Calculus of Variations and Geometric Measure Theory

V. Crismale - G. Scilla - F. Solombrino

A derivation of Griffith functionals from discrete finite-difference models

created by scilla on 02 Jan 2020
modified on 19 Oct 2020


Published Paper

Inserted: 2 jan 2020
Last Updated: 19 oct 2020

Journal: Calc. Var. Partial Differential Equations
Volume: 59
Number: 6
Pages: 193
Year: 2020
Doi: 10.1007/s00526-020-01858-7

ArXiv: 2001.00480 PDF


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.

Keywords: Brittle fracture, non-interpenetration, discrete approximations, finite difference methods, Griffith functional