Étude dans d’un modèle de plaques élastoplastiques comportant une non-linéarité géométrique
Evolutionary variational inequalities and applications in plasticity
An abstract theory of evolutionary variational inequalities and its applications to the traction boundary value problems of elastoplasticity are studied, using the penalty method to prove the existence of a solution.
Existence and stability of solutions for nonlinear Mecking-Lucke-Grilhe equations.
Existence of regular solutions for a one-dimensional simplified perfect-plastic problem with a unilateral gradient constraint.
Existence of regular solutions for a one-dimensional simplified perfect-plastic problem
Existence of the Displacements Field for an Elasto-Plastic Body Subject to Hencky's Law and von Mises Yield Condition.
Extremum theorem and convergence criterion for an iterative solution to the finite-step problem in elastoplasticity with mixed nonlinear hardening
For a class of elastic-plastic constitutive laws with nonlinear kinematic and isotropic hardening, the problem of determining the response to a finite load step is formulated according to an implicit backward difference scheme (stepwise holonomic formulation), with reference to discrete structural models. This problem is shown to be amenable to a nonlinear mathematical programming problem and a criterion is derived which guarantees monotonie convergence of an iterative algorithm for the solution...
Extremum theorems for finite-step back-ward-difference analysis of elastic-plastic nonlinearly hardening solids
For the finite-step, backward-difference analysis of elastic-plastic solids in small strains, a kinematic (potential energy) and a static (complementary energy) extremum property of the step solution are given under the following hypotheses: each yield function is the sum of an equivalent stress and a yield limit; the former is a positively homogeneous function of order one of stresses, the latter a nonlinear function of nondecreasing internal variables; suitable conditions of "material stability"...
Functional a posteriori error estimates for incremental models in elasto-plasticity
We consider incremental problem arising in elasto-plastic models with isotropic hardening. Our goal is to derive computable and guaranteed bounds of the difference between the exact solution and any function in the admissible (energy) class of the problem considered. Such estimates are obtained by an advanced version of the variational approach earlier used for linear boundary-value problems and nonlinear variational problems with convex functionals [24, 30]. They do no contain mesh-dependent constants...
General method of regularization. I: Functionals defined on BD space
The aim of this paper is to prove that the relaxation of the elastic-perfectly plastic energy (of a solid made of a Hencky material) is the lower semicontinuous regularization of the plastic energy. We find the integral representation of a non-locally coercive functional. In part II, we will show that the set of solutions of the relaxed problem is equal to the set of solutions of the relaxed problem proposed by Suquet. Moreover, we will prove the existence theorem for the limit analysis problem.
General method of regularization. II: Relaxation proposed by suquet
The aim of this paper is to prove that the relaxation of the elastic-perfectly plastic energy (of a solid made of a Hencky material) is the lower semicontinuous regularization of the plastic energy. We find the integral representation of a non-locally coercive functional. We show that the set of solutions of the relaxed problem is equal to the set of solutions of the relaxed problem proposed by Suquet. Moreover, we prove an existence theorem for the limit analysis problem.
General method of regularization. III: The unilateral contact problem
The aim of this paper is to prove that the relaxation of the elastic-perfectly plastic energy (of a solid made of a Hencky material with the Signorini constraints on the boundary) is the weak* lower semicontinuous regularization of the plastic energy. We consider an elastic-plastic solid endowed with the von Mises (or Tresca) yield condition. Moreover, we show that the set of solutions of the relaxed problem is equal to the set of solutions of the relaxed problem proposed by Suquet. We deduce that...
Generalised functions of bounded deformation
We introduce the space of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for , which leads to a compactness result for the space of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation of some variational...
Gradient theory for plasticity via homogenization of discrete dislocations
We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem. The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the -limit of this energy (suitably rescaled),...
Guaranteed and computable bounds of the limit load for variational problems with linear growth energy functionals
The paper is concerned with guaranteed and computable bounds of the limit (or safety) load, which is one of the most important quantitative characteristics of mathematical models associated with linear growth functionals. We suggest a new method for getting such bounds and illustrate its performance. First, the main ideas are demonstrated with the paradigm of a simple variational problem with a linear growth functional defined on a set of scalar valued functions. Then, the method is extended to...
Hencky-Prandtl nets and constant principal strain mappings with isolated singularities.
Implicit constitutive solution scheme for Mohr-Coulomb plasticity
This contribution summarizes an implicit constitutive solution scheme of the elastoplastic problem containing the Mohr-Coulomb yield criterion, a nonassociative flow rule, and a nonlinear isotropic hardening. The presented scheme builds upon the subdifferential formulation of the flow rule leading to several improvements. Mainly, it is possible to detect a position of the unknown stress tensor on the Mohr-Coulomb pyramid without blind guesswork. Further, a simplified construction of the consistent...
Inclusions in elasto-plastic solids under the influence of internal and external pressures.
Kinematic criteria of dynamic shakedown extended to nonassociative constitutive laws with saturation nonlinear hardening
The class of elastic-plastic material models considered allows for nonassociativity, nonlinear hardening and saturation in the sense that the static internal variables are constrained by a bounding surface described through convex bounding functions. With reference to finite element, generalized variables discretization in space, two dynamic shakedown criteria are established by a kinematic approach in Koiter's sense, based on weak constitutive restrictions and centered on two suitable definitions...
Kinetic and hydrodynamic equations for granular media
In this lecture i present some open mathematical problems concerning some PDE arising in the study of one-dimensional models for granular media.