  We analyze a finite-difference approximation of a functional of
Ambrosio-Tortorelli type in brittle fracture, in the discrete-to-continuum
limit. In a suitable regime between the competing scales, namely if the
discretization step $\delta$ is smaller than the ellipticity parameter
$\varepsilon$, we show the $\Gamma$-convergence of the model to the Griffith
functional, containing only a term enforcing Dirichlet boundary conditions and
no $L^p$ fidelity term. Restricting to two dimensions, we also address the case
in which a (linearized) constraint of non-interpenetration of matter is added
in the limit functional, in the spirit of a recent work by Chambolle, Conti and
Francfort.
