Non-local approximation of the Griffith functional

Abstract

An approximation, in the sense of $$\Gamma $$ Γ -convergence and in any dimension $$d\ge 1$$ d ≥ 1 , of Griffith-type functionals, with p-growth ($$p>1$$ 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.

On some non-local approximation of nonisotropic Griffith-type functionals

“…In this paper, we extend the focus of [22,23] by showing that general Griffithtype functionals of the form…”

“…With the following proposition, we prove the compactness statements in Theorem 3.1(i), and Theorem 3.2 (i), respectively. These results can be easily inferred by a comparison with non-local integral energies whose densities are averages of the gradient on balls with small radii, for which a compactness result has been provided in [23,Proposition 4.1]. In order to do that, we will only require assumption (N1) on the convolution kernel ρ.…”

“…The scope of this paper is to provide a generalization of recent results, obtained in [23], concerning the approximation of brittle fracture energies for linearly elastic materials, by means of nonlocal functionals defined on Sobolev spaces, which are easier to handle also from a computational point of view.…”

“…In [23] an approach originally devised by Braides and Dal Maso [6] for the approximation of the Mumford-Shah functional has been generalized to the linearly elastic context. Namely, it was shown that, for a given bounded increasing function f : R + → R + the energies…”

“…The proof strategy we devise is based on a localization method and involves nontrivial adaptions to the method used in [23], in particular when estimating the bulk term in the Γ-liminf inequality (Proposition 4.2). There, we have to impose (and this is the only point in the paper) an additional restriction on the convolution kernel ρ, namely of being nonincreasing with respect to the given norm φ (see Assumption (N2) below).…”