This paper presents a two-level nesting smoothed extended mesh-free method (S-XMM-N) for the static and dynamic fracture mechanics analysis in orthotropic materials. To capture the jump of displacement fields across the crack surface, as well as singularities of stress fields in the vicinity of the crack tip, both Heaviside function and asymptotic solutions from the orthotropic fracture modeling are incorporated into the radial point interpolation shape functions associated with the partition of unity approach. To improve the accuracy and accelerate the computation of the extended mesh-free method, the two-level nesting smoothing integration scheme based on the first and second-level triangular smoothing sub-domains is developed to perform the stiffness integration. Compared to the extended mesh-free method using the conventional Gaussian quadrature, the presented S-XMM-N uses even less integration sampling points. At the same time, the complex and time-consuming derivative calculation of extended shape functions is completely avoided. This leads to the dramatically improvement of efficiency. Moreover, the presented method gives very exactly numerical solutions by optimally combining the contributions from the two-level nesting smoothing sub-domains based on the Richardson extrapolation method. We use the interaction integral method in conjunction with the asymptotic near crack tip fields of orthotropic materials to extract the static and dynamic stress intensity factors. The numerical solutions obtained from the S-XMM-N are further compared with reference solutions from the literature to verify the accuracy of the proposed method.

本文提出了一种两级嵌套平滑扩展无网格方法（S-XMM-N），用于正交各向异性材料的静态和动态断裂力学分析。为了捕捉裂纹表面位移场的跳跃，以及裂纹尖端附近应力场的奇异性，将来自正交各向异性断裂建模的 Heaviside 函数和渐近解结合到与统一方法的划分。为提高扩展无网格方法的精度和加速计算，开发了基于一级和二级三角平滑子域的二级嵌套平滑积分方案进行刚度积分。与使用传统高斯正交的扩展无网格方法相比，所提出的 S-XMM-N 使用的积分采样点更少。同时，完全避免了扩展形函数复杂且耗时的导数计算。这导致效率的显着提高。此外，所提出的方法通过优化组合基于理查森外推法的两级嵌套平滑子域的贡献，给出了非常精确的数值解。我们使用相互作用积分法结合正交各向异性材料的渐近近裂纹尖端场来提取静态和动态应力强度因子。