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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.

MSC:
74R10 Brittle fracture
74-05 Experimental work for problems pertaining to mechanics of deformable solids
Software:
PETSc
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