Abstract
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 Ambrosio and Tortorelli (Comm. Pure Appl. Math., 43(8): 999–1036 (1990); Boll. Un. Mat. Ital. B (7), 6(1):105–123, 1992); second, our condition for crack growth, based on Griffith’s criterion, matches that of quasi-static settings Bourdin (Interfaces Free Bound 9(3): 411–430, 2007) 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 Larsen (Math Models Methods Appl Sci 20:1021–1048, 2010). 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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
Ambrosio L, Tortorelli VM (1990) Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm Pure Appl Math 43(8): 999–1036
Ambrosio L, Tortorelli VM (1992) On the approximation of free discontinuity problems. Boll Un Mat Ital B (7) 6(1): 105–123
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
Aranson IS, Kalatsky VA, Vinokur VM (2000) Continuum field description of crack propagation. Phys Rev Lett 85(1): 118–121
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
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
Bellettini G, Coscia A (1994) Discrete approximation of a free discontinuity problem. Numer Funct Anal Optim 15(3-4): 201–224
Bourdin B (2007) Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Bound 9(3): 411–430
Bourdin B, Francfort GA, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4): 797–826
Bourdin B, Francfort GA, Marigo J-J (2008) The variational approach to fracture. Springer
Braides A (1998) Approximation of free-discontinuity problems. Number 1694 in Lecture Notes in Mathematics. Springer
Braides A (2002) Γ-convergence for beginners, volume 22 of Oxford Lecture series in mathematics and its applications. Oxford University Press, Oxford
Bronsard L, Kohn RV (1990) On the slowness of phase boundary motion in one space dimension. Comm. Pure Appl Math 43(8): 983–997
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
Dal Maso G (1993) An introduction to Γ-convergence. Birkhäuser, Boston
Del Piero G, Lancioni G, March R (2007) A variational model for fracture mechanics: numerical experiments. J Mech Phys Solids 55: 2513–2537
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
Francfort GA, Larsen CJ (2003) Existence and convergence for quasi-static evolution in brittle fracture. Comm Pure Appl Math 56(10): 1465–1500
Francfort GA, Marigo J-J (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8): 1319–1342
Freddi F, Royer Carfagni G (2010) Regularized variational theories of fracture: a unified approach. J Mech Phys Solids 58(8): 1154–1174
Giacomini A (2005) Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calc Var Partial Differ Equ 22(2): 129–172
Grandinetti, L (eds) (2007) TeraGrid: analysis of organization, system architecture, and middleware enabling new types of applications, advances in parallel computing. IOS Press, Amsterdam
Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond 221: 163–198
Hakim V, Karma A (2009) Laws of crack motion and phase-field models of fracture. J Mech Phys Solids 57(2): 342– 368
Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 87(4): 045501
Karma A, Lobkovsky AE (2004) Unsteady crack motion and branching in a phase-field model of brittle fracture. Phys Rev Lett 92(24): 245510
Keyes DE, Reynolds DR, Woodward CS (2006) Implicit solvers for large-scale nonlinear problems. J Phys Conf Ser 46: 433–442
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
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
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
Lawn B (1993) Fracture of brittle solids. 2. Cambridge University Press, Cambridge
Marconi VI, Jagla EA (2005) Diffuse interface approach to brittle fracture. Phys Rev 71(3): 036110
Modica L, Mortola S (1977) Il limite nella Γ–convergenza di una famiglia di funzionali ellittici. Boll Un Mat Ital A (5) 14(3): 526–529
Modica L, Mortola S (1977) Un esempio di Γ–convergenza. Boll Un Mat Ital B (5) 14(1): 285–299
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bourdin, B., Larsen, C.J. & Richardson, C.L. A time-discrete model for dynamic fracture based on crack regularization. Int J Fract 168, 133–143 (2011). https://doi.org/10.1007/s10704-010-9562-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-010-9562-x