双材料界面裂纹的改进广义有限差分法

采用数值方法进行断裂力学分析时，裂纹尖端奇异区域处理的好坏直接关系到最终断裂力学参数的求解精度。与传统均匀介质不同，复合材料界面裂纹渐近位移和应力场表现出剧烈的振荡特性，许多用于表征经典的平方根和负平方根物理场渐近性的传统方法也因此失效。论文提出了一种改进的广义有限差分法，该方法基于多元函数泰勒级数展开和移动最小二乘法的思想，将节点变量的各阶导数由相邻点集函数的加权线性累加来近似，具有无网格、无数值积分、数据准备简单、稀疏矩阵快速求解等优点。为提高该方法求解断裂力学问题的计算精度和数值稳定性，论文引入了裂尖奇异区域局部点簇的自动创建技术和一种基于局部点簇几何尺寸的矩阵正则化算法。数值算例表明，所提算法稳定，效率高，在不增加计算量的前提下，显著提高了裂尖近场力学参量和断裂力学参数的求解精度和数值稳定性。

BIMATERIAL INTERFACE CRACK ANALYSIS BY USING AN IMPROVED GENERALIZED FINITE DIFFERENCE METHOD

Cracks on the interface of the bimaterial could be present either as a result of the manufacturing defects, and/or mismatch between the different material properties of the bimaterial. The asymptotic near-tip field for bimaterial interfacial cracks presents an oscillatory behavior which is very different from that for cracks in homogeneous materials. Due to the oscillatory behavior related to the complex eigenvalues, modeling such interface cracks by the conventional solution procedures designed for homogeneous materials is inadequate, and may not lead to accurate solutions by using the well-established and widely applied finite element method (FEM) or boundary element method (BEM), even when a very fine mesh near the crack-tip is employed. In the present paper, we document the attempt to apply the generalized finite difference method (GFDM) for fracture analysis of bimaterials containing interfacial cracks. The main idea of the method is based on the theories of local Taylor series expansion and moving-least square approximation. Since the method is meshless and no element connectivity is needed, it can be viewed as a competitive alternative for bimaterial interface crack analysis. In our calculations, a multi-domain technique is employed to handle the non-homogeneity of the dissimilar materials, and the displacement extrapolation method (DEM) is introduced to compute the complex stress intensity factors (SIFs) for cracked dissimilar materials. An improved GFDM formulation is proposed to further improve the accuracy and stability of the GFDM for fracture mechanics analysis. Two benchmark numerical examples are well studied to demonstrate the accuracy and stability of the present method for interface crack analysis of composite bimaterials. In the following-up works, other interesting and important problems, such as thermal effects and dynamic effects, and crack propagation and arrest should be considered. The present work provides an efficient alternative and the corresponding results form a solid basis for these interesting research topics.