Stochastic homogenization of plasticity equations

Abstract

In the context of infinitesimal strain plasticity with hardening, we derive a stochastic homogenization result. We assume that the coefficients of the equation are random functions: elasticity tensor, hardening parameter and flow-rule function are given through a dynamical system on a probability space. A parameter ε > 0 denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit ε → 0. The homogenization procedure is based on the fact that stochastic coefficients “allow averaging”: For one representative volume element, a strain evolution \hbox{$[0,T]\ni t\mapsto \xi(t) \in \symM$} induces a stress evolution \hbox{$[0,T]\ni t\mapsto \Sigma(\xi)(t) \in \symM$}. Once the hysteretic evolution law Σ is justified for averages, we obtain that the macroscopic limit equation is given by −∇·Σ(∇su) = f.