A density result in $GSBD^p$ with applications to the approximation of brittle fracture energies

TL;DR

This paper proves a density approximation result in $GSBD^p$ spaces, enabling better analysis of brittle fracture energies and their $ ext{Gamma}$-convergence, with applications to variational fracture models.

Contribution

It introduces a new density result in $GSBD^p$ spaces with applications to $ ext{Gamma}$-convergence of Griffith energies, improving approximation of fracture energies.

Findings

Functions in $GSBD^p$ can be approximated by $SBV$ functions with smooth jump sets.

The approximation preserves Griffith-type energies and converges in relevant norms.

Application to $ ext{Gamma}$-convergence of fracture energies with boundary conditions.

Abstract

We prove that any function in $GSBD^p(\Omega)$, with $\Omega$ a $n$-dimensional open bounded set with finite perimeter, is approximated by functions $u_k\in SBV(\Omega;\mathbb{R}^n)\cap L^\infty(\Omega;\mathbb{R}^n)$ whose jump is a finite union of $C^1$ hypersurfaces. The approximation takes place in the sense of Griffith-type energies $\int_\Omega W(e(u)) \,\mathrm{d}x +\mathcal{H}^{n-1}(J_u)$, $e(u)$ and $J_u$ being the approximate symmetric gradient and the jump set of $u$, and $W$ a nonnegative function with $p$-growth, $p>1$. The difference between $u_k$ and $u$ is small in $L^p$ outside a sequence of sets $E_k\subset \Omega$ whose measure tends to 0 and if $|u|^r \in L^1(\Omega)$ with $r\in (0,p]$, then $|u_k-u|^r \to 0$ in $L^1(\Omega)$. Moreover, an approximation property for the (truncation of the) amplitude of the jump holds. We apply the density result to deduce $\Gamma$-convergence approximation \emph{\`a la} Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are \emph{a priori} not even in $L^1(\Omega;\mathbb{R}^n)$.