As a model for the energy of a brittle elastic body we consider an integral functional consisting of two parts: a volume one (the usual linearly elastic energy) which is quadratic in the strain, and a surface part, which is concentrated along the fractures (i.e. on the discontinuities of the displacement function) and whose density depends on the jump part of the strain. We study the problem of the lower semicontinuous envelope of such a functional under the assumptions that the surface energy density...

Given a Borel function ψ defined on a bounded open set Ω with Lipschitz boundary and $\varphi \in L^1(\partial \Omega ,{\mathscr{H}}^{n-1})$, we prove an explicit representation formula for the L1 lower semicontinuous envelope of Mumford-Shah type functionals with the obstacle constraint $u^{+}\ge \psi$${\mathscr{H}}^{...}$