Quasistatic crack growth in finite elasticity with non-interpenetration
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, p. 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 : https://doi.org/10.1016/j.anihpc.2009.09.006
Classification:  35R35,  74R10,  74B20,  49J45,  49Q20,  35A35,  28B20
@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},
     publisher = {Elsevier},
     volume = {27},
     number = {1},
     year = {2010},
     pages = {257-290},
     doi = {10.1016/j.anihpc.2009.09.006},
     zbl = {1188.35205},
     mrnumber = {2580510},
     language = {en},
     url = {http://www.numdam.org/item/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. https://www.numdam.org/item/AIHPC_2010__27_1_257_0/

[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 1032614 | Zbl 0767.49001

[2] L. Ambrosio, On the lower semicontinuity of quasiconvex integrals in 𝑆𝐵𝑉(Ω, k ), Nonlinear Anal. 23 (1994), 405-425 | MR 1291580 | Zbl 0817.49017

[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 1857292 | Zbl 0957.49001

[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 50654 | Zbl 0047.28903

[6] B. Bourdin, G.A. Francfort, J.-J. Marigo, The variational approach to fracture, J. Elasticity 91 (2008), 5-148 | MR 2390547 | Zbl 1176.74018

[7] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. vol. 580, Springer-Verlag, Berlin, New York (1977) | MR 467310 | Zbl 0346.46038

[8] A. Chambolle, A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal. 167 (2003), 211-233 | MR 1978582 | Zbl 1030.74007

[9] P.G. Ciarlet, Mathematical Elasticity — vol. I: Three-Dimensional Elasticity, Stud. Math. Appl. vol. 20, North-Holland Publishing Co., Amsterdam (1988) | MR 936420 | Zbl 0648.73014

[10] P.G. Ciarlet, J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Ration. Mech. Anal. 97 (1987), 171-188 | MR 862546 | Zbl 0628.73043

[11] B. Dacorogna, Direct Methods in the Calculus of Variations, Appl. Math. Sci. vol. 78, Springer, New York (2008) | MR 2361288 | Zbl 1140.49001

[12] G. Dal Maso, G.A. Francfort, R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal. 176 (2005), 165-225 | MR 2186036 | Zbl 1064.74150

[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 2537276 | Zbl 1173.74037

[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 1897378 | Zbl 1042.74002

[16] I. Fonseca, G. Leoni, Modern Methods in the Calculus of Variations: L p Spaces, Springer, New York (2007) | MR 2341508 | Zbl 1153.49001

[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 1988896 | Zbl 1068.74056

[18] G.A. Francfort, J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998), 1319-1342 | MR 1633984 | Zbl 0966.74060

[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 2244798 | Zbl 1101.74015

[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 2218155 | Zbl 1125.49007

[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 2477450 | Zbl 1151.74034

[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 45.0444.01

[24] D. Knees, A. Mielke, Energy release rate for cracks in finite-strain elasticity, Math. Methods Appl. Sci. 31 (2008), 501-528 | MR 2394124 | Zbl 1132.74038

[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 2658341

[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 2182832

[28] A.P. Morse, Perfect blankets, Trans. Amer. Math. Soc. 61 (1947), 418-442 | MR 20618 | Zbl 0031.38702

[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 0257.73034

[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 0245.73032

[31] K. Yosida, Functional Analysis, Grundlehren Math. Wiss. vol. 123, Springer-Verlag, Berlin, New York (1980) | MR 617913 | Zbl 0152.32102