Asymptotic analysis of Ambrosio-Tortorelli energies in linearized elasticity

We provide an approximation result in the sense of $\Gamma$-convergence for energies of the form \[ \int_\Omega \Qq_1(e(u))\,dx+a\,\mathcal{H}^{n-1}(J_u)+b\,\int_{J_u}\Qq_0^{1/2}([u]\odot\nu_u)\,d\Hn, \] where $\Omega\subset\Rn$ is a bounded open set with Lipschitz boundary, $\Qq_0$ and $\Qq_1$ are coercive quadratic forms on $\Msym$, $a,\,b$ are positive constants, and $u$ runs in the space of fields $SBD^2(\Omega)$ , i.e., it's a special field with bounded deformation such that its symmetric gradient $e(u)$ is square integrable, and its jump set $J_u$ has finite $(n-1)$-Hausdorff measure in $\Rn$. The approximation is performed by means of Ambrosio-Tortorelli type elliptic regularizations, the prototype example being \[ \int_\Omega\Big(v|e(u)|^2+\frac{(1-v)^2}{\varepsilon}+{\cost\,\ve}|\nabla v|^2\Big)\,dx, \] where $(u,v)\in \Hu{\times} H^1(\Omega)$, $\ve\leq v\leq 1$ and $\gamma>0$.