Non-local approximation of the Griffith functional

TL;DR

This paper introduces a non-local approximation method for Griffith-type functionals with p-growth, using integral functionals based on local averages of symmetrized gradients, valid in any dimension.

Contribution

It provides the first $ ext{Gamma}$-convergence approximation of Griffith functionals via non-local integral functionals in arbitrary dimensions.

Findings

Established $ ext{Gamma}$-convergence of the non-local approximation

Applicable to any dimension $d\,geq1$

Works for functionals with $p$-growth in symmetrized gradients

Abstract

An approximation, in the sense of $\Gamma$-convergence and in any dimension $d\geq1$, of Griffith-type functionals, with $p-$growth ($p>1$) in the symmetrized gradient, is provided by means of a sequence of non-local integral functionals depending on the average of the symmetrized gradients on small balls.