Approximation results by difference schemes of fracture energies: the vectorial case.

We provide a variational approximation, in the sense of De Giorgi's $\Gamma$-convergence, by finite-difference schemes of functionals of the type \[ \int_\Omega \psi(\nabla u)\dx+\displaystyle\int_{J_u}g\left(u^+-u^-,\nu_u\right) dH^2 \] defined for $u\in SBV(\Om;\rr^N)$, where $\Om$ is an open set in $R^3$. These energies are related to variational models in fracture mechanics for non-linear elastic materials. The approximating functionals are of the form $$ \int_{\ticaleps\cap\Om}\psi_\e\left(\nabla u(x)\right)\dx $$ where $\psi_\e$ is an interaction potential taking into account a separation of scales, $\ticaleps$ is a suitable regular triangulation of $R^3$ and $u$ is affine on each element of the assigned triangulation.