A quasi-static evolution generated by local energy minimizers for an elastic material with a cohesive interface

We deal with a model for an elastic material with a cohesive crack along a prescribed fracture set. We consider two n-dimensional elastic bodies and a cohesive law, on their common interface, with incompenetrability constraint and general loading-unloading regimes. We first provide a time-discrete evolution by means of local minimizers of the energy with respect to the L-2-norm of the crack opening displacement. The choice of this norm is due to technical reasons (the lambda-convexity of the energy) and is in analogy with the classical approach in quasi-static brittle fracture, where the evolution of the system is condensed into the evolution of the crack. In the "time-continuous" limit we obtain a BV-evolution, in parametrized form, characterized by Karush-Kuhn-Tucker conditions, for the internal variable, equilibrium and energy identity.