Phase-Field Approximation of a Vectorial, Geometrically Nonlinear Cohesive Fracture Energy

We consider a family of vectorial models for cohesive fracture, which may incorporate \(\textrm{SO}(n)\)-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. We show that this type of functional 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.