裂纹尖端解析解与周边数值解联合求解应力强度因子

由于裂纹尖端位移和应力分布的复杂性，采用常规数值方法（如有限元法）的插值方式不易获得快速收敛的应力强度因子计算值。基于数值流形方法，提出将裂纹尖端Williams解析解与其周边高阶多项式级数的数值解联合应用以求解应力强度因子的新方法：在裂纹尖端所在网格的结点上采用Williams位移解析级数，并用结点自由度强制约束方式得到裂纹尖端区域的解析级数；在与之相邻的周边网格内将解析级数与多项式级数用形函数连接；给出应变矩阵和刚度矩阵的具体表达式及积分方式；利用数值流形方法的网格与材料边界分离的特性以及不连续覆盖技术，使裂纹可以在网格内穿过，给材料边界（包括裂纹边界）附近的网格划分带来很大的方便；通过典型算例验证了方法的有效性。考虑到Williams级数是对裂纹尖端位移场的最佳逼近，这种新方法相比扩展有限元等其他新方法而言将有更快的收敛性。

Compute Stress Intensity Factors via Combining Analytical Solutions around Crack Tips with Surrounding Numerical Solutions

Due to the complex distribution of the displacements and stresses around the crack tip, it is not easy to obtain the Stress Intensity Factor (SIF) with a rapid convergence when using conventional interpolation approaches of numerical methods such as Finite Element Method (FEM). On the basis of Numerical Manifold Method (NMM), a novel method is presented to compute the SIFs via combining analytical solutions with numerical solutions. The Williams expansion is used as the analytical solution, which is formed by applying the constraints of nodal freedoms in the mesh containing the crack tip. High-order polynomial functions are used as the numerical solutions which are connected with the analytical solution via shape functions in the surrounding meshes. Meanwhile, the meshes in NMM need not conform to the physical boundaries including the crack edges, and discontinuous covers are used to allow the cracks arbitrarily align within the meshes, providing the convenience of mesh generation. Numerical example shows the validity of the method. Considering that the Williams expansion is the best approximation for the displacement field around the crack tip, the method has a more rapid convergence than other new methods such as extended Finite Element Method (XFEM).