Quasistatic crack growth in finite elasticity with non-interpenetration
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 257-290.

We present a variational model to study the quasistatic growth of brittle cracks in hyperelastic materials, in the framework of finite elasticity, taking into account the non-interpenetration condition.

DOI : 10.1016/j.anihpc.2009.09.006
Classification : 35R35, 74R10, 74B20, 49J45, 49Q20, 35A35, 28B20
Mots clés : Variational models, Energy minimization, Free-discontinuity problems, Polyconvexity, Quasistatic evolution, Rate-independent processes, Brittle fracture, Crack propagation, Griffith's criterion, Finite elasticity, Non-interpenetration
@article{AIHPC_2010__27_1_257_0,
     author = {Dal Maso, Gianni and Lazzaroni, Giuliano},
     title = {Quasistatic crack growth in finite elasticity with non-interpenetration},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {257--290},
     publisher = {Elsevier},
     volume = {27},
     number = {1},
     year = {2010},
     doi = {10.1016/j.anihpc.2009.09.006},
     mrnumber = {2580510},
     zbl = {1188.35205},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.006/}
}
TY  - JOUR
AU  - Dal Maso, Gianni
AU  - Lazzaroni, Giuliano
TI  - Quasistatic crack growth in finite elasticity with non-interpenetration
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 257
EP  - 290
VL  - 27
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.006/
DO  - 10.1016/j.anihpc.2009.09.006
LA  - en
ID  - AIHPC_2010__27_1_257_0
ER  - 
%0 Journal Article
%A Dal Maso, Gianni
%A Lazzaroni, Giuliano
%T Quasistatic crack growth in finite elasticity with non-interpenetration
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 257-290
%V 27
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.006/
%R 10.1016/j.anihpc.2009.09.006
%G en
%F AIHPC_2010__27_1_257_0
Dal Maso, Gianni; Lazzaroni, Giuliano. Quasistatic crack growth in finite elasticity with non-interpenetration. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 257-290. doi : 10.1016/j.anihpc.2009.09.006. http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.006/

[1] L. Ambrosio, A compactness theorem for a new class of functions of bounded variation, Boll. Unione Mat. Ital. B 3 (1989), 857-881 | MR | Zbl

[2] L. Ambrosio, On the lower semicontinuity of quasiconvex integrals in 𝑆𝐵𝑉(Ω, k ), Nonlinear Anal. 23 (1994), 405-425 | MR | Zbl

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

[4] J.M. Ball, Some open problems in elasticity, P. Newton, P. Holmes, A. Weinstein (ed.), Geometry, Mechanics, and Dynamics, Springer, New York (2002), 3-59

[5] W.W. Bledsoe, A.P. Morse, Some aspects of covering theory, Proc. Amer. Math. Soc. 3 (1952), 804-812 | MR | Zbl

[6] B. Bourdin, G.A. Francfort, J.-J. Marigo, The variational approach to fracture, J. Elasticity 91 (2008), 5-148 | MR | Zbl

[7] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. vol. 580, Springer-Verlag, Berlin, New York (1977) | MR | Zbl

[8] A. Chambolle, A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal. 167 (2003), 211-233 | MR | Zbl

[9] P.G. Ciarlet, Mathematical Elasticity — vol. I: Three-Dimensional Elasticity, Stud. Math. Appl. vol. 20, North-Holland Publishing Co., Amsterdam (1988) | MR | Zbl

[10] P.G. Ciarlet, J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Ration. Mech. Anal. 97 (1987), 171-188 | MR | Zbl

[11] B. Dacorogna, Direct Methods in the Calculus of Variations, Appl. Math. Sci. vol. 78, Springer, New York (2008) | MR | Zbl

[12] G. Dal Maso, G.A. Francfort, R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal. 176 (2005), 165-225 | MR | Zbl

[13] G. Dal Maso, G.A. Francfort, R. Toader, Quasistatic Crack Growth in Finite Elasticity, SISSA, Trieste (2004), http://www.sissa.it/fa/

[14] G. Dal Maso, A. Giacomini, M. Ponsiglione, A variational model for quasi-static growth in nonlinear elasticity: Some qualitative properties of the solutions, Boll. Unione Mat. Ital. B 9 (2009), 371-390 | MR | Zbl

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

[16] I. Fonseca, G. Leoni, Modern Methods in the Calculus of Variations: L p Spaces, Springer, New York (2007) | MR | Zbl

[17] G.A. Francfort, C.J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math. 56 (2003), 1465-1500 | MR | Zbl

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

[19] G.A. Francfort, A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math. 595 (2006), 55-91 | MR | Zbl

[20] N. Fusco, C. Leone, R. March, A. Verde, A lower semi-continuity result for polyconvex functionals in SBV, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 321-336 | MR | Zbl

[21] A. Giacomini, M. Ponsiglione, Non interpenetration of matter for SBV-deformations of hyperelastic brittle materials, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 1019-1041 | MR | Zbl

[22] A.A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London A 221 (1921), 163-198

[23] H. Hahn, Über Annäherung an Lebesgue'sche Integrale durch Riemann'sche Summen, Sitzungsber. Math. Phys. Kl. K. Akad. Wiss. Wien 123 (1914), 713-743 | JFM

[24] D. Knees, A. Mielke, Energy release rate for cracks in finite-strain elasticity, Math. Methods Appl. Sci. 31 (2008), 501-528 | MR | Zbl

[25] D. Knees, C. Zanini, A. Mielke, Crack growth in polyconvex materials, Phys. D, doi:10.1016/j.physd.2009.02.008, in press | MR

[26] G. Lazzaroni, Quasistatic crack growth in finite elasticity with Lipschitz data, http://cvgmt.sns.it/ (2009)

[27] A. Mielke, Evolution of rate-independent systems, C.M. Dafermos, E. Feireisl (ed.), Evolutionary Equations — vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2005), 461-559 | MR

[28] A.P. Morse, Perfect blankets, Trans. Amer. Math. Soc. 61 (1947), 418-442 | MR | Zbl

[29] R.W. Ogden, Large deformation isotropic elasticity: On the correlation of theory and experiment for incompressible rubberlike solids, Proc. Roy. Soc. London A 326 (1972), 565-584 | Zbl

[30] R.W. Ogden, Large deformation isotropic elasticity: On the correlation of theory and experiment for compressible rubberlike solids, Proc. Roy. Soc. London A 328 (1972), 567-583 | Zbl

[31] K. Yosida, Functional Analysis, Grundlehren Math. Wiss. vol. 123, Springer-Verlag, Berlin, New York (1980) | MR | Zbl

Cité par Sources :