A density result in $GSB{D}^p$ with applications to the approximation of brittle fracture energies

Abstract

We prove that any function in $GSB{D}^p(\Omega )$, 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{à 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 $L^1(\Omega ;{\mathbb{R}}^n)$.

References in corpus (3)

• Approximation of a Brittle Fracture Energy with a Constraint of Non-Interpenetration
• Density of polyhedral partitions
• Approximation of fracture energies with $p$-growth via piecewise affine finite elements

Cited by in corpus (11)

• Existence of strong solutions to the Dirichlet problem for the Griffith energy
• Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity
• $\Gamma$-convergence for high order phase field fracture: continuum and isogeometric formulations
• Approximation of fracture energies with $p$-growth via piecewise affine finite elements
• Functionals defined on piecewise rigid functions: Integral representation and $\Gamma$-convergence
• Analysis of staggered evolutions for nonlinear energies in phase field fracture
• On the approximation of $SBD$ functions and some applications
• A unified model for stress-driven rearrangement instabilities
• Irreversibility and alternate minimization in phase field fracture: a viscosity approach
• Phase-field approximation of functionals defined on piecewise-rigid maps
• Non-local approximation of free-discontinuity problems in linear elasticity and application to stochastic homogenisation