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

We study the $\Gamma$-limit of Ambrosio-Tortorelli-type functionals $D_\varepsilon(u,v)$, whose dependence on the symmetrised gradient $e(u)$ is different in $\mathbb{A} u$ and in $e(u)-\mathbb{A} u$, for a $\mathbb{C}$-elliptic symmetric operator $\mathbb{A}$, in terms of the prefactor depending on the phase-field variable $v$. This is intermediate between an approximation for the Griffith brittle fracture energy and the one for a cohesive energy by Focardi and Iurlano. In particular we prove that $G(S)BD$ functions with bounded $\mathbb{A}$-variation are $(S)BD$.