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

Presented by: 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
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.

Received: 2016-03-04
Accepted: 2016-09-12
Published online: 2016-09-28

DOI: 10.1016/j.crma.2016.09.003

Author's affiliations:

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

@article{CRMATH_2016__354_11_1055_0,
     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},
}

TY  - JOUR
AU  - Sergio Conti
AU  - Matteo Focardi
AU  - Flaviana Iurlano
TI  - Existence of minimizers for the 2d stationary Griffith fracture model
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 1055
EP  - 1059
VL  - 354
IS  - 11
PB  - Elsevier
DO  - 10.1016/j.crma.2016.09.003
LA  - en
ID  - CRMATH_2016__354_11_1055_0
ER  -

%0 Journal Article
%A Sergio Conti
%A Matteo Focardi
%A Flaviana Iurlano
%T Existence of minimizers for the 2d stationary Griffith fracture model
%J Comptes Rendus. Mathématique
%D 2016
%P 1055-1059
%V 354
%N 11
%I Elsevier
%R 10.1016/j.crma.2016.09.003
%G en
%F CRMATH_2016__354_11_1055_0

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/

References
Cited by

[1] L. Ambrosio; V.M. Tortorelli Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence, Commun. Pure Appl. Math., Volume 43 (1990) no. 8, pp. 999-1036

[2] L. Ambrosio; A. Coscia; G. Dal Maso Fine properties of functions with bounded deformation, Arch. Ration. Mech. Anal., Volume 139 (1997) no. 3, pp. 201-238

[3] L. Ambrosio; N. Fusco; D. Pallara Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, UK, 2000

[4] G. Anzellotti; M. Giaquinta Existence of the displacement field for an elastoplastic body subject to Hencky's law and von Mises yield condition, Manuscr. Math., Volume 32 (1980) no. 1–2, pp. 101-136

[5] G. Bellettini; A. Coscia; G. Dal Maso Compactness and lower semicontinuity properties in $S B D (Ω)$ , Math. Z., Volume 228 (1998) no. 2, pp. 337-351

[6] B. Bourdin; G. Francfort; J.-J. Marigo The Variational Approach to Fracture, Springer, New York, 2008

[7] A. Chambolle An approximation result for special functions with bounded deformation, J. Math. Pures Appl. (9), Volume 83 (2004) no. 7, pp. 929-954

[8] A. Chambolle; A. Giacomini; M. Ponsiglione Piecewise rigidity, J. Funct. Anal., Volume 244 (2007) no. 1, pp. 134-153

[9] A. Chambolle; S. Conti; G. Francfort Korn–Poincaré inequalities for functions with a small jump set, Indiana Univ. Math. J., Volume 65 (2016) no. 4, pp. 1373-1399

[10] S. Conti; B. Schweizer Rigidity and Γ convergence for solid–solid phase transitions with $S O (2)$ -invariance, Commun. Pure Appl. Math., Volume 59 (2006), pp. 830-868

[11] S. Conti; M. Focardi; F. Iurlano Which special functions of bounded deformation have bounded variation?, Proc. R. Soc. Edinb. A (2015) (in press) | arXiv

[12] S. Conti; M. Focardi; F. Iurlano Integral representation for functionals defined on $S B D^{p}$ in dimension two, 2015 | arXiv

[13] S. Conti, M. Focardi, F. Iurlano, Existence result for the static Griffith fracture model in two dimensions, in preparation.

[14] G. Dal Maso Generalised functions of bounded deformation, J. Eur. Math. Soc., Volume 15 (2013) no. 5, pp. 1943-1997

[15] G. David Singular Sets of Minimizers for the Mumford–Shah Functional, Prog. Math., vol. 233, Birkhäuser Verlag, Basel, Switzerland, 2005

[16] E. De Giorgi; M. Carriero; A. Leaci Existence theorem for a minimum problem with free discontinuity set, Arch. Ration. Mech. Anal., Volume 108 (1989) no. 3, pp. 195-218

[17] C. De Lellis; M. Focardi Density lower bound estimates for local minimizers of the 2d Mumford–Shah energy, Manuscr. Math., Volume 142 (2013) no. 1–2, pp. 215-232

[18] G. De Philippis; F. Rindler On the structure of $A$ -free measures and applications, Ann. Math., Volume 184 (2016) no. 3, pp. 1017-1039

[19] G.A. Francfort; J.J. Marigo Revisiting brittle fractures as an energy minimization problem, J. Mech. Phys. Solids, Volume 46 (1998), pp. 1319-1342

[20] M. Friedrich A Korn type inequality in SBD for functions with small jump sets, 2015 | arXiv

[21] M. Friedrich A Korn–Poincaré-type inequality for special functions of bounded deformation, 2015 | arXiv

[22] M. Giaquinta; L. Martinazzi An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Appunti. Sc. Norm. Super. Pisa (N. S.), vol. 11, Edizioni della Normale, Pisa, 2012

[23] F. Iurlano A density result for GSBD and its application to the approximation of brittle fracture energies, Calc. Var. Partial Differ. Equ., Volume 51 (2014), pp. 315-342

[24] R. Kohn; R. Temam Dual spaces of stresses and strains, with applications to Hencky plasticity, Appl. Math. Optim., Volume 10 (1983) no. 1, pp. 1-35

[25] D. Mumford; J. Shah Optimal approximation by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math., Volume 17 (1989), pp. 577-685

[26] P.-M. Suquet Existence et régularité des solutions des équations de la plasticité, C. R. Acad. Sci. Paris, Sér. A–B, Volume 286 (1978), p. A1201-A1204

[27] R. Temam Problèmes mathématiques en plasticité, Méthodes Math. Inform., Math. Methods Inf. Sci., vol. 12, Gauthier–Villars, Montrouge, France, 1983

[28] R. Temam; G. Strang Functions of bounded deformation, Arch. Ration. Mech. Anal., Volume 75 (1980/1981), pp. 7-21

Cited by Sources:

Comments - Policy