This talk addresses the asymptotic analysis (in terms of the Gamma-convergence of the total energies) of the model of brittle damage introduced by Francfort and Marigo (1993), specified to the discrete setting in different regimes where we force the concentration of the damaged regions on vanishingly small sets. 

More specifically, we consider a linearly elastic material which exists in only two pure states: a sound phase, and a damaged phase whose elastic properties (i.e., rigidity) are weaker. Here, we introduce small parameters in Francfort-Marigo's total energy, thus forcing three phenomena which compete simultaneously as the parameters tend to 0: the concentration of the damaged phase onto Lebesgue-negligible sets, the degeneracy of the rigidity of the damaged material, and a spatial discretisation by adaptive finite elements (i.e., we restrict the domain of the total energy to continuous and piecewise affine displacements). According to the different scaling laws under consideration, we obtain five asymptotic mechanical models. I will present their specificities and try to explain the main arguments of the proof.