The modeling of cracks is an important topic - both in engineering as well as in mathematics. Since crack propagation is characterized by a free boundary value problem (the geometry of the crack is not known beforehand, but part of the solution), approximations of the underlying sharp-interface problem based on phase-field models are often considered. Focusing on a rate-independent setting, these models are defined by a unidirectional gradient-flow of an energy functional. Since this energy functional is non-convex, the evolution of the variables such as the displacement field and the phase-field variable might be discontinuous in time leading to so-called brutal crack growth. For this reason, solution concepts have to be carefully chosen in order to predict discontinuities that are physically reasonable. One such concept is that of Balanced Viscosity solutions (BV solutions). This concept predicts physically sound energy trajectories that do not jump across energy barriers. The paper deals with a time-adaptive finite element phase-field model for rate-independent fracture which converges to BV solutions. The model is motivated by constraining the pseudo-velocity of the crack tip. The resulting constrained minimization problem is solved by the augmented Lagrangian method. Numerical examples highlight the predictive capabilities of the model and furthermore show the efficiency and the robustness of the final algorithm.

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