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.

翻译：裂纹建模是工程与数学领域的重要课题。由于裂纹扩展表现为自由边界值问题（裂纹几何形状未知且需通过求解确定），常采用基于相场模型的尖锐界面近似方法。针对率无关框架，此类模型由能量泛函的单向梯度流定义。鉴于能量泛函的非凸性，位移场和相场变量等演变量可能呈现时间不连续性，导致所谓"脆性裂纹扩展"。因此需谨慎选择解的概念以预测物理合理的间断性，其中平衡黏性解（BV解）能避免跨越能垒的物理能量轨迹跳跃。本文提出一种收敛于BV解的率无关断裂自适应时间有限元相场模型，其动机源于约束裂纹尖端伪速度。通过增广拉格朗日方法求解带约束的最小化问题。数值算例展示了模型的预测能力，并验证了最终算法的效率与鲁棒性。