Energy release rate for non-smooth cracks in planar elasticity

Babadjian, Jean-François; Chambolle, Antonin; Lemenant, Antoine

Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 117-152.

Cet article est consacré à l’étude du taux de restitution d’énergie associé à une fissure fermée, connexe et de densité (de longueur) $1/2$ en pointe de fissure, sans autre hypothèse de régularité. Tout d’abord, la limite de blow-up du déplacement à la pointe est analysée, ainsi que la convergence vers une certaine fonction, positivement 1/2-homogène, explicite. Le taux de restitution d’énergie, qui est la dérivée de l’énergie élastique par rapport à un incrément infinitésimal de fissure, est alors obtenu comme solution d’un problème variationnel.

This paper is devoted to the characterization of the energy release rate of a crack which is merely closed, connected, and with (length) density $1/2$ at the tip, without further regularity assumptions. First, the blow-up limit of the displacement is analyzed, and the convergence to a (known) positively $1/2$-homogenous function in the cracked plane is established. Then, the energy release rate, which is the derivative of the elastic energy with respect to an infinitesimal additional crack increment, is obtained as the solution of a variational problem.

Classification : 74R10, 35J20, 49J45
Keywords: Elliptic problem, nonsmooth domain, blow-up limit, singular set, brittle fracture
Mot clés : Fissure, domaine non lisse, limite asymptotique, problème elliptique, ensemble singulier

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