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.

中文翻译：

---

适用于速率无关断裂力学的时间自适应有限元相场模型


裂缝建模是工程和数学中的一个重要主题。由于裂纹扩展的特点是自由边值问题（裂纹的几何形状事先未知，而是解的一部分），因此通常考虑基于相场模型的潜在锐界面问题的近似。这些模型着眼于与速率无关的设置，由能量泛函的单向梯度流定义。由于该能量泛函是非凸的，因此位移场和相场变量等变量的演化可能在时间上不连续，从而导致所谓的残酷裂纹扩展。因此，必须仔细选择解决方案概念，以便预测物理上合理的不连续性。其中一个概念是平衡粘度解决方案（BV 解决方案）。这个概念预测了物理上合理的能量轨迹，不会跨越能量障碍。本文讨论了与速率无关的断裂的时间自适应有限元相场模型，该模型收敛于 BV 解。该模型的动机是约束裂纹尖端的伪速度。由此产生的约束最小化问题通过增强拉格朗日方法解决。数值例子凸显了模型的预测能力，并进一步显示了最终算法的效率和鲁棒性。