Energy release rate for non-smooth cracks in planar elasticity
[Taux de restitution d’énergie pour des fissures non régulières en élasticité plane]
Babadjian, Jean-François1 ; Chambolle, Antonin2 ; Lemenant, Antoine3
1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie–Paris 6, CNRS Boîte courrier 187, 75252 Paris Cedex 05, France
2 CMAP, École Polytechnique, CNRS 91128 Palaiseau, France
3 Laboratoire Jacques-Louis Lions, Université Paris Diderot–Paris 7, CNRS Bâtiment Sophie Germain, 75205 Paris Cedex 13, France
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.

DOI : 10.5802/jep.19
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
Babadjian, Jean-François 1 ; Chambolle, Antonin 2 ; Lemenant, Antoine 3

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie–Paris 6, CNRS Boîte courrier 187, 75252 Paris Cedex 05, France
2 CMAP, École Polytechnique, CNRS 91128 Palaiseau, France
3 Laboratoire Jacques-Louis Lions, Université Paris Diderot–Paris 7, CNRS Bâtiment Sophie Germain, 75205 Paris Cedex 13, France 
@article{JEP_2015__2__117_0,
     author = {Babadjian, Jean-Fran\c{c}ois and Chambolle, Antonin and Lemenant, Antoine},
     title = {Energy release rate for non-smooth cracks in~planar elasticity},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique - Math\'ematiques},
     pages = {117--152},
     publisher = {Ecole polytechnique},
     volume = {2},
     year = {2015},
     doi = {10.5802/jep.19},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jep.19/}
}
TY  - JOUR
AU  - Babadjian, Jean-François
AU  - Chambolle, Antonin
AU  - Lemenant, Antoine
TI  - Energy release rate for non-smooth cracks in planar elasticity
JO  - Journal de l’École polytechnique - Mathématiques
PY  - 2015
SP  - 117
EP  - 152
VL  - 2
PB  - Ecole polytechnique
UR  - http://www.numdam.org/articles/10.5802/jep.19/
DO  - 10.5802/jep.19
LA  - en
ID  - JEP_2015__2__117_0
ER  - 
%0 Journal Article
%A Babadjian, Jean-François
%A Chambolle, Antonin
%A Lemenant, Antoine
%T Energy release rate for non-smooth cracks in planar elasticity
%J Journal de l’École polytechnique - Mathématiques
%D 2015
%P 117-152
%V 2
%I Ecole polytechnique
%U http://www.numdam.org/articles/10.5802/jep.19/
%R 10.5802/jep.19
%G en
%F JEP_2015__2__117_0
Babadjian, Jean-François; Chambolle, Antonin; Lemenant, Antoine. Energy release rate for non-smooth cracks in planar elasticity. Journal de l’École polytechnique - Mathématiques, Tome 2 (2015), pp. 117-152. doi : 10.5802/jep.19. http://www.numdam.org/articles/10.5802/jep.19/

[1] Adams, D. R.; Hedberg, L. I. Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften, 314, Springer-Verlag, Berlin, 1996, pp. xii+366 | DOI | MR | Zbl

[2] Alessandrini, G.; Morassi, A.; Rosset, E. The linear constraints in Poincaré and Korn type inequalities, Forum Math., Volume 20 (2008) no. 3, pp. 557-569 | DOI | MR | Zbl

[3] Bourdin, B.; Francfort, G. A.; Marigo, J.-J. The variational approach to fracture, Springer, New York, 2008, pp. x+164 | DOI | MR | Zbl

[4] Bucur, D.; Varchon, N. A duality approach for the boundary variation of Neumann problems, SIAM J. Math. Anal., Volume 34 (2002) no. 2, pp. 460-477 | DOI | MR | Zbl

[5] Chambolle, A. A density result in two-dimensional linearized elasticity, and applications, Arch. Rational Mech. Anal., Volume 167 (2003) no. 3, pp. 211-233 | DOI | MR | Zbl

[6] Chambolle, A.; Doveri, F. Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets, Comm. Partial Differential Equations, Volume 22 (1997) no. 5-6, pp. 811-840 | DOI | MR | Zbl

[7] Chambolle, A.; Francfort, G. A.; Marigo, J.-J. Revisiting energy release rates in brittle fracture, J. Nonlinear Sci., Volume 20 (2010) no. 4, pp. 395-424 | DOI | MR | Zbl

[8] Chambolle, A.; Giacomini, A.; Ponsiglione, M. Crack initiation in brittle materials, Arch. Rational Mech. Anal., Volume 188 (2008) no. 2, pp. 309-349 | DOI | MR | Zbl

[9] Chambolle, A.; Lemenant, A. The stress intensity factor for non-smooth fractures in antiplane elasticity, Calc. Var. Partial Differential Equations, Volume 47 (2013) no. 3-4, pp. 589-610 | DOI | MR

[10] Ciarlet, Ph. G. Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988, pp. xlii+451 (Three-dimensional elasticity) | MR | Zbl

[11] Costabel, M.; Dauge, M. Crack singularities for general elliptic systems, Math. Nachr., Volume 235 (2002), pp. 29-49 | DOI | MR | Zbl

[12] Dal Maso, G.; Francfort, G. A.; Toader, R. Quasistatic crack growth in nonlinear elasticity, Arch. Rational Mech. Anal., Volume 176 (2005) no. 2, pp. 165-225 | DOI | MR | Zbl

[13] Dal Maso, G.; Toader, R. A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Rational Mech. Anal., Volume 162 (2002) no. 2, pp. 101-135 | DOI | MR | Zbl

[14] Destuynder, P.; Djaoua, M. Sur une interprétation mathématique de l’intégrale de Rice en théorie de la rupture fragile, Math. Methods Appl. Sci., Volume 3 (1981) no. 1, pp. 70-87 | DOI | MR | Zbl

[15] Falconer, K. J. The geometry of fractal sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986, pp. xiv+162 | MR | Zbl

[16] Francfort, G. A.; Larsen, Ch. J. Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math., Volume 56 (2003) no. 10, pp. 1465-1500 | DOI | MR | Zbl

[17] Francfort, G. A.; Marigo, J.-J. Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, Volume 46 (1998) no. 8, pp. 1319-1342 | DOI | MR | Zbl

[18] Griffith, A. A. The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London, Volume 221A (1920), pp. 163-198

[19] Grisvard, P. Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, 24, Pitman, Boston, MA, 1985, pp. xiv+410 | MR | Zbl

[20] Grisvard, P. Singularités en elasticité, Arch. Rational Mech. Anal., Volume 107 (1989) no. 2, pp. 157-180 | DOI | MR | Zbl

[21] Henrot, A.; Pierre, M. Variation et optimisation de formes. Une analyse géométrique, Mathématiques & Applications, 48, Springer, Berlin, 2005, pp. xii+334 | DOI | MR | Zbl

[22] Knees, D.; Mielke, A. Energy release rate for cracks in finite-strain elasticity, Math. Methods Appl. Sci., Volume 31 (2008) no. 5, pp. 501-528 | DOI | MR | Zbl

[23] Kondrat’ev, V. A. Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč., Volume 16 (1967), pp. 209-292 | MR | Zbl

[24] Kondrat’ev, V. A.; Kopachek, I.; Lekveishvili, D. M.; Oleĭnik, O. A. Sharp estimates in Hölder spaces and the exact Saint-Venant principle for solutions of the biharmonic equation, Trudy Mat. Inst. Steklov., Volume 166 (1984), pp. 91-106 (Modern problems of mathematics. Differential equations, mathematical analysis and their applications) | MR | Zbl

[25] Kozlov, V. A.; Maz’ya, V. G.; Rossmann, J. Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs, 52, American Mathematical Society, Providence, RI, 1997, pp. x+414 | MR | Zbl

[26] Kozlov, V. A.; Maz’ya, V. G.; Rossmann, J. Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs, 85, American Mathematical Society, Providence, RI, 2001, pp. x+436 | MR | Zbl

[27] Lazzaroni, G.; Toader, R. Energy release rate and stress intensity factor in antiplane elasticity, J. Math. Pures Appl. (9), Volume 95 (2011) no. 6, pp. 565-584 | DOI | MR | Zbl

[28] Negri, M.; Ortner, Ch. Quasi-static crack propagation by Griffith’s criterion, Math. Models Methods Appl. Sci., Volume 18 (2008) no. 11, pp. 1895-1925 | DOI | MR | Zbl

[29] Šverák, V. On optimal shape design, J. Math. Pures Appl. (9), Volume 72 (1993) no. 6, pp. 537-551 | MR | Zbl

[30] Temam, R. Navier-Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977, pp. x+500 | MR | Zbl

[31] Temam, R. Problèmes mathématiques en plasticité, Méthodes Mathématiques de l’Informatique, 12, Gauthier-Villars, Montrouge, 1983, pp. vii+353 | Zbl

Cité par Sources :