We present a fourth-order phase field model for fracture behavior simulations of hyperelastic material undergoing finite deformation. Governing equations of the fourth-order phase field model consist of the biharmonic operator of the phase field, which requires the second-order derivatives of shape function. Therefore, a 5 × 5 Jacobian matrix of isoparametric transformation is constructed. Neo-Hooken model and Hencky model are adopted as the material constitutive models. The spectral decomposition of stored strain energy is used to distinguish the contributions of tension and compression, and the corresponding stress tensor and constitutive tensors are derived, and subsequently, the numerical framework of modeling fracture with the fourth-order phase field model is implemented in details. Several typical numerical examples are conducted to demonstrate the robustness and effectiveness of the fourth-order phase field model in simulating the fracture phenomenon of rubber-like materials.

中文翻译：

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用于模拟超弹性材料断裂的具有谱分解的四阶相场模型

我们提出了一个四阶相场模型，用于模拟经历有限变形的超弹性材料的断裂行为。四阶相场模型的控制方程由相场的双调和算子组成，它需要形状函数的二阶导数。因此，构造了一个 5 × 5 的等参变换雅可比矩阵。材料本构模型采用 Neo-Hooken 模型和 Hencky 模型。利用存储应变能谱分解区分拉伸和压缩的贡献，推导出相应的应力张量和本构张量，进而详细实现了用四阶相场模型模拟断裂的数值框架.