Crack growth in polyconvex materials. (English) Zbl 1201.49013

Knees, Dorothee; Zanini, Chiara; Mielke, Alexander

Physica D 239, No. 15, 1470-1484 (2010).

Summary: We discuss a model for crack propagation in an elastic body, where the crack path is described a priori. In particular, we develop in the framework of finite-strain elasticity a rate-independent model for crack evolution which is based on the Griffith fracture criterion. Due to the nonuniqueness of minimizing deformations, the energy-release rate is no longer continuous with respect to time and the position of the crack tip. Thus, the model is formulated in terms of the Clarke differential of the energy, generalizing the classical crack evolution models for elasticity with strictly convex energies. We prove the existence of solutions for our model and also the existence of special solutions, where only certain extremal points of the Clarke differential are allowed.

MSC:

49J52	Nonsmooth analysis
49J40	Variational inequalities
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35K90	Abstract parabolic equations
74B20	Nonlinear elasticity
74G65	Energy minimization in equilibrium problems in solid mechanics
74R10	Brittle fracture

Keywords:

rate-independent problems; energetic formulation; time-incremental minimization; parameterized solutions; energy-release rate; Griffith fracture criterion; finite-strain elasticity; local energetic solution

References:

[1]	Toader, R.; Zanini, C., An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital. (9), II, 1-35 (2009) · Zbl 1180.35521
[2]	Kočvara, M.; Mielke, A.; Roubíček, T., A rate-independent approach to the delamination problem, Math. Mech. Solids, 11, 423-447 (2006) · Zbl 1133.74038
[3]	Cagnetti, F., A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Math. Models Methods Appl. Sci. (M \({}^3\) AS), 18, 7, 1027-1071 (2008) · Zbl 1154.49005
[4]	Dal Maso, G.; Toader, R., A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12, 12, 1773-1799 (2002) · Zbl 1205.74149
[5]	Negri, M.; Ortner, C., Quasi-static crack propagation by Griffith’s criterion, Math. Models Methods Appl. Sci., 18, 11, 1895-1925 (2008) · Zbl 1155.74035
[6]	Knees, D.; Mielke, A.; Zanini, C., On the inviscid limit of a model for crack propagation, Math. Models Methods Appl. Sci., 18, 9, 1529-1569 (2008) · Zbl 1151.49014
[7]	Francfort, G.; Marigo, J.-J., Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46, 1319-1342 (1998) · Zbl 0966.74060
[8]	Dal Maso, G.; Francfort, G.; Toader, R., Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176, 165-225 (2005) · Zbl 1064.74150
[9]	Knees, D., Griffith formula and \(J\)-integral for a crack in a power-law hardening material, Math. Methods Applied Sci., 16, 11, 1723-1749 (2006) · Zbl 1121.74053
[10]	Chambolle, A.; Giacomini, A.; Ponsiglione, M., Crack initiation in brittle materials, Arch. Ration. Mech. Anal., 188, 309-349 (2008) · Zbl 1138.74042
[11]	Miehe, C.; Gürses, E., A robust algorithm for configurational-force-driven brittle crack propagation with \(R\)-adaptive mesh alignment, Internat. J. Numer. Methods Engrg., 72, 127-155 (2007) · Zbl 1194.74444
[12]	Knees, D.; Mielke, A., Energy release rate for cracks in finite-strain elasticity, Math. Methods Applied Sci., 31, 5, 501-528 (2008) · Zbl 1132.74038
[13]	Efendiev, M.; Mielke, A., On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13, 1, 151-167 (2006) · Zbl 1109.74040
[14]	A. Mielke, R. Rossi, G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Cont. Dynam. Systems A, 2008, WIAS preprint 1347 (in press); A. Mielke, R. Rossi, G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Cont. Dynam. Systems A, 2008, WIAS preprint 1347 (in press)
[15]	Mielke, A., Evolution in rate-independent systems (Ch. 6), (Dafermos, C.; Feireisl, E., Handbook of Differential Equations, Evolutionary Equations, vol.2 (2005), Elsevier B.V.: Elsevier B.V. Amsterdam), 461-559 · Zbl 1120.47062
[16]	Bauman, P.; Phillips, D.; Owen, N. C., Maximal smoothness of solutions to certain Euler-Lagrange equations from nonlinear elasticity, Proc. Roy. Soc. Edinburgh Sect. A, 119, 3-4, 241-263 (1991) · Zbl 0744.49008
[17]	Ball, J. M., Some open problems in elasticity, (Newton, P.; Holmes, P.; Weinstein, A., Geometry, Mechanics, and Dynamics (2002), Springer: Springer New York), 3-59 · Zbl 1054.74008
[18]	Francfort, G.; Mielke, A., Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math., 595, 55-91 (2006) · Zbl 1101.74015
[19]	Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63, 4, 337-403 (1977) · Zbl 0368.73040
[20]	Ciarlet, P. G., Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity (1988), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0648.73014
[21]	Antman, S. S., (Nonlinear Problems of Elasticity. Nonlinear Problems of Elasticity, Applied Mathematical Sciences, vol. 107 (1995), Springer-Verlag: Springer-Verlag New York) · Zbl 0820.73002
[22]	E. Spadaro, Non-uniqueness of minimizers for strictly polyconvex functionals, Mathematisch-naturwissenschaftliche Fakultät Universität Zürich Preprint 15, Zürich, 2007; E. Spadaro, Non-uniqueness of minimizers for strictly polyconvex functionals, Mathematisch-naturwissenschaftliche Fakultät Universität Zürich Preprint 15, Zürich, 2007 · Zbl 1170.49032
[23]	Dal Maso, G.; Toader, R., On a notion of unilateral slope for the Mumford-Shah functional, NoDEA Nonlinear Differential Equations Appl., 13, 5-6, 713-734 (2007) · Zbl 1119.49017
[24]	Dal Maso, G.; Zanini, C., Quasi-static crack growth for a cohesive zone model with prescribed crack path., Proc. Roy. Soc. Edinburgh Sect. A, Math., 137, 2, 253-279 (2007) · Zbl 1116.74004
[25]	G. Dal Maso, A. Giacomini, M. Ponsiglione, A variational model for quasistatic crack growth in nonlinear elasticity: Some qualitative properties of the solutions, Boll. Unione Mat. Ital. (2009) (in press); G. Dal Maso, A. Giacomini, M. Ponsiglione, A variational model for quasistatic crack growth in nonlinear elasticity: Some qualitative properties of the solutions, Boll. Unione Mat. Ital. (2009) (in press) · Zbl 1173.74037
[26]	G. Dal Maso, G. Lazzaroni, Quasistatic crack growth in finite elasticity with non-interpenetration, 2008 (Preprint SISSA 80/2008/M); G. Dal Maso, G. Lazzaroni, Quasistatic crack growth in finite elasticity with non-interpenetration, 2008 (Preprint SISSA 80/2008/M) · Zbl 1188.35205
[27]	Aubin, J.-P.; Frankowska, H., Set-Valued Analysis, Systems & Control: Foundations & Applications, vol. 2 (1990), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 0713.49021
[28]	Clarke, F. H., Optimization and Nonsmooth Analysis (1983), Wiley-Interscience · Zbl 0727.90045
[29]	Rockafellar, R. T.; Wets, R. J.-B., (Variational Analysis. Variational Analysis, Grundlehren der Mathematischen Wissenschaften, vol. 317 (1998), Springer: Springer Berlin) · Zbl 0888.49001
[30]	Mainik, A.; Mielke, A., Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci. (2008), Published online. doi:10.1007/s00332-008-9033-y · Zbl 1173.49013
[31]	T. Roubíček, Rate independent processes in viscous solids at small strains, Math. Methods Applied Sciences, 2008 (submitted for publication); T. Roubíček, Rate independent processes in viscous solids at small strains, Math. Methods Applied Sciences, 2008 (submitted for publication)