Stochastic homogenization of plasticity equations

TL;DR

This paper establishes a stochastic homogenization framework for infinitesimal strain plasticity equations with random coefficients, deriving effective macroscopic equations as the oscillation scale tends to zero.

Contribution

It introduces a novel stochastic homogenization approach for plasticity equations with random coefficients using the needle-problem method and averaging assumptions.

Findings

Derived effective equations for plasticity with random coefficients

Proved the homogenization limit as oscillation scale approaches zero

Validated the averaging property of stochastic coefficients

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.