Phase-field approximation for a class of cohesive fracture energies with an activation threshold

We study the Γ-limit of Ambrosio-Tortorelli-type functionals D ϵ (u, v) {D-{arepsilon}(u,v)}, whose dependence on the symmetrised gradient e (u) {e(u)} is different in u {mathbb{A}u} and in e (u)- u {e(u)-mathbb{A}u}, for a â., {mathbb{C}}-elliptic symmetric operator {mathbb{A}}, in terms of the prefactor depending on the phase-field variable v. The limit energy depends both on the opening and on the surface of the crack, and is intermediate between the Griffith brittle fracture energy and the one considered by Focardi and Iurlano [Asymptotic analysis of Ambrosio-Tortorelli energies in linearized elasticity, SIAM J. Math. Anal. 46 2014, 4, 2936-2955]. In particular, we prove that G(S)BD functions with bounded " {mathbb{A}}-variation are (S)BD.