A derivation of Griffith functionals from discrete finite-difference models

TL;DR

This paper demonstrates how a finite-difference approximation of an Ambrosio-Tortorelli type functional converges to the Griffith fracture model in the discrete-to-continuum limit, especially when the discretization scale is smaller than the ellipticity parameter.

Contribution

It establishes the $ ext{Gamma}$-convergence of a discrete finite-difference model to the Griffith functional, including boundary conditions and non-interpenetration constraints in 2D.

Findings

Finite-difference models converge to Griffith functional in the limit.

The convergence holds when discretization step is smaller than the ellipticity parameter.

In 2D, non-interpenetration constraints are incorporated in the limit.

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 $\delta$ 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.