img-6.6a'></div>
+<div id='img-6.6b'></div>
+<div id='img-6.6'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;"
 |-
-|
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-mesh0125_ux_legend_set-0_5cm-0_25cm-0cm-0_0cm-0_5cm-0_25cm-0cm-0_0cm.png|380px|''u'' formulation]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-mesh01_ux_legend_set-0_5cm-0_25cm-0cm-0_0cm-0_5cm-0_25cm-0cm-0_0cm.png|380px|''u-p'' formulation]]
+|- style="text-align: center; font-size: 75%;"
+| (a) ''u'' formulation
+| (b) ''u-p'' formulation
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"|  '''Figure 6.6:''' Rigid strip footing. Horizontal displacement
+|}
+<div id='img-6.7a'></div>
+<div id='img-6.7b'></div>
+<div id='img-6.7'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{\sigma } \Rightarrow \quad \left[\begin{array}{l}{\sigma _{x}} \\ {\sigma _{y}} \\ {\sigma _{z}} \\ {\tau _{y z}} \\ {\tau _{x z}} \\ {\tau _{x y}}\end{array}\right]; \quad \quad \boldsymbol{\varepsilon }=\nabla ^{s} \mathbf{\underline{u}} \Rightarrow \left[\begin{array}{c}{\varepsilon _{x}} \\ {\varepsilon _{y}} \\ {\varepsilon _{z}} \\ {\gamma _{y z}} \\ {\gamma _{x z}} \\ {\gamma _{x y}}\end{array}\right]=\left[\begin{array}{c}{\frac{\partial u_{x}}{\partial x}} \\ {\frac{\partial u_{y}}{\partial y}} \\ {\frac{\partial u_{z}}{\partial z}} \\ {\frac{\partial u_{y}}{\partial z}+\frac{\partial u_{z}}{\partial y}} \\ {\frac{\partial u_{x}}{\partial z}+\frac{\partial u_{x}}{\partial y}} \\ {\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial y}}\end{array}\right] </math>
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-mesh0125_uy_legend_set-0_5cm-0_25cm-0cm-0_0cm-0_5cm-0_25cm-0cm-0_0cm.png|380px|''u'' formulation]]
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-mesh01_uy_legend_set-0_5cm-0_25cm-0cm-0_0cm-0_5cm-0_25cm-0cm-0_0cm.png|380px|''u-p'' formulation]]
+|- style="text-align: center; font-size: 75%;"
+| (a) ''u'' formulation
+| (b) ''u-p'' formulation
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"|  '''Figure 6.7:''' Rigid strip footing. Vertical displacement
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.7)
+<div id='img-6.8a'></div>
+<div id='img-6.8b'></div>
+<div id='img-6.8'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;"
+|-
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-mesh0125_stressy-0_5cm-0_25cm-0cm-0_0cm-0_5cm-0_25cm-0cm-0_0cm.png|480px|''u'' formulation]]
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-mesh01_stressy-0_5cm-0_25cm-0cm-0_0cm-0_5cm-0_25cm-0cm-0_0cm.png|480px|''u-p'' formulation]]
+|- style="text-align: center; font-size: 75%;"
+| (a) ''u'' formulation
+| (b) ''u-p'' formulation
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"|  '''Figure 6.8:''' Rigid strip footing. Vertical Cauchy stress
 |}
-In the above equation, <math display="inline">\nabla ^{s} \mathbf{\underline{u}}</math> stands for the '''symmetric gradient''' operator, which in 3D Cartesian coordinates adopts the following form:
+Since the problem has no analytical solution, the numerical result of <span id='citeF-210'></span>[[#cite-210|[210]]], obtained through a sequential limit analysis formulation, is taken as reference solution. The problem is solved under the assumption of large deformations, hence, the bearing capacity of the soil is expected to be higher than the value of <math display="inline">2 + \pi </math> which corresponds to the small deformation case, for a given footing displacement. Under this hypothesis, the mobilized soil resistance does not reach an asymptotic value, but gradually increases, as explained in <span id='citeF-210'></span>[[#cite-210|[210]]]. In Figure [[#img-6.9|6.9]], the result obtained through the ''u-p'' formulation in terms of normalized bearing capacity of the soil, as a function of the normalized settlement, is depicted and compared with the benchmark solution.  It can be observed that, the obtained curve is in good agreement with the reference solution  <span id='citeF-210'></span>[[#cite-210|[210]]]. The discrepancy that is observed for the initial values of the settlement is the consequence of the chosen material elastic properties. The Young Modulus <math display="inline"> E </math> and the Poisson's ratio <math display="inline"> \nu </math> have values which correspond to an undrained bulk modulus of <math display="inline">K_u =  3,33\cdot 10^5 Pa </math>, which gives a ratio <math display="inline">K_u/c_u = 3,33\cdot 10^3  </math>. In <span id='citeF-118'></span>[[#cite-118|[118]]], the influence of this ratio on the normalised load-displacement curve is studied: the elastic response of the soil becomes less or more important and the bearing capacity of the soil can increase or decrease, for higher or lower values of this ratio, respectively. For this reason the numerical results plotted in Figure [[#img-6.9|6.9]] have an important elastic response and are deviating during the initial phase of the simulation from the perfectly rigid behaviour of the benchmark solution.
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+<div id='img-6.9'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;"
 |-
-|
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-load_disp_finer.png|360px|Rigid strip footing. Normalised load-displacement curve: comparison between reference solution taken from <span id='citeF-210'></span>[[#cite-210|[210]]] and the ''u-p'' formulation solution presented in this work. ]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"|  '''Figure 6.9:''' Rigid strip footing. Normalised load-displacement curve: comparison between reference solution taken from <span id='citeF-210'></span>[[#cite-210|[210]]] and the ''u-p'' formulation solution presented in this work.
+|}
+The example of the rigid footing on undrained soil has been validated using a stabilized mixed MPM formulation. The soil bearing capacity is well predicted and comparable with accurate numerical results from the literature. Moreover, a good description of the final deformation of the soil is achieved by using the MPM and its capability of solving large displacement and large deformation problems is equivalent, if not superior, to other techniques proposed in the literature <span id='citeF-208'></span><span id='citeF-209'></span>[[#cite-208|[208,209]]].
+==6.3 Discussion==
+In this Chapter, two test cases are proposed for the validation of the MPM strategy, by using an irreducible and a mixed formulation. The first test is represented by the granular column collapse, a classical example which is quite often considered for the validation of both the constitutive laws and the numerical techniques. Despite its simplicity, with this test it is possible to make an assessment of the robustness of the numerical model and understand to what extent the constitutive law can provide reliable results. It has been shown that the MPM code, object of study, is able to track with accuracy the configuration of the collapse at different time frames and the constitutive law, employed in this study, can provide good enough results, even if neither softening and density variable are considered as additional terms in the dissipation plastic process. In this regard, as future work, further constitutive models should be taken into account, in order to improve the prediction capability of this numerical strategy.    As a second example for validation, the rigid footing on undrained soil is considered. In this case, the good performance of the ''u-p'' formulation are tested also under the finite deformation regime: a higher accuracy of the displacement and stress fields are confirmed. Moreover, evaluating the bearing capacity as a function of the footing displacement, the load-displacement curve is obtained and used as a validation tool to be compared with a reference solution.
+=7 Application to an industrial case=
+In this Chapter, our Material Point Method formulation is applied in an industrial framework. Several laboratory tests, carried out at the  Nestlé Laboratories aiming at the characterization of flowability of different sugar powders, are reproduced numerically. The practical objective of this work lies on the assessment of flow performance of sugar powders, which can play an important role in product development. In case ingredient properties are not optimized a significant variability can be observed during the operation of filling of jars or sachets, which may be detrimental for the production line. In this study, we propose to discuss the performance of crystalline sugar. If the experimental investigation of food materials process is essential in this context, the capability of numerically modelling particulate processes might represent a complementary tool for a better and a more realistic understanding of the process at a pilot or industrial scale. It is experimentally found that the flow quality is strongly deteriorated below a critical particle size. On the other hand, in the numerical study the MPM strategy is employed and it is demonstrated how material parameters, such as, internal friction angle, dilatancy angle and apparent cohesion are important factors in the prediction of the macroscopic behaviour of a granular material and its flowability performance.
+==7.1 Introduction==
+Food Industry, as many other sectors dealing with powders, needs to keep a close look at the flowability properties of the raw materials or final powder mixes they develop. Especially, a good flow performance ensures a smooth movement of materials during operations, from raw materials reception to the final packing. For instance, dosing a few grams of granular matter in a small sachet is a challenge if the factory wants to keep a constant mass and ratio of all ingredients. For that reason, the variations of flowability performance with key materials properties must be mastered by product developers in order to know where to act in case of a problem.  Food powders often contain several ingredients, such as, crystalline particles or amorphous particles. Sucrose is a good example of a common crystalline material when processed in standard conditions. A common observation is the deterioration of flowability when increasing the quantity of fine particles <span id='citeF-211'></span><span id='citeF-212'></span><span id='citeF-213'></span><span id='citeF-214'></span><span id='citeF-215'></span>[[#cite-211|[211,212,213,214,215]]]. Some authors report a threshold size below which the deterioration of performance occurs <span id='citeF-211'></span><span id='citeF-213'></span>[[#cite-211|[211,213]]]. Unfortunately, this value is shown to be strongly dependent on the products tested in those two papers, where the authors considered only diameters from 50 <math display="inline">\mu m</math> to 600 <math display="inline">\mu m</math>. The effect of size is related to the increase of surface area per unit of mass, leading to more contact points and adhesion force between the particles <span id='citeF-215'></span>[[#cite-215|[215]]]. Cohesive forces acting between particles are mainly due to van der Waals and capillary forces associated with liquid bridging <span id='citeF-215'></span><span id='citeF-216'></span>[[#cite-215|[215,216]]]. The threshold size can be seen as a limit above which cohesive forces start having a decreasing effect <span id='citeF-211'></span>[[#cite-211|[211]]]. The shape of the particle is also an important factor affecting the flow <span id='citeF-211'></span><span id='citeF-217'></span><span id='citeF-218'></span>[[#cite-211|[211,217,218]]]. More elongated particles and particles with fewer corners tend to flow more difficultly due to higher friction forces. The shape effect is observed to be more relevant above the size threshold <span id='citeF-211'></span>[[#cite-211|[211]]].
+The importance of assuring a continuous industrial production process, e.g. without interruptions, is the main reason not only for a full experimental characterization of the material properties to be processed, as previously mentioned, but, during the last decades, also for a numerical investigation of the process from the laboratory to the real plant scale. Indeed, in the literature, many examples of several numerical techniques, applied in the industrial context, can be found. The most common numerical procedure, employed for the simulation of industrial granular flows, is represented by the Discrete Element Method (DEM) <span id='citeF-24'></span>[[#cite-24|[24]]], which since the beginning of its definition has been used for industrial oriented applications, such as, mixing and milling <span id='citeF-219'></span><span id='citeF-220'></span><span id='citeF-221'></span><span id='citeF-222'></span><span id='citeF-36'></span>[[#cite-219|[219,220,221,222,36]]], transport <span id='citeF-36'></span>[[#cite-36|[36]]] and hopper discharge <span id='citeF-223'></span><span id='citeF-224'></span><span id='citeF-225'></span>[[#cite-223|[223,224,225]]]. During the last decades, the potential of this technique to simulate more realistic systems evolved hand in hand with the increase of computers power <span id='citeF-36'></span>[[#cite-36|[36]]], achieving the capability of modelling three dimensional large scale systems with DEM. Despite the popularity of DEM, several aspects limit its usage at real scale. Firstly, it is well established that a long calibration procedure has to be performed in order to define all the DEM micro-parameters characterizing the material under study. Last, but not least, the simulation time of large systems of particles might be prohibitive, unless the analysis is performed in High Performance Computing (HPC) mode. For this reason, some attempts of numerically modelling particulate processes, which use continuous technique, i.e., the Finite Element Method (FEM), can be found in the literature, as well. For instance, in <span id='citeF-226'></span>[[#cite-226|[226]]] the simulation of the discharge silos is realized by using a FEM-based code, defined in an Eulerian framework. However, since the typical processes, interesting from the industrial world perspective, imply large displacement and large deformation of the medium and due to the ambiguous solid-like and fluid-like nature of granular materials, Lagrangian techniques might be preferable rather than Eulerian methods. In this regard, it is worth mentioning the Particle Finite Element Method (PFEM) and the Material Point Method (MPM), both already employed in the past for the simulation of industrial granular flows, such as, silos discharge <span id='citeF-95'></span><span id='citeF-227'></span><span id='citeF-117'></span>[[#cite-95|[95,227,117]]] and tumbling ball milling <span id='citeF-95'></span>[[#cite-95|[95]]]. In this study, we propose to compare the flow performance of different sugar powders observed through laboratory experiments and numerical tests performed using the MPM code, whose algorithm has been presented, verified and validated in the previous Chapters (see Chapter [[#2 Particle Methods|2]], [[#4 Irreducible formulation|4]] and [[#6 Validation|6]]).  In the next sections, firstly, the experimental analysis is presented, followed by the numerical simulations and, finally, some conclusions are drawn.
+==7.2 Experimental study==
+===7.2.1 Material===
+Four sucrose powders are selected for their differences in particles size and origin: Sugar White Fine Bulk (Schweizer Zucker AG, Switzerland), Sugar White Fine Special (Schweizer Zucker AG, Switzerland), Sugar EGII Fine (Agrana, Austria) and Sugar Icing (Central Sugar Refinery, Malaysia). For ease of reading, those powders are, respectively, named S1, S2, S3 and S4. Their particle size distribution is characterized by means of laser diffraction (Malvern Mastersizer 3000 fitted with Aero dispersion module set at a dispersion pressure of 2 bars). The particle size distribution gives access to different parameters, such as, <math display="inline">d_{10}</math>, <math display="inline"> d_{50} </math> or <math display="inline"> d_{90} </math>, respectively, e.g., the diameters at which 10%, 50% or 90% of the volume of particles is below this value. The ''span'' <math display="inline"> s=(d_{90}-d_{10})/d_{50}  </math>  describes the width of the particle size distribution. In order to obtain a wider range of particle sizes <math display="inline"> d_{50} </math>  and ''span s'', ten additional samples are generated by mixing or sieving the initial powders. Table [[#table-7.1|7.1]] summarizes the preparation method and size characteristics of the different sucrose powders used in this study.
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-7.1'></span>Table. 7.1 Name, preparation method, <math> d_{10} </math>, <math> d_{50} </math>, <math> d_{90} </math> and ''span s'' of the sucrose powders used in this study. ''Courtesy of Nestlé''
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  Reference name
+| style="border-left: 2px solid;border-right: 2px solid;" | Preparation method
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> d_{10} [\mu m] </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> d_{50} [\mu m]  </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> d_{90} [\mu m]  </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | s
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S1
+| style="border-left: 2px solid;border-right: 2px solid;" | Commercial powder
+| style="border-left: 2px solid;border-right: 2px solid;" | 244
+| style="border-left: 2px solid;border-right: 2px solid;" | 533
+| style="border-left: 2px solid;border-right: 2px solid;" | 977
+| style="border-left: 2px solid;border-right: 2px solid;" | 1.38
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S1_a </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | S1 > 800 <math display="inline"> [\mu m] </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 666
+| style="border-left: 2px solid;border-right: 2px solid;" | 994
+| style="border-left: 2px solid;border-right: 2px solid;" | 1590
+| style="border-left: 2px solid;border-right: 2px solid;" | 0.93
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S1_b </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 500 <math display="inline"> [\mu m] </math> > S1 > 800 <math display="inline"> [\mu m] </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 423
+| style="border-left: 2px solid;border-right: 2px solid;" | 633
+| style="border-left: 2px solid;border-right: 2px solid;" | 935
+| style="border-left: 2px solid;border-right: 2px solid;" | 0.81
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S1_c </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | S1 < 500 <math display="inline"> [\mu m] </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 203
+| style="border-left: 2px solid;border-right: 2px solid;" | 372
+| style="border-left: 2px solid;border-right: 2px solid;" | 602
+| style="border-left: 2px solid;border-right: 2px solid;" | 1.07
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S2
+| style="border-left: 2px solid;border-right: 2px solid;" | Commercial powder
+| style="border-left: 2px solid;border-right: 2px solid;" | 81
+| style="border-left: 2px solid;border-right: 2px solid;" | 182
+| style="border-left: 2px solid;border-right: 2px solid;" | 310
+| style="border-left: 2px solid;border-right: 2px solid;" | 1.26
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S3
+| style="border-left: 2px solid;border-right: 2px solid;" | Commercial powder
+| style="border-left: 2px solid;border-right: 2px solid;" | 71
+| style="border-left: 2px solid;border-right: 2px solid;" | 194
+| style="border-left: 2px solid;border-right: 2px solid;" | 352
+| style="border-left: 2px solid;border-right: 2px solid;" | 1.45
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S3_a </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | S3 < 200 <math display="inline"> [\mu m] </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 142
+| style="border-left: 2px solid;border-right: 2px solid;" | 261
+| style="border-left: 2px solid;border-right: 2px solid;" | 425
+| style="border-left: 2px solid;border-right: 2px solid;" | 1.09
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S3_b </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | S3 > 200 <math display="inline"> [\mu m] </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 55
+| style="border-left: 2px solid;border-right: 2px solid;" | 154
+| style="border-left: 2px solid;border-right: 2px solid;" | 269
+| style="border-left: 2px solid;border-right: 2px solid;" | 1.39
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S4
+| style="border-left: 2px solid;border-right: 2px solid;" | Commercial powder
+| style="border-left: 2px solid;border-right: 2px solid;" | 10
+| style="border-left: 2px solid;border-right: 2px solid;" | 54
+| style="border-left: 2px solid;border-right: 2px solid;" | 190
+| style="border-left: 2px solid;border-right: 2px solid;" | 3.36
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S4_a </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | S4 > 100 <math display="inline"> [\mu m] </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 38
+| style="border-left: 2px solid;border-right: 2px solid;" | 146
+| style="border-left: 2px solid;border-right: 2px solid;" | 280
+| style="border-left: 2px solid;border-right: 2px solid;" | 1.66
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S4_b </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | S4 < 100 <math display="inline"> [\mu m] </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 7
+| style="border-left: 2px solid;border-right: 2px solid;" | 32
+| style="border-left: 2px solid;border-right: 2px solid;" | 84
+| style="border-left: 2px solid;border-right: 2px solid;" | 2.42
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S1S3
+| style="border-left: 2px solid;border-right: 2px solid;" | Mix 50% S1 - 50% S1
+| style="border-left: 2px solid;border-right: 2px solid;" | 98
+| style="border-left: 2px solid;border-right: 2px solid;" | 279
+| style="border-left: 2px solid;border-right: 2px solid;" | 789
+| style="border-left: 2px solid;border-right: 2px solid;" | 2.47
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S2S3
+| style="border-left: 2px solid;border-right: 2px solid;" | Mix 50% S2 - 50% S3
+| style="border-left: 2px solid;border-right: 2px solid;" | 81
+| style="border-left: 2px solid;border-right: 2px solid;" | 195
+| style="border-left: 2px solid;border-right: 2px solid;" | 345
+| style="border-left: 2px solid;border-right: 2px solid;" | 1.35
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S1S4
+| style="border-left: 2px solid;border-right: 2px solid;" | Mix 30% S1 - 70% S4
+| style="border-left: 2px solid;border-right: 2px solid;" | 12
+| style="border-left: 2px solid;border-right: 2px solid;" | 86
+| style="border-left: 2px solid;border-right: 2px solid;" | 592
+| style="border-left: 2px solid;border-right: 2px solid;" | 6.74
+|}
+===7.2.2 Measurements procedure===
+All sucrose powders (Si) are characterized in terms of flowability using several techniques. At minimum, the measurements are performed twice for each technique.
+Firstly, the free-flow density, tap density and Carr Index are measured. In order to estimate the free-flow density, a 500 mL - stainless steel cylinder (Figure [[#img-7.1a|7.1a]]) is filled with powder using a funnel to increase pouring repeatability. After eliminating the excess powder by levelling the top of the cylinder with the flat blade of a pharmaceutical spatula, the free-flow density <math display="inline"> D_{free} </math> is directly deduced by dividing the weight of the powder by its volume. Tap density <math display="inline"> D_{tap} </math> is then obtained with the same procedure but after the powder has been subjected to a fixed number of taps. For this operation, an extension of the cylinder is used to start with more powder and the receptacle is placed on an electrical jolting density meter (100 jolts, total time 30 seconds, amplitude 8.5 mm). Finally, the Carr Index or compressibility <math display="inline"> I_C</math>  is calculated with the equation: <math display="inline"> I_C=(100 (D_{tap}-D_{free} ))/D_{tap} </math>.
+Second, the angle of repose is evaluated. A tailor-made apparatus (Figure [[#img-7.1b|7.1b]]) is used, inspired from ''ISO norm 4324'', but adapted to allow the measurement of both fluid and sticky powders. The determination of the angle of repose of a powder cone is obtained by passing 150 mL of the product through a special funnel placed at a fixed height (85 mm) above a completely flat and level surface. This surface is materialized by a 25 mm-high and 100 mm wide plastic cylinder that ensures the cone always has the same base diameter. Angle of repose (AR) is then calculated from the measurement of the cone height.
+Then, the flow behaviour through apertures of different sizes is quantified. For that purpose, the ''GranuFall apparatus'' (Aptis, Belgium, Figure [[#img-7.1c|7.1c]]) is used. It consists of a hollow cylinder with an internal diameter of 36.3 mm in which we introduce 200 mL of powder. At the bottom of the cylinder, a plate prevents the powder from falling. At the beginning of the experiment, a plate with a centred orifice is moved below the cylinder allowing powder to flow out of the vessel. A height detector located at the top of the cylinder measures the distance of the powder-air interface as a function of time, allowing flow rate calculation in mL/s. Hole diameter <math display="inline"> d_h </math> can be varied from 4 mm to 34 mm, with steps of 2 mm. In this study, for sugar powders the flow <math display="inline"> f_{18} </math> at <math display="inline"> d_h= 18 mm </math> and the last diameter where flow occurs <math display="inline"> d_h^{min} </math> were measured.
+Finally, the avalanche properties of powders are obtained using the ''Revolution powder analyzer'' (PS Prozesstechnik GmbH, Switzerland, Figure [[#img-7.1d|7.1d]]). It consists in observing the avalanche movement of a powder (100 mL) in a rotating cylinder with glass walls. Rotation speed is set to 0.3 rpm. The image analysis of the pictures acquired by the CCD camera allows extracting information about powder flowability properties such as avalanche energy <math display="inline"> E_a </math>, avalanche time <math display="inline"> t_a </math>, avalanche angle <math display="inline"> A_a </math>, rest angle <math display="inline"> R_a </math> or surface linearity <math display="inline"> S_a </math> by averaging those values over 150 avalanche events.
+<div id='img-7.1a'></div>
+<div id='img-7.1b'></div>
+<div id='img-7.1c'></div>
+<div id='img-7.1d'></div>
+<div id='img-7.1'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 85%;"
 |-
-| style="text-align: center;" | <math>\nabla ^{s} := \left[\begin{array}{lll}{\frac{\partial }{\partial x}} & {0} & {0} \\ {0} & {\frac{\partial }{\partial y}} & {0} \\ {0} & {0} & {\frac{\partial }{\partial z}} \\ {0} & {\frac{\partial }{\partial z}} & {\frac{\partial }{\partial y}} \\ {\frac{\partial }{\partial z}} & {0} & {\frac{\partial }{\partial x}} \\ {\frac{\partial }{\partial y}} & {\frac{\partial }{\partial x}} & {0}\end{array}\right] </math>
+|style="padding:10px;"| [[Image:Draft_Samper_987121664-monograph-free_flow_density_device.png|156px|]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-angle_of_repose_device.png|162px|]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granufall_device.png|156px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
+| (c)
+|-
+| colspan="3" style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-revolution_powder_device.png|366px|]]
+|- style="font-size: 75%;"
+| colspan="3" style="text-align: center;|(d)
+|- style="text-align: center; font-size: 75%;"
+| colspan="3" style="padding:10px;"| '''Figure 7.1:''' Flowability methods used in this study. (a) Free-flow density cylinder. (b) Angle of repose apparatus. (c) GranuFall. (d) Revolution powder analyzer. ''Courtesy of Nestlé''
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.8)
+===7.2.3 Experimental results===
+====7.2.3.1 Carr Index and bulk density====
+Density measurements on powder samples allowed collecting compressibility data. Bulk densities and Carr indexes are summarized in Figure [[#img-7.2|7.2]].  For sucrose, <math display="inline"> D_{free} </math> is found to increase with particle size. A range from 500 g/L at small diameters to approximately 850g/L at larger diameter is observed in Figure [[#img-7.2|7.2]]. Samples with larger span tend to have larger bulk densities as demonstrated by the sample S1S3 (outlier at 280 <math display="inline">\mu m </math>). <math display="inline"> I_C </math> follows a 1/x trend and reaches the value of 7 at large diameters which corresponds to good flowability according to literature <span id='citeF-228'></span>[[#cite-228|[228]]]. Then, the compressibility value starts to increase around 250 <math display="inline">\mu m </math> and reaches the maximum value of 30, which corresponds to poor flow.
+<div id='img-7.2'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-density_nestle.png|426px|Free-flow density and Carr Index results. Black and red solid lines are guides to the eye, respectively for density and Carr Index. ''Courtesy of Nestlé''.]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 7.2:''' Free-flow density and Carr Index results. Black and red solid lines are guides to the eye, respectively for density and Carr Index. ''Courtesy of Nestlé''.
 |}
-For the case of '''isotropic materials''' —that is, uniform properties in all directions—, it can be found that the elasticity matrix depends only on two coefficients, the '''Young's Modulus''' <math display="inline">E</math> (standing for the material stiffness or resistance to be elastically deformed) and the '''Poisson's coefficient''' <math display="inline">\nu </math>. The latter is the negative of the ratio of transverse strain (unit lateral contraction) to axial strain, and the responsible for the Possion effect: the phenomenon in which a given material tends to expand in directions perpendicular to the direction of compression, and to contract in the directions transverse to the direction of stretching <span id='citeF-16'></span>[[#cite-16|[16]]].
-The Young's Modulus has units of pressure, while the Poisson's coefficient is dimensionless, and theoretically limited to the interval <math display="inline">-1<\nu{<0},5</math>, although in reality only positive values of <math display="inline">\nu </math> are found. However, <math display="inline">\mathbf{C}</math> is commonly expressed as a function of the ''Lame'' parameters <math display="inline">\lambda </math> and <math display="inline">\mu </math>. In indicial form:
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-7.2'></span>Table. 7.2 Name, Carr Index and Bulk density [g/L] measured in this study. ''Courtesy of Nestlé''.
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  Reference name
+| style="border-left: 2px solid;border-right: 2px solid;" | Carr Index
+| style="border-left: 2px solid;border-right: 2px solid;" | Bulk density [g/L]
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S1
+| style="border-left: 2px solid;border-right: 2px solid;" | 7.1
+| style="border-left: 2px solid;border-right: 2px solid;" | 877
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S1_a </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 6.8
+| style="border-left: 2px solid;border-right: 2px solid;" | 826
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S1_b </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 7.4
+| style="border-left: 2px solid;border-right: 2px solid;" | 838
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S1_c </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 7.8
+| style="border-left: 2px solid;border-right: 2px solid;" | 848
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S2
+| style="border-left: 2px solid;border-right: 2px solid;" | 15
+| style="border-left: 2px solid;border-right: 2px solid;" | 746
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S3
+| style="border-left: 2px solid;border-right: 2px solid;" | 12.6
+| style="border-left: 2px solid;border-right: 2px solid;" | 804
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S3_a </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 8.7
+| style="border-left: 2px solid;border-right: 2px solid;" | 834
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S3_b </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 14.7
+| style="border-left: 2px solid;border-right: 2px solid;" | 768
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S4
+| style="border-left: 2px solid;border-right: 2px solid;" | 30.7
+| style="border-left: 2px solid;border-right: 2px solid;" | 585
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S4_a </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 16.9
+| style="border-left: 2px solid;border-right: 2px solid;" | 727
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S4_b </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 29
+| style="border-left: 2px solid;border-right: 2px solid;" | 494
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S1S3
+| style="border-left: 2px solid;border-right: 2px solid;" | 9.4
+| style="border-left: 2px solid;border-right: 2px solid;" | 912
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S2S3
+| style="border-left: 2px solid;border-right: 2px solid;" | 13.8
+| style="border-left: 2px solid;border-right: 2px solid;" | 775
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S1S4
+| style="border-left: 2px solid;border-right: 2px solid;" | 29.2
+| style="border-left: 2px solid;border-right: 2px solid;" | 695
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+|}
+====7.2.3.2 Angle of repose====
+The values of the angle of repose (AR) measured for sugars are summarized in Figure [[#img-7.3|7.3]], where the angle is plotted as a function of the particle size (represented here with <math display="inline"> d_{50} </math>). It is observed that the smaller the particle size, the higher the angle of repose. A plateau-like part appears at large diameters around 35<math display="inline">^{\circ }</math>, which correspond to a good flowability according literature <span id='citeF-229'></span>[[#cite-229|[229]]], while a strong increase of the AR is observed at small diameters. The increase starts around 250-350 <math display="inline">\mu m </math> and leads to 55<math display="inline">^{\circ }</math> at the smallest diameters, corresponding to poor flow.
+<div id='img-7.3'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
-|
+|style="padding-bottom:10px;"|[[File:Draft_Samper_987121664_9504_angle_of_repose_nestle.png|450px|Angle of repose results. Dashed lines are guides to the eye. ''Courtesy of Nestlé''.]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 7.3:''' Angle of repose results. Dashed lines are guides to the eye. ''Courtesy of Nestlé''.
+|}
+====7.2.3.3 GranuFall====
+GranuFall experiments for sugar powders are summarized in Figure [[#img-7.4|7.4]]. We plot <math display="inline"> f_{18} </math> and <math display="inline"> d_h^{min} </math> as a function of the particle size <math display="inline"> d_{50} </math>. It is observed that the powder does not flow for fine powders while the flow suddenly steps up to 70 mL/s around <math display="inline"> d_{50} </math>=200 <math display="inline">\mu m </math> before slowly diminishing down to 60 mL/s at larger diameters. One outlier with smaller flow is observed (sample S1S3), the same than in the density graph. Measurements of the critical diameter for flow show an opposite behavior. For fines, <math display="inline"> d_h^{min} </math> is larger than 34 mm (represented by an arbitrary point at 36 mm since powder did not flow at the maximum available hole diameter). Then, <math display="inline"> d_h^{min} </math> decreases with particle size probably reaching a minimum value below 4 mm which could not be determined with the apparatus (4 mm is the minimum available hole diameter).
+<div id='img-7.4'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;"
 |-
-| style="text-align: center;" | <math>C_{i j k l}=\lambda \delta _{i j} \delta _{k l}+\mu \left(\delta _{i k} \delta _{j l}+\delta _{i l} \delta _{j k}\right), \quad \lambda =\frac{\nu E}{(1+\nu )(1-2 \nu )} , \quad \mu =\frac{E}{2(1+\nu )} </math>
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granufall_nestle-6cm-6_0cm-9_0cm-6_0cm.png|500px|GranuFall measurements on sugar powders. Flow at 18 mm diameter and minimum diameter to flow, versus particle size  d₅₀ . Dashed lines are guides to the eye. ''Courtesy of Nestlé''.]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 7.4:''' GranuFall measurements on sugar powders. Flow at 18 mm diameter and minimum diameter to flow, versus particle size <math> d_{50} </math>. Dashed lines are guides to the eye. ''Courtesy of Nestlé''.
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.9)
+====7.2.3.4 Revolution Powder Analyzer====
+Finally, the last technique for flow characterization allowed obtaining the results presented in Figure [[#img-7.5|7.5]]. In this case, the results are presented in terms of angle of avalanche <math display="inline"> A_a </math>, corresponding to the angle between the powder-air interface and the horizontal plane before the avalanche occurrence, and the rest angle, measured immediately after the avalanche. Typically, <math display="inline"> A_a </math> varies from 30<math display="inline">^{\circ }</math> to 100<math display="inline">^{\circ }</math>, from free flow to very poor flow. By observing Figure [[#img-7.5|7.5]], in the case of sucrose, it is evident that both the avalanche and the rest angle are strongly influenced by the particle size below 300 <math display="inline">\mu m </math>; while above this critical value a plateau-like trend is noted for both the parameters under study.
+<div id='img-7.5'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;"
+|-
+|[[Image:Draft_Samper_987121664-monograph-avalanche_angle_nestle-9cm-5_0cm-7_0cm-6_0cm.png|500px|Revolution powder analyzer data. Sugar avalanche and rest angle, as a function of particle diameter  d₅₀ . Dashed lines are guides to the eye, respectively for avalanche and rest angle. ''Courtesy of Nestlé''.]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding-bottom:10px;"| '''Figure 7.5:''' Revolution powder analyzer data. Sugar avalanche and rest angle, as a function of particle diameter <math> d_{50} </math>. Dashed lines are guides to the eye, respectively for avalanche and rest angle. ''Courtesy of Nestlé''.
 |}
-In Voigt notation and for isotropic materials, Hooke's law adopts the form
+In Table [[#table-7.3|7.3]], the main experimental results in term of angle of repose, flow rate <math display="inline"> f_{18}</math> and avalanche angle are resumed. All the data are listed in descending order of <math display="inline"> d_{50}</math>.
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-7.3'></span>Table. 7.3 Name, <math> d_{50} [\mu m]  </math>, Angle of repose [<math>^{\circ }</math>], <math> f_{18} [mL/s] </math> and Avalanche angle [<math>^{\circ }</math>] measured in this study. ''Courtesy of Nestlé''.
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  Reference name
+| style="border-left: 2px solid;border-right: 2px solid;" | <math> d_{50} [\mu m]  </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | Angle of repose [<math display="inline">^{\circ }</math>]
+| style="border-left: 2px solid;border-right: 2px solid;" | flow rate <math display="inline"> f_{18} [mL/s] </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | Avalanche angle [<math display="inline">^{\circ }</math>]
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S1_a </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 994.4
+| style="border-left: 2px solid;border-right: 2px solid;" | 36.1
+| style="border-left: 2px solid;border-right: 2px solid;" | 59.55
+| style="border-left: 2px solid;border-right: 2px solid;" | 41.2
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S1_b </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 632.9
+| style="border-left: 2px solid;border-right: 2px solid;" | 34.8
+| style="border-left: 2px solid;border-right: 2px solid;" | 64.55
+| style="border-left: 2px solid;border-right: 2px solid;" | 40.7
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S1
+| style="border-left: 2px solid;border-right: 2px solid;" | 532.8
+| style="border-left: 2px solid;border-right: 2px solid;" | 34.4
+| style="border-left: 2px solid;border-right: 2px solid;" | 64.00
+| style="border-left: 2px solid;border-right: 2px solid;" | 39.4
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S1_c </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 372.3
+| style="border-left: 2px solid;border-right: 2px solid;" | 35.0
+| style="border-left: 2px solid;border-right: 2px solid;" | 69.35
+| style="border-left: 2px solid;border-right: 2px solid;" | 40.8
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S1S3
+| style="border-left: 2px solid;border-right: 2px solid;" | 279.3
+| style="border-left: 2px solid;border-right: 2px solid;" | 38.5
+| style="border-left: 2px solid;border-right: 2px solid;" | 57.90
+| style="border-left: 2px solid;border-right: 2px solid;" | 43.6
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S3_a </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 260.8
+| style="border-left: 2px solid;border-right: 2px solid;" | 38.8
+| style="border-left: 2px solid;border-right: 2px solid;" | 70.05
+| style="border-left: 2px solid;border-right: 2px solid;" | 40.0
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S2S3
+| style="border-left: 2px solid;border-right: 2px solid;" | 194.6
+| style="border-left: 2px solid;border-right: 2px solid;" | 42.6
+| style="border-left: 2px solid;border-right: 2px solid;" | 69.45
+| style="border-left: 2px solid;border-right: 2px solid;" | 54.7
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S3
+| style="border-left: 2px solid;border-right: 2px solid;" | 193.9
+| style="border-left: 2px solid;border-right: 2px solid;" | 42.3
+| style="border-left: 2px solid;border-right: 2px solid;" | 64.15
+| style="border-left: 2px solid;border-right: 2px solid;" | 51.3
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S2
+| style="border-left: 2px solid;border-right: 2px solid;" | 182
+| style="border-left: 2px solid;border-right: 2px solid;" | 41.3
+| style="border-left: 2px solid;border-right: 2px solid;" | 39.55
+| style="border-left: 2px solid;border-right: 2px solid;" | 57.3
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S3_b </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 154.1
+| style="border-left: 2px solid;border-right: 2px solid;" | 43.5
+| style="border-left: 2px solid;border-right: 2px solid;" | 62.65
+| style="border-left: 2px solid;border-right: 2px solid;" | 55.7
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S4_a </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 145.5
+| style="border-left: 2px solid;border-right: 2px solid;" | 42.5
+| style="border-left: 2px solid;border-right: 2px solid;" | 0.00
+| style="border-left: 2px solid;border-right: 2px solid;" | 61.2
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S1S4
+| style="border-left: 2px solid;border-right: 2px solid;" | 86.1
+| style="border-left: 2px solid;border-right: 2px solid;" | 54.1
+| style="border-left: 2px solid;border-right: 2px solid;" | 0.00
+| style="border-left: 2px solid;border-right: 2px solid;" | 67.1
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  S4
+| style="border-left: 2px solid;border-right: 2px solid;" | 53.6
+| style="border-left: 2px solid;border-right: 2px solid;" | 54.7
+| style="border-left: 2px solid;border-right: 2px solid;" | 0.00
+| style="border-left: 2px solid;border-right: 2px solid;" | 69.1
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> S4_b </math>
+| style="border-left: 2px solid;border-right: 2px solid;" | 31.8
+| style="border-left: 2px solid;border-right: 2px solid;" | 53.8
+| style="border-left: 2px solid;border-right: 2px solid;" | 0.00
+| style="border-left: 2px solid;border-right: 2px solid;" | 61.8
+|}
+==7.3 Numerical study==
+As it can be observed in Section [[#7.2 Experimental study|7.2]], the experimental results of the laboratory tests, performed for the assessment of the flow performance of sucrose, show a strong relation with the particle size. A clear trend is outlined: decreasing the size of particles, the flow quality tends to be more poor. In a continuum mechanics approach, using a standard Mohr-Coulomb law (see Chapter [[#3 Constitutive Models|3]]), the set of material parameters needed does not include micro-scale parameters, such as, e.g., the particle size. In the validation study, macro-parameters are considered, instead, and used for the numerical investigation of flowability properties of different types of sugar.
+It is observed that <math display="inline"> d_{50}=200\mu m</math> represents a critical value which markedly characterizes the flowability properties of sucrose. In this respect, in Table [[#table-7.3|7.3]] a classification of the experimental results can be found; three groups are individuated depending on whether the corresponding <math display="inline"> d_{50}</math> is above (''CASE 1''), around (''CASE 2'') or below (''CASE 3'') the value of <math display="inline">200\mu m</math>. The classification is made by considering only those samples showing the behaviour in line with the previously observed trends. For this reason, the outliers are not taken into account.  In what follows, the groups, listed in Table [[#table-7.3|7.3]], are used as guideline to the numerical study of the different behaviours observed.
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-7.4'></span>Table. 7.4 Classification of flowability behaviours
+|- style="border-top: 2px solid;font-size: 85%;"
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" | CASE
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" | <math>d_{50}</math>
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" | Bulk density
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" | Repose angle
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" | <math>f_{18}[mL/s]</math>
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
 |-
-|
+| Avalanche
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left[\begin{array}{l}{\sigma _{x}} \\ {\sigma _{y}} \\ {\sigma _{z}} \\ {\tau _{y z}} \\ {\tau _{x z}} \\ {\tau _{x y}}\end{array}\right]= \left[\begin{array}{cccccc}{2 \mu{+\lambda}} & {\lambda } & {\lambda } & {0} & {0} & {0} \\ {\lambda } & {2 \mu{+\lambda}} & {\lambda } & {0} & {0} & {0} \\ {\lambda } & {\lambda } & {2 \mu{+\lambda}} & {0} & {0} & {0} \\ {0} & {0} & {0} & {\mu } & {0} & {0} \\ {0} & {0} & {0} & {0} & {\mu } & {0} \\ {0} & {0} & {0} & {0} & {0} & {\mu }\end{array}\right]\left[\begin{array}{c}{\varepsilon _{x}} \\ {\varepsilon _{y}} \\ {\varepsilon _{z}} \\ {\gamma _{y z}} \\ {\gamma _{x z}} \\ {\gamma _{x y}}\end{array}\right] </math>
+| angle
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.10)
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  CASE 1
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>> 200 \mu m</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>> 800 [g/L]</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>\simeq 35^{\circ }</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>60:70</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>\simeq 40^{\circ }</math>
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  CASE 2
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>\simeq 200 \mu m</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>700:800</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>41^{\circ }:45^{\circ }</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>60:70</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>51^{\circ }:56^{\circ }</math>
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  CASE 3
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>< 200 \mu m</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>< 700 [g/L]</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>\simeq 54^{\circ }</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>0.0</math>
+| style="border-left: 2px solid;border-right: 2px solid;" | <math>61^{\circ }:69^{\circ }</math>
 |}
-The same can be deduced analogously for orthotropic and even anisotropic materials, with the difference that the constants <math display="inline">E</math> and <math display="inline">\nu </math> are no longer unique, but have a different value for each set of directions. Even thermal effects and other phenomena can be added into the equations describing the model. However, although Hooke's law is widely used, it presents great limitations as nonlinearities emerge when the deformations are big enough to be no longer considered small. Thus, nonlinear models have to be used, and the phenomena of '''plasticity''' has to be taken into account.
+Bulk density, <math display="inline"> \rho _{bulk} </math> (Figure [[#img-7.2|7.2]]), angle of repose <math display="inline">\phi </math> (Figure [[#img-7.3|7.3]]) and angle of avalanche <math display="inline">A_a</math> (Figure [[#img-7.5|7.5]]) are monotonic with <math display="inline"> d_{50} </math>.  The flow through the orifice with a diameter of 18 mm <math display="inline">f_{18}</math> (Figure [[#img-7.4|7.4]]) is monotonic with <math display="inline">d_{50}</math>, as well. However, it can be seen that the evolution of the flow rate looks like a step-wise function, varying between 0 mL/s for values of <math display="inline">d_{50}</math> below <math display="inline">  200 \mu m </math> and 70 mL/s for values of <math display="inline">d_{50}</math> just above the critical value of a few <math display="inline"> \mu m </math>.  In this regard, for example, by comparing the samples <math display="inline"> S3_b </math> and <math display="inline"> S4_a </math>, which have similar Particle Size Distribution, but a slightly different <math display="inline">d_{10}</math> (<math display="inline"> 55.3 \mu m </math> and <math display="inline"> 38 \mu m </math>, respectively), it is observed that <math display="inline">f_{18} = 62 mL/s</math> in the first case while no flow occurs in the second one. There might be the possibility that, when <math display="inline">d_{50}\simeq 200\mu m</math>, other parameters turn out to be leading factors in the determination of the flow rate in a flat bottom silo, such as, the <math display="inline">d_{10}</math> or the particle shape. Thus, for a better understanding of the physical mechanisms and physical parameters which can determine the flowability of a certain granular material, further experimental tests should be planned and performed, as future work, which include other factors not considered in the current study. Moreover, since a remarkable difference is observed in the flow rate values for sugar samples with a <math display="inline">d_{50}</math> close to <math display="inline">  200 \mu m </math>, only the samples with a flow rate which ranges between <math display="inline">60 mL/s</math> and <math display="inline"> 70mL/s </math> fall under ''CASE 2''.
-==A.2 Classical analysis methods==
+===7.3.1 Numerical models===
-The study of the vibration behaviour of rotatory blades is a well-known problem and the main subject of rotordynamics, a specialised branch of applied mechanics. In this section, a review on the main methods  and models used over the last century for the analysis of '''rotating structures''' will be addressed, following the key concepts presented in <span id='citeF-1'></span>[[#cite-1|[1]]].
+In this section, the computational models of angle of repose apparatus, GranuFall and Revolution Powder Analyzer, are presented. By referring to the description of the devises in Section [[#7.2.2 Measurements procedure|7.2.2]], models which use axi-symmetry and plane strain assumptions are introduced. For the solution of the axisymmetric problem, the formulation presented in Appendix [[#C Irreducible formulation in axisymmetric problems|C]] is employed, while in the plane strain case, the formulation presented in Chapter [[#4 Irreducible formulation|4]] is considered.
-===A.2.1 Energy methods===
+====7.3.1.1 Angle of repose====
-One of the most fundamentals principles in Physics states that energy in the Universe is conserved. Energy can adopt many forms, but in the study of rotordynamics the most relevant are kinetic and potential (strain) energy. If energy changes from potential to kinetic repetitively in time, then vibration appears. In this section, the main methods used to describe the beam behaviour from an energetic point of view will be briefly presented.  '''
+With respect to the angle of repose device, described in Section [[#7.2.2 Measurements procedure|7.2.2]], the geometry of its numerical model is depicted in Figure [[#img-7.6a|7.6a]]; while boundary conditions are depicted in Figure [[#img-7.6b|7.6b]] where the fixed displacement along x-direction is represented in blue colour  and in red the one along both x and y-directions. As it can be seen, the 3D geometry is reduced to a 2D axisymmetric model, in order to reduce the computational cost of the numerical simulation. The gray coloured area of Figure [[#img-7.6a|7.6a]] refers to the domain initially occupied by the material points, while in yellow is the remaining area of the computational domain, where the particles are free to move. For the solution of this case an unstructured triangular mesh with element size of ''1mm'' is adopted. The time step length can range between <math display="inline"> 10^{-4}s </math> and <math display="inline"> 10^{-5}s </math>, depending on the material properties adopted in each test case, and 6 material points are initially located in each element of the grid, which is found to be an optimal trade-off between accuracy of the results and computational cost.
-====A.2.1.1 Euler-Lagrange equations===='''
+<div id='img-7.6a'></div>
+<div id='img-7.6b'></div>
+<div id='img-7.6'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-angle_of_repose_model.png|234px|]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-angle_of_repose_bc.png|228px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 7.6:''' Angle of repose apparatus: geometry (a) and boundary conditions (b)
+|}
-The following equations take advantage of variational calculus to illustrate the behaviour of a one dimensional idealization of a '''cantilever beam'''. Let's consider a function that would represent the beam problem:
+This test is the most frequently used for different sizes and combinations of funneling methods (e.g., internal flow funnel and external flow funnel or a combination of both). In this case, with the method described in Section [[#7.2.3.2 Angle of repose|7.2.3.2]] it is measured the so-called external or poured angle of repose. In the literature, the angle of repose is often assumed to be equal to the residual internal friction angle or the constant volume angle in a critical state <span id='citeF-230'></span>[[#cite-230|[230]]]. However, this assumption is valid under very restricted conditions and assumptions, as shown in <span id='citeF-231'></span>[[#cite-231|[231]]], such as, uniform density, moisture content and particles size.
+It is important to highlight that, in the experimental characterization, a device, which allows the measurement of both fluid and sticky powders, is employed. In the numerical analysis of cohesive powders, if the original geometry is adopted (see Figure [[#img-7.6a|7.6a]]), with a very narrow outlet diameter, some phenomena of arching, which do not allow to complete the analysis, are observed. In order to overcome this aspect, the minimum diameter of the opening is found and used for the evaluation of the angle of repose.
+====7.3.1.2 GranuFall====
+In this section, the numerical model of the second device, used in the experimental study, is presented. By taking into consideration the description made in Section [[#7.2.2 Measurements procedure|7.2.2]], the geometry is shown in Figure [[#img-7.7a|7.7a]]; while the boundary conditions are depicted in Figure [[#img-7.7b|7.7b]] where the fixed displacement along x-direction is represented in blue colour  and in red the one along both x and y-directions. As done for the angle of repose model of Figure [[#img-7.6a|7.6a]], the 3D geometry is reduced to a 2D axisymmetric model. For the solution of this case an unstructured triangular mesh with element size of ''1mm'' is adopted, the time step length can range between <math display="inline"> 10^{-4}s </math> and <math display="inline"> 10^{-5}s </math>, depending on the material properties adopted in each test case, and 6 material points are initially located in each element of the grid, which is found to be an optimal trade-off between accuracy of the results and computational cost.
+<div id='img-7.7a'></div>
+<div id='img-7.7b'></div>
+<div id='img-7.7'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 55%;"
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granufall_model1.png|234px|]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-granufall_model2.png|114px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 7.7:''' Granufall apparatus: geometry (a) and boundary conditions (b)
+|}
+====7.3.1.3 Revolution Powder Analyzer====
+Finally, the model of the ''Revolution Powder Analyzer'' is introduced. According to the description of the device in Section [[#7.2.2 Measurements procedure|7.2.2]], the three-dimensional geometry is reduced to a 2D plane strain model, represented in Figure [[#img-7.8|7.8]]. The area coloured in gray refers to the domain initially occupied by the material points, while in yellow is the remaining area of the computational domain, where the particles are free to move. At the boundary of the rotating drum, prescribed displacement along the x and y direction (<math display="inline">u_x</math> and <math display="inline">u_y</math>) is applied in order to impose the rotational motion at an angular velocity of <math display="inline">\omega = 0.3[rpm] = \frac{\pi }{100}[rad/s]</math>:
-<span id="eq-A.11"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,447: / Line 5,933: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>I=\int _{x_{1}}^{x_{2}} F\left(x, y, y^{\prime }, y^{\prime \prime }\right)d x </math>
+| style="text-align: center;" | <math>\left\{     \begin{array}{l}u_x= 2R \sin \left(\frac{\theta }{2}\right)\cos \left(\alpha{+\phi}\right)\\       u_y= 2R \sin \left(\frac{\theta }{2}\right)\sin \left(\alpha{+\phi}\right)     \end{array}   \right. </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.11)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (7.1)
 |}
-with <math display="inline">x</math> as independent variable and <math display="inline">y</math> and its derivatives as an admissible path. Considering the varied path defined by <math display="inline">\breve{y}=y+\varepsilon \eta </math>, being <math display="inline">\epsilon </math> a small parameter and <math display="inline">\eta </math> a differentiable function zero valued at <math display="inline">x_1</math> and <math display="inline">x_2</math>, equation [[#eq-A.11|A.11]] over the varied path reads as:
+with <math display="inline">\theta = \omega t [rad]</math>, <math display="inline">\alpha = \frac{\pi - \theta }{2} [rad]</math> and <math display="inline">\phi=2\arcsin (\frac{chord}{2R}) [rad]</math>, where R is the radius of the drum and <math display="inline">chord</math> the distance between the point where the displacement is calculated (point P) and a reference point (point O), as depicted in Figure [[#img-7.8|7.8]]. For the solution of this case an unstructured triangular mesh with element size of ''2mm'' is adopted, with a time step of <math display="inline"> 10^{-5}s </math> and 6 material points are initially located in each element of the grid, which is found to be an optimal trade-off between accuracy of the results and computational cost.
-<span id="eq-A.12"></span>
+<div id='img-7.8'></div>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 40%;"
 |-
-|
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_model.png|246px|Revolution Powder Analyzer: geometry.]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding-bottom:10px;"| '''Figure 7.8:''' Revolution Powder Analyzer: geometry.
+|}
+Depending on angular velocity, diameter of the cylinder, filling level, friction between particles and the wall, and particle material characteristics, different regimes of granular flow are observed which have been termed slipping, slumping, rolling, cascading, cataracting, and centrifuging (see Figure [[#img-7.9|7.9]]). Except for slipping where the granulate assumes a solid state and slides against the wall, these regimes can be observed in dependence on rotation velocity while all other parameters are fixed. For low angular velocity, as in the present case, the flow is non-stationary in a slumping mode. Accordingly, the free surface of the granulate forms a plane whose tilt fluctuates between the avalanche angle and the rest angle, e.g., between the state of maximum and minimum potential energy, respectively.
+<div id='img-7.9'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
 |-
-| style="text-align: center;" | <math>\breve{I}=\int _{x_{1}}^{x_{2}} F\left(x, y+\varepsilon \eta , y^{\prime }+\varepsilon \eta ^{\prime }, y^{\prime \prime }+\varepsilon \eta ^{\prime \prime }\right)d x </math>
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_modes.png|468px|Modes of motion in a Revolution Powder Analyzer<span id='citeF-232'></span>[[#cite-232|[232]]].]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding-bottom:10px;"| '''Figure 7.9:''' Modes of motion in a Revolution Powder Analyzer <span id='citeF-232'></span>[[#cite-232|[232]]].
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.12)
+===7.3.2 Numerical results===
+In this section, a validation study is presented by comparing the numerical and experimental results in terms of angle of repose, flow rate and avalanche angle, for ''CASE 1'', ''CASE 2'' and ''CASE 3''. The problem is solved at the macroscale and a study of the influence of the Mohr-Coulomb parameters, such as, the internal friction angle <math display="inline">\phi </math>, the dilatancy angle <math display="inline">\psi </math> and the apparent cohesion <math display="inline">c</math>, on the sugar flowability performance is provided according to the classification previously performed. The internal friction angle describes the bulk friction during the incipient flow of a powder and it is determined from the linearised yield locus, as shown in Figure [[#img-3.2|3.2]]; while the apparent cohesion is the intercept from the linearised yield locus and it represents the strength of a powder under zero confining pressure. Even if these material parameters are representative of the behaviour of the bulk, in reality, they can be related to micro-scale parameters, such as, the inter-particle friction and the inter-particle cohesion, respectively. It is known that the inter-particle friction is due to the interlocking generated by the shape of the particles and the surface roughness; while the inter-particle cohesion, in the case of dry powder, is due to the van der Waals forces <span id='citeF-233'></span>[[#cite-233|[233]]]. The dilatancy is representative of the volume change of a granular material when it undergoes a shear deformation, before reaching the critical state. If the material is compacted, the grains are interlocked and do not have the freedom to move; however, when the sample is stressed, a lever motion occurs between the particles in contact and a bulk expansion of the material is generated.
+Since all the experimental results are single-valued function of <math display="inline">d_{50}</math>, the particle size corresponding to 50% of the sample's mass sieved, a classification is made according to this parameter. ''CASE 1'' corresponds to samples with the highest values of <math display="inline">d_{50}</math>, typical of the common granulated sugar type. On the other hand, much lower values of <math display="inline">d_{50}</math> fall under ''CASE 3'' and these samples are associated with a powdered sugar type. Finally, with ''CASE 2'' the case of transition, where flowability properties of sucrose drastically deteriorate in the transition from ''CASE 1'' to ''CASE 3'', is individuated.
+In what follows, only the influence of the internal friction angle, dilatancy angle and apparent cohesion, is investigated. The ranges of bulk density value are provided by the experimental results (see Table [[#table-7.2|7.2]]), while other material parameters, such as, the Young modulus and Poisson's ratio, are considered to be the same in all three cases. Their values have been found in the literature <span id='citeF-234'></span>[[#cite-234|[234]]].
+====7.3.2.1 <span id='lb-7.3.2.1'></span>Case 1: d₅₀ > 200 μm====
+''CASE 1'' represents all those samples which are characterized by a <math display="inline">d_{50}</math> much higher than <math display="inline">200 \mu m</math>. The granular material, object of study, is in dry condition and in this case the amount of fine particles is so low that cohesive forces do not appear, and the bulk behaviour results to be totally cohesionless <span id='citeF-234'></span>[[#cite-234|[234]]]. Thus, a zero apparent cohesion is used in all the investigated scenarios. On the other hand, internal friction angle and dilatancy angle may play a role in the flowability performance. In this regards, different cases have been investigated varying the values of <math display="inline">\phi </math> and <math display="inline">\psi </math>. For ''CASE 1'', a bulk density value of <math display="inline">830 kg/mc</math>, for all the scenarios to be investigated, is chosen following the classification made in Table [[#table-7.4|7.4]].
+The first experimental test to be analysed is the ''GranuFall'' test. Different values of <math display="inline">\phi </math> and <math display="inline">\psi </math> are considered: the internal friction angle ranges between <math display="inline">35^{\circ }</math> and <math display="inline">46^{\circ }</math>, while the dilatancy angle between <math display="inline">0^{\circ }</math> and <math display="inline">5^{\circ }</math>. In Figure [[#img-7.10|7.10]] it can be observed that by increasing the dilatancy angle by a few degrees, the flow rate through an orifice of 18mm drastically decreases. In particular, it is found that, for values of <math display="inline">\psi </math> between <math display="inline">1^{\circ }</math> and <math display="inline">3^{\circ }</math>, values of volumetric flow rate in the range of the experimental measurements are computed. These latter ones are depicted by the dot lines (Figure [[#img-7.10|7.10]]), that, hereinafter, they are used to represent the inferior and superior limit of the experimental range, as indicated in Table [[#table-7.4|7.4]].
+<div id='img-7.10'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
+|-
+|[[Image:Draft_Samper_987121664-monograph-granufall_CASE_I_b.png|498px|Case 1. GranuFall test: volumetric flow.]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 7.10:''' Case 1. GranuFall test: volumetric flow.
 |}
-If the function above is expanded:
+The second model to be investigated, in this validation study, is represented by the test of the angle of repose. The values of internal friction angle range between <math display="inline">35^{\circ }</math> and <math display="inline">46^{\circ }</math>; while the values of dilatancy angle are constrained to the values of <math display="inline">1^{\circ }</math>, <math display="inline">2^{\circ }</math> and <math display="inline">3^{\circ }</math>. In Figure [[#img-7.11|7.11]], the results in terms of angle of repose are presented. In this case, it is found that the dilatancy angle has not an important influence on the results; while a remarkable difference is given by the internal friction angle. It is observed that to higher values of <math display="inline">\phi </math> corresponds higher values of angle of repose and, as previously discussed, it is found that the angle of repose does not coincide with the values of internal friction angle. Further, it is seen that the numerical values are close to the experimental ones, depicted by the dot lines, representing the inferior and superior limit, as indicated in Table [[#table-7.4|7.4]], when the value of <math display="inline"> \phi </math> ranges between <math display="inline">42^{\circ }</math> and <math display="inline">46^{\circ }</math>.
-<span id="eq-A.13"></span>
+<div id='img-7.11'></div>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
-|
+|[[Image:Draft_Samper_987121664-monograph-angle_of_repose_CASE_I_b.png|498px|Case 1. Angle of repose test: angle of repose.]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 7.11:''' Case 1. Angle of repose test: angle of repose.
+|}
+The last test is represented by the ''Revolution Powder Analyzer'', described in Section [[#7.3.1.3 Revolution Powder Analyzer|7.3.1.3]]. In this case, only the scenarios with a dilatancy angle of <math display="inline">1^{\circ }</math>, <math display="inline">2^{\circ }</math> and <math display="inline">3^{\circ }</math> and an internal friction angle which ranges between <math display="inline">42^{\circ }</math> and <math display="inline">46^{\circ }</math> are considered. The results in terms of avalanche and rest angle are shown in Figures [[#img-7.12a|7.12a]] and [[#img-7.12b|7.12b]], respectively. It is found that the internal friction angle has some influence on the avalanche angle, while it is not observed a strong relation with the rest angle, since, in all the scenarios investigated, the value falls in a very narrow range: between <math display="inline">31^{\circ }</math> and <math display="inline">33^{\circ }</math>. With regards to the dilatancy angle, this parameters has less influence than the internal friction angle on the numerical results. Thus, for the range of <math display="inline">\psi </math> under study it is not possible to define any relation.
+<div id='img-7.12a'></div>
+<div id='img-7.12b'></div>
+<div id='img-7.12'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 90%;"
 |-
-| style="text-align: center;" | <math>\breve{I}-(\breve{I})_{\varepsilon=0}=\breve{I}-I=\left(\frac{\partial \breve{I}}{\partial \varepsilon }\right)_{\varepsilon=0} \varepsilon{+\left}(\frac{\partial ^{2} \breve{I}}{\partial \varepsilon ^{2}}\right)_{\varepsilon=0} \frac{\varepsilon ^{2}}{2 !}+\cdots  </math>
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_num_results_case1_avalanche_b.png|498px|]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_num_results_case1_rest_b.png|498px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 7.12:''' Case 1. Revolution Powder Analyzer: avalanche angle results (a) and rest angle results (b)
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.13)
+By looking at the results obtained with the three different models, the set of material parameters, which provide numerical results which fall into the range of the experimental data for ''CASE 1'', is found and listed in Table [[#table-7.5|7.5]]
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-7.5'></span>Table. 7.5 Case 1. Set of material properties.
+|- style="border-top: 2px solid;"
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
+|-
+| Young
+|-
+| Modulus
 |}
-Neglecting high order terms, one can state that <math display="inline">\left(\frac{\partial \breve{I}}{\partial \varepsilon }\right)_{\varepsilon=0} = 0</math>. Expanding and integrating by parts, the equation admits two solutions: or either <math display="inline">\eta </math> and <math display="inline">\eta ^{\prime }</math> are zero at <math display="inline">x_1</math> and <math display="inline">x_2</math> ('''Euler-Lagrange equation''', eq. [[#eq-A.14|A.14]]) or its value is prescribed (boundary conditions, eq.  [[#eq-A.15|A.15]]).
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
+|-
+| Poisson
+|-
+| ratio
+|}
-<span id="eq-A.14"></span>
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" | Density
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
 |-
-|
+| Internal friction
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\frac{d^{2}}{d x^{2}}\left(\frac{\partial F}{\partial y^{\prime \prime }}\right)-\frac{d}{d x}\left(\frac{\partial F}{\partial y^{\prime }}\right)+\frac{\partial F}{\partial y}=0} </math>
+| angle
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.14)
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
+|-
+| Dilatancy
+|-
+| angle
 |}
-<span id="eq-A.15"></span>
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" | Cohesion
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  1e6 Pa
+| style="border-left: 2px solid;border-right: 2px solid;" | 0.3
+| style="border-left: 2px solid;border-right: 2px solid;" | 830 kg/mc
+| style="border-left: 2px solid;border-right: 2px solid;" | 44 deg
+| style="border-left: 2px solid;border-right: 2px solid;" | 2-3 deg
+| style="border-left: 2px solid;border-right: 2px solid;" | 0 Pa
+|}
+In what follows, the results obtained by using the set of material parameters of Table [[#table-7.5|7.5]] are shown. In Figures [[#img-7.13a|7.13a]], [[#img-7.13b|7.13b]] and [[#img-7.13c|7.13c]] the repose angle, the avalanche and the rest angle, evaluated in the ''Revolution Powder Analyzer'', are depicted. It can be noted that the surfaces of the heap formed by the angle of repose of Figure [[#img-7.13a|7.13a]] and the material at rest after the avalanche of Figure [[#img-7.13c|7.13c]] are very smooth. With respect to the ''Revolution Powder Analyzer'', an avalanching shear layer is interacting with a quasi-static region. In Figure [[#img-7.14|7.14]], it is possible to observe that the surface flowing layer has a reduced depth in comparison to the quasi-static one, and, consequently, the mass moving during the collapse, as well.
+<div id='img-7.13a'></div>
+<div id='img-7.13b'></div>
+<div id='img-7.13c'></div>
+<div id='img-7.13'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;"
 |-
-|
+|style="padding-bottom:10px;"|[[Image:Draft_Samper_987121664-monograph-angle_of_repose_num_results_case1-0_1cm-0_1cm-0_1cm-1_0cm.png|140px|]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|style="padding-bottom:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case1-0_1cm-0_3cm-14_7cm-0_0cm.png|190px|]]
+| colspan="2" style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case1-14_1cm-0_3cm-0_7cm-0_0cm.png|190px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
+| (c)
+|- style="text-align: center; font-size: 75%;"
+| colspan="3" style="padding:10px;"| '''Figure 7.13:''' Case 1. Numerical results of repose angle test (a) and Revolution Powder Analyzer: before collapse (avalanche angle) (b) and after collapse (rest angle) (c). Black dot lines indicate the angle formed by the inclined surfaces.
+|}
+<div id='img-7.14a'></div>
+<div id='img-7.14b'></div>
+<div id='img-7.14c'></div>
+<div id='img-7.14d'></div>
+<div id='img-7.14'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 75%;"
 |-
-| style="text-align: center;" | <math>{\left(-\frac{d}{d x}\left(\frac{\partial F}{\partial y^{\prime \prime }}\right)+\frac{\partial F}{\partial y^{\prime }}\right)_{x_{1}}^{x_{2}}=0} </math>
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case1_time_instant-0_4cm-14cm-25cm-0_0cm.png|390px|t=24.6s]]
+|style="padding-right:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case1_time_instant-25cm-14cm-0_35cm-0_0cm.png|400px|t=24.7s]]
+|- style="text-align: center; font-size: 75%;"
+| (a) t=24.6s
+| (b) t=24.7s
+|-
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case1_time_instant-0_4cm-0_4cm-25cm-14cm.png|390px|t=24.8s]]
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case1_time_instant-25cm-0_4cm-0_4cm-14cm.png|400px|t=24.9s]]
+|- style="text-align: center; font-size: 75%;"
+| (c) t=24.8s
+| (d) t=24.9s
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 7.14:''' Case 1. Revolution Powder Analyzer: velocity field at different time instants.
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.15)
+====7.3.2.2 <span id='lb-7.3.2.2'></span>Case 2: d₅₀ ≃200 [μm]====
+''CASE 2'' represents the transition between ''CASE 1'' and ''CASE 3'', where the first signs of degradation of the flowability performance are observed. As it is observed in the experimental characterization, described in Section [[#7.2 Experimental study|7.2]], these types of sugar are characterized by a median particle size, <math display="inline">d_{50}</math>, which is around the critical value of <math display="inline">200\mu m</math>. In this case, it is assumed that the internal friction angle is still a material parameter which can affect the material behaviour; on the other hand, unlike ''CASE 1'', a zero dilatancy angle is considered due to the reduced particle size (as shown in <span id='citeF-234'></span>[[#cite-234|[234]]]). Moreover,   a reduction of the particle size generates an increase of inter-particle cohesion and, consequently, of the apparent cohesion <span id='citeF-233'></span>[[#cite-233|[233]]]. Thus, <math display="inline">c</math>  is included in the set of parameters of the numerical study. For ''CASE 2'', a bulk density value of <math display="inline">740 kg/mc</math>, for all the scenarios to be investigated, is chosen following the classification made in Table [[#table-7.4|7.4]]. With respect to the internal friction angle and the apparent cohesion, the ranges <math display="inline">35^{\circ }-47^{\circ }</math> and <math display="inline">2 Pa-18 Pa</math> are considered, respectively.
+The first model, object of study, is the ''GranuFall'' test. Different scenarios are analysed and the numerical results in terms of volumetric flow are shown in Figure [[#img-7.15|7.15]]. In this case, only the results, which fall into the range defined by the experimental measurements, are depicted. As can be observed, the greater the apparent cohesion, the lower the volumetric flow. Moreover, increasing the internal friction angle, the curve which describes the relation between the volumetric flow and the apparent cohesion, is shifted to the left part of the chart. Namely, this result represents the competition between the inter-particle cohesion and the inter-particle friction in the bulk friction behaviour of those samples which belong to ''CASE 2''.
+<div id='img-7.15'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
+|-
+|[[Image:Draft_Samper_987121664-monograph-granufall_CASE_II.png|420px|Case 2. GranuFall test: volumetric flow.]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 7.15:''' Case 2. GranuFall test: volumetric flow.
 |}
-These equations describe one of the simplest structures, a one dimensional '''idealization''' of a cantilever beam. The greatness of this equation lies in converting the complex problem of solving 15 coupled differential equations (the ''Elasticity Problem'') into a simple set of equations which are precise enough for engineering applications. For this particular case, the function <math display="inline">F</math> is found by computing the potential energy of the beam after a load is applied, and reads as:
+The same competition between the internal friction angle and the apparent cohesion is observed in Figure [[#img-7.16|7.16]], where the results of the angle of repose test, which fall in the range of the experimental data, are shown. It is found that the value of the angle of repose is affected by both <math display="inline">\phi </math> and <math display="inline">c</math>: it increases when higher values of <math display="inline">\phi </math> and <math display="inline">c</math> are adopted.  Unlike ''CASE 1'', in this case it has been necessary to modify the model, described in Section [[#7.3.1.1 Angle of repose|7.3.1.1]], in order to avoid phenomena of arching, which do not allow to complete the analyses. In ''CASE 2'', a variation of the opening diameter of a few <math display="inline">mm</math> is made, with the aim of reproducing the same dynamics during the pouring of the granular material, as in the experimental test.
-<span id="eq-A.16"></span>
+<div id='img-7.16'></div>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
-|
+|[[Image:Draft_Samper_987121664-monograph-angle_of_repose_CASE_II.png|426px|Case 2. Angle of repose test: angle of repose.]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 7.16:''' Case 2. Angle of repose test: angle of repose.
+|}
+Finally, the model of ''Revolution Powder Analyzer'' is analysed. The results in terms of avalanche and rest angle are shown in Figures [[#img-7.17a|7.17a]] and [[#img-7.17b|7.17b]], respectively. It is found that higher avalanche angles correspond to higher values of internal friction angle and apparent cohesion. Moreover, these results linearly depends on the apparent cohesion and the angular coefficient increases with the <math display="inline">\phi </math>. The same observations can be done concerning the rest angle. In both the charts of Figure [[#img-7.17|7.17]], one can note that the same final results can be achieved with different sets of <math display="inline">\phi </math> and <math display="inline">c</math>. Indeed, in this case there is not a unique set of material parameters which allows to predict the experimental data, but rather two sets are individuated and they are listed in Table [[#table-7.6|7.6]].
+<div id='img-7.17a'></div>
+<div id='img-7.17b'></div>
+<div id='img-7.17'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 90%;"
 |-
-| style="text-align: center;" | <math>F=\frac{1}{2} E I\left(\frac{d^{2} w}{d x^{2}}\right)^{2}-q w </math>
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_num_results_case2_avalanche_b.png|498px|]]
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_num_results_case2_rest_b.png|498px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 7.17:''' Case 2. Revolution Powder Analyzer: avalanche angle results (a) and rest angle results (b)
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.16)
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-7.6'></span>Table. 7.6 Case 2. Sets of material properties.
+|- style="border-top: 2px solid;"
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
+|-
+| Young
+|-
+| Modulus
 |}
-Including eq. [[#eq-A.16|A.16]] into eq. [[#eq-A.14|A.14]] yields to
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
+|-
+| Poisson
+|-
+| ratio
+|}
-<span id="eq-A.17"></span>
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" | Density
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
 |-
-|
+| Internal friction
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{d^{2}}{d x^{2}}\left[E I\left(\frac{d^{2} w}{d x^{2}}\right)\right]=q </math>
+| angle
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.17)
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
+|-
+| Dilatancy
+|-
+| angle
 |}
-Better known as the '''beam equation''': <math display="inline">w</math> states for lateral deflection, <math display="inline">x</math> for the independent coordinate along the beam axis, <math display="inline">E</math> for the Young's modulus, <math display="inline">I</math> for the second moment of area of the cross-section about the ''yy'' axis and <math display="inline">q</math> for a distributed load. An advantage of using energy methods instead of the Newtonian approach is that boundary conditions are completely fixed by equation   [[#eq-A.15|A.15]], whereas with classic methods they can be difficult to visualise.  '''
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" | Cohesion
-====A.2.1.2 Other methods===='''
+|- style="border-top: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  1e6 Pa
+| style="border-left: 2px solid;border-right: 2px solid;" | 0.3
+| style="border-left: 2px solid;border-right: 2px solid;" | 740 kg/mc
+| style="border-left: 2px solid;border-right: 2px solid;" | 39 deg
+| style="border-left: 2px solid;border-right: 2px solid;" | 0 deg
+| style="border-left: 2px solid;border-right: 2px solid;" | 16 Pa
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  1e6 Pa
+| style="border-left: 2px solid;border-right: 2px solid;" | 0.3
+| style="border-left: 2px solid;border-right: 2px solid;" | 740 kg/mc
+| style="border-left: 2px solid;border-right: 2px solid;" | 47 deg
+| style="border-left: 2px solid;border-right: 2px solid;" | 0 deg
+| style="border-left: 2px solid;border-right: 2px solid;" | 10 Pa
-There are plenty of methods that use conservation of energy as basic principle to describe the behaviour of beams. The '''Lagrange method''', for example, assumes that the solution of the equation satisfies the kinematic boundary conditions. For cantilever boundary conditions, in fact, a fourth grade polynomial is found to satisfy boundary conditions in ([[#eq-A.15|A.15]]). The solution is then found by minimizing total potential energy.
+|}
-Lagrange also studied time-dependent structures, that is, the phenomena of vibration. The conservation of energy for a dynamic system states as <math display="inline">\frac{d}{d t}\left(\frac{\partial T}{\partial \dot{q}}\right)-\frac{\partial T}{\partial q}+\frac{\partial U}{\partial q}=0</math>, <math display="inline">T</math> representing the kinetic energy and <math display="inline">U</math> the strain energy. If one uses <math display="inline">n</math> terms to define both kinetic and strain energy, the following matrix system is obtained, which leads to an eigenvalue problem that was not quite practical to solve until the <math display="inline">20^{th}</math> century:
+In addition, the numerical results obtained with an internal friction angle of <math display="inline">\phi=47^{\circ }</math> and an apparent cohesion of <math display="inline">c=10[Pa]</math> are shown in Figures [[#img-7.18a|7.18a]], [[#img-7.18b|7.18b]] and [[#img-7.18c|7.18c]], where the angle of repose, the avalanche and rest angle, evaluated in the ''Revolution Powder Analyzer'', are represented. It can be observed that the surface smoothness in Figure [[#img-7.18a|7.18a]] is retained; indeed, it is quite evident and easy to recognize the angle of repose, formed by the heap. On the other hand, in Figure [[#img-7.18c|7.18c]] the surface is not completely smooth due to the cohesive behaviour shown in ''CASE 2''. In addition, with respect to the ''Revolution Powder Analyzer'', by observing Figure [[#img-7.19|7.19]] the depth of the avalanching shear layer has increased, along with the amount of mass which is flowing during the collapse.
-<span id="eq-A.18"></span>
+<div id='img-7.18a'></div>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+<div id='img-7.18b'></div>
+<div id='img-7.18c'></div>
+<div id='img-7.18'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 75%;"
 |-
-|
+|[[Image:Draft_Samper_987121664-monograph-angle_of_repose_num_results_case2c-0_1cm-0_0cm-0_1cm-0_2cm.png|130px|]]
-{| style="text-align: left; margin:auto;width: 100%;"
+|style="padding:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case2c-0_1cm-0_3cm-14_7cm-0_0cm.png|150px|]]
+| colspan="2" style="padding-bottom:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case2c-14_1cm-0_3cm-0_7cm-0_0cm.png|150px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
+| (c)
+|- style="text-align: center; font-size: 75%;"
+| colspan="3" style="padding:10px;"| '''Figure 7.18:''' Case 2. Numerical results of repose angle test (a) and Revolution Powder Analyzer: before collapse (avalanche angle) (b) and after collapse (rest angle) (c). Black dot lines indicate the angle formed by the inclined surfaces.
+|}
+<div id='img-7.19a'></div>
+<div id='img-7.19b'></div>
+<div id='img-7.19c'></div>
+<div id='img-7.19d'></div>
+<div id='img-7.19'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 75%;"
 |-
-| style="text-align: center;" | <math>[M]\{ \ddot{q}\} +[K]\{ q\} =0 \; \; \; ,\; \; \; [M]=m \int _{0}^{l} \sum f_{i} f_{j} d z  \; \; \; ,\; \; \; [K]=E I \int _{0}^{l} \sum f_{i}^{\prime \prime } f_{j}^{\prime \prime } d z </math>
+|style="padding-left:5px;|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case2c_time_instant-0_4cm-14cm-25cm-0_0cm.png|380px|t=26.4s]]
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case2c_time_instant-25cm-14cm-0_0cm-0_0cm.png|392px|t=26.6s]]
+|- style="text-align: center; font-size: 75%;"
+| (a) t=26.4s
+| (b) t=26.6s
+|-
+|[[File:Draft_Samper_987121664_5645_Fig7_19c.png|380px|t=26.8s]]
+<!--[[Image:Draft_Samper_987121664-monograph-rotating_drum_case2c_time_instant-0_4cm-0_4cm-25cm-14cm.png|492px|t=26.8s]]
+-->
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case2c_time_instant-25cm-0_4cm-0_0cm-14cm.png|392px|t=27s]]
+|- style="text-align: center; font-size: 75%;"
+| (c) t=26.8s
+| (d) t=27s
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" | '''Figure 7.19:''' Case 2: Results of rotating drum test. Velocity field at different time instants.
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.18)
+====7.3.2.3 <span id='lb-7.3.2.3'></span>Case 3: d₅₀ < 200 μm====
+''CASE 3'' is representative of all those samples which show a very poor flowability performance. These granular materials are characterized by a <math display="inline">d_{50} < 200 \mu m</math> and by a high amount of fine particles, which make them very sticky and cohesive. When the particles are small, the inter-particle cohesion dominates the flow behaviour and enhances the shear resistance. In addition, these dry cohesive interactions result in the formation of clusters, which generate many voids within the bulk, thus, at the macroscale, resulting in a low bulk density. If the bulk density is low, there are free spaces for the particles to move and the geometrical inter-locking does not play an important role in this case. For this reason, a zero dilatancy angle is considered due to the reduced particle size (as shown in <span id='citeF-234'></span>[[#cite-234|[234]]]). On the other hand, several scenarios are investigated where a different set of internal friction angle and apparent cohesion are considered. For ''CASE 3'', a bulk density value of <math display="inline">600 kg/mc</math>, for all the scenarios to be investigated, is chosen in accordance with the classification made in Table [[#table-7.4|7.4]]. With respect to the internal friction angle and apparent cohesion, values which range between <math display="inline">35^{\circ }-45^{\circ }</math> and <math display="inline">0 Pa-40 Pa</math> are considered, respectively.
+As done already in the previous cases, the first test case to be considered is the ''GranuFall'' test. According to the experimental results listed in Table [[#table-7.3|7.3]], in this case no volumetric flow takes place. In this regard, from the numerical simulation it is observed that the minimum value of apparent cohesion for which no flow is observed corresponds to <math display="inline">20 Pa</math>. With regard to the internal friction angle, no influence is noted in the numerical solutions.
+The second model, under study, is represented by the angle of repose test. In this case, the value of apparent cohesion is high enough that it is not possible to perform the simulation with the original model, described in Section [[#7.3.1.1 Angle of repose|7.3.1.1]]. This is due to the fact that the material is very cohesive and the opening diameter is too narrow in order to see the material starting flowing. In the simulation of this case, the diameter has to be increased of several times its original dimension; however, it is observed that the material is flowing down at a very high velocity generating a mismatch with the dynamics, which have taken place during the experimental test. This can create a variation in the final numerical solution and, for this reason, it is decided that the results obtained from this model are not reliable and representative of the physical process, and, consequently, they are not included in this validation study.
+Finally, the third model, to be considered, is the ''Revolution Powder Analyzer''. In this case, the internal friction angle ranges between <math display="inline">35^{\circ }</math> and <math display="inline">45^{\circ }</math>, while the apparent cohesion between <math display="inline">20 Pa</math> and <math display="inline">40 Pa</math>. In Figures [[#img-7.20a|7.20a]] and [[#img-7.20b|7.20b]] the numerical results in terms of avalanche and rest angle are represented, respectively. With regard to the avalanche angle, it is seen that the internal friction angle has more influence in correspondence of lower values of cohesion; while the apparent cohesion has a strong influence in all the scenarios considered and a linear relation is observed, with an angular coefficient inversely proportional to the internal friction angle value. With respect to the rest angle, it seems that <math display="inline">\phi </math> does not have any influence on the final results; on the other hand, a linear relation between the rest angle and the apparent cohesion is noted, with an angular coefficient common to all the investigated cases.
+<div id='img-7.20a'></div>
+<div id='img-7.20b'></div>
+<div id='img-7.20'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+|-
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_num_results_case3_avalanche_b.png|498px|]]
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_num_results_case3_rest_b.png|498px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 7.20:''' Case 3. Revolution Powder Analyzer: avalanche angle results (a) and rest angle results (b)
 |}
+As can be observed in Figure [[#img-7.20|7.20]], the set of material parameters, which are able to provide results in the ranges defined by the experimental data, represented by the dot lines, are listed in Table [[#table-7.7|7.7]]. It is found that, in ''CASE 3'', the behaviour of the samples are not affected by the internal friction angle, but are mostly influenced by the apparent cohesion.
-'''Rayleigh's approach''', on the other hand, consisted in finding a method for calculating fundamental natural frequencies with a formulation that could be '''tabulated''' and used in industry as a routine numerical operations, so the designs could be faster developed. To do so, Rayleigh used the conservation principle which states that, for a vibrating conservative system in simple harmonic motion, the maximum potential energy at maximum displacement is equal to the maximum kinetic energy when the system passes trough its equilibrium with maximum velocity.
-Another great contributor to this field was Grigoryevich '''Galerkin''', who in 1915 published an article on how to solve differential equations boundary problems with an approximate solution method. Differential equations of the type of equation [[#eq-A.14|A.14]] cannot be solved except in a very few special cases. The key here is to multiply equation  [[#eq-A.14|A.14]] by <math display="inline">\eta (x)</math>, which is assumed to be the solution itself, and then integrating this product from <math display="inline">x_1</math> to <math display="inline">x_2</math>. The solution is found by making this integral zero. In the selection of <math display="inline">\eta (x)</math>, the Galerkin method uses <math display="inline">\breve{y}</math> itself. As in the Lagrange method, if one aims to solve the beam equation, it will be necessary to approximate the deflection first.
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+|+ style="font-size: 75%;" |<span id='table-7.7'></span>Table. 7.7 Case 3. Set of material properties.
+|- style="border-top: 2px solid;"
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
+|-
+| Young
+|-
+| Modulus
+|}
-Modern Finite Element Methods rely, however, on '''Hamilton's Principle''', the most general form for all vibrating systems, and one of the most powerful when applied to rotordynamics. A general equation for beam rotordynamics is found in terms of this principle for constant angular acceleration <math display="inline">\alpha </math>:
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
+|-
+| Poisson
+|-
+| ratio
+|}
-<span id="eq-A.19"></span>
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" | Density
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
 |-
-|
+| Internal friction
-{| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{array}{c}{E I_{x x} y^{\prime \prime \prime \prime }+\rho A \left[\begin{array}{c}{\ddot{y}-\left(\omega _{0}+\alpha t\right)^{2}\left\{y+R_{2} y^{\prime \prime }-(R+z) y^{\prime }\right\}+} \\ {+2\left(\omega _{0}+\alpha t\right)\left\{\dot{y} y^{\prime }-y^{\prime \prime } \int _{z}^{l} \dot{y} d z-\int _{0}^{z} y^{\prime } \dot{y}^{\prime } d z\right\}} \\ {+\alpha \left\{y y^{\prime }-\frac{1}{2} \int _{0}^{z} y^{\prime 2} d z-y^{\prime \prime } \int _{z}^{l} y d z\right\}}\end{array}\right]-}\\ \\ { -\left(\rho I_{x x}+\frac{k \rho E I_{x x}}{G}\right)\ddot{y}^{\prime \prime }+I_{x x} k \rho ^{2} \ddot{y}=-\rho A \alpha (R+z)} \end{array} </math>
+| angle
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (A.19)
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" |
+{|  style="text-align: left; margin: 1em auto;min-width:50%;font-size: 85%;"
+|-
+| Dilatancy
+|-
+| angle
 |}
-Neglecting the effect of shear (<math display="inline">k=0</math> for long blades, as in helicopters), the equation can be simplified and some terms identified. The following equation is solved using approximate methods, and the Galerkin method is a powerful tool to put it in terms of Finite Element Method. As can be seen, Coriolis forces  and angular acceleration add non-linearities to the equation. A stiffness '''stiffening''' and '''softening''' due to rotation can also be spotted.  <div id='img-A.2'></div>
+| rowspan='2' style="border-left: 2px solid;border-right: 2px solid;" | Cohesion
+|- style="border-top: 2px solid;border-bottom: 2px solid;font-size: 85%;"
+| style="border-left: 2px solid;border-right: 2px solid;" |  1e6 Pa
+| style="border-left: 2px solid;border-right: 2px solid;" | 0.3
+| style="border-left: 2px solid;border-right: 2px solid;" | 600 kg/mc
+| style="border-left: 2px solid;border-right: 2px solid;" | 35-45 deg
+| style="border-left: 2px solid;border-right: 2px solid;" | 0 deg
+| style="border-left: 2px solid;border-right: 2px solid;" | 40 Pa
+|}
+In what follows, the results obtained by using the set of material parameters of Table [[#table-7.7|7.7]] are shown. In Figures [[#img-7.21a|7.21a]] and [[#img-7.21b|7.21b]] the avalanche and the rest angle, evaluated in the ''Revolution Powder Analyzer'', are depicted. It can be noted that the results of ''CASE 3'' are not characterized by the surface smoothness, observed in the previous cases, due to the sticky behaviour of the material. Moreover, in Figure [[#img-7.22|7.22]] it is possible to observe a flowing layer with a depth comparable to the quasi-static one and a mass which is moving in a very cohesive way.
+<div id='img-7.21a'></div>
+<div id='img-7.21b'></div>
+<div id='img-7.21'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;"
+|-
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case3-0_1cm-0_3cm-14_7cm-0_0cm.png|290px|]]
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case3-14_1cm-0_3cm-0_7cm-0_0cm.png|290px|]]
+|- style="text-align: center; font-size: 75%;"
+| (a)
+| (b)
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 7.21:''' Case 3. Numerical results of Revolution Powder Analyzer: before collapse (avalanche angle) (a) and after collapse (rest angle) (b). Black dot lines indicate the angle formed by the inclined surfaces.
+|}
+<div id='img-7.22a'></div>
+<div id='img-7.22b'></div>
+<div id='img-7.22c'></div>
+<div id='img-7.22d'></div>
+<div id='img-7.22'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-Beam_Hamilton.png|390px|Simplified equation for long rotating beams <span id='citeF-1'></span>[[#cite-1|[1]]] ]]
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case3_time_instant-0_4cm-14cm-25cm-0_0cm.png|474px|t=30.8s]]
+|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case3_time_instant-25cm-14cm-0_0cm-0_0cm.png|474px|t=30.9s]]
+|- style="text-align: center; font-size: 75%;"
+| (a) t=30.8s
+| (b) t=30.9s
+|-
+|style="padding-bottom:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case3_time_instant-0_4cm-0_4cm-25cm-14cm.png|474px|t=31s]]
+|style="padding-bottom:10px;"|[[Image:Draft_Samper_987121664-monograph-rotating_drum_case3_time_instant-25cm-0_4cm-0_0cm-14cm.png|474px|t=31.1s]]
+|- style="text-align: center; font-size: 75%;"
+| (c) t=31s
+| (d) t=31.1s
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure A.2:''' Simplified equation for long rotating beams <span id='citeF-1'></span>[[#cite-1|[1]]]
+| colspan="2" style="padding:10px;"| '''Figure 7.22:''' Case 1: Results of rotating drum test. Velocity field at different time instants.
 |}
-===A.2.2 Graphical methods===
+==7.4 Discussion==
-At the beginning of the <math display="inline">20^{th}</math>, a rapid growth of rotating machinery made the need of rapid designs in the industrial environment arise. To handle this problem, graphical methods were used to determine natural frequencies of rotating structures. The goal was to create methods to be used by semi-skilled engineers and handled in the '''shortest possible time'''.
+In this Chapter, an application of the MPM strategy in the industrial framework is presented. The main aim of this study is the experimental and numerical characterization of flowability of different sugar powders. It is experimentally found that the flowability performance strictly depends on the particle size distribution, and, in particular, a strong relation with <math display="inline">d_{50}</math> is shown. In the numerical study, a macroscopic approach is followed. The phenomenological Mohr-Coulomb plastic law is employed and the study is performed in order to find a correlation between the behaviour of different types of sugar and the macroscopic material parameters of internal friction angle, dilatancy angle and apparent cohesion. The numerical solutions have shown that, depending on the class of flowability, the bulk friction behaviour can be influenced by different parameters. In those samples characterized by a good flowability and high values of <math display="inline">d_{50}</math> (''CASE 1''), it is numerically found that the parameters, which mainly affect the shear resistance, are represented by the internal friction angle and dilatancy angle; this is confirmed by those mechanisms which have been observed to take place at the microscale: the shear resistance is due to interlocking generated by the shape of the particles and their surface roughness. For those samples with very poor flowability and a sticky behaviour (''CASE 3''), a <math display="inline">d_{50}</math> much lower than <math display="inline">200\mu m</math> is experimentally estimated. In this case, it is numerically demonstrated that the apparent cohesion is the parameter which mostly governs the macroscopic behaviour of the material; while, at the microscale, due to the high amount of fine particles, cohesive forces appear between adjacent granules and, consequently, the inter-particle cohesion is the main mechanism in the shear resistance. In the case of transition (''CASE 2''), where the first signs of a deterioration of the flowability performance are visible, it is numerically observed that the results are affected in equal measure by the internal friction angle and apparent cohesion. This can be traduced, at the microscale, in a competition between inter-particle friction and inter-particle cohesion; this might be due to the presence of fine particles, even if their quantity is limited. In conclusion, all the numerical solutions, presented in the numerical study of Section [[#7.3 Numerical study|7.3]], as demonstrated, are representative of the mechanisms which are observed to take place at the particle level and the observations made are representative and totally in line with the experimental results of Section [[#7.2 Experimental study|7.2]].
-At that time, beam theory was already developed and used in many fields. A very common method to handle vibration problems consisted on drawing both the shear force and bending moment diagrams. With those, the deflection diagram could also be drawn and the integration was converted into graphical summation. This is a graphical way to interpret the '''Stodola-Viannello method''', which considered deflection itself as a proportional load that produces the deflection. However, this process of drawing the diagrams is rather slow and depends on the accuracy of the drawings. That is why the Stodola-Viannello method became popular in the industrial world not in its graphical form, but as a tabulated iterative process.
+=8 Conclusions and future work=
-In 1922, '''Holzer''' proposed another tabulated iterative method, based on Stodola's, which was able to capture higher torsional modes and its frequency. This method was widely used, and for instance a Railway Diesel-Electric Drive was designed. The main drawback of the Holzer method was the impossibility to predict deflection, slope, shear and bending moments.
+In this Chapter, the conclusions of the monograph are presented and an overview of the future lines of research is made.
-The first graphical method used in aircraft wings and turbine blades was '''Myklestad’s method''', widely spread during World War II. This tabular iterative method follows Holzer's in the way of capturing high torsional modes, but also computes deflection, slope, shear and bending moments. Phrol's method is quite similar to Myklestad’s, but addressed to shafts instead of aircraft structures. The main advantage of the latter was that the analysis included the rotatory inertia of the disks, thus adding the so-called '''gyroscopic effect'''.
+==8.1 Concluding remarks==
-This methods were widely used until the first half of the past century. With the advanced computational facilities, finite element methods gradually overtook these tabular and energy methods.
+The main aim of this work was the development of a numerical strategy for the simulation of quasi-static and dense granular flows in the industrial and engineering framework. These kinds of problems are characterized by a non-linear behaviour of the material and by large deformation of the continuum during the whole flow process. In order to perform a numerical investigation, a strategy, which is able to consider these non-linearities, is needed.  It is found that, among all the numerical methods, mostly used for the solution of granular flows problems, the Material Point Method (MPM) is the one which has shown the most suited capabilities for the cases targeted to study in this monograph (Chapter [[#2 Particle Methods|2]]). In the current work, an implicit MPM has been developed by the author in the multi-disciplinary Finite Element codes framework ''Kratos Multiphysics'' <span id='citeF-29'></span><span id='citeF-30'></span><span id='citeF-31'></span>[[#cite-29|[29,30,31]]] and in order to obtain a verified and validated numerical strategy the following points have been performed (Chapters [[#3 Constitutive Models|3]], [[#4 Irreducible formulation|4]], [[#5 Mixed formulation|5]] and [[#6 Validation|6]]):
-===A.2.3 Matrix methods===
+* Three phenomenological constitutive laws, implemented within the MPM numerical strategy, are introduced and their formulations are derived. In particular, a hyperelastic Neo-Hookean, a hyperelastic-plastic J2 and Mohr-Coulomb plastic laws, along with their limits of applicability, are discussed. All the constitutive materials are defined under the assumption of isotropy and finite strains. With the object of the current monograph in hand, the focus is mainly on the Mohr-Coulomb plastic law, where in this work, to the knowledge of the author, the return mapping is used for the first time under finite strain assumption. This phenomenological law is commonly adopted in the modelling of granular materials, since it shows a pressure-dependent behaviour. This constitutive law is used in the verification, validation and application of the MPM strategy.
-As it has already been addressed in equation [[#eq-A.18|A.18]], the Lagrange method enables the transformation of the differential equations governing the dynamics of free vibration systems into homogeneous algebraic equations. This leads to a well known '''eigenvalue problem''', were the key is to find the vibration mode shapes (eigenvectors) and the natural frequencies (eigenvalues). However, the computational needs for practical systems were not present until the late <math display="inline">20^{th}</math> century.
+* A variational displacement-based formulation, based on an Updated Lagrangian description, is presented and its derivation is described in detail. A verification of the MPM code is performed through some benchmark tests, typical in solid and geo-mechanics. Moreover, in the verification analysis, a comparison of the MPM code is done against the Galerkin Meshfree Method (GMM), a continuum particle-based technique. Since both the methods are implemented in the ''Kratos Multiphysics'' platform, it has been possible to perform a more objective comparison, which allows to better appreciate the main differences between these two techniques. In GMM the Eulerian background grid is replaced by a Lagrangian one and, unlike MPM, the shape functions are evaluated once the cloud of nodes of each material point is defined. The comparison is made with the aim of assessing the accuracy and robustness of the two methods in the simulation of cohesive-frictional materials, both in static and dynamic regimes and in problems dealing with large deformations. It is found that MPM leads to more accurate results and its robustness is proven. On the other hand, it is observed that the accuracy of GMM strictly depends on the choice of the basis functions and a modification of the algorithm has to be considered in large displacement cases. After this study, it is demonstrated that the MPM strategy represents a good choice to handle problems involving history-dependent materials and large deformations.
-However, some methods were developed to handle these kind of systems with relative ease: Jacobsen and Ayre's, Gräffe's, Prieb's and Hahn's are just a few examples of methods that dealt with matrix systems to solve structures.
+* A variational displacement and pressure-based <math display="inline">\boldsymbol{u}-p</math> formulation, based on an Updated Lagrangian description, is presented and its derivation is described in detail. The <math display="inline">\boldsymbol{u}-p</math> mixed formulation is developed with the aim of solving granular flow problems which undergo nearly-incompressible conditions. To the knowledge of the author, the treatment of the incompressibility constraint is relatively new in the context of MPM and this formulation represents an original solution among the works that one can find in the literature. A verification of the MPM code is performed through some benchmark tests, typical in solid mechanics. In the verification analysis, the results obtained through the mixed formulation are compared with those evaluated by means of the displacement-based one. As expected, it is noted that, in nearly-incompressible conditions, the typical issue of volumetric locking is overcome and pressure oscillations are avoided with the <math display="inline">\boldsymbol{u}-p</math> formulation. In addition, it is found that also in compressible cases the <math display="inline">\boldsymbol{u}-p</math> formulation provides more accurate results than the irreducible one. However, despite a higher accuracy in terms of displacement, equivalent plastic strains and vertical Cauchy stress fields, the mixed formulation, presented in this work, is not able to fix some other issues, such as, mesh independence and strain localization.
-As matrix methods proven to be a good promise for vibration problems, Duncan and Collar developed an iterative method (essentially an extension of Stodola-Vianello's) that enabled to handle relatively large matrices. '''Transfer matrix''' methods also gained popularity because they were easier to adapt for computerization, as they only involve simple matrix multiplications, and the overall transfer matrix is always small. The latter method was extensively used in rotor dynamics calculations for determining critical speeds and unbalanced response. This method was extended even for transient whirl analysis.
+* The MPM strategy, developed in the ''Kratos Multiphysics'' platform, is employed to solve typical problems of geo-mechanics. Two problems are considered in the validation study: the first one is represented by the typical test of column collapse of a dry cohesionless granular material and the second one by the evaluation of the bearing capacity of an undrained soil in the rigid strip footing test. In the first case, the irreducible formulation is employed and the numerical results are compared against experimental results. Different geometries of the column are investigated. It is both experimentally and numerically observed that the dynamic of the collapse strictly depends on the initial geometry. It is seen that for lower aspect ratio the MPM code is able to provide results in agreement with the experimental ones. On the other hand, in the case of higher aspect ratio, the MPM strategy underestimates the dissipation which takes place during the collapse of the column. In this regard, the disagreement might be mostly due to the employed constitutive law: the Mohr-Coulomb plastic law adopted in the current work is not able to predict the real energy dissipation that should have taken place during the failure mechanism. Indeed, as indicated in some similar works available in the literature, the evolution of some factors, such as, density and dilatancy, which play important role in defining the first failure surface, and, hence, the total mass that will move, are not here considered. In the second test, the <math display="inline">\boldsymbol{u}</math> and <math display="inline">\boldsymbol{u}-p</math> formulations are both employed. It is observed that the mixed formulation is able to provide better results in terms of displacement and stress field. By sampling the computed stress field at the edge, where the imposed displacement of the rigid footing is applied, it has been possible to define the curve which describes the bearing capacity of the soil. This result is compared with a curve obtained from a sequential limit analysis and a good agreement is observed between the two solutions.
-But as processing power of computers started to grow, tabular and matrix methods lost their place and nowadays have no use in modern designs.
+Finally, in Chapter [[#7 Application to an industrial case|7]] the MPM strategy is employed in an industrial framework, in the context of a collaboration with Nestlé. The numerical results are compared against unpublished experimental measurements performed for the assessment of the flowability performance of different types of sucrose. It is experimentally observed that the flowability performance is strictly dependent on the particle size distribution of the granular material and, in particular, on the median particle size <math display="inline">d_{50}</math>. On the other hand, the numerical study is performed by following a macroscopic approach and the flowability is studied according to macro-parameters, such as, the internal friction angle, the dilatancy angle and the apparent cohesion. In this study, the MPM strategy has been successfully employed and the advantage of its usage, as a complementary tool for a better understanding of the granular flow process, is demonstrated.
-=B Elemental matrix computation=
+==8.2 Future work==
-When the physical system of study is posed in terms of the FEM, spatial coordinates are discretised and the resulting equations are ruled by a set of global matrices (see in fact equation [[#eq-3.14|3.14]]). That is why matrix computation is always an important step of every FEM solver. However, those matrices involve integrals over the domain which have no closed solution unless they are split into local or elemental coordinates. Under those conditions, the global matrices are decomposed into smaller, '''elemental''' matrices. If mapped into the parent domain, the integrals can be approximated using the Gauss Quadrature method. The aim of this chapter is to review the process of assembly: how global matrices are built from elemental ones. The approximation of the latter will also be assessed.
+From the observation made and the conclusions drawn, the future lines of research are provided. It has been demonstrated that the MPM is not only a robust numerical tool for the simulation of problems involving large displacement and large deformation, but it is also an optimal platform for the implementation of complex constitutive laws. In this work, it is observed that the implemented Mohr-Coulomb plastic law is not able to predict the real energy dissipation that should have taken place during the failure mechanism. In the context of material modelling, a research should be focused on implementing constitutive laws with features, able to improve the prediction in terms of triggering of the collapse and amount of energy dissipated during the flow process. During the PhD, a collaboration with the Multiscale Mechanics group (MSM, University of Twente), lead by Prof. Stefan Luding, has started with the aim of implementing in the MPM strategy framework an elastic isotropic micro-based constitutive model for granular materials. This constitutive law, valid under quasi-static/elastic regime, has been developed by the researchers of the MSM group <span id='citeF-235'></span><span id='citeF-236'></span><span id='citeF-237'></span><span id='citeF-238'></span><span id='citeF-239'></span>[[#cite-235|[235,236,237,238,239]]], by performing a series of DEM simulations of isotropic compression and pure shear tests. By averaging some microscopic quantities, that can be directly retrieved from the DEM simulations, macro parameters like stress and fabric <span id="fnc-2"></span>[[#fn-2|<sup>1</sup>]] tensors can be evaluated. Then, by applying the definition of bulk and shear moduli, their expressions as a function of micro parameters, such as the volume fraction, pressure and coordination number, are obtained. Thus, the elastic mechanical properties are not considered constant, as it is usually assumed in a phenomenological model, but they vary accordingly with the evolution of the micro-structure.
-==B.1 The static stiffness matrix==
+With regard to the numerical formulation, other mixed variational formulations, which allow to overcome the issues of mesh dependence and strain localization, can be considered to improve the accuracy of the results.
-From equation [[#eq-3.13|3.13]], the expression of the stiffness matrix  for static structures <math display="inline">\mathbf{K}_{\hbox{ static}}</math> is, in terms of the FEM:
+In addition, in this work the applicability of the developed MPM strategy is limited to the simulation of dry granular flows. Other applications, where MPM is still a suited choice, is represented by granular flows interacting with rigid or deformable structures. Some examples can be found in the field of environmental engineering, such as, landslides interacting with systems of protective barriers or the quantitative risk assessment in landslide prone-area. For the development of the numerical strategy, a coupling between the MPM code and a FEM code is needed.  In this regard, a work, mainly focused on the imposition of boundary conditions and contact algorithm, is still missing. However, some promising results have been already done within the ''Kratos'' team and published in <span id='citeF-240'></span><span id='citeF-241'></span>[[#cite-240|[240,241]]], where algorithms for the imposition of non-conforming boundary conditions and frictional contact are presented and tested.
+Last but not least, the important aspect of parallelisation of the code, which is not addressed in the current monograph, has to be developed as future work. In the current state, the MPM strategy uses an OpenMP method, which is able to guarantee a sustainable computational cost for the simulation of real scale systems. However, in order to fully exploit the MPM capabilities and make the application competitive with other commercial and open-source software, a modification of the code in favour of a MPI parallelisation should be addressed.
+<span id="fn-2"></span>
+<span style="text-align: center; font-size: 75%;">([[#fnc-2|<sup>1</sup>]]) The fabric tensors is a tensor able to provide information which characterize the anisotropic architecture of the microstructure in a porous material</span>
+=Appendix A. Plastic flow rule in finite strains regime=
+In this section a general plastic flow rule within the framework of multiplicative plasticity is defined and it is demonstrated that if the ''specific strain energy function'' is expressed in terms of Hencky strains it is possible to recover the small strains format return mapping <span id='citeF-155'></span>[[#cite-155|[155]]]. In order to formulate a plastic flow rule in finite strains regime, it is convenient to introduce the kinematic quantities of rate of plastic deformation <math display="inline">\boldsymbol{D}^p</math> and the plastic spin tensor <math display="inline">\boldsymbol{W}^p</math>, defined as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,607: / Line 6,393: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}_{\hbox{ static}} = \int _{{\mho }}^{ } \mathbf{B}^T \mathbf{C}\;\mathbf{B} \; d\mho  </math>
+| style="text-align: center;" | <math>\boldsymbol{D}^p\equiv \mathrm{sym}\left[\boldsymbol{L}^p \right] </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.1)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.1)
 |}
-The above integral is to be evaluated over the domain <math display="inline">\mho </math>, which becomes infeasible unless local formulation is adopted. Using the formulation presented in section [[#eq-3.26|3.26]], the above integral can be splitted into elemental integrals as follows:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,619: / Line 6,403: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}_{\hbox{ static}}=\sum _{e=1}^{n_{e l}} \mathbf{L}^{e^T}\; \mathbf{K}^{e}_{\hbox{ static}} \;\mathbf{L}^{e}\;,\;\;\;\hbox{   with  }\;\;\;\mathbf{K}^{e}_{\hbox{ static}} =  \int _{{\mho }}^{ } \mathbf{B}^{e^T} \mathbf{C}\;\mathbf{B}^e \; d\mho ^e </math>
+| style="text-align: center;" | <math>\boldsymbol{W}^p\equiv \mathrm{skew}\left[ \boldsymbol{L}^p \right] </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.2)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.2)
 |}
-Where <math display="inline">\mathbf{L}^{e}</math> is a <math display="inline">n_{d} n^{e} \times n_{d} n_{pt}</math> matrix known as '''boolean''' connectivity matrix, consisting of the integers '''0''' and '''1''' relating element nodal variables <math display="inline">\boldsymbol{d}^{e} \in \mathbb{R}^{n_{d} n^{e}}</math> with the global nodal vector (<math display="inline">\mathbf{d}^{e}=\mathbf{L}^{e} \mathbf{d}</math>). With all, the problem of computing a single <math display="inline">{n}_{{d}} {n}_{{pt}} \times {n}_{{d}} {n}_{{pt}}</math> global matrix reduces to the computation of <math display="inline">\mathbf{n}_{\mathbf{el}}</math> <math display="inline">{n}_{{d}} {n}^{e} \times {n}_{{d}} {n}^{e}</math> elemental matrices. For the sake of simplicity, the assembly operation is expressed as:
+where <math display="inline">\boldsymbol{L}^p\equiv \dot{\boldsymbol{F}}^p \left(\boldsymbol{F}^p \right)^{-1}</math> is the plastic part of the velocity gradient <math display="inline">\boldsymbol{L}\equiv \nabla _x \boldsymbol{v}</math>. Since <math display="inline">\boldsymbol{D}^p</math> is a kinematic variables defined in the intermediate configuration, it is useful to perform the following rotation of <math display="inline">\boldsymbol{D}^p</math> in order to express it in the spatial configuration:
+<span id="eq-A.3"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,631: / Line 6,416: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}_{\hbox{ static}}=\overset{n_{el}}{\underset{e=1}{\mathbf{A}}} \mathbf{K}^{e}_{\hbox{ static}} </math>
+| style="text-align: center;" | <math>\boldsymbol{\tilde{D}}^p\equiv \boldsymbol{R}^e\boldsymbol{D}^p\boldsymbol{R}^{e^T}=\boldsymbol{R}^e\mathrm{sym}\left[\boldsymbol{L}^p \right]\boldsymbol{R}^{e^T}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.3)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.3)
 |}
-The only thing left is finding a reliable method to compute <math display="inline">\mathbf{K}^{e}_{\hbox{ static}}</math>. As integration boundaries may differ from one element to another, the best method to evaluate elemental matrices is by transforming them into the '''parent''' domain, where their geometry is normalised and thus Gauss Quadrature can be applied in the same way for all elements. Using the formulation presented in [[#3.3.1 The Gauss Quadrature integration method|3.3.1]]:
+with <math display="inline">\boldsymbol{R}^e</math> is the orthogonal tensor used in the polar decomposition
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,643: / Line 6,428: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}^{e}_{\hbox{ static}} =  \int _{{\mho }}^{ } \mathbf{B}^{e^T} \mathbf{C}\;\mathbf{B}^e \; d\mho ^e  =  \int _{{\mho }}^{ } <code>J</code>^{e} \;\mathbf{B}^{e^T} \mathbf{C}\;\mathbf{B}^e \; d\mho _{\xi } = \sum _{g=1}^{m} {w}_{g}\left(<code>J</code>^{e} \mathbf{B}^{e^{T}} \mathbf{C B}^{e}\right)_{\boldsymbol{\xi }=\boldsymbol{\xi }_g} </math>
+| style="text-align: center;" | <math>\boldsymbol{F}^e=\boldsymbol{V}^e\boldsymbol{R}^e </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.4)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.4)
 |}
-The number of '''Gauss points''' <math display="inline">m</math> required for the integration depends on the order of the employed shape functions, and thus is to be specified for each type of element.
+where <math display="inline">\boldsymbol{V}^e</math> is the left stretch tensor.
-==B.2 The force vector==
-In equation [[#eq-4.1|4.1]], <math display="inline">\mathbf{F}</math> stands for the force vector accounting for the effect of the body forces <math display="inline">\mathbf{F}_b</math>, boundary tractions <math display="inline">\mathbf{F}_t</math> and centrifugal forces <math display="inline">\mathbf{F}_{\hbox{ rot}}</math>. The methodology used to compute them is the same as in the previous case: splitting the global vector into elemental vectors and mapping them into the parent domain, where Gauss Quadrature can be applied.
-===B.2.1 Body forces===
+Given a general plastic potential <math display="inline">g(\boldsymbol{\tau })</math> defined in terms of the Kirchhoff stress tensor <math display="inline">\boldsymbol{\tau }</math> and the rate of the plastic multiplier <math display="inline">\dot{\gamma }</math>, the evolution of the plastic deformation gradient is defined by the following constitutive equation:
-Recall from equation [[#eq-3.13|3.13]] that body and rotation force vectors can be expressed in terms of the FEM as:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,663: / Line 6,442: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}_b = \int _{{\mho }} {\mathbf{N}}^{T} \boldsymbol{f} d \mho \;\;,\;\;\;\;\;\;\mathbf{F}_{\hbox{rot}} = -\int _{{\mho }}^{ } \mathbf{N}^T \rho \;(\widetilde{\Omega }^2 + \widetilde{\alpha })\;\mathbf{r}_i \; d\mho  </math>
+| style="text-align: center;" | <math>\boldsymbol{\tilde{D}}^p=\dot{\gamma }\frac{\partial g}{\partial \boldsymbol{\tau }} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.5)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.5)
 |}
-Where <math display="inline">\boldsymbol{f}</math> stands for the vector of external body forces acting on the structure. Notice that the shape of both <math display="inline">\mathbf{F}_b</math> and <math display="inline">\mathbf{F}_{\hbox{rot}}</math> is quite similar, and so they can be written as:
+by postulating a zero plastic spin <math display="inline">\boldsymbol{W}^p</math>, which is compatible with the assumption of plastic isotropy. With Equation [[#eq-A.3|A.3]] in hand and the following property <math display="inline">\boldsymbol{R}^{e^T}=\boldsymbol{R}^{e^{-1}}</math>, it is possible to rewrite the evolution equation as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,675: / Line 6,454: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}_{\hbox{(b + rot)}}  = \mathbf{F}_b + \mathbf{F}_{\hbox{rot}} = \int _{{\mho }} {\mathbf{N}}^{T} \left(\boldsymbol{f} - \rho \;(\widetilde{\Omega }^2 + \widetilde{\alpha })\;\mathbf{r}_i \right)d \mho \; = \int _{{\mho }} {\mathbf{N}}^{T}\; \; \widehat{\mathrm{f}} \; \; d \mho \;,  </math>
+| style="text-align: center;" | <math>\boldsymbol{L}^p\equiv \dot{\boldsymbol{F}}^p \left(\boldsymbol{F}^p \right)^{-1}=\dot{\gamma }\boldsymbol{R}^{e^T}\frac{\partial g}{\partial \boldsymbol{\tau }}\boldsymbol{R}^{e} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.6)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.6)
 |}
-Being <math display="inline">\widehat{\mathrm{f}}</math> the total contribution of body and rotation forces. As in the previous case, the global vector can be splitted into elemental force vectors using the assembly operator:
+In the definition of the plastic problem an isotropic perfectly plastic constitutive model is assumed. In this regard, the model is defined by postulating
+* a ''specific strain energy function'' <math display="inline">\Psi </math>, from which the hyperelastic law is derived;
+* a yield function <math display="inline">f</math> which defines when plastic flow starts;
+* a plastic potential <math display="inline">g</math>, from which the plastic flow rule is derived.
+Thus, the basic constitutive initial value problem states: ''given an initial value of <math>\boldsymbol{F}^p</math> at <math>t_0</math> and the history of the deformation gradient <math>\boldsymbol{F}(t)</math> for <math>t\in \left[t_0,T\right]</math>, find the functions <math>\boldsymbol{F}^p(t)</math> and <math>\dot{\gamma }(t)</math> which satisfy the flow rule''
+<span id="eq-A.7"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,687: / Line 6,474: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}_{\hbox{(b + rot)}}=\overset{n_{el}}{\underset{e=1}{\mathbf{A}}} \mathbf{F}_{\hbox{(b + rot)}}^{e} = \overset{n_{el}}{\underset{e=1}{\mathbf{A}}}   \int _{{\mho }} {\mathbf{N}}^{e^T}\; \; \widehat{\mathrm{f}} \; \; d \mho ^e </math>
+| style="text-align: center;" | <math>\dot{\boldsymbol{F}}^p(t) \left(\boldsymbol{F}^p(t) \right)^{-1}=\dot{\gamma }(t)\boldsymbol{R}^{e^T}(t)\frac{\partial g(t)}{\partial \boldsymbol{\tau }}\boldsymbol{R}^{e}(t)  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.7)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.7)
 |}
-The problem lies now in how to define the vector <math display="inline">\widehat{\mathrm{f}}</math> along each element. A common approach is to replace it by an '''approximation''' constructed by interpolation of its nodal values <math display="inline">\widehat{\mathrm{f}}\;^e</math> using the same shape functions as for the displacement field:
+''and the Kuhn-Tucker loading/unloading conditions''
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,699: / Line 6,486: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left.\widehat{\mathrm{f}}\;\right|_{\boldsymbol{x} \in \overline{\mho }^{\;e}} \approx N^{e} \;\;\widehat{\mathrm{f}}\;^e \;,\;\;\;\hbox{   with  }\;\;\; \widehat{\mathrm{f}}\;^e = \left[\begin{array}{c}{\widehat{\mathrm{f}}\;\left(\boldsymbol{x}_{1}^{e}\right)} \\ {\widehat{\mathrm{f}}\;\left(\boldsymbol{x}_{2}^{e}\right)} \\ {\vdots } \\ {\widehat{\mathrm{f}}\;\left(\boldsymbol{x}_{n^{e}}^{e}\right)}\end{array}\right] </math>
+| style="text-align: center;" | <math>\dot{\gamma }(t)\geq 0 \quad f(\boldsymbol{\tau }(t))\leq 0 \quad \dot{\gamma }(t) f(\boldsymbol{\tau }(t))=0 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.8)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.8)
 |}
-Note that this approximation can induce errors if, for example, two neighbouring elements have different densities, as density would be a nodal property rather than an elemental one. Using Gauss Quadrature, the sum of the elemental body and rotation force vectors is expressed as:
+with <math display="inline">f(\boldsymbol{\tau })</math> the yield function and <math display="inline">\boldsymbol{\tau }(t)</math> defined as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,711: / Line 6,498: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}_{\hbox{(b + rot)}}^e= \int _{{\mho }} {\mathbf{N}}^{e^T}\; \; (\mathbf{N}^{e} \;\;\widehat{\mathrm{f}}\;^e)\; d \mho ^e = \sum _{g=1}^{m} {w}_{g}\left(<code>J</code>^{e} \mathbf{N}^{e^{T}} \; (\mathbf{N}^{e} \;\;\widehat{\mathrm{f}}\;^e) \; \right)_{\boldsymbol{\xi }=\boldsymbol{\xi }_g} </math>
+| style="text-align: center;" | <math>\boldsymbol{\tau }(t)=\frac{\partial \Psi (t)}{\partial \boldsymbol{\epsilon }^e(t)} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.9)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.9)
 |}
-===B.2.2 Traction forces===
+where <math display="inline">\boldsymbol{\epsilon }^e(t)</math> is the elastic Hencky strain.
-When determining the external forces due to boundary traction conditions <math display="inline">\mathbf{F}_t</math>, it is convenient to distinguish between point loads <math display="inline">\mathbf{F}_{\hbox{pnt}}</math> and distributed loads <math display="inline">\mathbf{F}_{\hbox{dis}}</math>. The determination of <math display="inline">\mathbf{F}_{\hbox{pnt}}</math> is straightforward: if a point force <math display="inline">\mathbf{Q }\in \mathbb{R}^{n_{d}}</math> is applied to node <math display="inline">\mathbf{A}</math>, then <math display="inline">(\mathbf{F}_{\hbox{pnt}})_A = \mathbf{Q}</math>. Point loads are a particular case of distributed loads, where the force function is the Dirac delta operator: zero valued everywhere but '''A''', where it presents a spike.
+The algorithmic procedure to be established in order to solve the plastic problem is based on the discrete form of the evolution equation (see Equation [[#eq-A.7|A.7]]). Accordingly, a time stepping algorithm is performed by applying a backward exponential integrator on Equation [[#eq-A.7|A.7]], which leads to the updated formula for the plastic deformation gradient:
-The study of distributed loads requires from a more meticulous approach, and a new formulation regarding '''boundary elements''' is to be posed. To start with, let's define <math display="inline">\mathbf{E}_b^i</math> as the set of elements lying on the Neumann boundary <math display="inline">\Gamma _{\sigma }^{i}</math>, with <math display="inline">i=1,\;\cdots \;,n_d</math> :
+<span id="eq-A.10"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,727: / Line 6,513: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{E}_{b}^{i} :=\left\{e \in \left\{1,2 \ldots n_{e l}\right\}| \Gamma _{\sigma }^{e, i}=\Gamma ^{e} \cap \Gamma _{\sigma }^{i} \neq \varnothing \right\} </math>
+| style="text-align: center;" | <math>\boldsymbol{F}^p_{n+1}=\boldsymbol{R}^{e^{T}}_{n+1}\mathrm{exp}\left[\Delta \gamma \frac{\partial g_{n+1}}{\partial \boldsymbol{\tau }} \right]\boldsymbol{R}^e_{n+1}\boldsymbol{F}^p_{n}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.10)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.10)
 |}
-Being <math display="inline">n^{e}_{b}</math> the number of nodes per boundary element, and assuming for simplicity it is the same for all elements, one can define <math display="inline">\mathbf{CNb}^i</math> as the boundary connectivity matrix, a <math display="inline">{n}_{{el}} ^{{b,i}} \times {n}_{{b}} ^{{e}}</math> matrix containing, in each row, the global indices of the boundary nodes of each patch <math display="inline">\Gamma _{\sigma }^{e, i}\left(e \in E_{b}^{i}\right)</math>. For each boundary element <math display="inline">\Gamma _{\sigma }^{e, i}</math>, a matrix of boundary shape functions is defined:
+Equation [[#eq-A.10|A.10]] can be written in terms of the current elastic deformation gradient <math display="inline">\boldsymbol{F}^e_{n+1}</math> as follows
+<span id="eq-A.11"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,739: / Line 6,526: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\overline{N}^{e} :=\left[\begin{array}{cccc}{\overline{N}_{1}^{e}} & {\overline{N}_{2}^{e}} & {\cdots } & {\overline{N}_{n_{b}^{e}}^{e}}\end{array}\right] </math>
+| style="text-align: center;" | <math>\boldsymbol{F}^e_{n+1}=\boldsymbol{f}_{n+1}\boldsymbol{F}^e_{n}\boldsymbol{R}^{e^{T}}_{n+1}\mathrm{exp}\left[- \Delta \gamma \frac{\partial g_{n+1}}{\partial \boldsymbol{\tau }} \right]\boldsymbol{R}^e_{n+1}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.11)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.11)
 |}
-Notice that the boundary shape functions are different from the ones used to interpolate the displacements, as the dimension of the boundary is always '''smaller''' than the dimension of the elements, <math display="inline">\overline{N}_{a}^{e} : \Gamma _{\sigma }^{e, i} \rightarrow \mathbb{R}</math>.  Recall the definition of the global vector of prescribed boundary tractions (equation [[#eq-4.1|4.1]] ), whose decomposition into elemental integrals reads as:
+where the expressions of the incremental deformation gradient <math display="inline"> \boldsymbol{f}_{n+1}\equiv \boldsymbol{F}_{n+1}\boldsymbol{F}^{-1}_{n}</math> and Equation [[#eq-3.53|3.53]] are employed.
+In what follows, few steps are performed in order to express Equation [[#eq-A.11|A.11]] in terms of Hencky strains and it is demonstrated that the final form has the same format of the elastic strain update formula of the return mapping algorithm defined under the assumption of infinitesimal strains <span id='citeF-155'></span>[[#cite-155|[155]]].
+By post-multiplying Equation [[#eq-A.11|A.11]] by <math display="inline">\boldsymbol{R}^{e^{T}}</math>
+<span id="eq-A.12"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,751: / Line 6,543: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}_{\hbox{dis}} = \sum _{i=1}^{n_{ d}} \int _{\Gamma _{\sigma }^{i}} \overline{\mathbf{{N}}}_{i}^T \;\overline{t}^{\,i} d \Gamma = \sum _{i=1}^{n_{d}}\;\; \sum _{e\, \in \, \mathbf{E}_{b}^i}\left(\overline{\mathbf{L}}^{e, i}\right)^{T} \int _{\Gamma _{\sigma }^{e,i}}  \overline{\mathbf{{N}}}_{i}^T \;\overline{t}^{\,i} d \Gamma  </math>
+| style="text-align: center;" | <math>\boldsymbol{V}^{e}_{n+1}=\boldsymbol{f}_{n+1}\boldsymbol{F}^e_{n}\boldsymbol{R}^{e^{T}}_{n+1}\mathrm{exp}\left[- \Delta \gamma \frac{\partial g_{n+1}}{\partial \boldsymbol{\tau }} \right]  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.12)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.12)
 |}
-Where <math display="inline">\overline{\mathbf{L}}^{e, i}</math> is the Boolean operator linking the global <math display="inline">\left({\mathbf{c}} \in \mathbb{R}^{n_d \;n_b^e}\right)</math> and local <math display="inline">\left(\overline{\mathbf{c}}^{e, i} \in \mathbb{R}^{n_b^e}\right)</math>  vectors of test coefficients. The above expression can be expressed as:
+and moving the exponential to the left side of the equation
+<span id="eq-A.13"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,763: / Line 6,556: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}_{\hbox{dis}} = \sum _{i=1}^{n_{d}} \mathbf{F}_{\hbox{dis}}^{i} = \sum _{i=1}^{n_{d}} \;\;\sum _{e\, \in \, \mathbf{E}_{b}^{i}}\left(\overline{\mathbf{L}}^{e, i}\right)^{T} \mathbf{F}_{d i s}^{e, i} = \sum _{i=1}^{n_{d}}\;\; \sum _{e \,\in \, \mathbf{E}_{b}^i} \left(\overline{\mathbf{L}}^{e, i}\right)^{T} \int _{\Gamma _{\sigma }^{e,i}}  \overline{\mathbf{{N}}}_{i}^T \;\overline{\mathbf{t}}^{\,i} d \Gamma  </math>
+| style="text-align: center;" | <math>\boldsymbol{V}^{e}_{n+1}\mathrm{exp}\left[\Delta \gamma \frac{\partial g_{n+1}}{\partial \boldsymbol{\tau }} \right]=\boldsymbol{f}_{n+1}\boldsymbol{F}^e_{n}\boldsymbol{R}^{e^{T}}_{n+1}=\boldsymbol{F}^{e^{trial}}\boldsymbol{R}^{e^{T}}_{n+1}=\boldsymbol{V}^{e^{trial}}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.13)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.13)
 |}
-Where <math display="inline">\mathbf{F}_{{dis}}^{e, i} \in \mathbb{R}^{n_{b}^{e}}</math> is the elemental traction vector due to distributed loads along direction '''i''', <math display="inline">\mathbf{F}_{\hbox{dis}}^{i} \in \mathbb{R}^{n_{d} n_{p t}}</math> is the global vector due to distributed loads along direction '''i''' and <math display="inline">\mathbf{F}_{{dis}}\in \mathbb{R}^{n_{d} n_{p t}}</math> is the global vector due to distributed loads. As for the case of body forces, the input function <math display="inline">\overline{\mathbf{t}}^{\,i}</math> is  commonly replaced by an '''approximation''' constructed by interpolation of its nodal values <math display="inline">\overline{\mathbf{t}}^{\,e,i}</math> using the shape functions employed for interpolating the boundary test functions:
+Equation [[#eq-A.13|A.13]] is, then, multiplied on both sides by its transpose
+<span id="eq-A.14"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,775: / Line 6,569: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left.\overline{\mathrm{t}}^i\;\right|_{\boldsymbol{x} \in \Gamma _{\sigma }^{\;e,i} } \approx \overline{\mathbf{N}}^{e} \;\;\overline{\mathrm{t}}^{\;e,i} \;,\;\;\;\hbox{   with  }\;\;\; \overline{\mathrm{t}}^{\;e,i}  = \left[\begin{array}{c}{\overline{\mathrm{t}}^i\;\left(\boldsymbol{x}_{1}^{e}\right)} \\ {\overline{\mathrm{t}}^i\;\left(\boldsymbol{x}_{2}^{e}\right)} \\ {\vdots } \\ {\overline{\mathrm{t}}^i\;\left(\boldsymbol{x}_{n^{e}_b}^{e}\right)}\end{array}\right] </math>
+| style="text-align: center;" | <math>\boldsymbol{V}^{e}_{n+1}\mathrm{exp}\left[2\Delta \gamma \frac{\partial g_{n+1}}{\partial \boldsymbol{\tau }} \right]\boldsymbol{V}^{e}_{n+1}=\left(\boldsymbol{V}^{e^{trial}} \right)^2  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.14)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.14)
 |}
-Using this interpolation, and taking advantage of the assembly operator, the global vector due to distributed loads along direction '''i''' results in:
+and by rearranging the terms in Equation [[#eq-A.14|A.14]] and taking the square root of it gives
+<span id="eq-A.15"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,787: / Line 6,582: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}_{\hbox{dis}}^{i} = \underset{e \in \mathbf{E}_{b}^{i}}{\mathbf{A}} \mathbf{F}_{\hbox{dis}}^{e, i} = \underset{e \in \mathbf{E}_{b}^{i}}{\mathbf{A}}\;\; \int _{\Gamma _{\sigma }^{e, i}} \overline{\mathbf{N}}^{e^ T}\left(\overline{\mathbf{N}}^{e} \overline{\mathbf{t}}^{e, i}\right)d \Gamma  </math>
+| style="text-align: center;" | <math>\boldsymbol{V}^{e}_{n+1}=\boldsymbol{V}^{e^{trial}}\mathrm{exp}\left[-\Delta \gamma \frac{\partial g_{n+1}}{\partial \boldsymbol{\tau }} \right]  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.15)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.15)
 |}
-If mapped into the parent domain, the computation of the elemental traction vector due to distributed loads along direction '''i''' can be approximated using the Gauss Quadrature method:
+The final form is obtained by applying the tensor logarithm on both side of Equation [[#eq-A.15|A.15]]
+<span id="eq-A.16"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,799: / Line 6,595: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}_{\hbox{dis}}^{e, i} =  \int _{\Gamma _{\sigma }^{e, i}} \overline{\mathbf{N}}^{e^ T}\left(\overline{\mathbf{N}}^{e} \overline{\mathbf{t}}^{e, i}\right)d \Gamma = \int _{{\Gamma _{\sigma }}_{\xi }}<code>J</code>^{e,i}_b \; \overline{\mathbf{N}}^{e^ T}\left(\overline{\mathbf{N}}^{e} \overline{\mathbf{t}}^{e, i}\right)d \Gamma _{\xi } = \sum _{g=1}^{m_{b}} \overline{w}_{g}\left(<code>J</code>^{e,i}_b \; \overline{\boldsymbol{N}}^{e^T}\left(\overline{\boldsymbol{N}}^{e} \overline{\boldsymbol{t}}^{e, i}\right)\right)_{\overline{\xi }=\overline{\xi } g} </math>
+| style="text-align: center;" | <math>\boldsymbol{\epsilon }^{e}_{n+1}=\boldsymbol{\epsilon }^{e^{trial}}-\Delta \gamma \frac{\partial g_{n+1}}{\partial \boldsymbol{\tau }}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.16)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.16)
 |}
-As it has already been stressed out, <math display="inline">\Gamma _{\sigma }^{e, i} \subset \mathbb{R}^{n_{ d}-1}</math> and thus the shape functions and weights appearing in the above expressions are '''different''' from those computed when assessing the stiffness matrix and the body forces vector. For 3D problems, the boundary elements are 2D, and therefore a different Gauss rule is applied.
+In conclusion, Equation [[#eq-A.16|A.16]] represents the Hencky strain updated formula of the return mapping in finite strains.
-==B.3 The mass matrix==
+=Appendix B. Derivatives of the rank-one matrices principal direction=
-It is found from equation [[#eq-3.13|3.13]] that the mass matrix can be expressed in terms of the FEM as:
+In this section the expression of the spatial form of <math display="inline"> \mathrm{C}^{A, trial} </math> is derived
+<span id="eq-B.1"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,815: / Line 6,612: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M}= \int _{{\mho }}^{ } \mathbf{N}^T \boldsymbol{\rho } \;\mathbf{N} \; d\mho  </math>
+| style="text-align: center;" | <math>\mathrm{c}^{A, trial}=\frac{\partial \left(\boldsymbol{n}^{(A)}\otimes \boldsymbol{n}^{(A)}\right)}{\partial g}=\frac{\partial \boldsymbol{m}^{(A)}}{\partial g}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.17)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (B.1)
 |}
-The process of splitting the above integral into elemental ones, and approximating the latter using the Gauss Quadrature method is identical as for the static stiffness matrix case. Using the assembly operator:
+where <math display="inline"> \boldsymbol{n}^{A} </math> and <math display="inline"> \boldsymbol{m}^{A} </math> denote the eigenvectors and eigenbases associated with the eigenvalues <math display="inline"> \boldsymbol{\lambda }_A </math> of the left Cauchy-Green deformation tensor <math display="inline">\boldsymbol{b}^e</math>, respectively, and <math display="inline">\boldsymbol{g}</math>, the metric tensor in the current configuration. In order to find the expression of the closed-form, firstly, the relation of Equation [[#eq-B.1|B.1]] is transformed in material description by operating a pull-back transformation
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@@ Line 3,827: / Line 6,624: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M}=\overset{n_{el}}{\underset{e=1}{\mathbf{A}}} \mathbf{M}^{e} \;\;,\;\;\;\; \mathbf{M}^{e} = \int _{{\mho }}^{ } \mathbf{N}^{e^T} \boldsymbol{\rho } \;\mathbf{N}^e \; d\mho ^e </math>
+| style="text-align: center;" | <math>\frac{\partial \boldsymbol{M}^A}{\partial \boldsymbol{C}}=\boldsymbol{F}^{-1}\frac{\partial \boldsymbol{m}^{(A)}}{\partial g}\boldsymbol{F}^{-T} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.18)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (B.2)
 |}
-The elemental matrices are then mapped into the parent domain and approximated using Gauss Quadrature:
+with <math display="inline">\boldsymbol{C}</math> being the right Cauchy-Green strain tensor and <math display="inline"> \boldsymbol{M}^A </math> the eigenbases  associated to <math display="inline">\boldsymbol{C}</math>. The spectral decomposition of the right Cauchy-Green strain tensor <math display="inline">\boldsymbol{C}</math> which reads
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,839: / Line 6,636: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M}^{e} =  \int _{{\mho }}^{ } \mathbf{N}^{e^T} \boldsymbol{\rho } \;\mathbf{N}^e \; d\mho ^e=  \int _{{\mho }}^{ } <code>J</code>^{e} \; \mathbf{N}^{e^T} \boldsymbol{\rho } \;\mathbf{N}^e \; d\mho _{\xi } = \sum _{g=1}^{m} {w}_{g}\left(<code>J</code>^{e} \mathbf{N}^{e^{T}} \boldsymbol{\rho }\; \mathbf{N}^{e}\right)_{\boldsymbol{\xi }=\boldsymbol{\xi }_g} </math>
+| style="text-align: center;" | <math>\boldsymbol{C}:=\sum \limits _{A=1}^3\lambda _A^2\boldsymbol{N}^{(A)}\otimes \boldsymbol{N}^{(A)} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.19)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (B.3)
 |}
-==B.4 The static damping matrix==
+where <math display="inline"> \lambda _A </math> and <math display="inline"> \boldsymbol{N}^{(A)} </math> denotes eigenvalues and associated eigenvectors, is considered and from Serrin's representation <span id='citeF-242'></span>[[#cite-242|[242]]] it is possible to express <math display="inline">\boldsymbol{N}^{(A)}\otimes \boldsymbol{N}^{(A)}</math> in a closed-form in terms of <math display="inline">\boldsymbol{C}</math> as
-It is found from equation [[#eq-3.13|3.13]] that the static damping matrix can be expressed in terms of the FEM as:
+<span id="eq-B.4"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,853: / Line 6,649: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{D}\;_{\hbox{static}} = \int _{{\mho }}^{ } \mathbf{N}^T \overline{\mu }\;\mathbf{N}\;d\mho \;  + \; \int _{{\mho }}^{ } \mathbf{B}^T \overline{\mathbf{D}}\;\mathbf{B}\;d\mho  </math>
+| style="text-align: center;" | <math>\boldsymbol{N}^{(A)}\otimes \boldsymbol{N}^{(A)}=\lambda _A^2\frac{\boldsymbol{C}-\left(I_1 - \lambda _A^2 \right)\boldsymbol{I}+I_3\lambda _A^{-2}\boldsymbol{C}^{-1}}{D_A}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.20)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (B.4)
 |}
-Using the approximation presented in equation [[#eq-3.15|3.15]], the above integral is transformed into:
+where <math display="inline">I_1</math> and <math display="inline">I_3</math> are the first and third principal invariants of <math display="inline"> \boldsymbol{C} </math>. From Equation [[#eq-B.4|B.4]] it follows that the expression of <math display="inline">\boldsymbol{M}^A</math> reads
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,865: / Line 6,661: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{D}\;_{\hbox{static}} = \overline{\alpha } \;\mathbf{M} + \overline{\beta }\; \mathbf{K}\;_{\hbox{static}}  </math>
+| style="text-align: center;" | <math>\boldsymbol{M}^A=\lambda _A^{-2}\boldsymbol{N}^{(A)}\otimes \boldsymbol{N}^{(A)}=\frac{\boldsymbol{C}-\left(I_1 - \lambda _A^2 \right)\boldsymbol{I}+I_3\lambda _A^{-2}\boldsymbol{C}^{-1}}{D_A} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.21)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (B.5)
 |}
-Where <math display="inline">\mathbf{M}</math> and <math display="inline">\mathbf{K}\;_{\hbox{static}} </math> are known matrices, while <math display="inline">\overline{\alpha }</math> and <math display="inline">\overline{\beta }</math> are constant parameters.
+With the expression of <math display="inline"> \boldsymbol{M}^A </math> in hand, its derivative with respect to <math display="inline">\boldsymbol{C}</math> is
-==B.5 The rotation matrices==
-Recall that the rotation of the structure induces new matrices to appear in equation [[#eq-3.13|3.13]]:
+<span id="eq-B.6"></span>
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 |-
@@ Line 3,881: / Line 6,674: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}\;_{\hbox{rot}} = \int _{{\mho }}^{ } \mathbf{N}^T \boldsymbol{\rho }\;(\widetilde{\Omega }^2 + \widetilde{\alpha })\;\mathbf{N} \; d\mho \;\;,\;\;\;\;\mathbf{D}\;_{\hbox{rot}} = 2\int _{{\mho }}^{ } \mathbf{N}^T \boldsymbol{\rho }\;\widetilde{\Omega }\;\mathbf{N}\;d\mho  </math>
+| style="text-align: center;" | <math>\frac{\partial \boldsymbol{M}^A}{\partial \boldsymbol{C}} =\frac{1}{D_A}\frac{\partial \left(\boldsymbol{C}-\left(I_1 - \lambda _A^2 \right)\boldsymbol{I}+I_3\lambda _A^{-2}\boldsymbol{C}^{-1}\right)}{\partial \boldsymbol{C}}  + \frac{\partial D_A^{-1}}{\partial \boldsymbol{C}}\left(\boldsymbol{C}-\left(I_1 - \lambda _A^2 \right)\boldsymbol{I}+I_3\lambda _A^{-2}\boldsymbol{C}^{-1}\right)</math>
+|-
+| style="text-align: center;" | <math> =\frac{1}{D_A}(\mathrm{I}-\boldsymbol{I}\otimes \boldsymbol{I}+\lambda _A^2\boldsymbol{I}\otimes \boldsymbol{M}^A+ \lambda _A^{-2}I_3\boldsymbol{C}^{-1}\otimes \boldsymbol{C}^{-1}</math>
+|-
+| style="text-align: center;" | <math>  -I_3 \lambda _A^{-2}\boldsymbol{C}^{-1}\otimes \boldsymbol{M}^A-I_3\lambda _A^{-2}\mathrm{I}_{C^{-1}})-\frac{1}{D_A^2}\left(\boldsymbol{C}-\left(I_1 - \lambda _A^2 \right)\boldsymbol{I}+I_3\lambda _A^{-2}\boldsymbol{C}^{-1}\right)</math>
+|-
+| style="text-align: center;" | <math>  \left[\left(8\lambda _A^3-2I_1-2I_3\lambda _A^{-3} \right)\frac{1}{2}\lambda _A \boldsymbol{M}^A- \lambda _A^2\boldsymbol{I}+I_3\lambda _A^{-2}\boldsymbol{C}^{-1} \right]</math>
+|-
+| style="text-align: center;" | <math>  =\frac{1}{D_A}\left[\mathrm{I}-\boldsymbol{I}\otimes \boldsymbol{I}+ I_3 \lambda _A^{-2}\left(\boldsymbol{C}^{-1}\otimes \boldsymbol{C}^{-1}-\mathrm{I}_{C^{-1}} \right)\right]</math>
+|-
+| style="text-align: center;" | <math>  +\frac{1}{D_A}\left[\lambda _A^{2}\left(\boldsymbol{I}\otimes \boldsymbol{M}^A+\boldsymbol{M}^A\otimes \boldsymbol{I} \right)-\frac{1}{2}{D'}_A\lambda _A\boldsymbol{M}^A\otimes \boldsymbol{M}^A  \right]</math>
+|-
+| style="text-align: center;" | <math>  -\frac{1}{D_A}\left[I_3\lambda _A^{-2}\left(\boldsymbol{C}^{-1}\otimes \boldsymbol{M}^A + \boldsymbol{M}^A\otimes \boldsymbol{C}^{-1} \right)\right] </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.22)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (B.6)
 |}
-Its computation method is not so different from the mass matrix case:
+In order to obtain the final expression the following relations have been used:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,893: / Line 6,698: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}\;_{\hbox{rot}}=\overset{n_{el}}{\underset{e=1}{\mathbf{A}}} \mathbf{K}\;_{\hbox{rot}}^{e} \;\;\Rightarrow \;\;\;\; \mathbf{K}\;_{\hbox{rot}}^{e} = \int _{{\mho }}^{ } \mathbf{N}^{e^T} \boldsymbol{\rho }\;(\widetilde{\Omega }^2 + \widetilde{\alpha })\;\mathbf{N}^e \; d\mho ^e   </math>
+| style="text-align: center;" | <math>\frac{\partial \lambda _A}{\partial \boldsymbol{C}}=\frac{1}{2}\lambda _A\boldsymbol{M}^A </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.23)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (B.7)
 |}
@@ Line 3,903: / Line 6,708: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{D}\;_{\hbox{rot}}=\overset{n_{el}}{\underset{e=1}{\mathbf{A}}} \mathbf{D}\;_{\hbox{rot}}^{e} \;\;\Rightarrow \;\;\;\; \mathbf{D}\;_{\hbox{rot}}^{e} = 2\int _{{\mho }}^{ } \mathbf{N}^{e^T} \boldsymbol{\rho }\;\widetilde{\Omega }\;\mathbf{N}^e\;d\mho ^e </math>
+| style="text-align: center;" | <math>\frac{\partial I_1}{\partial \boldsymbol{C}}=\frac{\partial \mathrm{tr}\boldsymbol{C}}{\partial \boldsymbol{C}}=\boldsymbol{I} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.24)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (B.8)
 |}
-Using the Gauss Quadrature method to approximate the elemental integrals:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,915: / Line 6,718: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{K}\;_{\hbox{rot}}^{e} =   \int _{{\mho }}^{ } <code>J</code>^{e} \;\mathbf{N}^{e^T} \boldsymbol{\rho }\;(\widetilde{\Omega }^2 + \widetilde{\alpha })\mathbf{N}^e \; d\mho _{\xi } = \sum _{g=1}^{m} {w}_{g}\left(<code>J</code>^{e} \mathbf{N}^{e^{T}} \boldsymbol{\rho }\;(\widetilde{\Omega }^2 + \widetilde{\alpha })\; \mathbf{N}^{e}\right)_{\boldsymbol{\xi }=\boldsymbol{\xi }_g} </math>
+| style="text-align: center;" | <math>\frac{\partial I_3}{\partial \boldsymbol{C}}=\frac{\partial \mathrm{det}\boldsymbol{C}}{\partial \boldsymbol{C}}=\frac{\partial J^2}{\partial \boldsymbol{C}}=J^2\boldsymbol{C}^{-1} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.25)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (B.9)
 |}
@@ Line 3,925: / Line 6,728: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{D}\;_{\hbox{rot}}^{e} = 2  \int _{{\mho }}^{ } <code>J</code>^{e} \;\mathbf{N}^{e^T} \boldsymbol{\rho }\;\widetilde{\Omega } \mathbf{N}^e \; d\mho _{\xi } = 2 \sum _{g=1}^{m} {w}_{g}\left(<code>J</code>^{e} \mathbf{N}^{e^{T}} \boldsymbol{\rho }\;\widetilde{\Omega }\; \mathbf{N}^{e}\right)_{\boldsymbol{\xi }=\boldsymbol{\xi }_g} </math>
+| style="text-align: center;" | <math>{D'}_A=8\lambda _A^3-2I_1-2I_3\lambda _A^{-3} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (B.26)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (B.10)
 |}
-=C Example of vectorised assembly=
+By performing a push-forward operation on the final result of Equation [[#eq-B.6|B.6]], it is possible to obtain its spatial counterpart
-The FEM is able to solve boundary value problems along very complex geometries because it splits the domain into a finite number of smaller partitions called elements. In numerical terms, this implies that global matrices are not computed all at once, but element by element. Then, all the elemental matrices need to be assembled, but traditional assembly algorithms can be very slow.  For this reason a '''vectorised assembly algorithm''' has been used in the present project. The code was provided by ''prof. J. Hernández'' and adapted to the case of rotating structures. The goal of the present chapter is to illustrate how vectorised assembly works by developing, step by step, the assembly of the mass matrix for two linear quadrilateral elements.
-==C.1 Mass matrix vectorised assembly==
-To start with, recall the definition of the mass matrix and its decomposition into elemental components from the previous chapter:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,943: / Line 6,740: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{matrix}\mathbf{M}= \int _{{\mho }}^{ } \mathbf{N}^T \boldsymbol{\rho } \;\mathbf{N} \; d\mho \\ \\ \mathbf{M}=\overset{n_{el}}{\underset{e=1}{\mathbf{A}}} \mathbf{M}^{e} \;\;,\;\;\;\; \mathbf{M}^{e} = \int _{{\mho }}^{ } \mathbf{N}^{e^T} \boldsymbol{\rho } \;\mathbf{N}^e \; d\mho ^e\\ \\ \mathbf{M}^{e} =  \int _{{\mho }}^{ } \mathbf{N}^{e^T} \boldsymbol{\rho } \;\mathbf{N}^e \; d\mho ^e=  \int _{{\mho }}^{ } <code>J</code>^{e} \; \mathbf{N}^{e^T} \boldsymbol{\rho } \;\mathbf{N}^e \; d\mho _{\xi } = \sum _{g=1}^{m} {w}_{g}\left(<code>J</code>^{e} \mathbf{N}^{e^{T}} \boldsymbol{\rho }\; \mathbf{N}^{e}\right)_{\boldsymbol{\xi }=\boldsymbol{\xi }_g} \end{matrix} </math>
+| style="text-align: center;" | <math>\frac{\partial \boldsymbol{m}^{(A)}}{\partial g} =\boldsymbol{F}\frac{\partial \boldsymbol{M}^A}{\partial \boldsymbol{C}}\boldsymbol{F}^T  =\frac{1}{D_A}\left[\mathrm{I}_b-\boldsymbol{b}\otimes \boldsymbol{b}+ I_3 \lambda _A^{-2}\left(\boldsymbol{I}\otimes \boldsymbol{I}-\mathrm{I} \right)\right]</math>
+|-
+| style="text-align: center;" | <math>  +\frac{1}{D_A}\left[\lambda _A^{2}\left(\boldsymbol{b}\otimes \boldsymbol{m}^A+\boldsymbol{m}^A\otimes \boldsymbol{b} \right)-\frac{1}{2}{D'}_A\lambda _A\boldsymbol{m}^A\otimes \boldsymbol{m}^A  \right]</math>
+|-
+| style="text-align: center;" | <math>  -\frac{1}{D_A}\left[I_3\lambda _A^{-2}\left(\boldsymbol{I}\otimes \boldsymbol{m}^A + \boldsymbol{m}^A\otimes \boldsymbol{I} \right)\right] </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (C.1)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (B.11)
 |}
-To keep the example as simple as possible, let's consider the following geometry made from two linear quadrilateral elements: <div id='img-C.1'></div>
+=Appendix C. Irreducible formulation in axisymmetric problems=
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+In this section the matrix formulation for an axi-symmetrical finite element, undergoing finite deformations with respect to the spatial configuration, is presented.
+This formulation can be used in the case of 3D bodies, which have rotational symmetry, as depicted on the left side of Figure [[#img-C.1|C.1]], and, thus, can be reduced to a 2D axi-symmetrical model, right side of the picture.
+Hereinafter, it is assumed that the axis of symmetry coincides with the coordinate <math display="inline"> X_2 </math>.
+<div id='img-C.1'></div>
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-vectorised.png|300px|2D mesh composed by two quadrilateral elements.]]
+|style="padding:10px;|[[Image:Draft_Samper_987121664-monograph-axisymmetric_body.png|498px|Axisymmetric representation of a 3D body with rotational symmetry.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure C.1:''' 2D mesh composed by two quadrilateral elements.
+| colspan="1" style="padding-bottom:10px;| '''Figure C.1:''' Axisymmetric representation of a 3D body with rotational symmetry.
 |}
-Remember that for 2D elements, the numeration order is not important as well as it follows '''counter-clockwise''' direction. In the above figure, numbers inside a circle represent local numeration.  When the geometric domain is not as trivial as the presented one, keeping track of the global numeration is not an easy task. On the other hand, local numeration is always performed in the same way. Hence, one may suspect that some properties remain '''undisturbed''' in the local frame.
-Following the previous idea, the vectorised assembly takes advantage of the fact that in the parent domain, all the elements are geometrically identical and so are their shape functions. Hence, it is only necessary to compute the elemental matrix of one element and use it to construct a matrix including all the elemental shape functions matrices. The latter matrix, referred as <math display="inline">\mathbf{N}_{elem}</math>, is actually a <math display="inline">(n_{g}\; n_{d} \; n_{el} \times n^{e} \; n_{d})</math> matrix.
+Additionally, to the strains in the plane <math display="inline"> X_1, X_2 </math>, hoop strains (along <math display="inline"> X_3 </math> direction) occur in case of axi-symmetrical deformations. The deformation gradient, is, then, given, as:
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
+|
+{| style="text-align: left; margin:auto;width: 100%;"
+|-
+| style="text-align: center;" | <math>\boldsymbol{F} := d\boldsymbol{x}/d\boldsymbol{X} = \begin{bmatrix}\displaystyle\frac{\partial x_1}{\partial X_1} & \displaystyle\frac{\partial x_1}{\partial X_2} & 0\\ \displaystyle\frac{\partial x_2}{\partial X_1} & \displaystyle\frac{\partial x_2}{\partial X_2} & 0\\  0 & 0 & \displaystyle\frac{ x_1}{ X_1}\\  \end{bmatrix} </math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (C.1)
+|}
-Remember that, for vector fields, the elemental matrix of shape functions <math display="inline">\mathbf{N}^e</math> was defined as:
+By using linear shape functions, the symmetric gradient of the test functions reads
+<span id="eq-C.2"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,966: / Line 6,785: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{N}^e = \left\{   N_1\, \mathbf{I} \;\;\; N_2 \,\mathbf{I} \;\;\;\cdots \;\;\;N_{\,n^e}\, \mathbf{I} \right\} </math>
+| style="text-align: center;" | <math>(\nabla ^S\boldsymbol{w})^T = \left[w_{1,1}, w_{2,2}, \frac{w_1}{x_1}, (w_{1,2} + w_{2,1})\right]  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.2)
 |}
-Where <math display="inline">\mathbf{I}</math> is the <math display="inline">n_{d}\times n_{d}</math> identity matrix and <math display="inline"> N_A</math> is the scalar-valued shape function of node <math display="inline">A</math>. Let's define now <math display="inline">\mathbf{N}_{E}</math>, a <math display="inline">(n_{g} \; n_{d} \times n^{e} \; n_{d})</math> matrix that incorporates the value of <math display="inline">\mathbf{N}^e</math> at every Gauss Point:
+and this leads to <math display="inline"> \mathbf{B}_{I}^A </math>, the deformation matrix relative to node <math display="inline">I</math>, expressed here for a 2D axis-symmetrical problem as:
+<span id="eq-C.3"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,978: / Line 6,798: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{N}_E = \begin{bmatrix}\mathbf{N}^1_{\xi _1} & \mathbf{N}^2_{\xi _1} & \cdots  & \mathbf{N}^{\,n_e}_{\xi _1} \\ \\  \mathbf{N}^1_{\xi _2} & \mathbf{N}^2_{\xi _2} & \cdots  & \mathbf{N}^{\,n_e}_{\xi _2} \\ \vdots &\vdots & \ddots  & \vdots \\  \mathbf{N}^1_{\xi _{n_g}} &   \mathbf{N}^2_{\xi _{n_g}} & \cdots  &  \mathbf{N}^{\,n_e}_{\xi _{n_g}}  \end{bmatrix}  = \begin{bmatrix}\mathcal{}{N}^1\\  \mathcal{}{N}^2\\  \vdots \\  \mathcal{}{N}^{\,n_{pt}} \end{bmatrix} </math>
+| style="text-align: center;" | <math>\mathbf{B}_{I}^A = \begin{bmatrix}\displaystyle\frac{\partial N_I}{\partial x_1} & 0 \\ 0 & \displaystyle\frac{\partial N_I}{\partial x_2} \\ \displaystyle\frac{N_I}{ x_1} & 0 \\ \displaystyle\frac{\partial N_I}{\partial x_2} & \displaystyle\frac{\partial N_I}{\partial x_1} \\ \end{bmatrix}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.3)
 |}
-Where <math display="inline">\mathcal{}{N}</math> is a <math display="inline">n_{g} \; n_{d} \times n_{d}</math> matrix. Taking advantage of  the fact that <math display="inline">\mathbf{N}_{E}</math> is the same for all elements, <math display="inline">\mathbf{N}_{elem}</math> can be easily constructed by repeating <math display="inline">\mathbf{N}_{E}</math> a number of <math display="inline">n_{el}</math> times:
+Introduction of the Cauchy stress, written in a vector form <math display="inline"> \boldsymbol{\sigma }^T = \left[\sigma _{11}, \sigma _{22},\sigma _{33},\sigma _{12}\right]</math>, and multiplied by <math display="inline"> \nabla ^S\boldsymbol{w} </math>, expressed by Equation [[#eq-C.2|C.2]] leads to
+<span id="eq-C.4"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 3,990: / Line 6,811: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{N}_{elem} =\underset{n_g \; n_d \; n_{el}}{\underbrace{[\mathbf{N}_E \;\;\mathbf{N}_E \;\;\cdots \;\; \mathbf{N}_E}]^T} </math>
+| style="text-align: center;" | <math>\boldsymbol{\sigma } \cdot \nabla ^S\boldsymbol{w}\mid _{\Omega ^e} = \sum _{I=1}^n \boldsymbol{w}^T (\mathbf{B}_{I}^A)^T \boldsymbol{\sigma }  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.4)
 |}
-The next step is to built <math display="inline">\mathbf{N}_{st}</math>, a <math display="inline">(n_{g}\; n_{d} \;  n_{el} \times n_{pt} \; n_{d})</math> matrix that takes into account the global formulation and is constructed from <math display="inline">\mathbf{N}_{elem}</math> by using the connectivity matrix. For instance, for the presented geometry and with <math display="inline">n^{e} = 4</math> and <math display="inline">n_{g} = 4</math>, <math display="inline">\mathbf{N}_{st}</math> adopts the following form:
+By using the results of Equation [[#eq-C.4|C.4]], the virtual internal work of one element <math display="inline">\Omega ^e</math> is
+<span id="eq-C.5"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 4,002: / Line 6,824: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{matrix}\mathbf{N}_{st} = \begin{bmatrix}\mathcal{}{N}^1 & \mathcal{}{N}^2 & 0 &  0& \mathcal{}{N}^3 & \mathcal{}{N}^4\\  0 &  \mathcal{}{N}^1& \mathcal{}{N}^2 & \mathcal{}{N}^3 & \mathcal{}{N}^4 & 0 \end{bmatrix}\begin{matrix}e_1\\  e_2 \end{matrix} \\  \begin{matrix}\;\;\;\;\;\;\;\;\;\textrm{n}_1 & \;\textrm{n}_2 & \;\;\;\textrm{n}_3& \,\;\;\;\textrm{n}_4 & \;\;\;\textrm{n}_5 & \,\;\;\;\textrm{n}_6 \end{matrix} \end{matrix} </math>
+| style="text-align: center;" | <math>\sum _{I=1}^n (\boldsymbol{w}_I)^T 2 \pi \int _{\varphi (\Omega ^e)}(\mathbf{B}_{I}^A)^T \boldsymbol{\sigma } x_1 dv  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.5)
 |}
-The grace of this approach is that once <math display="inline">\mathbf{N}_E</math> is computed, the transformations that follow are performed using optimised in-built functions such as ''sparse'', ''reshape'' and ''repmat'' :  <pre>m = ndim*nelem*ngaus ; % Number of rows
+It is observed that the integration has to be performed over the coordinates <math display="inline">X_1</math> and <math display="inline">X_2</math> as in circumferential direction. Due to that, the coordinate <math display="inline">x_1</math> appears in Equation [[#eq-C.5|C.5]]. For the node <math display="inline">I</math> of element <math display="inline">\Omega ^e</math> the residual is defined as
-n = nnode*ndim ;          % Number of columns
-nzmaxLOC = m*nnodeE*ndim ;   % Maximum number of zeros
-Nst = sparse([],[],[],m,n,nzmaxLOC); % Allocating memory for Bst
-for inode = 1:nnodeE   % Loop over number of nodes at each element
-    setnodes = CONNECT(:,inode) ;  % Global numbering of the
-    %inode-th node of each element
-      for idime = 1:ndim   % loop over spatial dimensions
-        DOFloc = (inode-1)*ndim+idime ;  % Corresponding indices
-        s = Nelems(:,DOFloc) ;% Corresponding column
-        i = [1:m]' ;  % All rows
-        DOFglo = (setnodes-1)*ndim+idime ; %  b) Columns
-        j = repmat(DOFglo', ndim*ngaus,1) ;
-        j = reshape(j,length(i),1) ;
-        % Assembly process  (as sum of matrix with sparse representation)
-        Nst = Nst + sparse(i,j,s,m,n,m) ;
-    end
-end</pre>
-Until now, the Gauss '''weights''' and '''Jacobians''' were not considered. So let's define '''WSTs''', a <math display="inline"> n_{el}\times n_g</math> vector containing the product of weights and Jacobians at all Gauss points:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 4,035: / Line 6,836: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{WSTs} = \begin{bmatrix}s^1\\  s^2\\  \vdots \\  \hbox{s}^{\,n_{el}} \end{bmatrix} \;\;,\;\; s^{\,e} = \begin{bmatrix}w_1\;\cdot \;J_{\xi _1}^e\\  w_2\;\cdot \;J_{\xi _2}^e\\  \vdots \\  w_{\,n_g}\; \cdot \;J_{\xi _{\,n_g}}^e \end{bmatrix}  </math>
+| style="text-align: center;" | <math>r_I^A(\boldsymbol{u})= 2 \pi \int _{\varphi (\Omega ^e)} (\mathbf{B}_{I}^A)^T \boldsymbol{\sigma } x_1 dv </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.6)
 |}
-The previous matrix is to be transformed into diagonal. To do so, let's first define the following auxiliary matrices:
+and its linearization yields to the tangent matrix, sum of the ''geometric'' and ''material'' stiffness matrix, expressed as in Equation [[#eq-4.35|4.35]].
+The ''geometric'' stiffness matrix, whose definition is expressed by the first term of the integral of Equation [[#eq-4.25|4.25]], in its discretized form reads
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 4,047: / Line 6,850: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{w}_i^e = \left(w_i \;\cdot \;\mathbf{J}_{\xi _i}^e \right)\mathbf{I} \;\;,\;\; \mathbf{W}_d^e = \begin{bmatrix}\mathbf{w}_1^e& [\,0\,] &  [\,0\,]&[\,0\,] \\[] [\,0\,]&  \mathbf{w}_2^e& [\,0\,] & [\,0\,]\\  \vdots  & \vdots &  \ddots & \vdots \\[] [\,0\,]& \cdots  & \cdots  & \mathbf{w}_{n_g}^e&  \end{bmatrix} </math>
+| style="text-align: center;" | <math>\mathbf{K}^{G}=\bigcup _{p = 1}^{n_p}\sum ^{n}_{I=1}\sum ^{n}_{K=1}\boldsymbol{w}_{I}^T\left( \left(\nabla _x N_I \right)^T\boldsymbol{\sigma }\left(\nabla _x N_K \right)\mathbf{I} \right)V_p \delta \boldsymbol{u}_K </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.7)
 |}
-And now the diagonalised version of '''WSTs''' reads as:
+where <math display="inline">I </math> and <math display="inline"> K </math> are the indexes of the finite element's nodes, <math display="inline"> \nabla _x N_I </math> is the spatial gradient of the shape function evaluated at node <math display="inline">I </math>, <math display="inline"> V_p </math> is the volume relative to a single material point.
+In case of axi-symmetrical deformations, matrix form of the gradient of the test function <math display="inline">\boldsymbol{w}</math> takes the form:
+<span id="eq-C.8"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 4,059: / Line 6,865: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{W}_{diag} = \begin{pmatrix}\mathbf{W}_d^{\,1} & [\,0\,] & \cdots  &[\,0\,] \\ \\[] [\,0\,] &  \mathbf{W}_d^{\,2}& \cdots & [\,0\,]\\ \\  \vdots  & \vdots & \ddots  & \vdots \\ \\[] [\,0\,]&  [\,0\,]& \cdots  &\mathbf{W}_d^{\,n_{el}}  \end{pmatrix} </math>
+| style="text-align: center;" | <math>\nabla _x \boldsymbol{w} = \left\{\begin{matrix}\boldsymbol{w}_{1,1} \\ \boldsymbol{w}_{1,2} \\ \boldsymbol{w}_{3,3} \\ \boldsymbol{w}_{2,1} \\ \boldsymbol{w}_{2,2}  \end{matrix} \right\}= \sum _{I=1}^n \begin{bmatrix}N_{I,1} & 0 \\ N_{I,2} & 0 \\  \frac{N_I}{x_1} & 0 \\  0 & N_{I,1} \\ 0 & N_{I,2} \\ \end{bmatrix} \left\{\begin{matrix}\boldsymbol{w}_1 \\ \boldsymbol{w}_2 \\ \end{matrix} \right\}= \sum _{I=1}^n \overline{G}_I \boldsymbol{w}_I  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.8)
 |}
-To close down the section, let's take '''density''' into consideration. Recall that density is given as a vector input, having a value for each element:
+Using this relation of Equation [[#eq-C.8|C.8]], together with the expression of the Cauchy stress in the following matricial form:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 4,071: / Line 6,877: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\hbox{density} = \left\{\rho ^1 \; \rho ^2  \; \cdots \;\; \rho ^{\,n_{el}} \right\}^T </math>
+| style="text-align: center;" | <math>\boldsymbol{\widehat{\sigma }}=\begin{bmatrix}\sigma _{11} & \sigma _{12} & 0 & 0 & 0 \\ \sigma _{21} & \sigma _{22} & 0 & 0 & 0 \\ 0 & 0 & \sigma _{33} & 0 & 0 \\ 0 & 0 & 0 & \sigma _{11} & \sigma _{12} \\ 0 & 0 & 0 & \sigma _{21} & \sigma _{22} \\ \end{bmatrix} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.9)
 |}
-The next step is to transform the previous expression into a global matrix, accounting for each dimension and Gauss point. To do so, let's consider the following matrices:
+the ''geometric'' stiffness matrix in discretized form can be written as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 4,083: / Line 6,889: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\hbox{dens}^{\,e} =\rho ^{\,e }\;\mathbf{I} \;\;,\;\;{\hbox{dens}_{\,\mathbf{G}}}^{\,e} = \overset{(n_d\;n_d \;n_g)}{\overbrace{\begin{bmatrix}   \hbox{dens}^{\,e} &  [\,0\,]&\cdots   &  [\,0\,]\\[] [\,0\,]& \hbox{ dens}^{\,e}  & \cdots  & [\,0\,] \\    \vdots & \vdots  & \ddots  &\vdots  \\[] [\,0\,] & [\,0\,]  & \cdots  & \hbox{ dens}^{\,e}   \end{bmatrix}}}\begin{matrix}1\\  2\\  \vdots \\  n_g \end{matrix} </math>
+| style="text-align: center;" | <math>\mathbf{K}^{A, G}=\bigcup _{p = 1}^{n_p}\sum ^{n}_{I=1}\sum ^{n}_{K=1}\boldsymbol{w}_{I}^T 2 \pi \left( \overline{G}_I^T\boldsymbol{\widehat{\sigma }}\overline{G}_K^T \right)V_p \delta \boldsymbol{u}_K </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.10)
 |}
-And finally, the global density matrix is defined as:
+As explained in <span id='citeF-243'></span>[[#cite-243|[243]]], it is possible to defined the term <math display="inline"> \overline{G}_I^T\boldsymbol{\widehat{\sigma }}\overline{G}_K^T </math> in an explicit way as follows:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 4,095: / Line 6,901: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf\hbox{densglo} = \begin{bmatrix}{\hbox{dens}_{\,\mathbf{G}}}^{\,1}& [\,0\,] &  \cdots & [\,0\,]\\[] [\,0\,] &  {\hbox{dens}_{\,\mathbf{G}}}^{\,2}&\cdots   &[\,0\,] \\  \vdots &  \vdots & \ddots  &  \vdots \\[] [\,0\,]&  [\,0\,]& \cdots  & {\hbox{dens}_{\,\mathbf{G}}}^{\,n_{el}} \end{bmatrix} </math>
+| style="text-align: center;" | <math>\overline{G}_I^T\boldsymbol{\widehat{\sigma }}\overline{G}_K^T = \begin{bmatrix}A_{IK} + B_{IK} + C_{IK} & 0 \\ 0 & A_{IK} + B_{IK}  \end{bmatrix} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.11)
 |}
-After all this abstract formulation, the mass matrix can be simply '''assembled''' with no nested loops at all:
+if the terms
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 4,107: / Line 6,913: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{M} = {\mathbf{N}_{st}}^{\,\hbox{T}} \;\cdot \;\mathbf\hbox{densglo}\;\cdot \;\mathbf{W}_{diag}\;\cdot \;{\mathbf{N}_{st}} </math>
+| style="text-align: center;" | <math>A_{IK} = (N_{I,1} \sigma _{11} + N_{I,2} \sigma _{21})N_{K,1} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.12)
 |}
-==C.2 Generalisation to rotation matrices==
-During the development of the FEM method for rotating structures, it was found that new matrices emerged by the effect of rotation. These matrices have the form:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 4,121: / Line 6,923: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{P}= \int _{{\mho }}^{ } \mathbf{N}^T \boldsymbol{\rho } \;\mathbf{A}\;\mathbf{N} \; d\mho  </math>
+| style="text-align: center;" | <math>B_{IK} = (N_{I,1} \sigma _{12} + N_{I,2} \sigma _{22})N_{K,2} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.13)
 |}
-Where <math display="inline">\mathbf{A}</math> is a <math display="inline">n_d\times n_d</math> matrix accounting for the rotation effect, mostly Coriolis, centrifugal and tangential forces. The vectorised assembly of <math display="inline">\mathbf{P}</math> is almost straightforward, as most of the process is done exactly as in the case of the mass matrix. The only difference has to do with <math display="inline">\mathbf{A}</math>, which has to be globally defined:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 4,133: / Line 6,933: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\mathbf{A}_{\mathbf{G}}}^{\,e} = \overset{(n_d\;n_d \;n_g)}{\overbrace{\begin{bmatrix}   \mathbf{A}^{\,e} &  [\,0\,]&\cdots   &  [\,0\,]\\[] [\,0\,]& \mathbf{A}^{\,e}  & \cdots  & [\,0\,] \\    \vdots & \vdots  & \ddots  &\vdots  \\[] [\,0\,] & [\,0\,]  & \cdots  & \mathbf{A}^{\,e}   \end{bmatrix}}}\begin{matrix}1\\  2\\  \vdots \\  n_g \end{matrix} \;\;\;\;\;\;,\;\;\;\;\;\;\mathbf{A}_{\,glo} = \begin{bmatrix}{\mathbf{A}_{\mathbf{G}}}^{\,1}& [\,0\,]  & \cdots  &  [\,0\,]\\[] [\,0\,]&  {\mathbf{A}_{\mathbf{G}}}^{\,2}& \cdots  &  [\,0\,]\\  \vdots &  \vdots  & \ddots  & \vdots \\[] [\,0\,] &  [\,0\,] & \cdots & {\mathbf{A}_{\mathbf{G}}}^{\,n_{el}} \end{bmatrix} </math>
+| style="text-align: center;" | <math>C_{IK} = \frac{N_I}{x_1} \sigma _{33} \frac{N_K}{x_1} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.14)
 |}
-Finally, the global matrix is assembled as:
+are employed.
+With respect to the ''material'' stiffness matrix, the discretized form of the 3D case is expressed by Equation [[#eq-4.34|4.34]]. In the case of axi-symmetrical deformations, its expression reads
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 4,145: / Line 6,947: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{P} = {\mathbf{N}_{st}}^{\,\hbox{T}} \;\cdot \;\mathbf\hbox{densglo}\;\cdot \;\mathbf{A}_{\,glo}\;\cdot \;\mathbf{W}_{diag}\;\cdot \;{\mathbf{N}_{st}} </math>
+| style="text-align: center;" | <math>\mathbf{K}^{A, M}=\bigcup _{p = 1}^{n_p}\sum ^{n}_{I=1}\sum ^{n}_{K=1}\boldsymbol{w}_{I}^T 2 \pi \left( (\mathbf{B}_{I}^A)^T\mathbf{D}^A\mathbf{B}_{K}^A \right)V_p \delta \boldsymbol{u}_K </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (C.15)
 |}
-=D Raw simulation results=
+by taking into account the relations of <math display="inline">\mathbf{B}_{I}^A</math> (Equation [[#eq-C.3|C.3]]) and <math display="inline">\mathbf{D}^A</math>, matrix form of the incremental constitutive tensor of Equation [[#eq-4.24|4.24]] for axi-symmetric case.
-In this chapter, results obtained from the developed simulations will be displayed. The discussion and analysis of the present results has already been addressed in the report (sections [[#6 Analysis of rotating beams using the SVD|6]] and [[#7 Study of an aeronautics-related case|7]]).
+Finally, by considering the contribution of the terms <math display="inline">\mathbf{K}^{A, G}</math> and <math display="inline">\mathbf{K}^{A, M}</math>, the tangent stiffness matrix <math display="inline">\mathbf{K}^{A, tan}</math>, referred to the spatial configuration, can be expressed as follows:
-==D.1 Two-dimensional cantilever beam==
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
-===D.1.1 Natural modes and frequencies===
+|
+{| style="text-align: left; margin:auto;width: 100%;"
-<div id='img-D.1'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_987121664-monograph-8modes.png|600px|Natural modes associated to the eight lowest natural frequencies of a rectangular cantilever beam. ]]
+| style="text-align: center;" | <math>\mathbf{K}^{A, tan}=\bigcup _{p = 1}^{n_p}\sum ^{n}_{I=1}\sum ^{n}_{K=1}\boldsymbol{w}_{I}^T 2 \pi \left(\overline{G}_I^T\boldsymbol{\widehat{\sigma }}\overline{G}_K^T + (\mathbf{B}_{I}^A)^T\mathbf{D}^A\mathbf{B}_{K}^A \right)V_p \delta \boldsymbol{u}_K </math>
-|- style="text-align: center; font-size: 75%;"
+|}
-| colspan="1" | '''Figure D.1:''' Natural modes associated to the eight lowest natural frequencies of a rectangular cantilever beam.
+| style="width: 5px;text-align: right;white-space: nowrap;" | (C.16)
 |}
+===BIBLIOGRAPHY===
-{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
+<div id="cite-1"></div>
-|+ style="font-size: 75%;" |<span id='table-D.1'></span>'''Table. D.1''' The twenty lowest natural frequencies of a rectangular cantilever beam.
+'''[[#citeF-1|[1]]]''' Avi, L. and Ooi, J. Y. (2011) "Discrete element simulation: challenges in application and model calibration", Volume 13. Granular Matter 2 107&#8211;107
-|-
-|
-|}
-===D.1.2 Constant angular velocity===
+<div id="cite-2"></div>
+'''[[#citeF-2|[2]]]''' Utili, S. (2015) "International Symposium on Geohazards and Geomechanics (ISGG2015)", Volume 26. IOP Conference Series: Earth and Environmental Science 1 011001
-'''
+<div id="cite-3"></div>
+'''[[#citeF-3|[3]]]''' Nakagawa, M. and Luding, S. (2009) "Powders and grains 2009: Proceedings of the 6th international conference on micromechanics of granular media", Volume 1145. American Institute of Physics Conference Series
-====D.1.2.1 <span id='lb-D.1.2.1'></span>Case 1: Ω = 50 rad/s===='''
+<div id="cite-4"></div>
+'''[[#citeF-4|[4]]]''' Yu, A. and Dong, K.  and Yang, R. and Luding, S. (2013) "Powders and Grains 2013: Proceedings of the 7th International Conference on Micromechanics of Granular Media", Volume 1542. In American Institute of Physics Conference Series
-<div id='img-D.2'></div>
+<div id="cite-5"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-5|[5]]]''' Luding, S. and Tomas, J. (2014) "Particles, contacts, bulk-behavior", Volume 16. Granular Matter 3 279&#8211;280
-|-
-|[[Image:Draft_Samper_987121664-monograph-8modes50.png|600px|Left side vectors (predominant modes) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.2:''' Left side vectors (predominant modes) associated to the eight higher singular values.
-|}
-<div id='img-D.3'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-RSV_50_2.png|600px|Right side vectors (predominant oscillations) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.3:''' Right side vectors (predominant oscillations) associated to the eight higher singular values.
-|}
-'''
-====D.1.2.2 <span id='lb-D.1.2.2'></span>Case 2: Ω = 100 rad/s===='''
+<div id="cite-6"></div>
+'''[[#citeF-6|[6]]]''' Wu, C.Y. and T. Pöschel. (2013) "Micro-mechanics and dynamics of cohesive particle systems", Volume 15. Granular Matter 389&#8211;390
-<div id='img-D.4'></div>
+<div id="cite-7"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-7|[7]]]''' E.W. Merrow. (1988) "Estimating the startup times for solid processing plants", Volume 95. Chem. Eng. 15 89-92
-|-
-|[[Image:Draft_Samper_987121664-monograph-8modes100.png|600px|Left side vectors (predominant modes) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.4:''' Left side vectors (predominant modes) associated to the eight higher singular values.
-|}
-<div id='img-D.5'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-RSV_100.png|600px|Right side vectors (predominant oscillations) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.5:''' Right side vectors (predominant oscillations) associated to the eight higher singular values.
-|}
-'''
-====D.1.2.3 <span id='lb-D.1.2.3'></span>Case 3: Ω = 200 rad/s===='''
+<div id="cite-8"></div>
+'''[[#citeF-8|[8]]]''' B. J. Ennis and J. Green and R. Davies. (1994) "The legacy of neglect in the U.S.", Volume 90. Chem. Eng. Prog. 32 - 43
-<div id='img-D.6'></div>
+<div id="cite-9"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-9|[9]]]''' H. J. Feise. (2003) "Industrial perspective on the future of solids processing", Volume 81. Chemical Engineering Research and Design 837 - 841
-|-
-|[[Image:Draft_Samper_987121664-monograph-8modes200.png|600px|Left side vectors (predominant modes) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.6:''' Left side vectors (predominant modes) associated to the eight higher singular values.
-|}
-<div id='img-D.7'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-RSV_200.png|600px|Right side vectors (predominant oscillations) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.7:''' Right side vectors (predominant oscillations) associated to the eight higher singular values.
-|}
-'''
-====D.1.2.4 Comparative figures===='''
+<div id="cite-10"></div>
+'''[[#citeF-10|[10]]]''' Haff, P. K. (1983) "Grain flow as a fluid-mechanical phenomenon", Volume 134. Journal of Fluid Mechanics -1 401
-<div id='img-D.8'></div>
+<div id="cite-11"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-11|[11]]]''' Campbell, C. S. (2006) "Granular material flows - An overview", Volume 162. Powder Technology 3 208 - 229
-|-
-|[[Image:Draft_Samper_987121664-monograph-nsv_freq.png|300px|Mode intensities for increasing values of Ω]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.8:''' Mode intensities for increasing values of <math>\Omega </math>
-|}
-<div id='img-D.9'></div>
+<div id="cite-12"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-12|[12]]]''' Bathurst, R.J. and Rothenburg, L.L. (1988) "Micromechanical Aspects of Isotropic Granular Assemblies With Linear Contact Interactions", Volume 55. ASME. J. Appl. Mech. 1 17 - 23
-|-
-|[[Image:Draft_Samper_987121664-monograph-React_comparison.png|300px|Reaction torque for increasing values of Ω]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.9:''' Reaction torque for increasing values of <math>\Omega </math>
-|}
-<div id='img-D.10'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-ydisp.png|420px|Local transverse displacements at the tip of the beam for different values of Ω. Numeric results and its SVD approximation are compared.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.10:''' Local transverse displacements at the tip of the beam for different values of <math>\Omega </math>. Numeric results and its SVD approximation are compared.
-|}
-<div id='img-D.11a'></div>
+<div id="cite-13"></div>
-<div id='img-D.11b'></div>
+'''[[#citeF-13|[13]]]''' Goddard, J. D. (1990) "Nonlinear Elasticity and Pressure-Dependent Wave Speeds in Granular Media", Volume 430. Proceedings: Mathematical and Physical Sciences 1878 105 - 131
-<div id='img-D.11c'></div>
-<div id='img-D.11'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-spectral50.png|600px|]]
-|[[Image:Draft_Samper_987121664-monograph-spectral100.png|600px|]]
-|- style="text-align: center; font-size: 75%;"
-| (a)
-| (b)
-|-
-| colspan="2"|[[Image:Draft_Samper_987121664-monograph-spectral200.png|600px|]]
-|- style="text-align: center; font-size: 75%;"
-|  colspan="2" | (c)
-|- style="text-align: center; font-size: 75%;"
-| colspan="2" | '''Figure D.11:''' Power spectral density estimation of the sum of the right side vectors obtained for different rotation speeds: 50 (a), 100 (b) and 200 (c) rad/s.
-|}
-<div id='img-D.12a'></div>
+<div id="cite-14"></div>
-<div id='img-D.12b'></div>
+'''[[#citeF-14|[14]]]''' J. Duffy and R. D. Mindlin. (1957) "Stress-strain relations and vibrations of a granular medium", Volume 24. J. Appl. Mech. 585 - 593
-<div id='img-D.12c'></div>
-<div id='img-D.12'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-xdistip_50.png|600px|]]
-|[[Image:Draft_Samper_987121664-monograph-xdistip_100.png|600px|]]
-|- style="text-align: center; font-size: 75%;"
-| (a)
-| (b)
-|-
-| colspan="2"|[[Image:Draft_Samper_987121664-monograph-xdistip_200.png|600px|]]
-|- style="text-align: center; font-size: 75%;"
-|  colspan="2" | (c)
-|- style="text-align: center; font-size: 75%;"
-| colspan="2" | '''Figure D.12:''' Local horizontal displacements at the tip of the beam for different values of <math>\Omega </math>: 50 (a), 100 (b) and 200 (c) rad/s. Numeric results and its SVD approximation are compared.
-|}
+<div id="cite-15"></div>
+'''[[#citeF-15|[15]]]''' Bagnold, R.A. (1954) "Experiments on a Gravity-Free Dispersion of Large Solid Spheres in a Newtonian Fluid under Shear", Volume 225. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 1160 49&#8211;63
-{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
+<div id="cite-16"></div>
-|+ style="font-size: 75%;" |<span id='table-D.2'></span>'''Table. D.2''' Predominant frequencies and normalised singular values of the presented left side vectors.
+'''[[#citeF-16|[16]]]''' Goldhirsch, I. (2003) "RAPID GRANULAR FLOWS", Volume 35. Annual Review of Fluid Mechanics 1 267-293
-|-
-|
-|}
-'''
-====D.1.2.5 <span id='lb-D.1.2.5'></span>Deformed shapes and stress distribution for Ω = 50 rad/s ===='''
+<div id="cite-17"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-17|[17]]]''' H.P. Zhu and Y.H. Wu and A.B. Yu. (2005) "Discrete and continuum modelling of granular flow", Volume 3. China Particuology 6 354 - 363
-|-
-|[[Image:Draft_Samper_987121664-monograph-stress_x_plot_50.png|394px|]]
-|}
-<div id='img-D.13'></div>
+<div id="cite-18"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-18|[18]]]''' Nedderman, R. M. (1992) "Statics and Kinematics of Granular Materials". Cambridge University Press
-|-
-|[[Image:Draft_Samper_987121664-monograph-stress_x_plot_50_2.png|379px|Deformed shape (×500) and σ<sub>ςς</sub> distribution (colour map, in KPa) at 50 rad/s. Time from 0 to 14 ms. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.13:''' Deformed shape (<math>\times 500</math>) and <math>\sigma _{\varsigma \varsigma }</math> distribution (colour map, in KPa) at 50 rad/s. Time from 0 to 14 ms.
-|}
-===D.1.3 Constant angular acceleration===
+<div id="cite-19"></div>
+'''[[#citeF-19|[19]]]''' P. Jop and Y. Forterre and O. Pouliquen. (2006) "A constitutive law for dense granular flows", Volume 441. Nature 7094 727 - 730
-'''
+<div id="cite-20"></div>
+'''[[#citeF-20|[20]]]''' Vescovi, D. and Luding, S. (2016) "Merging fluid and solid granular behavior", Volume 12. Soft Matter 8616-8628
-====D.1.3.1 <span id='lb-D.1.3.1'></span>Case 1: α = 5 rad/s²===='''
+<div id="cite-21"></div>
+'''[[#citeF-21|[21]]]''' Chialvo, S. and Sun, J. and Sundaresan, S. (2012) "Bridging the rheology of granular flows in three regimes", Volume 85. American Physical Society. Phys. Rev. E 021305
-<div id='img-D.14'></div>
+<div id="cite-22"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-22|[22]]]''' Kamrin, K. and Koval, G. (2012) "Nonlocal Constitutive Relation for Steady Granular Flow", Volume 108. American Physical Society. Phys. Rev. Lett. 178301
-|-
-|[[Image:Draft_Samper_987121664-monograph-Modes_ACC_550.png|570px|Left side vectors (predominant modes) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.14:''' Left side vectors (predominant modes) associated to the eight higher singular values.
-|}
-<div id='img-D.15'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-RSVACC550.png|570px|Right side vectors (predominant oscillations) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.15:''' Right side vectors (predominant oscillations) associated to the eight higher singular values.
-|}
-'''
-====D.1.3.2 <span id='lb-D.1.3.2'></span>Case 2: α = 50 rad/s²===='''
+<div id="cite-23"></div>
+'''[[#citeF-23|[23]]]''' Singh, A. and Magnanimo, V. and Saitoh, K. and Luding, S. (2015) "The role of gravity or pressure and contact stiffness in granular rheology", Volume 17. New journal of physics 4 043028
-<div id='img-D.16'></div>
+<div id="cite-24"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-24|[24]]]''' P. A. Cundall and O. D. L. Strack. (1979) "A discrete numerical model for granular assemblies", Volume 29. Géotechnique 1 47-65
-|-
-|[[Image:Draft_Samper_987121664-monograph-Modes_ACC_5050.png|600px|Left side vectors (predominant modes) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.16:''' Left side vectors (predominant modes) associated to the eight higher singular values.
-|}
-<div id='img-D.17'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-RSVACC5050.png|600px|Right side vectors (predominant oscillations) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.17:''' Right side vectors (predominant oscillations) associated to the eight higher singular values.
-|}
-'''
-====D.1.3.3 Comparative figures===='''
+<div id="cite-25"></div>
+'''[[#citeF-25|[25]]]''' Ramkrishna, D. (2000) "Population Balances: Theory and Applications to Particulate Systems in Engineering". Academic press
-<div id='img-D.18'></div>
+<div id="cite-26"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-26|[26]]]''' Austin, L.G. (1971) "Introduction to the mathematical description of grinding as a rate process", Volume 5. Powder Technology 1 1 - 17
-|-
-|[[Image:Draft_Samper_987121664-monograph-normalssACC.png|420px|Mode intensities for increasing values of α. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.18:''' Mode intensities for increasing values of <math>\alpha </math>.
-|}
-<div id='img-D.19'></div>
+<div id="cite-27"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-27|[27]]]''' Herbst, J.A. and Fuerstenau, D.W. (1980) "Scale-up procedure for continuous grinding mill design using population balance models", Volume 7. International Journal of Mineral Processing 1 1 - 31
-|-
-|[[Image:Draft_Samper_987121664-monograph-y_disp_acc_1.png|450px|Temporal evolution of vertical displacements at the tip of a cantilever beam under constant angular acceleration.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.19:''' Temporal evolution of vertical displacements at the tip of a cantilever beam under constant angular acceleration.
-|}
-<div id='img-D.20'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-Reactions_50acc.png|450px|Reaction torque for increasing values of α.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.20:''' Reaction torque for increasing values of <math>\alpha </math>.
-|}
+<div id="cite-28"></div>
+'''[[#citeF-28|[28]]]''' Sulsky, D. and Zhou, S.-J. and Schreyer, H. L. (1995) "Application of a particle-in-cell method to solid mechanics", Volume 87. Computer Physics Communications 1-2 236&#8211;252
-{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
+<div id="cite-29"></div>
-|+ style="font-size: 75%;" |<span id='table-D.3'></span>'''Table. D.3''' Predominant frequencies and normalised singular values of the presented left side vectors.
+'''[[#citeF-29|[29]]]''' P. Dadvand. (2007) "A framework for developing finite element codes for multi-disciplinary applications.". PhD thesis: Universidad Politécnica de Cataluña
-|-
-|
-|}
-'''
-====D.1.3.4 <span id='lb-D.1.3.4'></span>Deformed shapes and stress distribution for α = 50 rad/s² ===='''
+<div id="cite-30"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-30|[30]]]''' P. Dadvand and R. Rossi and E. Oñate. (2010) "An object-oriented environment for developing finite element codes for multi-disciplinary applications", Volume 17. Archives of Computational Methods in Engineering 253-297
-|-
-|[[Image:Draft_Samper_987121664-monograph-stress_dis_150.png|424px|]]
-|}
-<div id='img-D.21'></div>
+<div id="cite-31"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-31|[31]]]''' Dadvand, P. and Rossi, R and Gil, M. and Martorell, X. and Cotela, J. and Juanpere, E. and Idelsohn, S.R. and Oñate, E. (2011) "Migration of a Generic Multi-Physics Framework to HPC Environments". Barcelona Supercomputing Center. Proceedings of the 23rd International Conference on Parallel Computational Fluid Dynamics
-|-
-|[[Image:Draft_Samper_987121664-monograph-stress_dis_250.png|425px|Deformed shape (×200) and σ<sub>ςς</sub> distribution (colour map, in KPa) at α = 50 rad/s² and Ω₀ = 50 rad/s. Time from 0 to 25 ms. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.21:''' Deformed shape (<math>\times 200</math>) and <math>\sigma _{\varsigma \varsigma }</math> distribution (colour map, in KPa) at <math>\alpha </math> = 50 rad/s<math>^2</math> and <math>\Omega _0</math> = 50 rad/s. Time from 0 to 25 ms.
-|}
-==D.2 Three-dimensional cantilever beam==
+<div id="cite-32"></div>
+'''[[#citeF-32|[32]]]''' Donea, J. and Giuliani, S. and Halleux, J.-P. (1982) "An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions", Volume 33. Elsevier. Computer methods in applied mechanics and engineering 1-3 689&#8211;723
-===D.2.1 Natural modes and frequencies===
+<div id="cite-33"></div>
+'''[[#citeF-33|[33]]]''' Li, S. and Liu, W.K. (2002) "Meshfree and particle methods and their applications", Volume 55. Applied Mechanics Review 54 1-34
-<div id='img-D.22'></div>
+<div id="cite-34"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-34|[34]]]''' Gröger, T. and Katterfeld, A. (2006) "On the numerical calibration of discrete element models for the simulation of bulk solids", Volume 21. 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering. Elsevier 533 - 538
-|-
-|[[Image:Draft_Samper_987121664-monograph-modes3D.png|600px|Natural modes associated to the eight lowest natural frequencies of a 3D cantilever beam. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.22:''' Natural modes associated to the eight lowest natural frequencies of a 3D cantilever beam.
-|}
-{|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
+<div id="cite-35"></div>
-|+ style="font-size: 75%;" |'''Table. D.4''' Fifty lowest natural frequencies of a 3D cantilever beam.
+'''[[#citeF-35|[35]]]''' Luding, S. (2008) "Introduction to discrete element methods", Volume 12. Taylor  Francis. European Journal of Environmental and Civil Engineering 7-8 785-826
-|-
-|
-|}
-===D.2.2 Constant angular velocity===
+<div id="cite-36"></div>
+'''[[#citeF-36|[36]]]''' P. W. Cleary. (2009) "Industrial particle flow modelling using discrete element method", Volume 26. Engineering Computations 6 698-743
-'''
+<div id="cite-37"></div>
+'''[[#citeF-37|[37]]]''' Labra, C. and Rojek, J. and Oñate, E. and Zarate, F. (2008) "Advances in discrete element modelling of underground excavations", Volume 3. Acta Geotechnica 317&#8211;322
-====D.2.2.1 <span id='lb-D.2.2.1'></span>Case 1: Ω = 25 rad/s===='''
+<div id="cite-38"></div>
+'''[[#citeF-38|[38]]]''' Casas, G. and Mukherjee, D. and Celigueta, M.A. and Zohdi, T. and Oñate, E. (2017) "A modular, partitioned, discrete element framework for industrial grain distribution systems with rotating machinery", Volume 4. Computer Methods in Applied Mechanics and Engineering 181&#8211;198
-<div id='img-D.23'></div>
+<div id="cite-39"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-39|[39]]]''' Guo, Y. and Curtis, J. S. (2015) "Discrete Element Method Simulations for Complex Granular Flows", Volume 47. Annual Review of Fluid Mechanics 1 21-46
-|-
-|[[Image:Draft_Samper_987121664-monograph-lsv25.png|540px|Left side vectors (predominant modes) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.23:''' Left side vectors (predominant modes) associated to the eight higher singular values.
-|}
-<div id='img-D.24'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-rsv25.png|540px|Right side vectors (predominant oscillations) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.24:''' Right side vectors (predominant oscillations) associated to the eight higher singular values.
-|}
-'''
-====D.2.2.2 <span id='lb-D.2.2.2'></span>Case 2: Ω = 50 rad/s===='''
+<div id="cite-40"></div>
+'''[[#citeF-40|[40]]]''' J. F. Favier  and M. H. Abbaspour-Fard  and M. Kremmer. (2001) "Modeling Nonspherical Particles Using Multisphere Discrete Elements", Volume 127. Journal of Engineering Mechanics 10 971-977
-<div id='img-D.25'></div>
+<div id="cite-41"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-41|[41]]]''' Azéma, E. and  Radjaï, F. and Peyroux, R. and Richefeu, V. and Saussine, G. (2008) "Short-time dynamics of a packing of polyhedral grains under horizontal vibrations", Volume 26. Springer. The European Physical Journal E 3 327&#8211;335
-|-
-|[[Image:Draft_Samper_987121664-monograph-lsv50.png|600px|Left side vectors (predominant modes) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.25:''' Left side vectors (predominant modes) associated to the eight higher singular values.
-|}
-<div id='img-D.26'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-rsv50.png|540px|Right side vectors (predominant oscillations) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.26:''' Right side vectors (predominant oscillations) associated to the eight higher singular values.
-|}
-'''
-====D.2.2.3 <span id='lb-D.2.2.3'></span>Case 3: Ω = 100 rad/s===='''
+<div id="cite-42"></div>
+'''[[#citeF-42|[42]]]''' Podlozhnyuk, A. and Pirker, S. and Kloss, C. (2017) "Efficient implementation of superquadric particles in Discrete Element Method within an open-source framework", Volume 4. Computational Particle Mechanics 1 101&#8211;118
-<div id='img-D.27'></div>
+<div id="cite-43"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-43|[43]]]''' Luding, S. (1998) "Collisions and Contacts between Two Particles", Volume 350. Springer. Physics of Dry Granular Media. NATO ASI Series (Series E: Applied Sciences)
-|-
-|[[Image:Draft_Samper_987121664-monograph-lsv100.png|600px|Left side vectors (predominant modes) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.27:''' Left side vectors (predominant modes) associated to the eight higher singular values.
-|}
-<div id='img-D.28'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-rsv100.png|540px|Right side vectors (predominant oscillations) associated to the eight higher singular values. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.28:''' Right side vectors (predominant oscillations) associated to the eight higher singular values.
-|}
-'''
-====D.2.2.4 Campbell diagram===='''
+<div id="cite-44"></div>
+'''[[#citeF-44|[44]]]''' Walton, O. R.  and Johnson, S. M. (2009) "Simulating the Effects of Interparticle Cohesion in Micron-Scale Powders", Volume 1145. AIP Conference Proceedings 1 897-900
-<div id='img-D.29'></div>
+<div id="cite-45"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-45|[45]]]''' Thakur, S.C. and Morrissey, J.P. and Sun, J. and Chen, J.F. and Ooi, J.Y. (2014) "Micromechanical analysis of cohesive granular materials using the discrete element method with an adhesive elasto-plastic contact model", Volume 16. Granular Matter
-|-
-|[[Image:Draft_Samper_987121664-monograph-campbell2.png|353px|Campbell diagram: comparison between modal prediction and SVD predominant frequencies. Subscripts a and b refer to pairs of equal modes that activate either on ϱ or z. ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.29:''' Campbell diagram: comparison between modal prediction and SVD predominant frequencies. Subscripts <math>a</math> and <math>b</math> refer to pairs of equal modes that activate either on <math>\varrho </math> or <math>z</math>.
-|}
-'''
-====D.2.2.5 Comparative figures ===='''
+<div id="cite-46"></div>
+'''[[#citeF-46|[46]]]''' Walton, O.R. (1993) "Particulate Two-Phase Flow". Butterworth-Heinemann
-<div id='img-D.30'></div>
+<div id="cite-47"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-47|[47]]]''' Brilliantov, N.V. and Spahn, F. and Hertzsch, J.M. and Poschel T. (1996) "Model for collisions in granular gases", Volume 53. Physical review E 5392
-|-
-|[[Image:Draft_Samper_987121664-monograph-ydisp3d.png|420px|Temporal evolution of ϱ displacements at the tip of a cantilever beam under constant angular velocity.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.30:''' Temporal evolution of <math>\varrho </math> displacements at the tip of a cantilever beam under constant angular velocity.
-|}
-<div id='img-D.31'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-zdisp3d.png|480px|Temporal evolution of z displacements at the tip of a cantilever beam under constant angular velocity.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.31:''' Temporal evolution of <math>z</math> displacements at the tip of a cantilever beam under constant angular velocity.
-|}
-<div id='img-D.32'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:Draft_Samper_987121664-monograph-reaction3D.png|420px|Reaction torque for increasing values of Ω.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.32:''' Reaction torque for increasing values of <math>\Omega </math>.
-|}
-'''
-====D.2.2.6 <span id='lb-D.2.2.6'></span>Deformed shapes and stress distribution for Ω = 50 rad/s ===='''
+<div id="cite-48"></div>
+'''[[#citeF-48|[48]]]''' Thornton, C. and Ning, Z. (1998) "A theoretical model for the stick/bounce behaviour of adhesive, elastic-plastic spheres", Volume 99. Powder Technology 2 154 - 162
-<div id='img-D.33'></div>
+<div id="cite-49"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-49|[49]]]''' Lucy, L. B. (1977) "A numerical approach to the testing of the fission hypothesis", Volume 82. The Astronomical Journal 1013-1024
-|-
-|[[Image:Draft_Samper_987121664-monograph-3dstress.png|383px|Deformed shape (×500) and σ<sub>ςς</sub> distribution (colour map, in KPa) at Ω₀ = 50 rad/s. Time from 0 to 17 ms (up-down, left-right). ]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.33:''' Deformed shape (<math>\times 500</math>) and <math>\sigma _{\varsigma \varsigma }</math> distribution (colour map, in KPa) at <math>\Omega _0</math> = 50 rad/s. Time from 0 to 17 ms (up-down, left-right).
-|}
-==D.3 Starting of a rotor blade==
+<div id="cite-50"></div>
+'''[[#citeF-50|[50]]]''' Gingold, R.A. and Monaghan, J.J. (1977) "Smoothed particle hydrodynamics-theory and application to non-spherical stars", Volume 181. Monthly Notices of the Royal Astronomical Society 375&#8211;389
-<div id='img-D.34'></div>
+<div id="cite-51"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-51|[51]]]''' Monaghan, J.J. (2005) "Smoothed particle hydrodynamics", Volume 68. Reports on Progress in Physics 8 1703&#8211;1759
-|-
-|[[Image:Draft_Samper_987121664-monograph-xvel.png|600px|Temporal evolution of the angular speed and the local axial displacements at the tip.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.34:''' Temporal evolution of the angular speed and the local axial displacements at the tip.
-|}
-<div id='img-D.35'></div>
+<div id="cite-52"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-52|[52]]]''' Dalrymple, R.A. and Rogers, B.D. (2006) "Numerical modeling of water waves with the SPH method", Volume 53. Elsevier. Coastal engineering 2 141&#8211;147
-|-
-|[[Image:Draft_Samper_987121664-monograph-reactacc.png|600px|Temporal evolution of the angular acceleration and elastic reaction torque.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.35:''' Temporal evolution of the angular acceleration and elastic reaction torque.
-|}
-<div id='img-D.36'></div>
+<div id="cite-53"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-53|[53]]]''' Hultman, J. and Pharayn, A. (1999) "Hierarchical, dissipative formation of elliptical galaxies: Is thermal instability the key mechanism? Hydrodynamical simulations including supernova feedback multi-phase gas and metal enrichment in CDM: Structure and dynamics of elliptical galaxies", Volume 347. Astron. Astrophys. 769 - 798
-|-
-|[[Image:Draft_Samper_987121664-monograph-stress_blade.png|600px|Temporal evolution of the maximum values of stress.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.36:''' Temporal evolution of the maximum values of stress.
-|}
-<div id='img-D.37'></div>
+<div id="cite-54"></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+'''[[#citeF-54|[54]]]''' Monaghan, J.J. and J.C. Lattanzio. (1991) "A simulation of the collapse and fragmentation of cooling molecular clouds", Volume 375. Astrophys. J. 177&#8211;189
-|-
-|[[Image:Draft_Samper_987121664-monograph-rsvblade.png|600px|Right side vectors of the four most predominant oscillations.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure D.37:''' Right side vectors of the four most predominant oscillations.
-|}
-===BIBLIOGRAPHY===
+<div id="cite-55"></div>
+'''[[#citeF-55|[55]]]''' Levasseur, S. and Malécot, Y. and Boulon, M. and Flavigny, E. and Malecot, Y. (1998) "Newtonian hydrodynamics of the coalescence of black holes with neutron stars ii. tidally locked binaries with a soft equation of state", Volume 360. Mon. Not. R. Astron. Astrophys. 780&#8211;794
-<div id="cite-1"></div>
+<div id="cite-56"></div>
-'''[[#citeF-1|[1]]]''' Rao, J. S. (2011) "History of rotating machinery dynamics". Springer 358
+'''[[#citeF-56|[56]]]''' Bui, H. H. and Fukagawa, R. and Sako, K. and Ohno, S. (2008) "Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic&#8211;plastic soil constitutive model", Volume 32. Wiley Online Library. International Journal for Numerical and Analytical Methods in Geomechanics 12 1537&#8211;1570
-<div id="cite-2"></div>
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