This study aims to find a solution for crack propagation in 2D brittle elastic material using the local radial basis function collocation method. The staggered solution of the fourth-order phase field and mechanical model is structured with polyharmonic spline shape functions augmented with polynomials. Two benchmark tests are carried out to assess the performance of the method. First, a non-cracked square plate problem is solved under tensile loading to validate the implementation by comparing the numerical and analytical solutions. The analysis shows that the iterative process converges even with a large loading step, whereas the non-iterative process requires smaller steps for convergence to the analytical solution. In the second case, a single-edge cracked square plate subjected to tensile loading is solved, and the results show a good agreement with the reference solution. The effects of the incremental loading, length scale parameter, and mesh convergence for regular and scattered nodes are demonstrated. This study presents a pioneering attempt to solve the phase field crack propagation using a strong-form meshless method. The results underline the essential role of the represented method for an accurate and efficient solution to crack propagation. It also provides valuable insights for future research towards more sophisticated material models.

中文翻译：

---

采用强形式无网格法的脆性断裂四阶相场建模


本研究旨在使用局部径向基函数搭配方法寻找二维脆性弹性材料中裂纹扩展的解。四阶相场和力学模型的交错解是用多项式增强的多谐波样条形状函数构建的。执行了两个基准测试来评估该方法的性能。首先，在拉伸载荷下求解无裂纹方板问题，通过比较数值解和解析解来验证实现。分析表明，即使加载步骤较大，迭代过程也会收敛，而非迭代过程需要较小的步骤才能收敛到解析解。在第二种情况下，求解了受拉载荷作用的单边开裂方板，结果与参考解吻合较好。演示了增量载荷、长度比例参数和网格收敛对规则节点和分散节点的影响。本研究提出了使用强形式无网格方法解决相场裂纹扩展的开创性尝试。结果强调了所表示方法在准确有效地求解裂纹扩展方面的重要作用。它还为未来对更复杂材料模型的研究提供了有价值的见解。