Homogenization of Griffith’s Criterion for Brittle Laminates

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 evolutionis governed again by Griffith's criterion, in terms of the energy release (of the ho-mogenized elastic energy) and an effective toughness, which, in general, differsfrom the weak & lowast;limit of the periodic toughness. We provide a variational character-ization of the effective toughness and, by the energy identity, we link the tougheningeffect (in the limit) to the micro-instabilities of the evolution (in the periodic lami-nate). Finally, we provide a couple of explicit calculations of the effective toughnessin the anti-plane setting, showing, in particular, an example of toughening by elasticcontrast.