Inserted: 11 dec 2016
Last Updated: 11 dec 2016
We consider a model for an elastic material with a cohesive crack along a prescribed fracture set. In the framework of in-plane elasticity we consider a cohesive law with incompenetrability constraint and general loading-unloading regimes. We provide first 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-Tuker conditions for the internal variable, equilibrium and energy identity.