Fracture mechanics in perforated domains: a variational approach 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. We study the asymptotic behaviour of free-discontinuity energies on periodically perforated domains with Neumann boundary conditions on the perforations in terms of Gamma-convergence as the size of the cell of the underlying lattice vanishes. As a fīrst (non-trivial) step we show that the domain of any limit functional is in SBV despite the degeneracies introduced by the perforations. Then we provide explicit formulas 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.