祁勇峰 苏海东 龚亚琦

摘 要：为保证在裂纹尖端具有最佳的逼近，裂纹尖端的解析解与其周边数值解联合应用的新型流形法可用来求解应力强度因子，但仅限于平面问题的Ⅰ型和Ⅱ型裂纹。沿用解析解与数值解联合应用的思路，以三维穿透直裂纹为研究对象，在裂纹尖端引入Williams解析解级数，应用解析覆盖与周边数值覆盖联合求解三维应力强度因子，相对于其他数值方法而言，计算精度高。推导相应的刚度矩阵和应变矩阵的表达式，通过典型算例验证了三维应力强度因子精确求解方法的有效性。

关键词：三维应力强度因子;数值流形方法;裂纹尖端Williams解析级数;解析覆盖;数值覆盖

Abstract： This paper proposes a new manifold method combined with the analytical solutions of the crack tip and its surrounding numerical solutions is used to solve the stress intensity factor to ensure the best approximation at the crack tip. But this method is limited to type I and type II cracks in the plane problem. This method follows the idea of combined application of analytical and numerical solutions. Taking the three-dimensional penetrating straight crack as the research object， the Williams analytical solution series is introduced at the crack tip， and the analytical covers and the surrounding numerical covers are combined to solve the 3D stress intensity factor. Compared with other numerical methods， this method has higher computational precision. Meanwhile， the corresponding formulas of the stiffness matrix and the strain matrix have been deduced，and typical numerical examples shows the validity of the method in solving the exact solution of the 3D stress intensity factor.

Keywords： 3D stress intensity factor; numerical manifold method; Williams series for crack tip; analytical covers; numerical covers

应力强度因子准则[1]是目前最常用的结构裂缝扩展准则之一，基于线弹性断裂力学的应力强度因子（Stress Intensity Factor，简称SIF）是用来表征裂纹尖端附近应力场和应变场强度，是控制裂纹尖端应力场和应变场强度的关键参数，在裂纹扩展分析中有着极其重要的地位。由于应力强度因子取决于外力的大小和分布、结构的几何条件以及裂缝的形状和位置，实际上只有少数问题存在解析解，对于复杂几何形状和加载条件的问题，只能通过数值方法来计算。目前裂缝扩展分析的主流数值计算方法有有限元法、扩展有限元法、数值流形法等。

有限元法[2-4]是目前分析裂缝等不连续问题的主要方法，为体现裂纹尖端（下简称裂尖）周围的应力集中和奇异性，往往需要在裂尖附近的复杂应力区布置高密度的单元网格，导致单元数目非常庞大;另外，在模拟裂缝扩展过程时，需不断重构网格，因此，有限元法对网格的要求和依赖性极大地影响了计算效率。……