## Ambrosio-Tortorelli approximation of cohesive fracture models in linearized elasticity

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##### Date

2013-05

##### Authors

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

SISSA

##### Abstract

We provide an approximation result in the sense of $\Gamma$-convergence for
cohesive fracture energies of the form
\[
\int_\Omega \mathscr{Q}_1(e(u))\,dx+a\,\mathcal{H}^{n-1}(J_u)+b\,\int_{J_u}\mathscr{Q}_0^{1/2}([u]\odot\nu_u)\,d\mathcal{H}^{n-1},
\]
where $\Omega\subset{\mathbb R}^n$ is a bounded open set with Lipschitz boundary, $\mathscr{Q}_0$ and $\mathscr{Q}_1$ are coercive
quadratic forms on ${\mathbb M}^{n\times n}_{sym}$, $a,\,b$ 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 ${\mathbb R}^n$.
The approximation is performed by means of Ambrosio-Tortorelli type elliptic regularizations, the prototype example being
\[
\int_\Omega\Big(v|e(u)|^2+\frac{(1-v)^2}{\varepsilon}+{\gamma\,\varepsilon}|\nabla v|^2\Big)\,dx,
\]
where $(u,v)\in H^1(\Omega,{\mathbb R}^n){\times} H^1(\Omega)$, $\varepsilon\leq v\leq 1$ and $\gamma>0$.