Threshold-based Quasi-static Brittle Damage Evolution

We introduce models for static and quasi-static damage in elastic materials, based on a strain threshold, and then investigate the relationship between these threshold models and the energy-based models introduced in Francfort and Marigo (Eur J Mech A Solids 12:149–189, 1993) and Francfort and Garroni (Ration Mech Anal 182(1):125–152, 2006). A somewhat surprising result is that, while classical solutions for the energy models are also threshold solutions, this is shown not to be the case for nonclassical solutions, that is, solutions with microstructure. A new and arguably more physical definition of solutions with microstructure for the energy-based model is then given, in which the energy minimality property is satisfied by sequences of sets that generate the effective elastic tensors, rather than by the tensors themselves. We prove existence for this energy-based problem, and show that these solutions are also threshold solutions. A by-product of this analysis is that all local minimizers, in both the classical setting and for the new microstructure definition, are also global minimizers.