A DENSITY RESULT IN $GSBD^p$ WITH APPLICATIONS TO THE APPROXIMATION OF BRITTLE FRACTURE ENERGIES - Archive ouverte HAL Access content directly
Journal Articles Archive for Rational Mechanics and Analysis Year : 2019

A DENSITY RESULT IN $GSBD^p$ WITH APPLICATIONS TO THE APPROXIMATION OF BRITTLE FRACTURE ENERGIES

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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 $\int_\Omega W(e(u)) dx +\mathcal{H}^{n-1}(J_u)$, $e(u)$ 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 à la Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are a priori not even in $L^1(\Omega;\mathbb{R}^n)$.
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Dates and versions

hal-01573936 , version 1 (11-08-2017)

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  • HAL Id : hal-01573936 , version 1

Cite

Antonin Chambolle, Vito Crismale. A DENSITY RESULT IN $GSBD^p$ WITH APPLICATIONS TO THE APPROXIMATION OF BRITTLE FRACTURE ENERGIES. Archive for Rational Mechanics and Analysis, 2019, 232 (3), pp.1329--1378. ⟨hal-01573936⟩
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