A derivation of Griffith functionals from discrete finite-difference models

2020/01/02 by Crismale, Vito, Scilla, Giovanni, Solombrino, Francesco
#49J45 #49M25 #65M06 #74R10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2001.00480

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 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 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.

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