A simple damage-gradient enhanced elastoplastic formulation and its numerical implementation

This paper presents a ductile damage-gradient based nonlocal and fully coupled elastoplastic constitutive equations by adding a Helmholtz equation to regularize the initial and boundary value problem (IBVP) exhibiting some damage induced softening. First, a thermodynamically-consistent formulation of gradient-regularized plasticity fully coupled with isotropic ductile damage and accounting for mixed non linear isotropic and kinematic hardening is presented. For the sake of simplicity, only a simplified version of this model based on von Mises isotropic yield function and accounting for the single nonlinear isotropic hardening is studied and implemented numerically using an in house FE code. An additional partial differential equation governing the evolution of the nonlocal isotropic damage is added to the equilibrium equations and the associated weak forms derived to define the IBVP (initial and boundary value problem). After the time and space discretization, two algebraic equations: one highly nonlinear associated with the equilibrium equation and the second purely linear associated with the damage non locality equation are obtained. Over a typical load increment, the first equation is solved iteratively thanks to the Newton-Raphson scheme and the second equation is solved directly to compute the nonlocal damage \hbox{$\Bar{{D}}$} at each node. All the constitutive equations are “strongly” affected by this nonlocal damage variable transferred to each integration point. Some applications show the ability of the proposed approach to obtain a mesh independent solution for a fixed value of the length scale parameter. Comparisons between fully local and nonlocal solutions are given. Dans cet article, nous présentons la régularisation d’un modèle élastoplastique à écrouissage isotrope non-linéaire couplé à un endommagement ductile isotrope non local solution d’une équation de Helmholtz en gradient d’endommagement. Dans un premier temps, nous présentons la formulation thermodynamique du modèle élastoplastique à écrouissages cinématique et isotrope non-linéaires couplé à l’endommagement non local. Par simple souci d’abréviation, une version simplifiée de ce modèle basée sur le critère de von Mises avec un écrouissage purement isotrope non-linéaire couplé à l’endommagement non local est traité numériquement. Une équation différentielle aux dérivées partielles régissant la non localité de l’endommagement est ajoutée à l’équation d’équilibre pour en déduire deux fo