Existence of strong minimizers for the Griffith static fracture model in dimension two

Conti, Sergio; Focardi, Matteo; Iurlano, Flaviana

doi:10.1016/j.anihpc.2018.06.003

Conti, Sergio ; Focardi, Matteo ; Iurlano, Flaviana

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 455-474.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization to the vectorial situation of the decay estimate by De Giorgi, Carriero, and Leaci. This is based on replacing the coarea formula by a method to approximate $S B D^{p}$ functions with small jump set by Sobolev functions, and is restricted to two dimensions. The other two ingredients will appear in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy $G S B D^{p}$ functions by $S B V^{p}$ ones.

DOI : 10.1016/j.anihpc.2018.06.003

Mots-clés : Calculus of variations, Functions of bounded deformation, Existence, Fracture mechanics, Griffith's model

@article{AIHPC_2019__36_2_455_0,
     author = {Conti, Sergio and Focardi, Matteo and Iurlano, Flaviana},
     title = {Existence of strong minimizers for the {Griffith} static fracture model in dimension two},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {455--474},
     publisher = {Elsevier},
     volume = {36},
     number = {2},
     year = {2019},
     doi = {10.1016/j.anihpc.2018.06.003},
     mrnumber = {3913194},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2018.06.003/}
}

TY  - JOUR
AU  - Conti, Sergio
AU  - Focardi, Matteo
AU  - Iurlano, Flaviana
TI  - Existence of strong minimizers for the Griffith static fracture model in dimension two
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2019
SP  - 455
EP  - 474
VL  - 36
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2018.06.003/
DO  - 10.1016/j.anihpc.2018.06.003
LA  - en
ID  - AIHPC_2019__36_2_455_0
ER  -

%0 Journal Article
%A Conti, Sergio
%A Focardi, Matteo
%A Iurlano, Flaviana
%T Existence of strong minimizers for the Griffith static fracture model in dimension two
%J Annales de l'I.H.P. Analyse non linéaire
%D 2019
%P 455-474
%V 36
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2018.06.003/
%R 10.1016/j.anihpc.2018.06.003
%G en
%F AIHPC_2019__36_2_455_0

Conti, Sergio; Focardi, Matteo; Iurlano, Flaviana. Existence of strong minimizers for the Griffith static fracture model in dimension two. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 455-474. doi : 10.1016/j.anihpc.2018.06.003. http://www.numdam.org/articles/10.1016/j.anihpc.2018.06.003/

Bibliographie
Cité par

[1] Acerbi, E.; Fonseca, I.; Fusco, N. Regularity of minimizers for a class of membrane energies, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 25 (1997), pp. 11–25 (1998. Dedicated to Ennio De Giorgi) | Numdam | MR | Zbl

[2] Acerbi, E.; Fonseca, I.; Fusco, N. Regularity results for equilibria in a variational model for fracture, Proc. R. Soc. Edinb., Sect. A, Volume 127 (1997), pp. 889–902 | DOI | MR | Zbl

[3] Acerbi, E.; Fusco, N. Regularity for minimizers of nonquadratic functionals: the case $1 < p < 2$ , J. Math. Anal. Appl., Volume 140 (1989), pp. 115–135 | DOI | MR | Zbl

[4] Ambrosio, L.; Coscia, A.; Dal Maso, G. Fine properties of functions with bounded deformation, Arch. Ration. Mech. Anal., Volume 139 (1997), pp. 201–238 | DOI | MR | Zbl

[5] Ambrosio, L.; Fusco, N.; Pallara, D. Functions of Bounded Variation and Free Discontinuity Problems, Oxf. Math. Monogr., The Clarendon Press, Oxford University Press, New York, 2000 | MR | Zbl

[6] Babadjian, J.-F. Traces of functions of bounded deformation, Indiana Univ. Math. J., Volume 64 (2015), pp. 1271–1290 | MR

[7] Bellettini, G.; Coscia, A.; Dal Maso, G. Compactness and lower semicontinuity properties in $S B D (Ω)$ , Math. Z., Volume 228 (1998), pp. 337–351 | DOI | MR | Zbl

[8] Bourdin, B.; Francfort, G.A.; Marigo, J.-J. The variational approach to fracture, J. Elast., Volume 91 (2008), pp. 5–148 | DOI | MR | Zbl

[9] Bucur, D.; Luckhaus, S. Monotonicity formula and regularity for general free discontinuity problems, Arch. Ration. Mech. Anal., Volume 211 (2014), pp. 489–511 | DOI | MR | Zbl

[10] Carriero, M.; Leaci, A. $S^{k}$ -valued maps minimizing the $L^{p}$ -norm of the gradient with free discontinuities, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 18 (1991), pp. 321–352 | Numdam | MR | Zbl

[11] Chambolle, A. An approximation result for special functions with bounded deformation, J. Math. Pures Appl. (9), Volume 83 (2004), pp. 929–954 | DOI | MR | Zbl

[12] Chambolle, A. Addendum to: “An approximation result for special functions with bounded deformation” [11], J. Math. Pures Appl., Volume 9 (2005) no. 84, pp. 137–145 | MR | Zbl

[13] Chambolle, A.; Conti, S.; Francfort, G. Korn–Poincaré inequalities for functions with a small jump set, Indiana Univ. Math. J., Volume 65 (2016), pp. 1373–1399 | DOI | MR

[14] Chambolle, A.; Conti, S.; Iurlano, F. Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy, 2017 (preprint) | arXiv | MR

[15] Chambolle, A.; Giacomini, A.; Ponsiglione, M. Piecewise rigidity, J. Funct. Anal., Volume 244 (2007), pp. 134–153 | DOI | MR | Zbl

[16] Conti, S.; Focardi, M.; Iurlano, F. Approximation of fracture energies with p-growth via piecewise affine finite elements, ESAIM Control Optim. Calc. Var. (2017) (in press) | DOI | Numdam | MR

[17] Conti, S.; Focardi, M.; Iurlano, F. A note on the Hausdorff dimension of the singular set of solutions to elasticity type systems, Comm. Contemp. Math. (2018) (in press, 2018) | arXiv | MR

[18] Conti, S.; Focardi, M.; Iurlano, F. Existence of minimizers for the 2d stationary Griffith fracture model, C. R. Acad. Sci. Paris, Volume 354 (2016), pp. 1055–1059 | DOI | MR

[19] Conti, S.; Focardi, M.; Iurlano, F. Integral representation for functionals defined on $S B D^{p}$ in dimension two, Arch. Ration. Mech. Anal., Volume 223 (2017), pp. 1337–1374 | DOI | MR

[20] Conti, S.; Focardi, M.; Iurlano, F. Which special functions of bounded deformation have bounded variation?, Proc. R. Soc. Edinb., Sect. A, Volume 148 (2018), pp. 33–50 | DOI | MR

[21] Conti, S.; Schweizer, B. Rigidity and Gamma convergence for solid-solid phase transitions with SO(2) invariance, Commun. Pure Appl. Math., Volume 59 (2006), pp. 830–868 | DOI | MR | Zbl

[22] Dal Maso, G. Generalised functions of bounded deformation, J. Eur. Math. Soc. (JEMS), Volume 15 (2013), pp. 1943–1997 | DOI | MR | Zbl

[23] Dal Maso, G.; Morel, J.-M.; Solimini, S. A variational method in image segmentation: existence and approximation results, Acta Math., Volume 168 (1992), pp. 89–151 | MR | Zbl

[24] Dal Maso, G.; Toader, R. A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal., Volume 162 (2002), pp. 101–135 | DOI | MR | Zbl

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

[26] De Giorgi, E.; Carriero, M.; Leaci, A. Existence theorem for a minimum problem with free discontinuity set, Arch. Ration. Mech. Anal., Volume 108 (1989), pp. 195–218 | DOI | MR | Zbl

[27] De Lellis, C.; Focardi, M. Density lower bound estimates for local minimizers of the 2d Mumford–Shah energy, Manuscr. Math., Volume 142 (2013), pp. 215–232 | MR | Zbl

[28] De Philippis, G.; Rindler, F. On the structure of $A$ -free measures and applications, Ann. Math. (2), Volume 184 (2016), pp. 1017–1039 | DOI | MR

[29] Diening, L.; Ettwein, F. Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math., Volume 20 (2008), pp. 523–556 | DOI | MR | Zbl

[30] Diening, L.; Kaplický, P. $L^{q}$ theory for a generalized Stokes system, Manuscr. Math., Volume 141 (2013), pp. 333–361 | DOI | MR | Zbl

[31] Focardi, M. Fine regularity results for Mumford–Shah minimizers: porosity, higher integrability and the Mumford–Shah conjecture, Free Discontinuity Problems, CRM Monogr. Ser., vol. 19, Ed. Norm, Pisa, 2016, pp. 1–68 | MR

[32] Fonseca, I.; Fusco, N. Regularity results for anisotropic image segmentation models, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 24 (1997), pp. 463–499 | Numdam | MR | Zbl

[33] Francfort, G.A.; Marigo, J.-J. Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, Volume 46 (1998), pp. 1319–1342 | DOI | MR | Zbl

[34] Friedrich, M. A Korn-Poincaré-type inequality for special functions of bounded deformation, 2015 (preprint) | arXiv

[35] Friedrich, M. A Korn-type inequality in SBD for functions with small jump sets, Math. Models Methods Appl. Sci., Volume 27 (2017), pp. 2461–2484 | DOI | MR

[36] Friedrich, M.; Solombrino, F. Quasistatic crack growth in 2d-linearized elasticity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 35 (2018), pp. 27–64 | DOI | Numdam | MR

[37] Fuchs, M.; Seregin, G. Variational Methods for Problems From Plasticity Theory and for Generalized Newtonian Fluids, Lect. Notes Math., vol. 1749, Springer-Verlag, Berlin, 2000 | DOI | MR | Zbl

[38] Fusco, N.; Mingione, G.; Trombetti, C. Regularity of minimizers for a class of anisotropic free discontinuity problems, J. Convex Anal., Volume 8 (2001), pp. 349–367 | MR | Zbl

[39] Giaquinta, M.; Martinazzi, L. 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 | MR | Zbl

[40] Giusti, E. Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003 | DOI | MR | Zbl

[41] Hutchinson, J.W. A Course on Nonlinear Fracture Mechanics, Department of Solid Mechanics, Techn. University of Denmark, 1989

[42] Iurlano, F. 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 | DOI | MR | Zbl

[43] Maddalena, F.; Solimini, S. Lower semicontinuity properties of functionals with free discontinuities, Arch. Ration. Mech. Anal., Volume 159 (2001), pp. 273–294 | DOI | MR | Zbl

[44] Temam, R. Problèmes mathématiques en plasticité, Méthodes Math. Inform., Mathematical Methods of Information Science, vol. 12, Gauthier-Villars, Montrouge, 1983 | MR | Zbl

Cité par Sources :