Homogenization of Griffith's Criterion for brittle Laminates

TL;DR

This paper investigates the homogenization of Griffith's criterion for brittle crack evolution in periodic laminates, revealing how microstructure influences effective toughness and crack behavior in the limit of vanishing layer size.

Contribution

It provides a variational characterization of the effective toughness and links micro-instabilities to toughening effects in homogenized brittle laminates.

Findings

Effective toughness can differ from the periodic toughness due to homogenization.

Toughening can occur through elastic contrast in the laminate.

Explicit calculations demonstrate microstructure-induced toughening effects.

Abstract

We consider a periodic, linear elastic laminate with a brittle crack, evolving along a prescribed path according to Griffith's criterion. We study the homogenized limit of this evolution, as the size of the layers vanishes. The limit evolution is governed again by Griffith's criterion, in terms of the energy release (of the homogenized elastic energy) and an effective toughness, which in general differs from the weak* limit of the periodic toughness. We provide a variational characterization of the effective toughness and, by the energy identity, we link the toughening effect (in the limit) to the micro-instabilities of the evolution (in the periodic laminate). Finally, we provide a couple of explicit calculations of the effective toughness in the anti-plane setting, showing in particular an example of toughening by elastic contrast.