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 $\eps>0$ denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit $\eps\to 0$. The homogenization limit is based on the needle-problem approach: We verify that the stochastic coefficients "allow averaging": In average, a strain evolution $[0,T]\ni t\mapsto \xi(t) \in \symM$ induces a stress evolution $[0,T]\ni t\mapsto \Sigma(\xi)(t) \in \symM$. With the abstract result of [9] we obtain the stochastic homogenization limit.