Stochastic homogenisation of free-discontinuity functionals in random perforated domains

TL;DR

This paper investigates the asymptotic behavior of random free-discontinuity energies in perforated domains, revealing how porous materials with cracks behave as the scale parameter approaches zero.

Contribution

It introduces a novel extension technique to handle non-equi-coerciveness in stochastic homogenisation of free-discontinuity functionals in perforated domains.

Findings

Identification of limit energy densities through stochastic convergence

Development of an extension result for perforated domains

Analysis of crack development in porous materials

Abstract

In this paper we study the asymptotic behaviour of a family of random free-discontinuity energies $E_\varepsilon$ defined on a randomly perforated domain, as $\varepsilon$ goes to zero. The functionals $E_\varepsilon$ model the energy associated to displacements of porous random materials that can develop cracks. To gain compactness for sequences of displacements with bounded energies, we need to overcome the lack of equi-coerciveness of the functionals. We do so by means of an extension result, under the assumption that the random perforations cannot come too close to one another. The limit energy is then obtained in two steps. As a first step we apply a general result of stochastic convergence of free-discontinuity functionals to a modified, coercive version of $E_\varepsilon$. Then the effective volume and surface energy densities are identified by means of a careful limit procedure.