A density result for GSBD and its application to the approximation of brittle fracture energies

Abstract

We present an approximation result for functions $u:\Omega \rightarrow \mathbb {R}^n$ belonging to the space $GSBD(\Omega )\cap L^2(\Omega ,{{\mathbb R}}^n)$ with $e(u)$ square integrable and $\fancyscript{H}^{n-1}(J_u)$ finite. The approximating functions $u_k$ are piecewise continuous functions such that $u_k\rightarrow u$ in $L^2(\Omega ,\mathbb {R}^n)$, $e(u_k)\rightarrow e(u)$ in $L^2(\Omega ,\mathbb {M}^{n{\times }n}_{sym})$, $\fancyscript{H}^{n-1}(J_{u_k}\triangle J_u)\rightarrow 0$, and $\int _{J_{u_k}\cup J_u}|u_k^\pm -u^\pm |\wedge 1d{\fancyscript{H}}^{n-1}\rightarrow 0$.  As an application, we provide the extension to the vector-valued case of the $\Gamma$-convergence result in $GSBV(\Omega )$ proved by Ambrosio and Tortorelli (Commun Pure Appl Math 43:999–1036, 1990; Boll. Un. Mat. Ital. B (7) 6:105–123, 1992).