Abstract

This paper presents the application of stabilized mixed explicit strain/displacement formulation (MEX-FEM) [23, 24] for solving non-linear plasticity problems in solid mechanics with strain localization. In order to use the same linear interpolation order for displacements and strains, the formulation uses the variational subscales method. Compared to the standard irreducible formulation, the proposed formulation yields improved strain and stress fields, and it is capable of addressing nearly incompressible situations. This work investigates the effects of the improved strain and stress fields in problems involving strain softening and localization leading to failure for the Mohr-Coulomb and Drucker Prager plasticity models. Numerical examples validate the ability of the proposed formulation to correctly predict failure mechanisms with localized patterns of strain, virtually free of mesh dependence and without using tracking algorithm.

Abstract

Abstract

This work investigates systematically strain localization and failure mechanics for elastoplastic damage solids. Two complementary methodologies, i.e., traction-based discontinuities localized in an elastic solid and strain localization of a stress-based inelastic softening solid, are addressed. In the former it is assumed a priori that the discontinuity (band) forms with a continuous stress field and along the known orientation. A traction-based failure criterion is introduced to characterize the discontinuity (band) and the orientation is determined from Mohr’s maximization postulate. If the (apparent) displacement jumps are retained as independent variables, the strong/regularized discontinuity approaches follow, requiring constitutive models for both the bulk and discontinuity (band). Elimination of the displacement jumps at the material point level results in the embedded/smeared discontinuity approaches in which an overall inelastic constitutive model fulfilling the static constraint suffices. The second methodology is then adopted to check whether the assumed strain localization can occur and identify its consequences on the resulting approaches. The kinematic constraint guaranteeing stress boundedness/continuity upon strain localization is established for general inelastic softening solids. Application to a unified elastoplastic damage model naturally yields all the ingredients of a localized model for the discontinuity (band), justifying the first methodology. Two dual but not necessarily equivalent approaches, i.e., the traction-based elastoplastic damage model and the stress-based projected discontinuity model, are identified. The former is equivalent to the embedded/smeared discontinuity approaches, whereas in the later the discontinuity orientation and associated failure criterion, not given a priori, are determined consistently from the kinematic constraint. The bi-directional connections and equivalence conditions between the traction- and stress-based approaches are classified. Closed-form 2D results under plane stress condition are also given, with the classical Rankine, Mohr-Coulomb, von Mises and Drucker-Prager criteria analyzed as the illustrative examples. A generic failure criterion of either elliptic, parabolic or hyperbolic type, is then considered in a unified manner, resulting in many failure criteria frequently employed in practice.

Abstract

En el penúltim paràgraf (p. 566) de l’Informe del BIRF de 1962 sobre l’economia espanyola es pot llegir que *las corporaciones locales deberían ser también estimuladas a mancomunarse......con fines de planificación y establecimiento de servicios sobre una base metropolitana*. Segurament es tracta de les poques recomanacions que no van ser mai implementades, atesa la seva força històrica i sentimental arreu.
 
El gruix de les competències locals i els recursos els gestionen els ajuntaments actuals que són uns 8000 a l’Estat espanyol i 948 a Catalunya, que té més de 150 anys de vida des de la implantació de les províncies de Javier de Burgos. Mentrestant la majoria dels països europeus han reduït el nombre d’administracions per tal de fer-les econòmicament més eficients (UK, Alemanya i àdhuc França). La més recent és la de Dinamarca (d’extensió i població homologable amb Catalunya), organitzada en 98 ajuntaments i 5 regions, reduint els prop de 400 ajuntaments que va implantar la reforma del 1975, quan n’hi havia uns 1300. Aquest podria ser el referent més adient per a Catalunya atesa la seva població de prop de 5 milions de residents  i la seva extensió semblant a la de CAT.
 
A Catalunya, la gairebé perpètua *discussió comarcal* des del 1930, però especialment després del 1978, té el rerefons de la millora de la gestió de l’administració local, malgrat que Pau Vila i Josep Iglésies dubtaven dels *continents* més adequats perquè no havien rebut l’encàrrec de quins *continguts* havien de servir.
 
Per tal de millorar l’eficiència de la gestió dels serveis municipals, les lleis d’organització territorial del 1984  van crear 39 consells comarcals (avui 42). Malgrat també intentar la supressió de les 4 Diputacions en les mateixes lleis,  el TC la va tombar en una de les primeres sentències sonades de calat unionista. Així doncs l’organització de l’administració local de Catalunya té 3 nivells, a més a més de l’Àrea Metropolitana de Barcelona, que va ser aprovada per unanimitat fa uns anys. En termes d’experiència comparada sembla una organització massa complexa i per això sovint s’ha parlat de la necessitat de la seva simplificació.
 
Es disposen de pocs estudis sobre les corbes de costos unitaris de cada servei municipal (els quals es poden dividir entre serveis d’administració, de territori i personals) en funció de la grandària del municipi, i la seva qualitat i accessibilitat respecte de l’usuari. En canvi, però, tenim a Catalunya l’experiència relativament recent d’haver creat exnovo els serveis de proximitat dels Mossos d’Esquadra i del Bombers, sense condicionants previs, i de fet es situen a l’entorn de les 125 comissaries i 150 parcs de diferent dimensió i dotacions. La seva implantació segurament ha seguit la lògica que indicava el geògraf Casassas quan deia que *el territori mana*, i no ha estat condicionant de forma especial per les divisions administratives o la geografia municipal.
 
De fet, no sembla que des del punt de vista de la lluita contra la criminalitat les policies municipals podrien integrar-se a les comissaries dels Mossos d’Esquadra i disposa d’un sol nivell de policia com és el cas de Dinamarca.
 
L’objecte principal de la comunicació és aportar l’estudi de la lògica territorial i econòmica dels serveis dels Mossos d’Esquadra i de Bombers, per tal d’obtenir un referent del que podria ser la reorganització de l’administració local a Catalunya, si escau.
 
Referències històriques.
Casassas, Ll, i Clusa, J. (1981)
*L’Organització Territorial de Catalunya*
Llibres de la Fundació Jaume Bofill
B.40078-1981
 
Casassas, Ll, i Clusa, J. (1981)
*L’Organització Territorial i Administrativa de Barcelona*
Ajuntament de Barcelona
D.L: B-24130/81

Abstract

The expression ‘finite calculus’ refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes, forces, etc. in a space–time domain of finite size. The governing equations resulting from this approach are different from those of infinitesimal calculus theory and they incorporate new terms which depend on the dimensions of the balance domain. The new governing equations allow the derivation of naturally stabilized numerical schemes using any discretization procedure. The paper discusses the possibilities of the finite calculus method for the finite element solution of convection–diffusion problems with sharp gradients, incompressible fluid flow and incompressible solid mechanics problems and strain localization situations. 

Abstract

This paper describes a procedure for the solution of problems involving tensile cracking using the so-called smeared crack approach, that is, standard finite elements with continuous displacement fields and a standard local constitutive model with strain-softening. An isotropic Rankine damage model is considered. The softening modulus is adjusted according to the material fracture energy and the element size. The resulting continuum and discrete mechanical problems are analyzed and the question of predicting correctly the direction of crack propagation is deemed as the main difficulty to be overcome in the discrete problem. It is proposed to use a crack tracking technique to attain the desired stability and convergence properties of the corresponding formulation. Numerical examples show that the resulting procedure is well-posed, stable and remarkably robust; the results obtained do not seem to suffer from spurious mesh-size or mesh-bias dependence.

Abstract

This paper deals with the question of strain localization associated with materials which exhibit softening due to tensile straining. A standard local isotropic Rankine damage model with strain-softening is used as exemplary constitutive model. Both the irreducible and mixed forms of the problem are examined and stability and solvability conditions are discussed. Lack of uniqueness and convergence difficulties related to the strong material nonlinearities involved are also treated. From this analysis, the issue of local discretization error in the pre-localization regime is deemed as the main difficulty to be overcome in the discrete problem. Focus is placed on low order finite elements with continuous strain and displacement fields (triangular P1P1 and quadrilateral Q1Q1), although the presented approach is very general. Numerical examples show that the resulting procedure is remarkably robust: it does not require the use of auxiliary tracking techniques and the results obtained do not suffer from spurious mesh-bias dependence.

Abstract

This paper studies the phenomenon of strain bifurcation and localization in J2plasticity under plane stress and plane strain conditions. Necessary conditions for the outcome of bifurcation, localization, stress boundedness and decohesion are analytically established. It is shown that the explicit consideration of these conditions allows for the determination of localization angles in certain situations of interest that can be used to conduct benchmark tests on finite element formulations. The relative merits of irreducible, (stabilized) mixed and (displacement and/or strain) enhanced formulations are discussed. Numerical examples show that the mixed displacement/pressure formulation is to be preferred to the standard irreducible schemes in order to predict correct failure mechanisms with localized patterns of plastic deformation. Mixed elements are shown to be practically free from mesh directional bias dependence.

Abstract

This paper presents the application of a stabilized mixed strain/displacement finite element formulation for the solution of nonlinear solid mechanics problems involving compressible and incompressible plasticity. The variational multiscale stabilization introduced allows the use of equal order interpolations in a consistent way. Such formulation presents two advantages when compared to the standard, displacement based, irreducible formulation: (a) it provides enhanced rate of convergence for the strain (and stress) field and (b) it is able to deal with incompressible situations. The first advantage also applies to the comparison with the mixed pressure/displacement formulation. The paper investigates the effect of the improved strain and stress fields in problems involving strain softening and localization leading to failure, using low order finite elements with continuous strain and displacement fields (P1P1 triangles or tetrahedra and Q1Q1quadrilaterals, hexahedra, and triangular prisms) in conjunction with an associative frictional Drucker–Prager plastic model. The performance of the strain/displacement formulation under compressible and nearly incompressible deformation patterns is assessed and compared to a previously proposed pressure/displacement formulation. Benchmark numerical examples show the capacity of the mixed formulation to predict correctly failure mechanisms with localized patterns of strain, virtually free from any dependence of the mesh directional bias. No auxiliary crack tracking technique is necessary.

Abstract

This paper studies the phenomenon of structural size effect and strain localization in J2 plasticity. Size effect is here understood as the change in the response of a given structure when the spatial dimensions are scaled up or down while the geometry and other relevant properties of the structure are preserved. The work exploits the advantages of the mixed displacement–pressure formulation in incompressible or quasi-incompressible situations. Elasto-J2-plastic constitutive behaviour with regularized softening is considered. Stability issues are discussed to ensure existence and uniqueness of the solution of the corresponding discrete finite element formulation. Numerical examples show that the formulation derived is able to solve a wide range of structural scales, including real life engineering applications. The results obtained do not suffer from spurious mesh dependence. Furthermore, the formulation includes the classical theories of perfect plasticity and linear fracture mechanics as limit cases.