Calculus of Variations and Geometric Measure Theory

X. Pellet - L. Scardia - C. I. Zeppieri

Stochastic homogenisation of free-discontinuity functionals in randomly perforated domains

created by zeppieri1 on 04 Feb 2020
modified on 13 Jul 2023


Accepted Paper

Inserted: 4 feb 2020
Last Updated: 13 jul 2023

Journal: Adv. Calc. Var.
Year: 2023


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.

Keywords: Brittle fracture, $\Gamma$-convergence, Free-discontinuity problems, homogenisation, Neumann boundary conditions, porous materials, randomly perforated domains