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Phase-field approximation of a vectorial, geometrically nonlinear cohesive fracture energy

We consider a family of vectorial models for cohesive fracture, which may incorporate $\mathrm{SO}(n)$-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 $Γ$-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.