Calculus of Variations and Geometric Measure Theory

M. Focardi - F. Iurlano

Asymptotic analysis of Ambrosio-Tortorelli energies in linearized elasticity

created by iurlano on 10 May 2013
modified on 06 Jan 2015


Published Paper

Inserted: 10 may 2013
Last Updated: 6 jan 2015

Journal: SIAM Journal on Mathematical Analysis
Volume: 46
Pages: 2936--2955
Year: 2014


We provide an approximation result in the sense of $\Gamma$-convergence for 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 $\Omega\subset{\mathbb R}$ 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 ${\mathbb R}$.

The approximation is performed by means of Ambrosio-Tortorelli type elliptic regularizations, the prototype example being \[ \int_\Omega\Big(v
^2+\frac{(1-v)^2}{\varepsilon} +{\gamma\,\varepsilon}
\nabla v
^2\Big)\,dx, \] where $(u,v)\in H^1(\Omega,\mathbb{R}^n){\times} H^1(\Omega)$, $\varepsilon\leq v\leq 1$ and $\gamma>0$.

Keywords: free discontinuity problems, Functions of Bounded Deformation, fracture