A derivation of Griffith functionals from discrete finite-difference models

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 δ is smaller than the ellipticity parameter ε, we show the Γ-convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no Lp 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.