Existence of Solutions to a Regularized Model of Dynamic Fracture

Abstract

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.

Numerical investigation of dynamic brittle fracture via gradient damage models

“…10 we plot the crack length evolution as well as the conventional energy release rate G α t both for the dynamic model and the first-order quasi-static model. It is recalled that the static G α t can be simply obtained by settingu t andü t to zero in (21). We observe that these two solutions coincide, and both present a time-continuous crack evolution (initiation and propagation) conforming to the asymptotic Griffith's law (23).…”

“…The irreversibility condition will be automatically enforced during the bound-constrained minimization process. The time-discrete model which we describe below should converge to the continuous one when the time increment becomes small, see [21]. In particular, the energy balance condition (9) will be hence automatically satisfied.…”

“…When a relatively small time-step is used, it is expected that the damage increment α n+1 − α n is bounded and the staggered time-discrete problem will converge to the continuous one, cf. [21]. Introducing the displacement prediction at time t = t n+1…”

Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition

“…In the context of quasi-static evolution, such a principle is minimality, but minimality is not a useful principle when the displacement is following elastodynamics, instead of minimization. An exception to this is given by phase-field models, which were formulated and studied in [2,14]. These papers led to the maximal dissipation approach proposed in [13].…”

Existence for wave equations on domains with arbitrary growing cracks

“…We note that, based on the success of numerical methods for quasi-static fracture using the Ambrosio-Tortorelli approximation (see, e.g., [3]), a numerical algorithm was proposed in [5] for dynamic fracture, which was shown in [18] to converge, as the time step tends to zero, to a solution obeying the appropriate elastodynamics, the total energy (stored elastic, kinetic, and the surface energy of the crack set) is conserved, and the field modeling the crack set satisfies a minimality analogous to that in the quasi-static setting. For these phasefield models, this minimality provides the principle iii), requiring the "crack" to run so as to maintain minimality.…”