We consider a family of vectorial models for cohesive fracture, which may incorporate SO(n)-invariance. The deformation belongs to the space of (generalized) functions of bounded variation and the energy contains an (elastic) volume energy, an opening-dependent jump energy concentrated on the fractured surface, and a Cantor part representing diffuse damage. In recent work, together with Sergio Conti (U. Bonn) and Flaviana Iurlano (U. Sorbonne), we have shown that this type of functionals can be naturally obtained as Gamma-limit of an appropriate phase-field model.

The energy densities entering the limiting functional can be expressed, in a partially implicit way, in terms of those appearing in the phase-field approximation.

Along the talk, we will comment on some phase-field models that have been introduced and analyzed since the seminal works of Ambrosio and Tortorelli and that finally have led to those we have proposed.