Non-local approximation of free-discontinuity problems in linear elasticity and application to stochastic homogenisation

TL;DR

This paper studies the convergence of non-local functionals to local free-discontinuity models in elasticity, providing a stochastic homogenisation framework with explicit bulk term characterization.

Contribution

It introduces a novel analysis of non-local convolution functionals converging to free-discontinuity problems, including a stochastic homogenisation approach with explicit bulk term formulas.

Findings

Established $mbda$-convergence of non-local functionals

Derived an explicit asymptotic cell formula for the bulk term

Applied results to stochastic homogenisation in elasticity

Abstract

We analyse the $\Gamma$-convergence of general non-local convolution type functionals with varying densities depending on the space variable and on the symmetrized gradient. The limit is a local free-discontinuity functional, where the bulk term can be completely characterized in terms of an asymptotic cell formula. From that, we can deduce an homogenisation result in the stochastic setting.