Existence of minimizers for the 2d stationary Griffith fracture model

Sergio Conti; Matteo Focardi; Flaviana Iurlano

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

Sergio Conti ¹ ; Matteo Focardi ² ; Flaviana Iurlano ¹

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

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

Abstract (VO)
Résumé

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.

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.

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 :

Sergio Conti ¹ ; Matteo Focardi ² ; Flaviana Iurlano ¹

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

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     author = {Sergio Conti and Matteo Focardi and Flaviana Iurlano},
     title = {Existence of minimizers for the 2d stationary {Griffith} fracture model},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1055--1059},
     publisher = {Elsevier},
     volume = {354},
     number = {11},
     year = {2016},
     doi = {10.1016/j.crma.2016.09.003},
     language = {en},
}

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Sergio Conti; Matteo Focardi; Flaviana Iurlano. Existence of minimizers for the 2d stationary Griffith fracture model. Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1055-1059. doi : 10.1016/j.crma.2016.09.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.09.003/

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Shokhrukh Y. Kholmatov; Paolo Piovano Existence of minimizers for the SDRI model in 2d: Wetting and dewetting regime with mismatch strain, Advances in Calculus of Variations, Volume 17 (2024) no. 3, p. 673 | DOI:10.1515/acv-2022-0053
Shokhrukh Yu. Kholmatov; Paolo Piovano A Unified Model for Stress-Driven Rearrangement Instabilities, Archive for Rational Mechanics and Analysis, Volume 238 (2020) no. 1, p. 415 | DOI:10.1007/s00205-020-01546-y
Sergio Conti; Matteo Focardi; Flaviana Iurlano Existence of strong minimizers for the Griffith static fracture model in dimension two, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 36 (2019) no. 2, p. 455 | DOI:10.1016/j.anihpc.2018.06.003
Sergio Conti; Matteo Focardi; Flaviana Iurlano A note on the Hausdorff dimension of the singular set of solutions to elasticity type systems, Communications in Contemporary Mathematics, Volume 21 (2019) no. 06, p. 1950026 | DOI:10.1142/s0219199719500263
Sergio Conti; Matteo Focardi; Flaviana Iurlano Approximation of fracture energies withp-growthviapiecewise affine finite elements, ESAIM: Control, Optimisation and Calculus of Variations, Volume 25 (2019), p. 34 | DOI:10.1051/cocv/2018021
Antonin Chambolle; Sergio Conti; Gilles A. Francfort Approximation of a Brittle Fracture Energy with a Constraint of Non-interpenetration, Archive for Rational Mechanics and Analysis, Volume 228 (2018) no. 3, p. 867 | DOI:10.1007/s00205-017-1207-z
Sergio Conti; Matteo Focardi; Flaviana Iurlano Integral Representation for Functionals Defined on SBDp in Dimension Two, Archive for Rational Mechanics and Analysis, Volume 223 (2017) no. 3, p. 1337 | DOI:10.1007/s00205-016-1059-y

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