Variational methods applied to discrete models in brittle damage (Elise Bonhomme, Université Libre de Bruxelles)

The purpose of this talk is to study a brittle damage model, in different regimes where the damaged zone concentrates on vanishingly small sets, and try to identify the nature of the effective limit models obtained by means of an asymptotic analysis based on the $\Gamma$-convergence of the total energies. First, I will introduce (succinctly) the mechanical model of brittle damage that was introduced by Francfort and Marigo. Then, I will address the question of the asymptotic analysis of these brittle damage energies in the discrete setting, where the energies are restricted to piecewise affine continuous displacements. I will exhibit different effective limit models according to the related regimes' scaling laws (some of these regimes are still in progress). In particular, I will show a discrete adaptive finite element approximation result for the isotropic two-dimensional Griffith energy arising in fracture mechanics.