Abstract
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{} 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.
Highlights
Introduction and state of the artIn the framework of conventional plasticity, materials with induced strain softening lead to initial and boundary value problem (IBVP)’s exhibiting a solution highly sensitive to the mesh size
As flow localization takes place and the cracks being initiated in the material, the mechanical fields as well as some material properties become highly inhomogeneous leading to a non-simple material as defined by
Higher gradients of the displacement and/or of some state variables are required to define the mechanical state of each material point of the body
Summary
In the framework of conventional (local) plasticity, materials with induced strain softening lead to IBVP’s exhibiting a solution highly sensitive to the mesh size. Where P is the residual term for a material point of inhomogeneous material and is regarded as the influence of mechanical variables at neighboring points In nonlocal theories, it was defined by Maugin [12] that the integral of P over a representative volume element Ω should be zero i.e.:. Where is a kind of internal length scale related to the material under concern [6,7,13,15,16] It has been shown in [6] that, the Equation (10) can be obtained as a particular case of the integral form proposed for example in [15,16] to regularize some damage related internal variables. The extension to more realistic constitutive equations with kinematic hardening, induced volume variation under finite plastic strains and using the framework of generalized continuum theory (micromorphic theory) is developed in [25, 26]
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