A derivation of Griffith functionals from discrete finite-difference models

Abstract

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 $$\delta $$ δ is smaller than the ellipticity parameter $$\varepsilon $$ ε , we show the $$\varGamma $$ Γ -convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no $$L^p$$ 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.

Non-local approximation of the Griffith functional

“…For a suitable fine mesh, with size δ = δ(ε) small enough, these numerical approximations Γ-converge, as ε → 0, to the Mumford-Shah functional (see [6,8], and [7] for the case of a stochastic lattice). A similar result for the energy (1.1) has been recently provided in [25]. For other discrete approaches based on finite differences or finite elements we may mention [16,21,39], in the context of the Mumford-Shah functional, and [1,36] for the Griffith model.…”

From atomistic systems to linearized continuum models for elastic materials with voids

“…We will present here only the case d = 3, since the case d = 2 is completely analogous, and some arguments along the proofs can actually even be simplified. The passage from atomistic to continuum models (see [7]) via Γ-convergence [6,20] has been carried out in various contexts, including elasticity [1,8,10,11,44], fracture [2,9,19,29,30,31,41,43], or more general problems containing free discontinuities [4,39,42].…”

On some non-local approximation of nonisotropic Griffith-type functionals

“…As a final remark, it would be desirable to get rid on the structural assumption (N2) on the convolution kernels, which is used only in Proposition 4.2. It is our opinion that this is going to require quite a delicate abstract analysis of the Γ-limit of nonlocal functionals which approximate free-discontinuity problems in GSBD, possibly including also finite-difference models which are well suited to numerical approximations (see [15] for a recent discrete finite-difference approximation of some Griffith-type functionals in GSBD). A similar analysis for the SBV setting has been performed in [11], where integral representation formulas for the limit energy have been provided.…”