# Stochastic homogenization of rate-dependent models of monotone type in plasticity

## TL;DR

This paper studies the stochastic homogenization of rate-dependent monotone models in plasticity, using Fitzpatrick functions and two-scale convergence to analyze the asymptotic behavior of inelastic materials with microstructure.

## Contribution

It introduces a novel approach combining Fitzpatrick functions and stochastic two-scale convergence to analyze rate-dependent monotone models in plasticity.

## Key findings

- Established existence of weak solutions for the models.
- Derived homogenization results using stochastic two-scale convergence.
- Connected homogenization with classical $\Gamma$-convergence theory.

## Abstract

In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical $\Gamma$-convergence theory.