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 Ωε ⊂ ℝN (ε being an internal scale representing the size of the periodically distributed perforations), we will consider a total energy of the type Fε(u):= ∫Ωε|∇ u(x)|2dx + H N-1(Su). Here u is in SBV(Ωε) (the space of special functions of bounded variation), Su is the set of discontinuities of u, which is identified with a macroscopic crack in the porous medium Ωε, and HN-1(Su) stands for the (N - 1)-Hausdorff measure of the crack Su. We study the asymptotic behavior of the functionals Fε 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 surface energy densities of the Γ-limit, representing in our model the effective elastic and brittle properties of the porous medium, respectively. © 2009 World Scientific Publishing Company.