Abstract
This paper presents a simple damage-gradient based elastoplastic model with non linear isotropic hardening in order to regularize the associated initial and boundary value problem (IBVP). Using the total energy equivalence hypothesis, fully coupled constitutive equations are used to describe the non local damage induced softening leading to a mesh independent solution. An additional partial differential equation governing the evolution of the non local isotropic damage is added to the classical equilibrium equations and associated weak forms derived. This leads to discretized IBVP governed by two algebric systems. The first one, associated with equilibrium equations, is highly non linear and can be solved by an iterative Newton Raphson method. The second one, related to the non local damage, is a linear algebric system and can be solved directly to compute the non local damage variable at each load increment. Two fields, linear interpolation triangular element with additional degree of freedom is terms of the non local damage variable is constructed. The non local damage variable is then transferred from mesh nodes to the quadrature (or Gauss) points to affect strongly the elastoplastic behavior. Two simple 2D examples are worked out in order to investigate the ability of proposed approach to deliver a mesh independent solution in the softening stage.
Highlights
It is well known that the local constitutive equation exhibiting an induced strain softening which succeeds to the positive strain hardening, leads inevitably to more or less strain localization
This paper presents a simple damage-gradient based elastoplastic model with non linear isotropic hardening in order to regularize the associated initial and boundary value problem (IBVP)
An additional partial differential equation governing the evolution of the non local isotropic damage is added to the classical equilibrium equations and associated weak forms derived
Summary
It is well known that the local constitutive equation exhibiting an induced strain softening which succeeds to the positive strain hardening, leads inevitably to more or less strain localization. Strain localization refers to the emergence of finite narrow bands inside which the plastic flow localizes while the remaining part of the deforming body is elastically unloaded It has been shown in several published works, that the numerical solution (using FEM) of this class of dissipation problems exhibits a high sensitivity to the space and time discretization [1,2,3,4,5,6,7]. To regularize the solution of these problems, leads to incorporate some effects of characteristic lengths of the materials microstructure into constitutive models via the mechanics of generalized continue as the higher grade continues [9,10,11,12] or higher order continua [13,14,15] In these approaches, the stress at a given material point depends on additional degrees of freedom as well their higher order spatial derivatives or gradients.
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