A time-discrete model for dynamic fracture based on crack regularization

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 quasistatic settings Bourdin (Interfaces Free Bound 9(3): 411–430, 2007) where Griffith originally stated his criterion; third, solutions to our model converge, as B. Bourdin (B) Department of Mathematics and Center for Computation & Technology, Louisiana State University, Baton Rouge, LA 70803, USA e-mail: [email protected] C. J. Larsen Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280, USA e-mail: [email protected] C. L. Richardson Department of Mathematics, Worcester Polytechnic Institute, Worcester, USA Present Address: C. L. Richardson Center for Imaging Science, Clark 319B, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA e-mail: [email protected] 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.