The fracture of flexoelectric materials involves strain gradients, which pose challenges for theoretical and numerical analysis. The phase-field model (PFM) is highly effective for simulating crack propagation. However, PFM within the finite element method (FEM) framework faces certain challenges in simulating the fracture behavior of flexoelectric materials since the conventional FEM can only provide C0 continuity. In this study, an adaptive PFM within multi-patch isogeometric analysis using polynomial splines over hierarchical T-meshes (PHT-splines) is proposed to simulate electromechanical fracture in flexoelectric materials. The PHT-splines functions feature higher-order continuity and can effectively discretize the strain gradient. All computational models are accurately modeled using multiple PHT-splines patches. The continuity of field variables such as displacement, electric potential, and phase field at the coupling edge is ensured using Nitsche’s method. To effectively compute the crack-driving force, the generalized Miehe decomposition method is employed. To alleviate the computational burden, a mesh refinement adaptive scheme based on user-defined thresholds for the phase field is used. The proposed method’s accuracy, reliability, and robustness are demonstrated using several fracture simulations.

中文翻译：

---

使用多面片等几何分析对柔性电材料中机电断裂进行自适应相场建模

柔性电材料的断裂涉及应变梯度，这给理论和数值分析带来了挑战。相场模型 （PFM） 对于模拟裂纹扩展非常有效。然而，由于传统的有限元法只能提供 C0 连续性，因此有限元方法 （FEM） 框架内的 PFM 在模拟柔性电材料的断裂行为方面面临一定的挑战。在本研究中，提出了一种在多面片等几何分析中使用分层 T 网格上的多项式样条（PHT 样条）的自适应 PFM 来模拟柔性电材料中的机电断裂。PHT 样条函数具有高阶连续性，可以有效地离散应变梯度。所有计算模型都使用多个 PHT 样条补丁进行精确建模。使用 Nitsche 方法确保了耦合边缘的位移、电势和相场等场变量的连续性。为了有效地计算裂纹驱动力，采用了广义的 Miehe 分解方法。为了减轻计算负担，使用了基于用户定义的相场阈值的网格细化自适应方案。通过多次裂缝模拟证明了所提出的方法的准确性、可靠性和鲁棒性。