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.
@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}, volume = {27}, year = {2010}, pages = {257-290}, doi = {10.1016/j.anihpc.2009.09.006}, mrnumber = {2580510}, zbl = {1188.35205}, language = {en}, url = {http://dml.mathdoc.fr/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) pp. 257-290. doi : 10.1016/j.anihpc.2009.09.006. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_1_257_0/
[1] 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] On the lower semicontinuity of quasiconvex integrals in , Nonlinear Anal. 23 (1994), 405-425 | MR 1291580 | Zbl 0817.49017
,[3] 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] Some open problems in elasticity, , , (ed.), Geometry, Mechanics, and Dynamics, Springer, New York (2002), 3-59
,[5] Some aspects of covering theory, Proc. Amer. Math. Soc. 3 (1952), 804-812 | MR 50654 | Zbl 0047.28903
, ,[6] The variational approach to fracture, J. Elasticity 91 (2008), 5-148 | MR 2390547 | Zbl 1176.74018
, , ,[7] Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. vol. 580, Springer-Verlag, Berlin, New York (1977) | MR 467310 | Zbl 0346.46038
, ,[8] A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal. 167 (2003), 211-233 | MR 1978582 | Zbl 1030.74007
,[9] Mathematical Elasticity — vol. I: Three-Dimensional Elasticity, Stud. Math. Appl. vol. 20, North-Holland Publishing Co., Amsterdam (1988) | MR 936420 | Zbl 0648.73014
,[10] Injectivity and self-contact in nonlinear elasticity, Arch. Ration. Mech. Anal. 97 (1987), 171-188 | MR 862546 | Zbl 0628.73043
, ,[11] Direct Methods in the Calculus of Variations, Appl. Math. Sci. vol. 78, Springer, New York (2008) | MR 2361288 | Zbl 1140.49001
,[12] Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal. 176 (2005), 165-225 | MR 2186036 | Zbl 1064.74150
, , ,[13] Quasistatic Crack Growth in Finite Elasticity, SISSA, Trieste (2004), http://www.sissa.it/fa/
, , ,[14] 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] 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] Modern Methods in the Calculus of Variations: Spaces, Springer, New York (2007) | MR 2341508 | Zbl 1153.49001
, ,[17] Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math. 56 (2003), 1465-1500 | MR 1988896 | Zbl 1068.74056
, ,[18] Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998), 1319-1342 | MR 1633984 | Zbl 0966.74060
, ,[19] 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] 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] 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] The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London A 221 (1921), 163-198
,[23] 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] 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] Quasistatic crack growth in finite elasticity with Lipschitz data, http://cvgmt.sns.it/ (2009)
,[27] Evolution of rate-independent systems, , (ed.), Evolutionary Equations — vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2005), 461-559 | MR 2182832
,[28] Perfect blankets, Trans. Amer. Math. Soc. 61 (1947), 418-442 | MR 20618 | Zbl 0031.38702
,[29] 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] 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] Functional Analysis, Grundlehren Math. Wiss. vol. 123, Springer-Verlag, Berlin, New York (1980) | MR 617913 | Zbl 0152.32102
,