Calculus of Variations and Geometric Measure Theory

M. Focardi - M. S. Gelli - M. Ponsiglione

Fracture mechanics in perforated domains: a variational model for brittle porous media

created by gelli on 11 Dec 2007
modified by focardi on 19 Dec 2011


Published Paper

Inserted: 11 dec 2007
Last Updated: 19 dec 2011

Journal: Math. Models Methods Appl. Sci.
Volume: 19
Number: 11
Pages: 2065-2100
Year: 2009


This paper deals with fracture mechanics in periodically perforated domains. Our aim is to provide a variational model for brittle porous media. For the sake of simplicity we will restrict our analysis to the case of anti-planar elasticity.

We study the asymptotic behavior of the energy functionals involving a bulk and a surface term in terms of Gamma-convergence as the size of the perforations tends to zero. As a first (non-trivial) 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.