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This paper deals with fracture mechanics in periodically perforated domains. Our aim is to provide a variational model for brittle porous media in the case of anti-planar elasticity.

Given the perforated domain Ω_ε ⊂ ℝ^N (ε being an internal scale representing the size of the periodically distributed perforations), we will consider a total energy of the type Here u is in SBV(Ω_ε) (the space of special functions of bounded variation), S_u is the set of discontinuities of u, which is identified with a macroscopic crack in the porous medium Ω_ε, and stands for the (N - 1)-Hausdorff measure of the crack S_u.

We study the asymptotic behavior of the functionals in terms of Γ-convergence as ε → 0. As a first (nontrivial) step we show that the domain of any limit functional is SBV(Ω) despite the degeneracies introduced by the perforations. Then we provide explicit formula for the bulk and …