Stochastic homogenization of a scalar viscoelastic model exhibiting stress–strain hysteresis

Abstract

Motivated by rate-independent stress–strain hysteresis observed in filled rubber, this article considers a scalar viscoelastic model in which the constitutive law is random and varies on a lengthscale which is small relative to the overall size of the solid. Using a variant of stochastic two-scale convergence as introduced by Bourgeat et al., we obtain the homogenized limit of the evolution, and demonstrate that under certain hypotheses, the homogenized model exhibits hysteretic behaviour which persists under asymptotically slow loading. These results are illustrated by means of numerical simulations in a particular one-dimensional instance of the model.