Stochastic homogenization of plasticity equations

¹ Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany.
² TU Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, 44227 Dortmund, Germany.
ben.schweizer@tu-dortmund.de

Received: 27 April 2016
Accepted: 31 January 2017

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 $\mathrm{\left[}\mathrm{0}\mathit{,T}\mathrm{\right]}\mathrm{\ni }\mathit{t}\mathrm{}\mathrm{\to }\mathit{\xi }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\in }{R}_{\mathit{s}}^{\mathit{d}\mathrm{\times }\mathit{d}}$ induces a stress evolution $\mathrm{\left[}\mathrm{0}\mathit{,T}\mathrm{\right]}\mathrm{\ni }\mathit{t}\mathrm{}\mathrm{\to }\mathit{\Sigma }\mathrm{\left(}\mathit{\xi }\mathrm{\right)}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\in }{R}_{\mathit{s}}^{\mathit{d}\mathrm{\times }\mathit{d}}$. Once the hysteretic evolution law Σ is justified for averages, we obtain that the macroscopic limit equation is given by −∇·Σ(∇^su) = f.