Stochastic homogenization of plasticity equations

Heida, Martin; Schweizer, Ben

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 153-176.

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 $\epsilon >0$ denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit $\epsilon \to 0$. The homogenization procedure is based on the fact that stochastic coefficients “allow averaging”: For one representative volume element, a strain evolution $[0,T]\ni t\mapsto \xi \left(t\right)\in$ induces a stress evolution $[0,T]\ni t\mapsto \Sigma \left(\xi \right)\left(t\right)\in$. Once the hysteretic evolution law $\Sigma$ is justified for averages, we obtain that the macroscopic limit equation is given by $-\nabla ·\Sigma \left(\nabla {}^{s}u\right)=f$.

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