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

TL;DR

This paper studies a scalar viscoelastic model with random, small-scale variations to understand how hysteresis emerges in the homogenized limit, supported by numerical simulations.

Contribution

It introduces a stochastic homogenization approach for a viscoelastic model exhibiting hysteresis, extending the understanding of rate-independent effects in random media.

Findings

Homogenized model exhibits persistent hysteresis under slow loading.

Stochastic two-scale convergence effectively captures the limit behavior.

Numerical simulations validate theoretical predictions.

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 stochastic two-scale convergence as introduced by Bourgeat, Mikelic and Wright, 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.