FRACTURE MECHANICS IN PERFORATED DOMAINS: A VARIATIONAL MODEL FOR BRITTLE POROUS MEDIA MATTEO FOCARDI, MARIA STELLA GELLI, AND MARCELLO PONSIGLIONE Abstract. 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. Given the perforated domain Ωε ⊂ R N (ε being an internal scale representing the size of the periodically distributed perforations), we will consider a total energy of the type Fε(u) := Z Ωε |∇u(x)| 2 dx + 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 H N-1 (Su) stands for the (N − 1)-Hausdorff measure of the crack Su. Critical points of the total energy are, according to Griffith’s theory, stable configurations of the cracked body. We study the asymptotic behavior of the functionals Fε in terms of Γ-convergence as ε → 0. As a first (non-trivial) 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. Keywords: Brittle fracture, Homogenization, Perforated domains, Composite and mix- ture properties, Cavities, Variational methods, Special functions of bounded variation. 2000 Mathematics Subject Classification: 74R10, 74Q15, 74E30, 35R05, 76B10, 35J25, 26A45. Contents 1. Introduction 2 2. Preliminaries 4 2.1. Rectifiable Sets and Coarea Formula 5 3. Formulation of the problem 5 3.1. The perforated domain 6 3.2. The energy functionals 6 4. Compactness 6 4.1. BV -compactness 7 4.2. Compactness in SBV 2 (Ω): the case N =2 9 4.3. Compactness in SBV 2 (Ω): the general case 13 4.4. L 1 -compactness 14 5. The Γ-convergence result 15 6. Matching boundary conditions 24 7. Further results 25 1
FRACTURE MECHANICS IN PERFORATED DOMAINS: A VARIATIONAL MODEL FOR BRITTLE POROUS MEDIA
May 29, 2023
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