We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires $$C^1$$ C 1 -continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized- $$\alpha $$ α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching.

中文翻译：

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基于LR NURBS的脆性壳动态断裂自适应时空相场公式

我们提出了一种用于脆壳动态断裂的自适应时空相场公式。它们的变形由使用曲线表面描述的 Kirchhoff-Love 薄壳理论表征。所有运动学对象都在壳的中平面上定义。相场的演化方程由基于格里菲斯脆性断裂理论的能量函数的最小化确定。膜和弯曲对断裂过程的贡献分别建模，并为后者建立了厚度积分。耦合系统由两个非线性四阶偏微分方程组成，所有量都定义在一个演化的二维流形上。由于弱形式需要 $$C^1$$ C 1 -连续性，所以使用等几何形状函数。使用局部可精炼 (LR) NURBS 基于相场自适应地细化网格。时间基于使用自适应时间步长的广义 $$\alpha $$ α 方法进行离散化，离散化的耦合系统使用整体 Newton-Raphson 方案求解。表面变形和裂纹演化之间的相互作用通过几个显示动态裂纹扩展和分支的数值例子来证明。