Stochastic homogenization of rate-independent systems

Abstract

We study the stochastic and periodic homogenization 1-homogeneous convex functionals. We proof some convergence results with respect to stochastic twoscale convergence, which are related to classical Gamma-convergence results. The main result is a general lim inf-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rate-independent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandlt-Reuss plasticity, Coulomb friction on a macroscopic surface and Coulomb friction on microscopic fissures.