Calculus of Variations and Geometric Measure Theory
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V. Crismale - G. Scilla - F. Solombrino

A derivation of Griffith functionals from discrete finite-difference models

created by scilla on 02 Jan 2020
modified by solombrin on 25 Mar 2020

[BibTeX]

Submitted Paper

Inserted: 2 jan 2020
Last Updated: 25 mar 2020

Year: 2020

ArXiv: 2001.00480 PDF

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.

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


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