Existence of minimizers for the 2d stationary Griffith fracture model

Conti, Sergio; Focardi, Matteo; Iurlano, Flaviana

doi:10.1016/j.crma.2016.09.003

Mathematical analysis/Partial differential equations

Existence of minimizers for the 2d stationary Griffith fracture model
[Existence de déformations minimisant le modèle de Griffith des fractures 2d-stationnaires]

Présenté par : Haïm Brézis

Conti, Sergio ¹ ; Focardi, Matteo ² ; Iurlano, Flaviana ¹

¹ Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany
² DiMaI, Università di Firenze, 50134 Firenze, Italy

Comptes Rendus. Mathématique, Tome 354 (2016) no. 11, pp. 1055-1059.

Résumé
Abstract

Nous considérons la formulation variationnelle du modèle de fracture de Griffith en dimension spatiale 2. Nous montrons l'existence de champs de déformation continûment différentiables hors d'un ensemble fermé de sauts, minimisant l'énergie relevante. Pour ce faire, nous montrons que les déformations minimisant la formulation faible du problème, dont l'existence est bien connue, placés dans l'espace des fonctions $G S B D^{2}$ , minimisent de fait la formulation forte. Nous suivons l'approche développée par De Giorgi, Carriero et Leach dans le cadre scalaire correspondant du problème de Mumford–Shah.

We consider the variational formulation of the Griffith fracture model in two spatial dimensions and prove the existence of strong minimizers, that is deformation fields that are continuously differentiable outside a closed jump set and that minimize the relevant energy. To this aim, we show that minimizers of the weak formulation of the problem, set in the function space $G S B D^{2}$ and whose existence is well known, are actually strong minimizers following the approach developed by De Giorgi, Carriero, and Leaci in the corresponding scalar setting of the Mumford–Shah problem.

Reçu le : 2016-03-04
Accepté le : 2016-09-12
Publié le : 2016-09-28

DOI : 10.1016/j.crma.2016.09.003

Affiliations des auteurs :

Conti, Sergio ¹ ; Focardi, Matteo ² ; Iurlano, Flaviana ¹

¹ Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany
² DiMaI, Università di Firenze, 50134 Firenze, Italy

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     author = {Conti, Sergio and Focardi, Matteo and Iurlano, Flaviana},
     title = {Existence of minimizers for the 2d stationary {Griffith} fracture model},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1055--1059},
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Conti, Sergio; Focardi, Matteo; Iurlano, Flaviana. Existence of minimizers for the 2d stationary Griffith fracture model. Comptes Rendus. Mathématique, Tome 354 (2016) no. 11, pp. 1055-1059. doi : 10.1016/j.crma.2016.09.003. http://www.numdam.org/articles/10.1016/j.crma.2016.09.003/

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