Effects of oscillation scales in discrete brittle damage models

Abstract

This paper is concerned with the asymptotic analysis of a sequence of variational models of brittle damage in the context of linearized elasticity in the two-dimensional discrete setting. We consider a discrete version of Francfort and Marigo’s brittle damage model, where the total energy is restricted to continuous and piecewise affine vectorial displacements, within different regimes where the damaged regions concentrate on vanishingly small sets while the stiffness of the damaged material degenerates to $0$. In this setting, the convergence of the space discretization, the concentration of the damaged regions, and the decay of the elastic properties of the damaged phase all compete simultaneously in non-trivial ways according to the scaling law under consideration. The mesh size turns out to be a crucial feature of the analysis, as it induces a minimal scale of spatial oscillations for admissible displacements. This study was motivated by the numerical investigations performed in Allaire–Jouve–Van Goethem (2011) on the one hand, where the authors have shown that forcing the stiffness decay on sets of arbitrarily small measure seems to lead to concentration phenomena such as in brittle fracture, and by the static analysis performed in Babadjian–Iurlano–Rindler (2021) on the other hand, where the authors addressed the rigorous asymptotic analysis of this observation in terms of the $\Gamma$-convergence of the total energies, in the continuous setting in space. Surprisingly, they showed that fracture-type models were not obtained asymptotically, thus raising the question of the dependence of the effective models with respect to the scaling of a spatial discretization in Francfort and Marigo’s model. This is the content of the present work, where we show that concentration phenomena are only captured asymptotically in regimes where the mesh size and the concentration of damaged regions are of the same order.