[Taux de restitution d’énergie pour des fissures non régulières en élasticité plane]
Cet article est consacré à l’étude du taux de restitution d’énergie associé à une fissure fermée, connexe et de densité (de longueur) 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 at the tip, without further regularity assumptions. First, the blow-up limit of the displacement is analyzed, and the convergence to a (known) positively -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.
Accepté le :
DOI : 10.5802/jep.19
Keywords: Elliptic problem, nonsmooth domain, blow-up limit, singular set, brittle fracture
Mot clés : Fissure, domaine non lisse, limite asymptotique, problème elliptique, ensemble singulier
Jean-François Babadjian 1 ; Antonin Chambolle 2 ; Antoine Lemenant 3
CC-BY-ND 4.0
@article{JEP_2015__2__117_0,
author = {Jean-Fran\c{c}ois Babadjian and Antonin Chambolle and Antoine Lemenant},
title = {Energy release rate for non-smooth cracks in~planar elasticity},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {117--152},
publisher = {\'Ecole polytechnique},
volume = {2},
year = {2015},
doi = {10.5802/jep.19},
mrnumber = {3366673},
zbl = {1325.74126},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.19/}
}
TY - JOUR AU - Jean-François Babadjian AU - Antonin Chambolle AU - Antoine Lemenant 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 - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.19/ DO - 10.5802/jep.19 LA - en ID - JEP_2015__2__117_0 ER -
%0 Journal Article %A Jean-François Babadjian %A Antonin Chambolle %A Antoine Lemenant %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 École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.19/ %R 10.5802/jep.19 %G en %F JEP_2015__2__117_0
Jean-François Babadjian; Antonin Chambolle; Antoine Lemenant. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.19/
[1] - Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften, vol. 314, Springer-Verlag, Berlin, 1996 | DOI | MR
[2] - “The linear constraints in Poincaré and Korn type inequalities”, Forum Math. 20 (2008) no. 3, p. 557-569 | DOI | Zbl
[3] - The variational approach to fracture, Springer, New York, 2008 | DOI | Zbl
[4] - “A duality approach for the boundary variation of Neumann problems”, SIAM J. Math. Anal. 34 (2002) no. 2, p. 460-477 | DOI | MR | Zbl
[5] - “A density result in two-dimensional linearized elasticity, and applications”, Arch. Rational Mech. Anal. 167 (2003) no. 3, p. 211-233 | DOI | MR | Zbl
[6] - “Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets”, Comm. Partial Differential Equations 22 (1997) no. 5-6, p. 811-840 | DOI | MR | Zbl
[7] - “Revisiting energy release rates in brittle fracture”, J. Nonlinear Sci. 20 (2010) no. 4, p. 395-424 | DOI | MR | Zbl
[8] - “Crack initiation in brittle materials”, Arch. Rational Mech. Anal. 188 (2008) no. 2, p. 309-349 | DOI | MR | Zbl
[9] - “The stress intensity factor for non-smooth fractures in antiplane elasticity”, Calc. Var. Partial Differential Equations 47 (2013) no. 3-4, p. 589-610 | DOI | MR | Zbl
[10] - Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988, Three-dimensional elasticity | MR
[11] - “Crack singularities for general elliptic systems”, Math. Nachr. 235 (2002), p. 29-49 | DOI | MR | Zbl
[12] - “Quasistatic crack growth in nonlinear elasticity”, Arch. Rational Mech. Anal. 176 (2005) no. 2, p. 165-225 | DOI | MR | Zbl
[13] - “A model for the quasi-static growth of brittle fractures: existence and approximation results”, Arch. Rational Mech. Anal. 162 (2002) no. 2, p. 101-135 | DOI | MR | Zbl
[14] - “Sur une interprétation mathématique de l’intégrale de Rice en théorie de la rupture fragile”, Math. Methods Appl. Sci. 3 (1981) no. 1, p. 70-87 | DOI | Zbl
[15] - The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986 | MR | Zbl
[16] - “Existence and convergence for quasi-static evolution in brittle fracture”, Comm. Pure Appl. Math. 56 (2003) no. 10, p. 1465-1500 | DOI | MR | Zbl
[17] - “Revisiting brittle fracture as an energy minimization problem”, J. Mech. Phys. Solids 46 (1998) no. 8, p. 1319-1342 | DOI | MR | Zbl
[18] - “The phenomena of rupture and flow in solids”, Philos. Trans. Roy. Soc. London 221A (1920), p. 163-198
[19] - Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman, Boston, MA, 1985 | MR | Zbl
[20] - “Singularités en elasticité”, Arch. Rational Mech. Anal. 107 (1989) no. 2, p. 157-180 | DOI | Zbl
[21] - Variation et optimisation de formes. Une analyse géométrique, Mathématiques & Applications, vol. 48, Springer, Berlin, 2005 | DOI | Zbl
[22] - “Energy release rate for cracks in finite-strain elasticity”, Math. Methods Appl. Sci. 31 (2008) no. 5, p. 501-528 | DOI | MR | Zbl
[23] - “Boundary value problems for elliptic equations in domains with conical or angular points”, Trudy Moskov. Mat. Obšč. 16 (1967), p. 209-292 | MR | Zbl
[24] - “Sharp estimates in Hölder spaces and the exact Saint-Venant principle for solutions of the biharmonic equation”, Trudy Mat. Inst. Steklov. 166 (1984), p. 91-106, Modern problems of mathematics. Differential equations, mathematical analysis and their applications | Zbl
[25] - Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs, vol. 52, American Mathematical Society, Providence, RI, 1997 | MR | Zbl
[26] - Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society, Providence, RI, 2001 | MR | Zbl
[27] - “Energy release rate and stress intensity factor in antiplane elasticity”, J. Math. Pures Appl. (9) 95 (2011) no. 6, p. 565-584 | DOI | MR | Zbl
[28] - “Quasi-static crack propagation by Griffith’s criterion”, Math. Models Methods Appl. Sci. 18 (2008) no. 11, p. 1895-1925 | DOI | MR | Zbl
[29] - “On optimal shape design”, J. Math. Pures Appl. (9) 72 (1993) no. 6, p. 537-551 | MR | Zbl
[30] - Navier-Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977 | MR | Zbl
[31] - Problèmes mathématiques en plasticité, Méthodes Mathématiques de l’Informatique, vol. 12, Gauthier-Villars, Montrouge, 1983 | Zbl
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