zbMATH — the first resource for mathematics

A time-discrete model for dynamic fracture based on crack regularization. (English) Zbl 1283.74055
Summary: We propose a discrete time model for dynamic fracture based on crack regularization. The advantages of our approach are threefold: first, our regularization of the crack set has been rigorously shown to converge to the correct sharp-interface energy L. Ambrosio and V. M. Tortorelli [Commun. Pure Appl. Math. 43, No. 8, 999–1036 (1990; Zbl 0722.49020); Boll. Unione Mat. Ital., VII. Ser., B 6, No. 1, 105–123 (1992; Zbl 0776.49029)]; second, our condition for crack growth, based on Griffith’s criterion, matches that of quasi-static settings B. Bourdin [Interfaces Free Bound. 9, No. 3, 411–430 (2007; Zbl 1130.74040)] where Griffith originally stated his criterion; third, solutions to our model converge, as the time-step tends to zero, to solutions of the correct continuous time model C. J. Larsen et al. [Math. Models Methods Appl. Sci. 20, No. 7, 1021–1048 (2010; Zbl 1425.74418)]. Furthermore, in implementing this model, we naturally recover several features, such as the elastic wave speed as an upper bound on crack speed, and crack branching for sufficiently rapid boundary displacements. We conclude by comparing our approach to so-called “phase-field” ones. In particular, we explain why phase-field approaches are good for approximating free boundaries, but not the free discontinuity sets that model fracture.

74R10 Brittle fracture
74-05 Experimental work for problems pertaining to mechanics of deformable solids
Full Text: DOI
[1] Ambrosio L, Braides A (1995) Energies in SBV and variational models in fracture mechanics. In: Homogenization and applications to material sciences (Nice, 1995), volume 9 of GAKUTO Internat. Ser Math Sci Appl Gakkōtosho, Tokyo, pp 1–22 · Zbl 0904.73045
[2] Ambrosio L, Tortorelli VM (1990) Approximation of functionals depending on jumps by elliptic functionals via \(\Gamma\)-convergence. Comm Pure Appl Math 43(8): 999–1036 · Zbl 0722.49020 · doi:10.1002/cpa.3160430805
[3] Ambrosio L, Tortorelli VM (1992) On the approximation of free discontinuity problems. Boll Un Mat Ital B (7) 6(1): 105–123 · Zbl 0776.49029
[4] Amor H, Marigo J-J, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8): 1209–1229 · Zbl 1426.74257 · doi:10.1016/j.jmps.2009.04.011
[5] Aranson IS, Kalatsky VA, Vinokur VM (2000) Continuum field description of crack propagation. Phys Rev Lett 85(1): 118–121 · doi:10.1103/PhysRevLett.85.118
[6] Balay S, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Smith BF, Zhang H (2008) PETSc users manual. ANL-95/11-Revision 3.0.0, Argonne National Laboratory
[7] Balay S, Buschelman K, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Smith BF, Zhang H (2009) PETSc Web page, http://www.mcs.anl.gov/petsc
[8] Bellettini G, Coscia A (1994) Discrete approximation of a free discontinuity problem. Numer Funct Anal Optim 15(3-4): 201–224 · Zbl 0806.49002 · doi:10.1080/01630569408816562
[9] Bourdin B (2007) Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Bound 9(3): 411–430 · Zbl 1130.74040 · doi:10.4171/IFB/171
[10] Bourdin B, Francfort GA, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4): 797–826 · Zbl 0995.74057 · doi:10.1016/S0022-5096(99)00028-9
[11] Bourdin B, Francfort GA, Marigo J-J (2008) The variational approach to fracture. Springer · Zbl 1176.74018
[12] Braides A (1998) Approximation of free-discontinuity problems. Number 1694 in Lecture Notes in Mathematics. Springer
[13] Braides A (2002) \(\Gamma\)-convergence for beginners, volume 22 of Oxford Lecture series in mathematics and its applications. Oxford University Press, Oxford
[14] Bronsard L, Kohn RV (1990) On the slowness of phase boundary motion in one space dimension. Comm. Pure Appl Math 43(8): 983–997 · Zbl 0761.35044 · doi:10.1002/cpa.3160430804
[15] Corson F, Adda-Bedia M, Henry H, Katzav E (2009) Thermal fracture as a framework for crack propagation law. Int J Fract 158: 1–14 · Zbl 1293.74372 · doi:10.1007/s10704-009-9361-4
[16] Dal Maso G (1993) An introduction to \(\Gamma\)-convergence. Birkhäuser, Boston · Zbl 0816.49001
[17] Del Piero G, Lancioni G, March R (2007) A variational model for fracture mechanics: numerical experiments. J Mech Phys Solids 55: 2513–2537 · Zbl 1166.74413 · doi:10.1016/j.jmps.2007.04.011
[18] Eastgate LO, Sethna JP, Rauscher M, Cretegny T, Chen C-S, Myers CR (2002) Fracture in mode I using a conserved phase-field model. Phys Rev 65(3): 036117
[19] Francfort GA, Larsen CJ (2003) Existence and convergence for quasi-static evolution in brittle fracture. Comm Pure Appl Math 56(10): 1465–1500 · Zbl 1068.74056 · doi:10.1002/cpa.3039
[20] Francfort GA, Marigo J-J (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8): 1319–1342 · Zbl 0966.74060 · doi:10.1016/S0022-5096(98)00034-9
[21] Freddi F, Royer Carfagni G (2010) Regularized variational theories of fracture: a unified approach. J Mech Phys Solids 58(8): 1154–1174 · Zbl 1244.74114 · doi:10.1016/j.jmps.2010.02.010
[22] Giacomini A (2005) Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calc Var Partial Differ Equ 22(2): 129–172 · Zbl 1068.35189 · doi:10.1007/s00526-004-0269-6
[23] Grandinetti, L (eds) (2007) TeraGrid: analysis of organization, system architecture, and middleware enabling new types of applications, advances in parallel computing. IOS Press, Amsterdam
[24] Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond 221: 163–198 · doi:10.1098/rsta.1921.0006
[25] Hakim V, Karma A (2009) Laws of crack motion and phase-field models of fracture. J Mech Phys Solids 57(2): 342– 368 · Zbl 1421.74089 · doi:10.1016/j.jmps.2008.10.012
[26] Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 87(4): 045501 · doi:10.1103/PhysRevLett.87.045501
[27] Karma A, Lobkovsky AE (2004) Unsteady crack motion and branching in a phase-field model of brittle fracture. Phys Rev Lett 92(24): 245510 · doi:10.1103/PhysRevLett.92.245510
[28] Keyes DE, Reynolds DR, Woodward CS (2006) Implicit solvers for large-scale nonlinear problems. J Phys Conf Ser 46: 433–442 · doi:10.1088/1742-6596/46/1/060
[29] Lancioni G, Royer-Carfagni G (2009) The variational approach to fracture: a practical application to the french Panthéon. J Elast 95(1–2): 1–30 · Zbl 1166.74029 · doi:10.1007/s10659-009-9189-1
[30] Larsen CJ, Ortner C, Süli E (2010) Existence of solutions to a regularized model of dynamic fracture. Math Models Methods Appl Sci 20: 1021–1048 · Zbl 1425.74418 · doi:10.1142/S0218202510004520
[31] Larsen CJ (2010) Models for dynamic fracture based on Griffith’s criterion. In: Hackl K (eds) IUTAM symposium on variational concepts with applications to the mechanics of materials. Springer, Berlin, pp 131–140
[32] Lawn B (1993) Fracture of brittle solids. 2. Cambridge University Press, Cambridge
[33] Marconi VI, Jagla EA (2005) Diffuse interface approach to brittle fracture. Phys Rev 71(3): 036110
[34] Modica L, Mortola S (1977) Il limite nella \(\Gamma\)–convergenza di una famiglia di funzionali ellittici. Boll Un Mat Ital A (5) 14(3): 526–529
[35] Modica L, Mortola S (1977) Un esempio di \(\Gamma\)–convergenza. Boll Un Mat Ital B (5) 14(1): 285–299
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.