$\Gamma$-convergence for free-discontinuity problems in linear elasticity: Homogenization and relaxation

TL;DR

This paper studies the $ ext{Gamma}$-convergence of free-discontinuity functionals in linear elasticity, providing a framework for homogenization and relaxation of models involving fractures, damage, and voids.

Contribution

It introduces a representation of the $ ext{Gamma}$-limit in integral form on $GSBD^p$, including homogenization and relaxation results for free-discontinuity problems.

Findings

Established compactness and integral representation of $ ext{Gamma}$-limits.

Identified asymptotic cell formulas for integrands.

Proved convergence of minima and minimizers in boundary value problems.

Abstract

We analyze the $\Gamma$-convergence of sequences of free-discontinuity functionals arising in the modeling of linear elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. We prove compactness with respect to $\Gamma$-convergence and represent the $\Gamma$-limit in an integral form defined on the space of generalized special functions of bounded deformation ($GSBD^p$). We identify the integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions. Eventually, we investigate sequences of corresponding boundary value problems and show convergence of minimum values and minimizers. In particular, our techniques allow to characterize relaxations of functionals on $GSBD^p$, and cover the classical case of periodic homogenization.