FRACTURE MECHANICS IN PERFORATED DOMAINS: A VARIATIONAL MODEL FOR BRITTLE POROUS MEDIA

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 Omega(epsilon) subset of R-N (epsilon being an internal scale representing the size of the periodically distributed perforations), we will consider a total energy of the type F-epsilon(u) := integral Omega(epsilon) vertical bar del u(x)vertical bar(2) dx + HN-1(S-u). Here u is in SBV (Omega(epsilon)) (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 Omega(epsilon), and HN-1(S-u) stands for the (N - 1)-Hausdorff measure of the crack S-u. We study the asymptotic behavior of the functionals F-epsilon in terms of Gamma-convergence as epsilon -> 0. As a first (nontrivial) step we show that the domain of any limit functional is SBV (Omega) despite the degeneracies introduced by the perforations. Then we provide explicit formula for the bulk and surface energy densities of the Gamma-limit, representing in our model the effective elastic and brittle properties of the porous medium, respectively.