Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation
Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation
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DOI：
10.1007/s10704-012-9753-8
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发表时间：
2012-11-01
影响因子：
2.5
通讯作者：
Miehe, Christian
中科院分区：
文献类型：
--
作者：
Hofacker, Martina;Miehe, Christian
基于尖锐裂纹不连续性的断裂导致的固体失效机制建模在复杂裂纹拓扑（包括分支）的情况下会受到影响。这一缺点可以通过 Miehe 等人提出的基于引入裂纹相场的扩散裂纹建模来克服。 （ComputMethods Appl Mech Eng 19：2765-2778，2010a；Int J Numer Meth Eng 83：1273-1311，2010b），Hofacker 和 Miehe（Int J Numer Meth Eng，2012）。在这项工作中，我们总结了准静态和动态条件下热力学一致、基于变分的扩散裂纹扩展模型的基本要素。结果表明，所有耦合场方程，特别是动量平衡方程和裂纹相场的梯度型演化方程，都遵循混合速率型变分原理的欧拉方程，其中裂缝驱动力作为混合场变量。这一原理使得所提出的公式极其紧凑，并为有限元实现提供了完美的基础。然后，我们引入一个局部历史场，其中包含在变形历史中获得的最大能量裂纹源。它驱动裂纹相场的演化。这允许构建极其鲁棒的算子分割方案，该方案在典型的时间步长中更新历史场、裂纹相场以及最终的位移场。我们通过代表性的数值例子展示了断裂相场公式的性能，这些例子显示了动态载荷下复杂裂纹模式的演变。
The modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations of complex crack topologies including branching. This drawback can be overcome by a diffusive crack modeling based on the introduction of a crack phase field as proposed in Miehe et al. (Comput Methods Appl Mech Eng 19:2765-2778, 2010a; Int J Numer Meth Eng 83:1273-1311, 2010b), Hofacker and Miehe (Int J Numer Meth Eng, 2012). In this work, we summarize basic ingredients of a thermodynamically consistent, variational-based model of diffusive crack propagation under quasi-static and dynamic conditions. It is shown that all coupled field equations, in particular the balance of momentum and the gradient-type evolution equation for the crack phase field, follow as the Euler equations of a mixed rate-type variational principle that includes the fracture driving force as the mixed field variable. This principle makes the proposed formulation extremely compact and provides a perfect basis for the finite element implementation. We then introduce a local history field that contains a maximum energetic crack source obtained in the deformation history. It drives the evolution of the crack phase field. This allows for the construction of an extremely robust operator split scheme that updates in a typical time step the history field, the crack phase field and finally the displacement field. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples, which show the evolution of complex crack patterns under dynamic loading.