Nonlinear Analysis of Orthotropic Membrane and Shell Structures Including Fluid-structure Interaction


=Resumen=

Problemas de interacción fluido-estructura representan hoy en día un gran desafío en diferentes áreas de ingeniería y ciencias aplicadas. Dentro de las aplicaciones en ingeniería civil, el flujo del viento y los movimientos estructurales pueden ocasionar inestabilidades aeroelásticas en construcciones tales como puentes de gran luz, rascacielos y cubiertas estructurales ligeras. Por otro lado, aplicaciones en biomecánica están interesadas en el estudio de hemodinámica, por ejemplo: flujo sanguíneo en arterias, donde grandes deformaciones de las venas interactúan con fluidos.

En la parte estructural de este trabajo, una nueva metodología para el análisis geométricamente no-lineal ortótropo de membranas y láminas sin grados de libertad de rotación es desarrollada basándose en la orientación de la fibra principal del material. Una consecuencia directa de la estrategia de orientación de fibras es la posibilidad de analizar membranas y láminas pretensadas cuya configuración inicial está fuera del plano. Por otra parte, ya que la teoría convencional de membranas permite que existan tensiones de compresión, un modelo de arrugado basado en la modificación de la ecuación constitutiva se presenta. El desarrollo estructural es modelado con elementos finitos provenientes de las ecuaciones de la elastodinámica.

La parte de fluidos de este trabajo está gobernada por las ecuaciones de Navier-Stokes para flujos incompresibles, las cuales son modeladas por interpolaciones estabilizadas de elementos finitos.  Ya que la solución monolítica de dichas ecuaciones tiene la desventaja que consumen mucho tiempo en la solución de grandes sistemas de ecuaciones, el método de pasos fraccionados se usa para aprovechar las ventajas computacionales que brinda gracias al desacoplamiento de la presión del campo de las velocidades. Además, el esquema <math display="inline">\alpha </math>-generalizado para integración en el tiempo para fluidos es adaptado para que se use con la técnica de los pasos fraccionados.

El problema de interacción fluido-estructura es formulado como un sistema de tres campos: la estructura, el fluido y el movimiento de la malla. El movimiento del dominio del fluido es tomado en cuenta mediante la formulación arbitraria Lagrangiana-Euleriana, para la cual se usan dos estrategias de movimiento de malla. Para el acoplamiento entre el fluido y la estructura se usa un acoplamiento fuerte por bloques usando la técnica de Gauss-Seidel. Debido a que la interacción entre el fluido y la estructura es altamente no-lineal, se implementa el método de relajación basado en la técnica de Aitken, la cual acelera la convergencia del problema.

Finalmente varios problemas se presentan en los diferentes campos, los cuales verifican la eficiencia de los algoritmos implementados.

=Agradecimientos=

Agradezco a mis directores de tesis, Eugenio Oñate y Juan Miquel Canet, por la oportunidad y buena disposición para llevar a cabo este trabajo. Gracias a ellos y especialmente a Salvador Botello he descubierto el  maravilloso mundo de la mecánica computacional, destacado por la pasión,  dedicación y coraje que se necesita para lograr nuevas metas.

Asimismo quiero agradecer a Carlos Arnold y Sergio Silva, profesores de la  facultad de Ingeniería Civil de la Universidad  de Guanajuato por haber depositado en mi su confianza y apoyarme para  conseguir la beca PROMEP.  También agradezco al CIMNE por la financiación prestada que me permitió  concluir este trabajo.

A todas aquellas personas que han hecho que este trabajo sea posible  les  agradezco infinitamente su buena disposición, en especial al grupo de fluidos del CIMNE dirigido por Ramón Codina y a mis compañeros  Herbert Owen y Javier Principe. También agradezco a  Sergio Oller, Fernando Flores, Francisco Zárate, Michele  Chiumenti,  Miguel Cervera, Carlos Agelet, Juan Carlos Cante, Javier Oliver,   Gabriel Bugeda y Robert L. Taylor, porque siempre me resolvieron más de una duda. Asimismo agradezco al personal del CIMNE y del departamento de resistencia de materiales y estructuras en ingeniería RMEE por su siempre buena disposición y amables atenciones, en especial a Rosa María Olea. Debo agradecer también a Mariano Vázquez y Guillaume Houzeaux por las correcciones y  recomendaciones hechas a este trabajo.

Quiero agradecer profundamente a Dorian Luis Linero, Martha Liliana Carreño, Sergio Blanco, Carlos González Ferrari, Guillermo Díaz y  Mabel Cristina Marulanda, mis compañeros del despacho 111, por su compañía, compresión,   y gran amistad nacida durante estos años. Asimismo, quiero agradecer a Noel Hernández, Fernando Guzmán, Gelacio Juárez, Humberto Esqueda e Ivo Dias por esos grandes momentos dedicados al estudio y reir siempre a la hora de la comida.  No me puedo olvidar de Roberto Clemente, Fernando Rastellini y Jeovan Faleiro con quienes compartí ideas y momentos difíciles durante el doctorado, así como a Pablo Mata, José Manuel González, Joaquín Hernández, Riccardo Rossi, Pere Andreu, Santiago Badía y Xavier Martínez.

A las personas que me enseñaron el camino a seguir en Barcelona cuando recién llegue para comenzar mis estudios les estaré por siempre agradecido,  en especial a mis queridos e inolvidables amigos Quino, Ysabel y por supuesto Elizabeth, que más puedo  decir, simplemente gracias por todo.  También agradezco a mis amigos de la generación de salida con quienes compartí muchos  agradables momentos, a Guillaume, Ana,  Eduardo y Esteban. También agradezco a las nuevas generaciones del doctorado por esos momentos que hemos podido compartir, a Pablo, Maritzabel, Christian, Juan Carlos,  Flavia y María Laura.

Quiero agradecer a la gente del futbol de los viernes por la noche por los momentos  inolvidables durante y después del partido, en especial a Manolo, Lucho, Antonio, Alex y Gigio.

No puedo dejar de mencionar en este trabajo a mis grandes amigos de toda la vida, Luis Eugenio, Victor, Ismael, Carlos y Julio César, asi como a mis compañeros y  tambien grandes amigos de la universidad, Alberto, Heriberto, Miguel Eugenio,  Julio, Lalo, Vero, Ana  y en especial a Sandra, gracias por las palabras de ánimo en los momentos más difíciles  cuando no veía la salida.

Y por último, quiero agradecer a toda mi familia, empezando por mi hermano Fernando y mis padres Irma y Jesús por estar siempre ahí, apoyándome y dándome palabras de aliento que me dieron la fuerza necesaria para salir adelante. Gracias a mis primos, tíos y todos mis familiares, y muy especialmente quiero agradecer a mis abuelos  José, Jesús y Lolo porque de ellos proviene mi escencia y de quienes siempre me acuerdo. Sé que algunos de ellos me vieron partir pero no pudieron verme regresar, pero saben que siempre los llevaré en mi corazón y les estaré eternamente agradecido.

Jesús Gerardo Valdés Vázquez

Octubre de 2007

=1 Introduction=

==1.1 Motivation==

Modeling of structural elements, such as membranes and thin shells, is widely used  in many engineering fields. In civil engineering applications, their elegance, effectiveness and optimal material  usage make these light weight structures an ideal construction element.

The introduction of new fiber materials, such as glass, carbon or aramide fibers with orthotropic material behavior   have motivated a deep study of such elements  which are used to build membrane and thin shell structures.

<div id='img-1'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Content_586240065-Munich_1.png|600px|Membrane long span structure]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' Membrane long span structure
|}

Moreover membrane and thin shell structures are characterized by their low flexural  stiffness. Consequently these elements should not resist any compression at all. Therefore the usage of such construction materials is performed by introducing a prestressed force to the structure. Figs. [[#img-1|1]] and [[#img-2|2]] show different applications of membranes in civil engineering structures.

<div id='img-2'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Content_586240065-Munich_2.png|600px|Prestressed membrane structure]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 2:''' Prestressed membrane structure
|}

Other examples where light weight structures can be found include aircraft  and spacecraft applications, parachutes, automobile airbags, sails,  windmills and human tissues.

Since this kind of structures are highly flexible systems, they are susceptible  to wind loads applied to them.  Wind flow and structural motion may lead to aeroelastic instabilities which may  cause important damage or even collapse of the structure.

Maybe one of the most important examples of aeroelastic instabilities is the disaster of the Tacoma Narrows suspension bridge  that took place in the U.S.A. on November 7, 1940. The collapse of the bridge was due to wind-induced vibrations that at the beginning produced large transverse and rotational oscillations, as can be seen in Fig. [[#img-3|3]].

<div id='img-3'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Content_586240065-Tacoma_1.png|600px|Aeroelastic instabilities of the Tacoma Narrows suspension bridge, U.S.A.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 3:''' Aeroelastic instabilities of the Tacoma Narrows suspension bridge, U.S.A.
|}

In general many physical problems of different fields belong to multiphysic problems. In particular, numerical simulation of fluid-structure interaction problems have gained great interest from the industry community in order to reduce development time and cost in coupled systems. This kind of problems are complex because they consist of structural nonlinear boundary conditions imposed on fluid moving domains where the position is part of the solution.

Recently biomechanical applications are interested in the numerical simulation of  hemodynamics, which study the blood flow through veins and arteries. In this problem, large structural deformations of the arteries interact with viscous  blood flow as a consequence of each heart beat. Another field where fluid-structure interaction plays an important role is the aerospace engineering which study wind flow around flexible wings of aircrafts.

==1.2 Background==

In this work, membrane and shell structures with large deformations are studied.  A numerical solution for membranes may be constructed using the finite element  method, which solution for small deformations can be found in <span id='citeF-1'></span>[[#cite-1|[1]]],  <span id='citeF-2'></span>[[#cite-2|[2]]] or <span id='citeF-3'></span>[[#cite-3|[3]]].

Theory for large deformations can proceed following the presentations  of <span id='citeF-4'></span>[[#cite-4|[4]]], <span id='citeF-5'></span>[[#cite-5|[5]]], <span id='citeF-6'></span>[[#cite-6|[6]]] or <span id='citeF-7'></span>[[#cite-7|[7]]].  In particular a large displacement formulation of membrane elements composed by three-node triangular finite elements based on rectangular Cartesian coordinates is proposed by <span id='citeF-8'></span>[[#cite-8|[8]]].

This last formulation together with the study of <span id='citeF-9'></span>[[#cite-9|[9]]] form the basis for  the development of the membrane formulation given in this work, which includes orthotropic material behavior and prestressed forces.

Several studies have been carried out to study the geometrically nonlinear behavior of shells, for example the works of <span id='citeF-4'></span>[[#cite-4|[4]]], <span id='citeF-5'></span>[[#cite-5|[5]]],  <span id='citeF-10'></span>[[#cite-10|[10]]], among many others. Since shell analysis requires a lot of memory and cpu-time to compute, several authors have tried to derive plate and shell elements with displacements as the only nodal variables.

In this area, <span id='citeF-11'></span>[[#cite-11|[11]]] proposed a general procedure based on finite  volume concepts for deriving linear thin plate elements of triangular and  quadrilateral shapes with the nodal deflections as the only degree of freedom. <span id='citeF-12'></span>[[#cite-12|[12]]] proposed a different approach to compute the constant curvature field within each triangle in terms of the six-node displacement of a macro-element. This triangular element was successfully applied to nonlinear shell analysis using an explicit dynamic approach. <span id='citeF-13'></span>[[#cite-13|[13]]] continue with the study of rotation free elements of <span id='citeF-11'></span>[[#cite-11|[11]]] developing new triangular elements. This formulation applied to large deformations with an explicit dynamic procedure was presented by <span id='citeF-14'></span>[[#cite-14|[14]]]. <span id='citeF-15'></span>[[#cite-15|[15]]] proposed the same element that <span id='citeF-14'></span>[[#cite-14|[14]]] but applied to metal forming processes.

As an alternative formulation for large strain plasticity, the BST shell  element was introduced by <span id='citeF-16'></span>[[#cite-16|[16]]]. This formulation constitutes the starting point for the development of the  rotation-free shell formulation developed in this work, which includes orthotropic material behavior and prestressed forces.

Besides the structural developments, to perform a fluid-structure interaction study the fluid flow for incompressible problems has to be implemented. Finite element analysis of fluids present potential numerical  instabilities that emerge for incompressible flow problems. To circumvent these problems, different stabilization techniques  have been proposed. One of the stabilization techniques that has been extensively used is the ''streamline-upwind/Petrov-Galerkin'' SUPG method.  Here numerical oscillations could be avoided by introducing numerical diffusion  only along the streamlines as explained in the work of <span id='citeF-17'></span>[[#cite-17|[17]]] for advection-diffusion equation.  The use of the streamline diffusion in the context of weighted residual methods is given in <span id='citeF-18'></span>[[#cite-18|[18]]]. Another kind of stabilization is the ''pressure-stabilizing/Petrov-Galerkin'' PSPG method.  In <span id='citeF-19'></span>[[#cite-19|[19]]], the SUPG and PSPG stabilization methods are used together with equal-order interpolations.

With the idea to better understand the origins of stabilized methods, which can be derived from a firm theoretical foundation and a precise definition of the intrinsic time scale parameter, <span id='citeF-20'></span>[[#cite-20|[20]]] developed the ''subgrid scale'' method. In the context of these methods, the ''orthogonal sub-scales'' OSS method was introduced by <span id='citeF-21'></span>[[#cite-21|[21]]]. The stabilization methods described require the addition of some artificial diffusion terms. However another technique where the stabilization terms emerge from the governing equations of the problem is the ''finite calculus'' FIC method given by <span id='citeF-22'></span>[[#cite-22|[22]]]. An application of the FIC for incompressible viscous flow problems can be found  in <span id='citeF-23'></span>[[#cite-23|[23]]].

The monolithic coupled equations for incompressible fluid problems  have the disadvantage that take great computer effort to solve the algebraic system for each time step in a transient analysis. Since the original works of <span id='citeF-24'></span>[[#cite-24|[24]]] and <span id='citeF-25'></span>[[#cite-25|[25]]], fractional step methods for the incompressible Navier-Stokes equations have earned widespread popularity because of the computational efficiency given by the uncoupling of the pressure from the velocity field.  A detailed stability analysis of fractional step methods for incompressible flows is given in <span id='citeF-26'></span>[[#cite-26|[26]]]. In this work, the FIC and OSS stabilization techniques are implemented to study the coupled problem of large structural deformations and incompressible fluids flow using pressure segregation methods.

With the structural and fluid problem introduced, the remaining task to study  is the coupled problem between both of them. The implementation of a coupled problem can be done using two different global  strategies, which are the ''monolithic'' methods and the  ''partitioned'' methods. In partitioned methods application of existing appropriate and sophisticated solvers  for either structural or fluid subsystems will be used.  Partitioned methods were introduced by <span id='citeF-27'></span>[[#cite-27|[27]]]. The key idea for  these methods is clearly described in <span id='citeF-28'></span>[[#cite-28|[28]]]. Partitioned solutions with staggered coupling algorithms are developed by <span id='citeF-29'></span>[[#cite-29|[29]]] to be used in aeroelastic wing problems.

Applications of strongly partitioned algorithms to large displacements  structural problems coupled to viscous incompressible fluids are given by <span id='citeF-30'></span>[[#cite-30|[30]]] and <span id='citeF-31'></span>[[#cite-31|[31]]]. Other large displacements structural problems interacting with incompressible fluids  are detailed in <span id='citeF-32'></span>[[#cite-32|[32]]], <span id='citeF-33'></span>[[#cite-33|[33]]] and <span id='citeF-34'></span>[[#cite-34|[34]]]. More sophisticated developments on strong partitioned method for FSI problems can be  found in <span id='citeF-35'></span>[[#cite-35|[35]]], <span id='citeF-36'></span>[[#cite-36|[36]]] and <span id='citeF-37'></span>[[#cite-37|[37]]]. A study on strong coupling partitioned methods for fluid-structure interaction problems applied to hemodynamic can be found in <span id='citeF-38'></span>[[#cite-38|[38]]], <span id='citeF-39'></span>[[#cite-39|[39]]],  and <span id='citeF-40'></span>[[#cite-40|[40]]]. In this work, strong coupling partitioned methods are used for fluid-structure interaction problems.

==1.3 Objectives==

Two general objectives in this work are pursued. The first of them is the improvement  of prior developments made at CIMNE related to nonlinear membrane and shell analysis.  The second general objective is the study of fluid-structure interaction problems using fluid flow pressure segregation methods and stabilization techniques developed at CIMNE.

The two general objectives must be implemented in one efficient, robust and accurate computational tool that use finite element technology to solve the problem in study,  which also must have possibilities to analyze each subproblem, i.e. the solid or fluid  part, independently as a highly developed software that exchange data for the  solution of the coupled problem with other separate solvers.

The following particular objectives belong to the structural part:

* To review the state of the art for geometrically nonlinear membrane and shell          finite elements.
* To improve the membrane and rotation-free shell finite elements developed         at CIMNE in prior studies.
* To develop a new methodology to analyze orthotropic material behavior of membrane         and shell finite elements, including the wrinkling phenomena to avoid compression          stresses in membranes.
* To propose a new strategy to study prestressed membrane elements.
* To explore existing time integration schemes for structural dynamic problems and         to work with the best choice for long time analysis periods.
* To implement the membrane and rotation-free shell finite elements together with          their new developments in the COMET program, which is a software for coupled          contact, mechanical and thermal analysis built at CIMNE.

The particular objectives belonging to the fluid-structure interaction problem  part are:

* To implement in the COMET program the finite element fluid problem based on the          incompressible Navier-Stokes equations and to use the stabilization techniques          developed at CIMNE.
* To implement the mesh update strategy developed at CIMNE for the motion of the         fluid domain.
* To explore existing time integration schemes for fluid dynamic problems and         to work with the best choice for long time analysis periods.
* To review the state of the art for fluid-structure interaction problems.
* To implement a general and simple methodology to perform the study of coupled          problems using the COMET program.

All these objectives are oriented to improve and to combine  the developments made ''in-the-house'' in the solid mechanics field,  i.e. membrane and thin shell finite elements, and the fluid dynamics field,  i.e. stabilization techniques and mesh moving algorithms, to perform fluid-structure interaction of problems involving large  structural displacements and incompressible fluid flows.

==1.4 Outline==

Since this work deals with different topics, a detailed review of the  state-of-the-art is provided at the beginning of each theme  to be developed.  The work presented is organized as follows:

''Chapter 2''. Since the computational models emerged in this  work are developed from the mechanics of a continuous medium, in this chapter the kinematics, stress and strain measures for solids and fluids,  conservation equations, and the constitutive and Navier-Stokes equations used in this work are given. The remainder of this work is referred  to these equations.

''Chapter 3''. In this chapter a review of the total Lagrangian formulation for geometrically nonlinear solid mechanics is made.  From the continuum mechanics theory, the general formulation of each  subject is presented which later is discretized with finite elements. Here the concept of principal fiber orientation is introduced. Next a new formulation for membrane elements is developed based on the fiber orientation,  including orthotropic material behavior and initially out-of-plane prestressed conditions. Also a basic wrinkling algorithm to avoid compression stresses is studied. Then the rotation-free shell elements are derived, using the fiber orientation to yield a new formulation to study prestressed and orthotropic material behavior of shells. Later a review of some time integration schemes for solids are addressed. The solution strategy used in this work and linearization of the semi-discrete equations of motion is explained. Finally some example problems are presented to validate the finite element implementations in the COMET code.

''Chapter 4''. This chapter deals with the fluid dynamic equations that solve the incompressible flow problem. Next the governing equations which yield the weak form and the finite element  discretization for fluids are explained. Later a review of some time integration schemes for fluids are studied. Two different pressure segregation methods to solve the incompressible fluid  equations problem are presented. Then three stabilization techniques for the incompressible fluid problem are introduced. The solution strategy used in this work to solve the nonlinear algebraic equations is shown. Finally some benchmark problems are solved and compared to validate the finite element implementation in the COMET program.

''Chapter 5''. In this chapter the coupling strategies for fluid-structure interaction problems are explained. The finite element formulation described in chapter 4 dealing with the incompressible flow problem is extended to account for moving fluid domains by means of the arbitrary Lagrangian-Eulerian formulation. Next the governing equations which yield the weak form of the coupled problem are obtained. Then different partitioned methods to solve the fluid-structure interaction problem are studied. Later two different mesh update techniques are presented. A detailed algorithm for the strong coupling with relaxation used in this  work is given. Finally some example problems are presented to validate the finite element implementations in the COMET code.

''Chapter 6''. Here the conclusions and achievements of  this work are presented. This monograph concludes with some suggestions for future research lines to be developed as a direct consequence of this work.

=2 Continuum Mechanics=

This work deals with the study of structures with large deformations interacting with incompressible fluids. As mentioned in <span id='citeF-41'></span>[[#cite-41|[41]]], the distinction between solids and fluids is not a sharp one.  While a solid has a definite shape which changes are small when there is a  small change in the external conditions, a fluid does not have a preferred  shape and the relative positions of their elements change by an amount which  is not small even tough the applied forces are small.

The macroscopic behavior of fluids, solids and structures is given by models emerged from the mechanics of a continuous medium. In this chapter a summary of the continuum mechanics used in this work for fluid-structure interaction problems is presented. A general and more detailed description of continuum mechanics can be found in <span id='citeF-42'></span>[[#cite-42|[42]]], <span id='citeF-43'></span>[[#cite-43|[43]]],  <span id='citeF-44'></span>[[#cite-44|[44]]], <span id='citeF-41'></span>[[#cite-41|[41]]], <span id='citeF-45'></span>[[#cite-45|[45]]], and <span id='citeF-46'></span>[[#cite-46|[46]]] among many other well known books.

This chapter begins with the kinematics that involved motion of a body. Next the concepts of stress and strain related to nonlinear solid mechanics and fluids are described. Then the conservation equations, also known as balance equations, are presented. Finally the constitutive and Navier-Stokes equations are given.

==2.1 Kinematics==

Kinematics is the study of motion and deformation of a body without regard to the forces responsible for such action. A body <math display="inline">\mathcal{B}</math> can be imagined as a composition of a set of particles which are called  ''material points''. This body is in an initial state when time <math display="inline">t = 0</math> as shown in Fig.&nbsp;[[#img-4|4]]. The domain of the body in this state is denoted by <math display="inline">\Omega _0</math> which occupies a region in space and is known as the ''initial configuration''.  To describe the  kinematics of a body another configuration is needed where equations are referred to and is called the ''reference configuration''. Most of the times the initial  configuration is used as the referenced configuration, unless we specify otherwise.

<div id='img-4'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Content_586240065-Fig2.1.png|600px|Configurations of a body]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 4:''' Configurations of a body
|}

Now we assume that the domain <math display="inline">\Omega _0</math> moves to a new region <math display="inline">\Omega </math> which is occupied by the body <math display="inline">\mathcal{B}</math> for any subsecuent time <math display="inline">t > 0</math>. At this time the configuration of the body is called the ''current configuration'', also known as the ''deformed configuration''. The boundary of the domain, in this case for the current configuration, is denoted by <math display="inline">\Gamma </math>. The dimension of any model is denoted by <math display="inline">n_\textrm{dime}</math> denoting the number of space dimensions of the body <math display="inline">\mathcal{B}</math>.

The position vector of a material point in the reference configuration is defined by  '''X''', where

<span id="eq-1"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{X} = X_i\hbox{e}_i = \sum ^{n_\textrm{dime}}_{i=1}X_i\hbox{e}_i </math>
|}
| style="width: 5px;text-align: right;" | (1)
|}

where <math display="inline">X_i</math> are the components of '''X''' in the reference configuration and <math display="inline">\hbox{e}_i</math> are the unit base vectors for a rectangular Cartesian coordinate system.  The variable vectors '''X''' are called ''material coordinates'' or ''Lagrangian coordinates''. The ''motion'' of the body <math display="inline">\mathcal{B}</math> is given by

<span id="eq-2"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{x} = \phi\mbox{ } ( \hbox{X} ,t ) = \hbox{x} ( \hbox{X} ,t ) </math>
|}
| style="width: 5px;text-align: right;" | (2)
|}

where

<span id="eq-3"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{x} = x_i\hbox{e}_i = \sum ^{n_\textrm{dime}}_{i=1}x_i\hbox{e}_i </math>
|}
| style="width: 5px;text-align: right;" | (3)
|}

is the position of the material point '''X''' in the current configuration. The variable vectors '''x''' are called  ''spatial coordinates'' or  ''Eulerian coordinates'', and the function <math display="inline">\boldsymbol{\phi}(\hbox{X},t)</math> is a mapping of the reference configuration onto the current configuration.

When describing the kinematics of a continuum two approaches can be used. First, if we take material coordinates <math display="inline">X_i</math> and time <math display="inline">t</math> as the independent variables, the description is called ''material description'' or  ''Lagrangian description''. On the other hand, if the independent variables are the spatial coordinates <math display="inline">x_i</math> and time <math display="inline">t</math>, we are taking about a ''spatial description'' or  ''Eulerian description''. In general solid mechanics  use Lagrangian descriptions while fluid mechanics use Eulerian descriptions.

The difference for a material point between its current and reference configuration gives the ''displacement'' which in material description is

<span id="eq-4"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{u} ( \hbox{X},t ) = \hbox{x} - \hbox{X}  </math>
|}
| style="width: 5px;text-align: right;" | (4)
|}

Using Eq. ([[#eq-1|1]]) and Eq. ([[#eq-2|2]]) into Eq. ([[#eq-4|4]]) yields

<span id="eq-5"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{u} ( \hbox{X},t ) =  \phi\mbox{ } ( \hbox{X} ,t ) -                                  \phi\mbox{ } ( \hbox{X} ,0 ) =                                 \phi\mbox{ } ( \hbox{X} ,t ) - \hbox{X}  </math>
|}
| style="width: 5px;text-align: right;" | (5)
|}

since for <math display="inline">t=0</math>, <math display="inline">\hbox{x}=\boldsymbol{\phi}(\hbox{X},0) = \hbox{X}</math>  which means that at reference configuration <math display="inline">\hbox{x}=\hbox{X}</math>.  Given <math display="inline">(\hbox{x},t)</math> as the independent variables, the inverse mapping of the motion is defined as

<span id="eq-6"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{X} = \phi\mbox{ }^{-1} ( \hbox{x} ,t ) = \hbox{X} ( \hbox{x} ,t ) </math>
|}
| style="width: 5px;text-align: right;" | (6)
|}

meaning that the material point '''X''' is associated with the place '''x''' at time ''t''. For a material  point the ''velocity'' is the rate of change, or derivative,  of the position vector. When '''X''' is held constant then the derivative is called ''material time derivative'' or ''total time derivative''. Using Eq. ([[#eq-2|2]]) and Eq. ([[#eq-5|5]]) the material velocity is given by

<span id="eq-7"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{v} ( \hbox{X},t ) =  \frac{\partial \hbox{x} ( \hbox{X} ,t )}{\partial t}                              =  \frac{\partial \hbox{u} ( \hbox{X} ,t )}{\partial t}                            =  \dot{\hbox{u}} ( \hbox{X} ,t ) </math>
|}
| style="width: 5px;text-align: right;" | (7)
|}

The material ''acceleration'' is the rate of change of the velocity, or the material time derivative of the velocity expressed by

<span id="eq-8"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{a} ( \hbox{X},t ) =  \frac{\partial \hbox{v} ( \hbox{X} ,t ) }{\partial t}                            =  \dot{\hbox{v}} ( \hbox{X} ,t ) = \ddot{\hbox{u}} ( \hbox{X} ,t ) </math>
|}
| style="width: 5px;text-align: right;" | (8)
|}

For expressions given in spatial description, i.e. the velocity <math display="inline">\hbox{v} (\hbox{x},t) = \hbox{v} ( \hbox{x} (\hbox{X} ,t) ,t)</math> where we use Eq. ([[#eq-2|2]]),  its material time derivative  can be found using

<span id="eq-9"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{Dv_{i} ( \hbox{x},t ) }{Dt} = \frac{\partial v_{i} ( \hbox{x},t ) }{\partial t} +                                       \frac{\partial v_{i} ( \hbox{x},t ) }{\partial x_j} \cdot                                         \frac{\partial x_j ( \hbox{X},t ) }{\partial t} =                                        \frac{\partial v_{i}}{\partial t} +                                       \frac{\partial v_{i}}{\partial x_j} v_j                                     </math>
|}
| style="width: 5px;text-align: right;" | (9)
|}

where <math display="inline">{\partial v_{i} (\hbox{x},t) }/{\partial t}</math> is the ''spatial time derivative'' and the second term in the right hand side is the  ''convective'' term, where <math display="inline">{\partial v_{i}} / {\partial x_j}</math> is the ''right gradient'' of the velocity vector field with respect to  the spatial coordinates, which in indicial form is <math display="inline">v_{i,j}</math> or in tensor notation is <math display="inline">\hbox{v} \nabla </math>. Using the inverse mapping of the motion,  Eq. ([[#eq-6|6]]) to express the velocity in spatial description, Eq. ([[#eq-9|9]]) can be written as

<span id="eq-10"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{D\hbox{v} ( \hbox{x},t ) }{Dt} = \frac{\partial \hbox{v} ( \hbox{x},t ) }{\partial t} +                                    \hbox{v} ( \hbox{x},t ) \cdot \nabla \hbox{v} ( \hbox{x},t ) </math>
|}
| style="width: 5px;text-align: right;" | (10)
|}

where <math display="inline">\nabla \hbox{v}</math> is the ''left gradient''  of the velocity vector field with respect to the spatial coordinates, which in indicial form is <math display="inline">\partial _j v_i</math>. It is important to see that

<span id="eq-11"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{D\hbox{v} ( \hbox{x},t ) }{Dt} =  \frac{\partial \hbox{v} ( \hbox{X},t ) }{\partial t} </math>
|}
| style="width: 5px;text-align: right;" | (11)
|}

In general the material time derivative of any function, vector or tensor given in spatial variables '''x''' and time ''t'' can be obtained with

<span id="eq-12"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{D ( \bullet ) }{Dt} = \frac{\partial ( \bullet ) }{\partial t} +                                          \hbox{v} \cdot \nabla ( \bullet )  </math>
|}
| style="width: 5px;text-align: right;" | (12)
|}

When a continuum body moves from the reference configuration <math display="inline">\Omega _0</math> to the current configuration <math display="inline">\Omega </math>, it changes its size and shape giving a ''deformation''.  A primary measure of deformation in nonlinear mechanics is the material ''deformation gradient'' tensor given by

<span id="eq-13"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{F} = \frac{\partial \hbox{x}}{\partial \hbox{X}} \textrm{      or     }   F_{ij} = \frac{\partial \phi _i}{\partial X_j} = \frac{\partial x_i}{\partial X_j} </math>
|}
| style="width: 5px;text-align: right;" | (13)
|}

which relates a quantity in the reference configuration to its corresponding quantity in the current  configuration. For example consider an infinitesimal line segment ''d'' '''X''' in the reference  configuration, then using Eq. ([[#eq-13|13]]) the resulting line segment ''d'' '''x''' in the current configuration is

<span id="eq-14"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>d\hbox{x} = \hbox{F} \cdot d\hbox{X} \textrm{      or     }  dx_i = F_{ij}dX_j </math>
|}
| style="width: 5px;text-align: right;" | (14)
|}

The deformation gradient '''F''' is also known as the  ''Jacobian matrix''. Another important quantity related to '''F''' is the  ''Jacobian determinant'' given by

<span id="eq-15"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>J = \textrm{det} ( \hbox{F} ) </math>
|}
| style="width: 5px;text-align: right;" | (15)
|}

The Jacobian determinant is useful to relate integrals in the reference configuration to its counterpart in the current configuration.

Using '''x''' from Eq. ([[#eq-4|4]]) in Eq. ([[#eq-13|13]]), the deformation gradient tensor can be  expressed by

<span id="eq-16"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>F_{ij} = \frac{\partial u_i}{\partial X_j} + \frac{\partial X_i}{\partial X_j}          = \frac{\partial u_i}{\partial X_j} + \delta _{ij} </math>
|}
| style="width: 5px;text-align: right;" | (16)
|}

where <math display="inline">\partial u_i</math>/<math display="inline">\partial X_j</math> is the material ''displacement gradient'' tensor and <math display="inline">\delta _{ij}</math> is the Kronecker delta which values are

<span id="eq-17"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\delta _{ij} =      \displaystyle 1 </math>
| style="text-align: right;" | <math> \mbox{when } i = j </math>
|-
| style="text-align: center;" | <math>                     0 </math>
| style="text-align: right;" | <math> \mbox{otherwise }     </math>
|}
| style="width: 5px;text-align: right;" | (17)
|}

==2.2 Strain Measures==

In the behavior of materials, the strain measures the geometrical deformation caused by the forces applied on a continuum body <math display="inline">\mathcal{B}</math>. The strain is computed as the change between the undeformed initial configuration of the body and its final deformed configuration. Therefore, the strain expresses itself the motion and deformation of a body.

There are many kinematic measures of strain in continuum mechanics. For Lagrangian descriptions the most essential strain is the  ''Green-Lagrange strain''  tensor defined by

<span id="eq-18"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{E} = \frac{1}{2} \left(\hbox{F}^T \cdot \hbox{F} - \hbox{I} \right)\textrm{      or     }       E_{ij} = \frac{1}{2} \left(F^T_{ik} F_{kj} - \delta _{ij} \right) </math>
|}
| style="width: 5px;text-align: right;" | (18)
|}

which also can be expressed as a function of the displacement gradient tensor yielding

<span id="eq-19"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>E_{ij} = \frac{1}{2} \left(\frac{\partial u_i}{\partial X_j} + \frac{\partial u_j}{\partial X_i}                        + \frac{\partial u_k}{\partial X_i} \frac{\partial u_k}{\partial X_j} \right) </math>
|}
| style="width: 5px;text-align: right;" | (19)
|}

For linear strain problems, the ''infinitesimal strain tensor''  can be found from Eq. ([[#eq-19|19]]) by neglecting the nonlinear terms giving

<span id="eq-20"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\varepsilon _{ij} = \frac{1}{2} \left(\frac{\partial u_i}{\partial X_j} +                                           \frac{\partial u_j}{\partial X_i} \right) </math>
|}
| style="width: 5px;text-align: right;" | (20)
|}

Now we define the spatial ''velocity gradient'' tensor by

<span id="eq-21"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{l} = \frac{\partial \hbox{v}}{\partial \hbox{x}} \textrm{      or     }    l_{ij} = \frac{\partial v_i}{\partial x_j} </math>
|}
| style="width: 5px;text-align: right;" | (21)
|}

which can be decomposed into its symmetric and skew-symmetric parts using

<span id="eq-22"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{l} = \frac{1}{2} \left(\hbox{l} + \hbox{l} ^{T} \right)+                \frac{1}{2} \left(\hbox{l} - \hbox{l} ^{T} \right) </math>
|}
| style="width: 5px;text-align: right;" | (22)
|}

where the spatial ''rate of deformation'' tensor '''d''',  also known as the ''velocity strain'' tensor or ''strain rate'' tensor is given by the symmetric part of the velocity gradient according to

<span id="eq-23"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{d} = \frac{1}{2} \left(\hbox{l} + \hbox{l} ^{T} \right)   \textrm{      or     }   d_{ij} = \frac{1}{2} \left(\frac{\partial v_i}{\partial x_j} +                               \frac{\partial v_j}{\partial x_i} \right) </math>
|}
| style="width: 5px;text-align: right;" | (23)
|}

The spatial ''rate of rotation'' tensor '''w''',  also known as the ''spin'' tensor is given by the skew-symmetric part of  '''l''' yielding

<span id="eq-24"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{w} = \frac{1}{2} \left(\hbox{l} - \hbox{l} ^{T} \right)   \textrm{      or     }   w_{ij} = \frac{1}{2} \left(\frac{\partial v_i}{\partial x_j} -                               \frac{\partial v_j}{\partial x_i} \right) </math>
|}
| style="width: 5px;text-align: right;" | (24)
|}

Taking the material time derivative of the deformation gradient tensor, Eq. ([[#eq-13|13]]), gives

<span id="eq-25"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\dot{\hbox{F}} = \frac{\partial \hbox{v}}{\partial \hbox{X}}   \textrm{      or     }   \dot{F}_{ij} = \frac{\partial v_i}{\partial X_j} </math>
|}
| style="width: 5px;text-align: right;" | (25)
|}

and now Eq. ([[#eq-21|21]]) can be written as

<span id="eq-26"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{l} = \dot{\hbox{F}} \cdot \hbox{F}^{-1}   \textrm{      or     }   l_{ij} = \dot{F}_{ik} F^{-1}_{kj} </math>
|}
| style="width: 5px;text-align: right;" | (26)
|}

where we use the spatial ''deformation gradient'' tensor

<span id="eq-27"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{F}^{-1} = \frac{\partial \hbox{X}}{\partial \hbox{x}}   \textrm{      or     }   F^{-1}_{kj} = \frac{\partial X_k}{\partial x_j} </math>
|}
| style="width: 5px;text-align: right;" | (27)
|}

If we take the material time derivative of the Green-Lagrange strain tensor, Eq. ([[#eq-18|18]]), we get

<span id="eq-28"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\dot{\hbox{E}} = \frac{1}{2} \left(\hbox{F}^T \cdot \dot{\hbox{F}} +                     \dot{\hbox{F}}^T \cdot \hbox{F} \right)=                     \hbox{F}^T \cdot \hbox{d} \cdot \hbox{F} </math>
|}
| style="width: 5px;text-align: right;" | (28)
|}

==2.3 Stress Measures==

The motion and deformation of a continuum body <math display="inline">\mathcal{B}</math> gives rise to ''forces'' emerging from interactions between interior parts of the body  or between the body and its environment.  Physically, the stress measures a force per unit area within a body. Let ''P'' be a point on the boundary <math display="inline">\Gamma </math> of the body, '''n''' the outward  ''normal'' unit vector for ''P'' and ''d''<math display="inline">\Gamma </math> the part  of the surface on the body where ''P'' is contained. Then <math display="inline">d \hbox{f}_s</math> is  the ''surface force'' acting at  ''P'' that depends of '''n''' and ''d''<math display="inline">\Gamma </math>. Henceforth the  ''surface traction'' '''t''' at point ''P'' on the surface with normal '''n''' is defined by

<span id="eq-29"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{t} = \hbox{t} (\hbox{n}) = \lim _{d\Gamma \rightarrow 0} \frac{d\hbox{f}_s}{d\Gamma } </math>
|}
| style="width: 5px;text-align: right;" | (29)
|}

where '''t''' does not necessarily coincide in direction with '''n'''. It is important to see that the surface traction has units of force per unit area.     There exists a spatial tensor field }\sigma\mbox{ called the ''Cauchy stress'' tensor such that for each unit vector '''n'''

<span id="eq-30"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{t} =  \hbox{n}  \cdot  \sigma\mbox{ } = \sigma\mbox{ }^T \cdot \hbox{n}   \textrm{      or     }   t_i =  \sigma_{ji} n_j </math>
|}
| style="width: 5px;text-align: right;" | (30)
|}

which is also known as the ''Cauchy's theorem''. Since the Cauchy stress tensor involves the normal to the current surface and the traction on the current surface too, this tensor is also known as the ''true physical'' stress tensor and has the property that is symmetric, see section  '''[[#2.4.2 Conservation of Linear and Angular Momentum|2.4.2]]'''. In the reference configuration the counterpart of Eq. ([[#eq-30|30]]) is

<span id="eq-31"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{t}_0 = \hbox{n}_0 \cdot \hbox{P}   \textrm{      or     }   t^0_i =  P_{ji} n^0_j </math>
|}
| style="width: 5px;text-align: right;" | (31)
|}

where '''P''' is the ''nominal'' stress tensor and <math display="inline">\hbox{t}_0</math> and <math display="inline">\hbox{n}_0</math> is the traction force and unit normal respectively in  the reference configuration. Unlike the Cauchy stress tensor, the nominal stress tensor is not symmetric and is important to see that the normal is to the left. The transpose of the nominal stress tensor is the ''first Piola-Kirchhoff'' stress tensor. The ''second Piola-Kirchhoff'' stress tensor '''S''' is defined by

<span id="eq-32"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{F}^{-1} \cdot \hbox{t}_0 = \hbox{n}_0 \cdot \hbox{S} </math>
|}
| style="width: 5px;text-align: right;" | (32)
|}

where the transformation of the forces by <math display="inline">\hbox{F}^{-1}</math> makes it a symmetric tensor. The transformation between these stresses is given by

<span id="eq-33"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma\mbox{ } = J^{-1} \hbox{F} \cdot \hbox{P}                              = J^{-1} \hbox{F} \cdot \hbox{S} \cdot \hbox{F}^{T} </math>
|}
| style="width: 5px;text-align: right;" | (33)
|}

<span id="eq-34"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{P} = J \hbox{F}^{-1} \cdot \sigma\mbox{ }              = \hbox{S} \cdot \hbox{F}^{T} </math>
|}
| style="width: 5px;text-align: right;" | (34)
|}

<span id="eq-35"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{S} = J \hbox{F}^{-1} \cdot \sigma\mbox{ } \cdot \hbox{F}^{-T}              = \hbox{P} \cdot \hbox{F}^{-T} </math>
|}
| style="width: 5px;text-align: right;" | (35)
|}

==2.4 Conservation Equations==

The ''conservation equations''  reflect some physical quantity for a continuum medium which always must be satisfied and that are not restricted in their application to any material. Applying the conservation equations to the domain <math display="inline">\Omega </math> of a body <math display="inline">\mathcal{B}</math> leads to an integral relation. Since the integral relation must hold for any subdomain of the body, then the conservation equations can be expressed as ''partial differential equations''.

Before continuing with the conservation equations, the material time derivative of an integral relation for any spatial property is defined by

<span id="eq-36"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{D}{Dt} \int _\Omega ( \bullet ) =                  \int _\Omega \left(\frac{D ( \bullet ) }{Dt} +                 ( \bullet ) \nabla \cdot \hbox{v} \right)d\Omega  </math>
|}
| style="width: 5px;text-align: right;" | (36)
|}

which is the Reynold's transport theorem.  The ''divergence'' <math display="inline">\nabla \cdot ( \bullet )</math> taken respect to current coordinates can also be expressed as <math display="inline">\textrm{div}(\hbox{v})</math> or in indicial form <math display="inline">v_{i,i}</math>.

===2.4.1 Mass Conservation===

Consider the domain <math display="inline">\Omega </math> of a body <math display="inline">\mathcal{B}</math>  bounded by the surface <math display="inline">\Gamma </math> which is filled with a constant material ''density'' <math display="inline">\rho (\hbox{X},t)</math>. Then the ''mass'' of the body is given by

<span id="eq-37"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>m = \int _{\Omega } \rho (\hbox{X},t) d\Omega      = \int _{\Omega _0} \rho (\hbox{X},t) J d\Omega _0     = \int _{\Omega _0} \rho _0 (\hbox{X}) d\Omega _0 </math>
|}
| style="width: 5px;text-align: right;" | (37)
|}

where we use Eq. ([[#eq-15|15]]) to relate integrals in the reference and current configurations. ''Mass conservation'' requires  that the mass of any material domain be constant. Consequently the material time derivative of the mass must be zero, giving

<span id="eq-38"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{Dm}{Dt} = \frac{D}{Dt} \int _{\Omega } \rho d\Omega  = 0 </math>
|}
| style="width: 5px;text-align: right;" | (38)
|}

which leads to the following integral relation using Eq. ([[#eq-36|36]])

<span id="eq-39"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{Dm}{Dt} = \int _{\Omega } \left(\frac{D \rho }{Dt} + \rho                    \nabla \cdot \hbox{v}  \right)d\Omega = 0 </math>
|}
| style="width: 5px;text-align: right;" | (39)
|}

where the quantities used are expressed in spatial coordinates.  Since the above  holds for any subdomain <math display="inline">\Omega </math>, the mass conservation yields the following first-order partial differential equation

<span id="eq-40"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{D \rho }{Dt} + \rho \nabla \cdot \hbox{v} = 0     \textrm{      or     }     \dot{\rho } + \rho v_{i,i} = 0 </math>
|}
| style="width: 5px;text-align: right;" | (40)
|}

The above equation is also known as the ''continuity equation''. Using the definition of material time derivative Eq. ([[#eq-12|12]]) in first term of Eq. ([[#eq-40|40]]),  the continuity equation can be written as

<span id="eq-41"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{\partial \rho }{\partial t} + (\rho v_i)_{,i}  = 0 </math>
|}
| style="width: 5px;text-align: right;" | (41)
|}

which is known as the ''conservative form'' of the mass conservation equation. When a material is said to be ''incompressible'', the density keeps constant in time so that its material time derivative vanishes and the continuity equation becomes

<span id="eq-42"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\nabla \cdot \hbox{v} = 0     \textrm{      or     }     v_{i,i}  = 0 </math>
|}
| style="width: 5px;text-align: right;" | (42)
|}

This is the continuity equation used in this work for fluid problems which always are considered as incompressible.

For Lagrangian descriptions, the mass conservation equation Eq. ([[#eq-38|38]]) can be integrated in time to obtain an algebraic equation for the density in the form of Eq. ([[#eq-37|37]]) yielding

<span id="eq-43"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho (\hbox{X},t) J = \rho _0 (\hbox{X})   </math>
|}
| style="width: 5px;text-align: right;" | (43)
|}

which is the Lagrangian description for the mass conservation equation.

===2.4.2 Conservation of Linear and Angular Momentum===

The ''conservation of linear momentum'' states that the rate of change of its linear momentum is equal to the total force applied to it. The conservation of linear momentum is also known as the ''balance of momentum'' principle or simply the ''momentum  conservation'' principle. If we consider an arbitrary domain <math display="inline">\Omega </math> with boundary <math display="inline">\Gamma </math> in the current configuration subjected to ''body forces'' <math display="inline">\rho \hbox{b}</math> and surface tractions '''t''', where '''b''' is a force per unit mass, then the total force '''f''' is given by

<span id="eq-44"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{f}(t) = \int _\Omega \rho \hbox{b} (\hbox{x},t) d\Omega +                      \int _\Gamma \hbox{t} (\hbox{x},t) d\Gamma  </math>
|}
| style="width: 5px;text-align: right;" | (44)
|}

The linear momentum is given by the product of the density <math display="inline">\rho </math> and the velocity '''v''' over the domain <math display="inline">\Omega </math> in the form

<span id="eq-45"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{p}(t) = \int _\Omega \rho \hbox{v} (\hbox{x},t) d\Omega  </math>
|}
| style="width: 5px;text-align: right;" | (45)
|}

Hence the conservation of linear momentum is expressed by

<span id="eq-46"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{D}{Dt} \int _\Omega \rho \hbox{v} (\hbox{x},t) d\Omega =                \int _\Omega \rho \hbox{b} (\hbox{x},t) d\Omega +                \int _\Gamma \hbox{t} (\hbox{x},t) d\Gamma  </math>
|}
| style="width: 5px;text-align: right;" | (46)
|}

Using Eq. ([[#eq-36|36]]) and Eq. ([[#eq-40|40]]) in Eq. ([[#eq-46|46]]), the rate of change of the linear momentum is found to be

<span id="eq-47"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{D}{Dt} \int _\Omega \rho \hbox{v} (\hbox{x},t) d\Omega =   \int _\Omega \rho \frac{D \hbox{v} (\hbox{x},t)}{Dt}  d\Omega  </math>
|}
| style="width: 5px;text-align: right;" | (47)
|}

The boundary integral in Eq. ([[#eq-46|46]]) can be transformed to a domain integral using Eq. ([[#eq-30|30]]) and Gauss' divergence theorem yielding

<span id="eq-48"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\int _\Gamma \hbox{t} (\hbox{x} ,t) d\Gamma =     \int _\Omega \nabla \cdot \sigma (\hbox{x},t) d\Omega  </math>
|}
| style="width: 5px;text-align: right;" | (48)
|}

Substituting Eq. ([[#eq-47|47]]) and ([[#eq-48|48]]) into ([[#eq-46|46]]) gives

<span id="eq-49"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\int _\Omega \left(\rho \frac{D\hbox{v}}{Dt} - \rho \hbox{b} -         \nabla \cdot \sigma \right)d\Omega = 0 </math>
|}
| style="width: 5px;text-align: right;" | (49)
|}

and since it holds for any arbitrary domain we find

<span id="eq-50"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho \frac{D\hbox{v}}{Dt} = \nabla \cdot \sigma + \rho \hbox{b}     \textrm{      or     }    \rho \frac{Dv_i}{Dt} = \frac{\partial \sigma _{ij}}{\partial x_j} + \rho b_i </math>
|}
| style="width: 5px;text-align: right;" | (50)
|}

This is called the ''momentum equation''. In an Eulerian description for the momentum equation, the material time derivative of the velocity in  Eq. ([[#eq-50|50]]) is developed using Eq. ([[#eq-12|12]]) yielding

<span id="eq-51"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho \left(\frac{\partial \hbox{v}}{\partial t} + \hbox{v} \cdot \nabla \hbox{v} \right)=     \nabla \cdot \sigma + \rho \hbox{b}     \textrm{      or     }    \rho \left(\frac{\partial v_i}{\partial t} + v_j \partial _j v_i \right)=     \frac{\partial \sigma _{ij}}{\partial x_j} + \rho b_i </math>
|}
| style="width: 5px;text-align: right;" | (51)
|}

where all the quantities are given is spatial coordinates. In this work, Eq. ([[#eq-51|51]]) is the one to be used in the fluid mechanics problem. For fluid finite elements this equation is called ''Eulerian formulation''.

The momentum equation Eq. ([[#eq-50|50]]) can also be written in a  Lagrangian description where all the quantities are expressed in material coordinates, giving

<span id="eq-52"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho \frac{\partial \hbox{v}}{\partial t} =     \nabla \cdot \sigma + \rho \hbox{b}     \textrm{      or     }    \rho \frac{\partial v_i}{\partial t} =     \frac{\partial \sigma _{ij}}{\partial x_j} + \rho b_i </math>
|}
| style="width: 5px;text-align: right;" | (52)
|}

Since Eq. ([[#eq-50|50]]) is in the current configuration, the divergence term is taken respect to spatial coordinates and therefore <math display="inline">\sigma ( \hbox{X},t )</math> is expressed by <math display="inline">\sigma \left(\phi^{-1}  (\hbox{x},t),t \right)</math> so that the spatial gradient of the stress field can be  evaluated. For nonlinear solid finite elements, Eq. ([[#eq-52|52]]) is called ''updated Lagrangian formulation''.

The ''conservation of angular momentum'' is obtained by taking the cross product of the current vector position '''x''' by each term of the linear momentum equation Eq. ([[#eq-46|46]]), yielding

<span id="eq-53"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{D}{Dt} \int _\Omega \hbox{x} \times \rho \hbox{v} (\hbox{x},t) d\Omega =                \int _\Omega \hbox{x} \times \rho \hbox{b} (\hbox{x},t) d\Omega +                \int _\Gamma \hbox{x} \times \hbox{t} (\hbox{x},t) d\Gamma  </math>
|}
| style="width: 5px;text-align: right;" | (53)
|}

where it can be shown that leads to the following result

<span id="eq-54"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma = \sigma^{T} </math>
|}
| style="width: 5px;text-align: right;" | (54)
|}

The acceleration term in Eq. ([[#eq-50|50]]) can be neglected when the loads are applied  slowly so that the inertial forces become insignificant. Then we can write

<span id="eq-55"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\nabla \cdot \sigma + \rho \hbox{b} = 0     \textrm{      or     }    \frac{\partial \sigma _{ij}}{\partial x_j} + \rho b_i = 0 </math>
|}
| style="width: 5px;text-align: right;" | (55)
|}

which is known as the ''equilibrium equation''.  Problems that use Eq. ([[#eq-55|55]]) are called ''static problems''.

The conservation of linear momentum can also be expressed in the reference configuration.  Consider an arbitrary domain <math display="inline">\Omega _0</math> with boundary <math display="inline">\Gamma _0</math> in the reference configuration subjected to body forces <math display="inline">\rho _0 \hbox{b}</math> and surface tractions <math display="inline">\hbox{t}_0</math>, then the total force '''f''' is given by

<span id="eq-56"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{f}(t) = \int _{\Omega _0} \rho _0 \hbox{b} (\hbox{X},t) d\Omega _0 +                      \int _{\Gamma _0} \hbox{t}_0 (\hbox{X},t) d\Gamma _0 </math>
|}
| style="width: 5px;text-align: right;" | (56)
|}

The linear momentum is given by

<span id="eq-57"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{p}(t) = \int _{\Omega _0} \rho _0 \hbox{v} (\hbox{X},t) d\Omega _0 </math>
|}
| style="width: 5px;text-align: right;" | (57)
|}

Hence the conservation of linear momentum is expressed by

<span id="eq-58"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{d}{dt} \int _{\Omega _0} \rho _0 \hbox{v} (\hbox{X},t) d\Omega _0 =                \int _{\Omega _0} \rho _0 \hbox{b} (\hbox{X},t) d\Omega _0 +                \int _{\Gamma _0} \hbox{t}_0 (\hbox{X},t) d\Gamma _0 </math>
|}
| style="width: 5px;text-align: right;" | (58)
|}

The boundary integral in Eq. ([[#eq-58|58]]) can be transformed to a domain integral using Eq. ([[#eq-31|31]]) and Gauss' divergence theorem yielding

<span id="eq-59"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\int _{\Gamma _0} \hbox{t}_0 (\hbox{X} ,t) d\Gamma _0 =     \int _{\Omega _0} \nabla _0 \cdot \hbox{P} (\hbox{X},t) d\Omega _0 </math>
|}
| style="width: 5px;text-align: right;" | (59)
|}

where <math display="inline">\nabla _0 \cdot ( \bullet )</math> is the divergence taken respect  to material coordinates. Leaving the derivation of the conditions which follow  from Eq. ([[#eq-58|58]]), the conservation of linear momentum in reference configuration for Lagrangian coordinates is

<span id="eq-60"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho _0 \frac{\partial \hbox{v}}{\partial t} =     \nabla _0 \cdot \hbox{P} + \rho _0 \hbox{b}     \textrm{      or     }    \rho _0 \frac{\partial v_i}{\partial t} =     \frac{\partial P_{ji}}{\partial X_j} + \rho _0 b_i </math>
|}
| style="width: 5px;text-align: right;" | (60)
|}

This is called the Lagrangian form of the momentum equation. For nonlinear solid finite elements, Eq. ([[#eq-60|60]]) is called ''total Lagrangian formulation''. The corresponding equilibrium equation for this description is

<span id="eq-61"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\nabla _0 \cdot \hbox{P} + \rho _0 \hbox{b} = 0    \textrm{      or     }    \frac{\partial P_{ji}}{\partial X_j} + \rho _0 b_i = 0 </math>
|}
| style="width: 5px;text-align: right;" | (61)
|}

As a consequence of the conservation of angular momentum Eq. ([[#eq-53|53]]) and Eq. ([[#eq-54|54]]) the nominal stress tensor yields

<span id="eq-62"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{F} \cdot \hbox{P} = \hbox{P}^T \cdot \hbox{F}^T </math>
|}
| style="width: 5px;text-align: right;" | (62)
|}

which is in general not symmetric. The number of conditions imposed by angular momentum conservation are usually imposed directly on the constitutive equation. Using Eq. ([[#eq-34|34]]) in Eq. ([[#eq-62|62]]) we obtain that for the  second Piola-Kirchhoff stress

<span id="eq-63"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{S} = \hbox{S}^T </math>
|}
| style="width: 5px;text-align: right;" | (63)
|}

is a symmetric tensor.

===2.4.3 Conservation of Energy===

The ''kinetic energy'' of a material is given by

<span id="eq-64"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathcal{E}^{\textrm{kin}} = \int _\Omega \frac{1}{2} \rho \hbox{v} \cdot \hbox{v} d \Omega  </math>
|}
| style="width: 5px;text-align: right;" | (64)
|}

which for a continuum body <math display="inline">\mathcal{B}</math> is only part of the total energy. The remainder energy is called the ''internal energy'' that is expressed by <math display="inline">\hbox{w}^{\textrm{int}}</math> per unit mass. The internal energy per unit volume is denoted by

<span id="eq-65"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathcal{E}^{\textrm{int}} = \int _\Omega \rho \hbox{w}^{\textrm{int}} d \Omega  </math>
|}
| style="width: 5px;text-align: right;" | (65)
|}

The total energy is then expressed by <math display="inline">\mathcal{E}^{\textrm{tot}} = \mathcal{E}^{\textrm{int}} + \mathcal{E}^{\textrm{kin}}</math>. Then the ''conservation of energy'' requires that the power of the total energy equals the power of the applied forces plus the power at which ''other energy'' enters in the domain. The other energy may take different forms, but the most important is the energy due to heat sources and heat flux across <math display="inline">\mathcal{B}</math>. Other energy sources arise from radiation, chemical changes, electromagnetic fields, etc. We consider thermomechanical processes only.

The power of the total energy is given by

<span id="eq-66"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathcal{P}^{\textrm{tot}} = \mathcal{P}^{\textrm{int}} + \mathcal{P}^{\textrm{kin}}                              = \frac{D}{Dt} \int _\Omega \rho \hbox{w}^{\textrm{int}} d \Omega +                                \frac{D}{Dt} \int _\Omega \frac{1}{2} \rho \hbox{v} \cdot \hbox{v} d \Omega  </math>
|}
| style="width: 5px;text-align: right;" | (66)
|}

while the power of the applied forces is expressed by

<span id="eq-67"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathcal{P}^{\textrm{ext}} = \int _\Omega \hbox{v} \cdot \rho \hbox{b} d \Omega +                                \int _\Gamma \hbox{v} \cdot \hbox{t} d \Gamma  </math>
|}
| style="width: 5px;text-align: right;" | (67)
|}

The power supplied by heat sources ''s'' and the heat flux '''q''' is

<span id="eq-68"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathcal{P}^{\textrm{heat}} = \int _{\Omega } \rho s d \Omega -                                  \int _{\Gamma } \hbox{n} \cdot \hbox{q} d \Gamma  </math>
|}
| style="width: 5px;text-align: right;" | (68)
|}

The conservation of energy states that

<span id="eq-69"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathcal{P}^{\textrm{tot}} = \mathcal{P}^{\textrm{ext}} + \mathcal{P}^{\textrm{heat}} </math>
|}
| style="width: 5px;text-align: right;" | (69)
|}

which is known as the ''first law of thermodynamics''. Replacing Eqs. ([[#eq-66|66]])-([[#eq-68|68]]) into Eq. ([[#eq-69|69]]) yields the equation of conservation of energy

<span id="eq-70"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{D}{Dt} \int _\Omega \rho </math>
| style="text-align: right;" | <math> \hbox{w}^{\textrm{int}} d \Omega               +     \frac{D}{Dt} \int _\Omega \frac{1}{2} \rho \hbox{v} \cdot \hbox{v} d \Omega =     \int _\Omega \hbox{v} \cdot \rho \hbox{b} d \Omega                          + </math>
|-
| style="text-align: center;" | 
| style="text-align: right;" | <math>     \int _\Gamma \hbox{v} \cdot \hbox{t} d \Gamma                               +     \int _{\Omega } \rho s d \Omega                                                  -      \int _{\Gamma } \hbox{n} \cdot \hbox{q} d \Gamma    </math>
|}
| style="width: 5px;text-align: right;" | (70)
|}

The equation which emerges from the above integral form leads to the following Eulerian partial differential equation of energy conservation

<span id="eq-71"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho \frac{D \hbox{w}^{\textrm{int}}}{Dt} =   \sigma : \hbox{d} - \nabla \cdot \hbox{q} + \rho s </math>
|}
| style="width: 5px;text-align: right;" | (71)
|}

For a purely mechanical process, the above equation becomes

<span id="eq-72"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho \frac{D \hbox{w}^{\textrm{int}}}{Dt} = \sigma : \hbox{d} </math>
|}
| style="width: 5px;text-align: right;" | (72)
|}

which is no longer a partial differential equation. As a consequence of Eq. ([[#eq-72|72]]) we can say that the Cauchy stress tensor <math display="inline">\sigma</math> and the rate of deformation tensor '''d''' are ''conjugate in power''.

The conservation of energy can also be expressed in Lagrangian coordinates and in the reference configuration, where the counterpart of Eq. ([[#eq-70|70]]) gives

<span id="eq-73"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{d}{dt} \int _{\Omega _0} </math>
| style="text-align: right;" | <math> \left(\rho _0 \hbox{w}^{\textrm{int}}           +                  \frac{1}{2} \rho _0 \hbox{v} \cdot \hbox{v} \right)d \Omega _0 =      \int _{\Omega _0} \hbox{v} \cdot \rho _0 \hbox{b} d \Omega _0                  + </math>
|-
| style="text-align: center;" | 
| style="text-align: right;" | <math>     \int _{\Gamma _0} \hbox{v} \cdot \hbox{t}_0 d \Gamma _0                       +     \int _{\Omega _0} \rho _0 s d \Omega _0                                            -      \int _{\Gamma _0} \hbox{n}_0 \cdot \hbox{q} d \Gamma _0   </math>
|}
| style="width: 5px;text-align: right;" | (73)
|}

which gives the Lagrangian partial differential equation of energy conservation

<span id="eq-74"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho _0 \dot{\hbox{w}}^{int} =    \hbox{P} : \dot{\hbox{F}}^T -\nabla _0 \cdot \hbox{q} + \rho _0 s </math>
|}
| style="width: 5px;text-align: right;" | (74)
|}

For a purely mechanical process, the Lagrangian energy conservation is

<span id="eq-75"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho _0 \dot{\hbox{w}}^{int} = \hbox{P} : \dot{\hbox{F}}^T  </math>
|}
| style="width: 5px;text-align: right;" | (75)
|}

showing that the nominal stress tensor is conjugate in power to the material time derivative of the deformation gradient tensor. Using Eq. ([[#eq-34|34]]) in Eq. ([[#eq-75|75]]) we obtain the energy conservation equation in terms of the second Piola-Kirchhoff stress tensor

<span id="eq-76"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho _0 \dot{\hbox{w}}^{int} = \hbox{S} : \dot{\hbox{E}}   </math>
|}
| style="width: 5px;text-align: right;" | (76)
|}

which shows that the second Piola-Kirchhoff stress tensor is conjugate in power to the rate of the Green-Lagrange strain tensor.

==2.5 Constitutive Equations==

The equations given so far are still insufficient to describe the mechanical behavior of any material. Therefore we need additional equations called ''constitutive equations'' which complete the set of equations specifying the mechanical properties of a material. For a purely mechanical process, the constitutive equation of a material specifies the dependence of the stress tensor in terms of kinematic variables such as the strain tensor.

===2.5.1 Linear Elasticity===

Engineering materials such as metals or concrete usually undergo very small changes of shape when they are subjected to the forces which they are exposed. They also have an initial shape to which they will return if the forces applied are removed.  Since the changes of shape are very small, there is no difference between the reference  and current configuration.

The ''linear elasticity'' theory gives an excellent model for the behavior of such materials. The infinitesimal strain tensor <math display="inline">\varepsilon</math> is used to measure strains while the Cauchy stress tensor <math display="inline">\sigma</math> measures the stresses. For the linear elasticity theory the energy conservation equation takes the form

<span id="eq-77"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho _0 \dot{\hbox{w}}^{int} = \sigma : \dot{\varepsilon} =   \sigma_{ij} \dot{\varepsilon}_{ij} </math>
|}
| style="width: 5px;text-align: right;" | (77)
|}

where <math display="inline">\sigma</math> and <math display="inline">\dot{\varepsilon}</math> are conjugate in power.

It is conventional to denote the internal energy per unit volume <math display="inline">\rho _0 w^{\textrm{int}}</math> by <math display="inline">W^{\textrm{int}}</math> which is called the ''strain energy function''. For a linear elastic material the strain energy function depends only of the components <math display="inline">\varepsilon_{ij}</math> and is a quadratic function of the form

<span id="eq-78"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>W^{\textrm{int}} = \frac{1}{2} \mathbb{C}_{ijkl} \varepsilon_{ij} \varepsilon_{kl}     \textrm{      or     }   W^{\textrm{int}} = \frac{1}{2} \varepsilon : C : \varepsilon </math>
|}
| style="width: 5px;text-align: right;" | (78)
|}

where <math display="inline">\mathbb{C}_{ijkl}</math> are called ''elastic constants''.  Since the elastic constants possess symmetry of the form

<span id="eq-79"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbb{C}_{ijkl} = \mathbb{C}_{jikl} = \mathbb{C}_{ijlk} = \mathbb{C}_{klij} </math>
|}
| style="width: 5px;text-align: right;" | (79)
|}

then for an ''isotropic material'', its properties are  the same in all directions.

Since <math display="inline">W^{\textrm{int}}</math>  depends only on <math display="inline">\varepsilon_{ij}</math>, the material time derivative of Eq. ([[#eq-78|78]])  gives

<span id="eq-80"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{\partial W^{\textrm{int}}}{\partial t} =      \frac{\partial W^{\textrm{int}}}{\partial \varepsilon_{ij}}   \frac{\partial \varepsilon_{ij}}{\partial t} =     \frac{\partial W^{\textrm{int}}}{\partial \varepsilon_{ij}}\dot{\varepsilon}_{ij} </math>
|}
| style="width: 5px;text-align: right;" | (80)
|}

where the symmetry of the material has been used. Substituting  Eq. ([[#eq-80|80]]) into Eq. ([[#eq-77|77]]) gives

<span id="eq-81"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma_{ij} = \frac{\partial W^{\textrm{int}}}{\partial \varepsilon_{ij}} </math>
|}
| style="width: 5px;text-align: right;" | (81)
|}

However from Eq. ([[#eq-78|78]]) and Eq. ([[#eq-79|79]])

<span id="eq-82"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{\partial W^{\textrm{int}}}{\partial \varepsilon_{ij}} = \mathbb{C}_{ijkl} \varepsilon_{kl} </math>
|}
| style="width: 5px;text-align: right;" | (82)
|}

Substituting Eq. ([[#eq-82|82]]) into Eq. ([[#eq-81|81]]) yields

<span id="eq-83"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma_{ij} = \mathbb{C}_{ijkl} \varepsilon_{kl}   \textrm{      or     }   \sigma = C : \varepsilon </math>
|}
| style="width: 5px;text-align: right;" | (83)
|}

which is the constitutive equation that relates stresses and strains. The constitutive equation complete the equations to describe the mechanical behavior of linear elastic materials. For an isotropic material <math display="inline">\mathbb{C}_{ijkl}</math> takes the form

<span id="eq-84"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbb{C}_{ijkl} = \lambda \delta _{ij} \delta _{kl} + \mu \left( \delta _{ik} \delta _{jl} +              \delta _{il} \delta _{jk}  \right)   \textrm{      or     }   C = \lambda \hbox{I} \otimes \hbox{I} + 2\mu I </math>
|}
| style="width: 5px;text-align: right;" | (84)
|}

where only two constants <math display="inline">\lambda </math> and <math display="inline">\mu </math> of the original 81 of the fourth-order tensor survived after the restrictions of material isotropy and stress symmetry. The two independent material constants <math display="inline">\lambda </math> and <math display="inline">\mu </math> are called the ''Lamé constants'', '''I''' is the second-order identity tensor and <math display="inline">I</math> is the fourth-order symmetric identity tensor given by <math display="inline">\frac{1}{2} \left(\delta _{ik} \delta _{jl} + \delta _{il} \delta _{jk} \right)</math>. The constitutive equation  Eq. ([[#eq-83|83]]) becomes

<span id="eq-85"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma_{ij} = \lambda \varepsilon_{kk} \delta _{ij} + 2 \mu \varepsilon_{ij}   \textrm{      or     }   \sigma = \lambda \textrm{tr}(\varepsilon) \hbox{I} + 2 \mu \varepsilon </math>
|}
| style="width: 5px;text-align: right;" | (85)
|}

where <math display="inline">\textrm{tr}(\varepsilon)</math> is the trace of <math display="inline">\varepsilon=\varepsilon_{kk}</math>.

===2.5.2 Nonlinear Elasticity===

Engineering applications also involved small strains and large deformations, where these effects arise from large displacements and large rotations of the structure. The response of such materials may be modeled with a ''Saint Venant-Kirchhoff'' material  which is a generalization of the linear theory to large deformations giving the ''nonlinear elasticity'' theory.

The strain energy function for a nonlinear elastic material is a generalization of  Eq. ([[#eq-78|78]]) and is given by

<span id="eq-86"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>W^{\textrm{int}} = \frac{1}{2} \mathbb{C}_{ijkl} E_{ij} E_{kl}     \textrm{      or     }   W^{\textrm{int}} = \frac{1}{2} \hbox{E} : C : \hbox{E} </math>
|}
| style="width: 5px;text-align: right;" | (86)
|}

where the stress is

<span id="eq-87"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>S_{ij} = \frac{\partial W^{\textrm{int}}}{\partial E_{ij}} </math>
|}
| style="width: 5px;text-align: right;" | (87)
|}

The counterpart of Eq. ([[#eq-83|83]]) in the nonlinear theory yields

<span id="eq-88"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>S_{ij} = \mathbb{C}_{ijkl} E_{kl}   \textrm{      or     }   \hbox{S} = C : \hbox{E} </math>
|}
| style="width: 5px;text-align: right;" | (88)
|}

where <math display="inline">\mathbb{C}_{ijkl}</math> is given by Eq. ([[#eq-84|84]]). Finally, the constitutive equation for nonlinear elastic materials is

<span id="eq-89"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>S_{ij} = \lambda E_{kk} \delta _{ij} + 2 \mu E_{ij}   \textrm{      or     }   \hbox{S} = \lambda \textrm{tr}(\hbox{E}) \hbox{I} + 2 \mu \hbox{E} </math>
|}
| style="width: 5px;text-align: right;" | (89)
|}

The Lamé constants <math display="inline">\lambda </math> and <math display="inline">\mu </math> can be expressed in terms of other physical  measurements given by

<span id="eq-90"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mu = \frac{E}{2(1+\upsilon )} </math>
|}
| style="width: 5px;text-align: right;" | (90)
|}

<span id="eq-91"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\lambda = \frac{\upsilon E}{(1+\upsilon )(1-2\upsilon )} </math>
|}
| style="width: 5px;text-align: right;" | (91)
|}

<span id="eq-92"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>K = \lambda + \frac{2}{3} \mu  </math>
|}
| style="width: 5px;text-align: right;" | (92)
|}

where <math display="inline">E</math> is the Young's modulus, <math display="inline">\upsilon </math> is the Poisson's ratio and <math display="inline">K</math> is the bulk modulus.

===2.5.3 Newtonian Fluid===

An equation that linearly relates the stress tensor to the rate of strain tensor in a fluid medium is called the constitutive equation for ''Newtonian fluids''.

In a static fluid there are only normal components of the stress tensor on a boundary, so the stress tensor for a fluid at rest is isotropic and takes the form

<span id="eq-93"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma_{ij} = -p \delta _{ij} </math>
|}
| style="width: 5px;text-align: right;" | (93)
|}

where ''p'' is the ''thermodynamic pressure''  related to the density <math display="inline">\rho </math> and the temperature ''T''. A moving fluid develops additional components of stress due to viscosity yielding

<span id="eq-94"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma_{ij} = -p \delta _{ij} + \sigma^{\textrm{dev}}_{ij} </math>
|}
| style="width: 5px;text-align: right;" | (94)
|}

where the deviatoric stress tensor <math display="inline">\sigma^{\textrm{dev}}_{ij}</math> is linearly related to the strain rate tensor by

<span id="eq-95"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma^{\textrm{dev}}_{ij} = \mathbb{C}_{ijkl} d_{kl}  </math>
|}
| style="width: 5px;text-align: right;" | (95)
|}

Substituting Eq. ([[#eq-23|23]]) and Eq. ([[#eq-84|84]]) into Eq. ([[#eq-95|95]]) and this new equation in Eq. ([[#eq-94|94]]), the resulting equation is

<span id="eq-96"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma_{ij} = -p \delta _{ij} + \lambda d_{kk} \delta _{ij} + 2 \mu d_{ij} </math>
|}
| style="width: 5px;text-align: right;" | (96)
|}

where <math display="inline">d_{kk} = \nabla \cdot \hbox{v}</math> is the volumetric strain rate. If the Stokes assumption, <math display="inline">\lambda + \frac{2}{3} \mu =0</math>, is used in Eq. ([[#eq-96|96]]) to relate <math display="inline">\lambda </math> and <math display="inline">\mu </math> the new equation is

<span id="eq-97"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma_{ij} = -\left(p + \frac{2}{3} \mu \nabla \cdot \hbox{v} \right)\delta _{ij} + 2 \mu d_{ij}   \textrm{     or    }   \sigma = -\left(p + \frac{2}{3} \mu \nabla \cdot \hbox{v} \right)\hbox{I} + 2 \mu \hbox{d}   </math>
|}
| style="width: 5px;text-align: right;" | (97)
|}

which is the constitutive equation for Newtonian fluids. For incompressible fluids  the continuity equation Eq.([[#eq-42|42]]) is substituted into Eq. ([[#eq-97|97]]) and the constitutive equation for incompressible fluids takes the simple form

<span id="eq-98"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma_{ij} = -p \delta _{ij} + 2 \mu d_{ij}   \textrm{     or    }   \sigma = -p \hbox{I} + 2 \mu \hbox{d}   </math>
|}
| style="width: 5px;text-align: right;" | (98)
|}

where ''p'' is the pressure. For incompressible fluids, ''p'' is called mechanical pressure.

==2.6 Navier-Stokes Equation==

The equation of motion for a Newtonian fluid is obtained by substituting the constitutive equation for Newtonian fluids Eq. ([[#eq-97|97]]) into the momentum equation in Eulerian description Eq. ([[#eq-51|51]]) to obtain

<span id="eq-99"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho \left(\frac{\partial v_i}{\partial t} + v_j \partial _j v_i \right)=     -\frac{\partial p}{\partial x_i} + \rho b_i + \frac{\partial }{\partial x_j}    \left(2 \mu d_{ij} - \frac{2}{3} \mu ( \nabla \cdot \hbox{v} ) \delta _{ij} \right) </math>
|}
| style="width: 5px;text-align: right;" | (99)
|}

Equation ([[#eq-99|99]]) is the general form of the ''Navier-Stokes'' equation. If <math display="inline">\mu </math> is taken as a constant, the derivative in the right hand side term can be written as

<span id="eq-100"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{\partial }{\partial x_j} \left(2 \mu d_{ij} - \frac{2}{3} \mu ( \nabla \cdot     \hbox{v} ) \delta _{ij} \right)= \mu \left(\nabla ^2 v_i + \frac{1}{3}     \frac{\partial }{\partial x_i} ( \nabla \cdot \hbox{v} ) \right) </math>
|}
| style="width: 5px;text-align: right;" | (100)
|}

where <math display="inline">\nabla ^2 v_i</math> is the ''Laplacian''<span id="fnc-1"></span>[[#fn-1|<sup>1</sup>]] of <math display="inline">v_i</math>. For incompressible fluids the continuity equation Eq.([[#eq-42|42]]) is substituted into Eq. ([[#eq-100|100]]) and the Navier-Stokes equation reduces to

<span id="eq-101"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho \left(\frac{\partial v_i}{\partial t} + v_j \partial _j v_i \right)=     -\frac{\partial p}{\partial x_i} + \rho b_i + \mu \nabla ^2 v_i </math>
|}
| style="width: 5px;text-align: right;" | (101)
|}

If viscous effects are negligible, which may occur far from boundaries of the flow field,

<span id="eq-102"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho \left(\frac{\partial v_i}{\partial t} + v_j \partial _j v_i \right)=     -\frac{\partial p}{\partial x_i} + \rho b_i </math>
|}
| style="width: 5px;text-align: right;" | (102)
|}

and the ''Euler'' equation is obtained. Now consider given a characteristic velocity scale <math display="inline">v_c</math> and a characteristic length scale <math display="inline">l_c</math>, then the Reynolds number is defined as <math display="inline">\textrm{Re}=\rho v_c l_c / \mu </math>. When the Reynolds number  for the flow is very low, the convective term can be neglected yielding

<span id="eq-103"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho  \frac{\partial v_i}{\partial t}      + \frac{\partial p}{\partial x_i} - \mu \nabla ^2 v_i = \rho b_i </math>
|}
| style="width: 5px;text-align: right;" | (103)
|}

which is known as the Stokes flow.  In the literature, it is common to express Eq. ([[#eq-103|103]]) without the inertial term.

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