Quantitative stochastic homogenization of nonlinearly elastic, random laminates

TL;DR

This paper develops a quantitative framework for stochastic homogenization of nonlinear elastic laminates, demonstrating smoothness of the homogenized energy and providing error estimates for RVE approximations with exponential moment bounds.

Contribution

It introduces a new approach to analyze the regularity of the homogenized energy and quantifies RVE approximation errors with optimal scaling and exponential moment bounds.

Findings

Homogenized energy function is $C^3$ near rotations.

Explicit stochastic corrector representations for stress and moduli.

Error estimates for RVE approximations with exponential integrability.

Abstract

In this paper we study quantitative stochastic homogenization of a nonlinearly elastic composite material with a laminate microstructure. We prove that for deformations close to the set of rotations the homogenized stored energy function $W_{\rm hom}$ is $C^3$ and that $W_{\rm hom}$, the stress-tensor $DW_{\rm hom}$, and the tangent-moduli $D^2W_{\rm hom}$ can be represented with help of stochastic correctors. Furthermore, we study the error of an approximation of these quantities via representative volume elements. More precisely, we consider periodic RVEs obtained by periodizing the distribution of the random material. For materials with a fast decay of correlations on scales larger than a unit scale, we establish error estimates on the random and systematic error of the RVE with optimal scaling in the size of the RVE and with a multiplicative random constant that has exponential moments.