Discrete approximation of the Griffith functional by adaptive finite elements

TL;DR

This paper demonstrates that adaptive finite element methods can effectively approximate the Griffith fracture energy in 2D, with the mesh adapting as part of the solution to capture fracture surfaces accurately.

Contribution

It introduces a novel adaptive finite element approximation framework for the Griffith energy using $ ext{Γ}$-convergence, where the mesh adapts as part of the problem to model isotropic surface energies.

Findings

Proves $ ext{Γ}$-convergence of discrete functionals to the Griffith energy.

Shows the mesh adapts as part of the solution, enabling accurate fracture modeling.

Provides a flexible approach to approximate isotropic surface energies in fracture mechanics.

Abstract

This paper is devoted to show a discrete adaptive finite element approximation result for the isotropic two-dimensional Griffith energy arising in fracture mechanics. The problem is addressed in the geometric measure theoretic framework of generalized special functions of bounded deformation which corresponds to the natural energy space for this functional. It is proved to be approximated in the sense of $\Gamma$-convergence by a sequence of discrete integral functionals defined on continuous piecewise affine functions. The main feature of this result is that the mesh is part of the unknown of the problem, and it gives enough flexibility to recover isotropic surface energies.