Abstract Phase-field models for fracture have gained exceptional popularity in the last couple of decades and have been extended into areas well beyond brittle quasi-static fracture propagation to ductile, dynamic, or hydraulic fracturing. Despite the significant theoretical advancements in these more complex physical settings, little attention has been paid to the quantification of crack opening displacement to date as most applications do not explicitly require the crack opening displacement for the morphological evolution of cracks. However, one of the exemptions would be hydraulic fracturing where the crack propagation is driven by the fluid pressure which strongly depends on the crack opening displacement. In this study, we look into two known approaches, a line integral and a level-set method, for crack opening computation mainly from an implementation point of view. Firstly, we derive an approximation of a discontinuous function field in the variational phase-field setting which is then applied to obtain the crack opening (displacement jump) and verify the approximation against a closed solution. We then propose a “certain distance from the crack” required for the level-set function using a one-dimensional analysis. Finally, we compare these approaches under several different conditions such as crack alignment to the mesh or under loading (asymmetric displacement field) and investigate the convergence with respect to the mesh size.

中文翻译：

---

断裂变相场模型中裂纹张开的计算

摘要 裂缝的相场模型在过去几十年中获得了异常的普及，并且已经扩展到远远超出脆性准静态裂缝扩展到韧性、动态或水力压裂的领域。尽管在这些更复杂的物理环境中取得了重大的理论进步，但迄今为止很少有人关注裂纹张开位移的量化，因为大多数应用并没有明确要求裂纹张开位移用于裂纹的形态演变。然而，其中一种例外情况是水力压裂，其中裂纹扩展由流体压力驱动，流体压力在很大程度上取决于裂纹张开位移。在这项研究中，我们研究了两种已知的方法，线积分和水平集方法，主要是从实现的角度来计算开裂。首先，我们在变分相场设置中推导出不连续函数场的近似值，然后将其应用于获得裂纹开口（位移跳跃）并针对封闭解验证近似值。然后，我们使用一维分析提出了水平集函数所需的“距裂缝的特定距离”。最后，我们比较了几种不同条件下的这些方法，例如裂纹与网格对齐或在加载（非对称位移场）下，并研究关于网格尺寸的收敛性。我们在变分相场设置中推导出不连续函数场的近似值，然后将其应用于获得裂纹开口（位移跳跃）并针对封闭解验证近似值。然后，我们使用一维分析提出了水平集函数所需的“距裂缝的特定距离”。最后，我们比较了几种不同条件下的这些方法，例如裂纹与网格对齐或在加载（非对称位移场）下，并研究关于网格尺寸的收敛性。我们在变分相场设置中推导出不连续函数场的近似值，然后将其应用于获得裂纹开口（位移跳跃）并针对封闭解验证近似值。然后，我们使用一维分析提出了水平集函数所需的“距裂缝的特定距离”。最后，我们比较了几种不同条件下的这些方法，例如裂纹与网格对齐或在加载（非对称位移场）下，并研究关于网格尺寸的收敛性。