Two-Scale Homogenization for Evolutionary Variational Inequalities via the Energetic Formulation

Abstract

This paper is devoted to two-scale homogenization for a class of rate-independent systems described by the energetic formulation or equivalently by an evolutionary variational inequality. In particular, we treat the classical model of linearized elastoplasticity with hardening. The associated nonlinear partial differential inclusion has periodically oscillating coefficients, and the aim is to find a limit problem for the case in which the period tends to 0. Our approach is based on the notion of energetic solutions, which is phrased in terms of a stability condition and an energy balance of an energy-storage functional and a dissipation functional. Using the recently developed method of weak and strong two-scale convergence via periodic unfolding, we show that these two functionals have a suitable two-scale limit, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. Moreover, relying on an abstract theory of $\Gamma$-convergence for the energetic formulation using so-called joint recovery sequences, it is possible to show that the solutions of the problem with periodicity converge to the energetic solution associated with the limit functionals.