Investigation of an elastoplastic material model coupled to nonlocal damage in an implicit‐gradient framework

Abstract

In this work, an elastoplastic material model coupled to nonlocal damage is discussed which is based on an implicit gradientenhanced approach. Combined nonlinear isotropic and kinematic hardening as well as continuum damage of Lemaitre-type are considered. The model is a direct nonlocal extension of a corresponding local model which was presented earlier (see e. The problem at hand is defined in the domain of interest Ω by the usual balance equations of linear momentum and a partial differential equation (PDE) which reflects the gradient-extension of the material model (see Eq. (1)). The gradient-extension is implicit in the sense that an additional PDE has to be solved for the scalar 'nonlocal' damage variableD, an approach inspired by the work of [4]. Suitable boundary conditions for u andD (e. g. Eqns. (2) and (3)) have to be provided on the boundary Γ = Γ t ∪ Γ u , Γ t ∩ Γ u = ∅, which leads to the following strong form of the problem:The gradient influence ofD within the material is expressed by means of the parameter c which introduces an internal length into the model. The corresponding weak form of the problem needed in order to work e. g. with finite elements is then obtained in the usual way by multiplying Eqns.(1) of the model individually by suitable test functions, integrating over the domain and applying Gauss' theorem (not shown here for brevity). The outcome is a problem in which both the displacements u and the nonlocal damageD are treated as independent field variables. It can be proven that implicit gradient-extended models are largely equivalent to specific nonlocal integral-type formulations which justifies their classification as 'true' nonlocal material models [5]. 3D elastoplasticity model coupled to nonlocal damageThe coupled elastoplastic damage model discussed here is a direct nonlocal extension of a corresponding local model which was presented earlier and discussed in detail e. g. by