Globally stable quasistatic evolution for strain gradient plasticity coupled with damage

Abstract

Abstract. We consider evolutions for a material model which couples scalar damage with strain gradient plasticity, in small strain assumptions. For strain gradient plasticity, we follow the Gurtin-Anand formulation [19]. The aim of the present model is to account for different phenomena: on the one hand the elastic stiffness reduces and the plastic yield surface shrinks due to material's degradation, on the other hand the dislocation density affects the damage growth. The main result of this paper is the existence of a globally stable quasistatic evolution (in the socalled energetic formulation). Furthermore we study the limit model as the strain gradient terms tend to zero. Under stronger regularity assumptions, we show that the evolutions converge to the ones for the coupled elastoplastic-damage model studied in [8].

Damage model for plastic materials at finite strains

“…Apart from providing a convenient approach to constitutive modeling, the framework of generalized standard materials naturally leads to the notion of energetic solutions -a solution concept for rate-independent problems developed by MIELKE and co-workers [38,41] that characterizes the evolution of state variables by conditions of global stability and energy conservation. The existence of an energetic solution for small-strain damage-plastic models has recently been shown by CRISMALE, [13] who further extended his result to gradient plasticity coupled with damage [14] (see also [15,16]). However, existence results for finite-strain models are currently lacking, although finite-strain damage [40] and gradient plasticity [34,36,37,39,42] were successfully addressed within the energetic solution concept.…”

“…Let us now give an estimate for the rotation in the decomposition (14). We use decomposition (5) to estimatê…”

“…In this contribution the authors show that, as the regularization term coefficients vanish, the solutions approach a quasistatic evolution to perfect plasticity . The Gurtin‐Anand model has been coupled with damage in [], where the regularization term for the z variable controls its H 1 ‐norm; this is necessary to ensure lower semicontinuity of the plastic potential. In spirit of [], the quasistatic perfectly plastic limit is achieved with the aid of a higher order regularization for the damage, which was later improved in [].…”

A model of isotropic damage with strain-gradient effects: existence and uniqueness of weak solutions for progressive damage processes

“…For appropriately smooth solutions (assuming that they exist) we can compute the first variation of the energy functional, obtaining pointwise relations under the constraint that the damage field decreases in time. Such a point of view has been followed for isotropic damage [6,10,11], hence represented by a scalar variable, so that the constraint requires that the time derivative of the damage variables is less than or equal to zero. If we would describe anisotropic damage, the consequent introduction of a tensor-valued damage field would require additional care in describing the above-mentioned constraint.…”

Dynamic perfect plasticity and damage in viscoelastic solids

“…We refer, e.g., to [42,43,45,50,57,59] and the references therein for an overview of the main results.The interplay between plasticity and damage has been already extensively investigated, prominently in the quasistatic framework. The interaction between damage and strain gradient plasticity is addressed in [21] whereas a perfect-plastic model has been proposed in [1], where the one-dimensional response is also studied. Existence results in general dimensions have been obtained in [20,22], see also [23] for some recent associated lower semicontinuity results.…”