This paper investigates the mathematical well-posedness of the variational model of quasi-static growth for a brittle crack proposed by Francfort and Marigo in [15]. The starting point is a time-discretized version of that evolution which results in a sequence of minimization problems of Mumford-Shah-type functionals. The natural weak setting is that of special functions of bounded variation, and the main difficulty in showing existence of the time-continuous, quasi-static growth is to pass to the limit as the time discretization step tends to 0. This is performed with the help of a jump transfer theorem that permits, under weak convergence assumptions for a sequence {u n } of SBV functions to its BV limit u, the transference of the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence {u n }. In particular, it is shown that the notion of minimizer of a Mumford-Shah-type functional for its own jump set is stable under weak convergence assumptions. Furthermore, our analysis justifies numerical methods used for computing the time-continuous, quasi-static evolution.

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.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.

Existence and convergence results are proved for a regularized model of dynamic brittle fracture based on the AmbrosioÀTortorelli approximation. We show that the sequence of solutions to the time-discrete elastodynamics, proposed by Bourdin, Larsen & Richardson as a semidiscrete numerical model for dynamic fracture, converges, as the time-step approaches zero, to a solution of the natural time-continuous elastodynamics model, and that this solution satis¯es an energy balance. We emphasize that these models do not specify crack paths a priori, but predict them, including such complicated behavior as kinking, crack branching, and so forth, in any spatial dimension.

We introduce models for static and quasi-static damage in elastic materials, based on a strain threshold, and then investigate the relationship between these threshold models and the energy-based models introduced in Francfort and Marigo (Eur J Mech A Solids 12:149-189, 1993) and Francfort and Garroni (Ration Mech Anal 182(1):125-152, 2006). A somewhat surprising result is that, while classical solutions for the energy models are also threshold solutions, this is shown not to be the case for nonclassical solutions, that is, solutions with microstructure. A new and arguably more physical definition of solutions with microstructure for the energy-based model is then given, in which the energy minimality property is satisfied by sequences of sets that generate the effective elastic tensors, rather than by the tensors themselves. We prove existence for this energy-based problem, and show that these solutions are also threshold solutions. A by-product of this analysis is that all local minimizers, in both the classical setting and for the new microstructure definition, are also global minimizers

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