计算应力强度因子的离散分离变量法

提出一种用于数值求解带有一条边界裂纹的多角形区域上的Navier's方程组边值问题的半离散方法。做一个适当的坐标变换后,将原边值问题化为半无限长条上的不连续系数问题。将其半离散化以后,等价于一个常系数常微分方程组的边值问题。进一步,用直接法来求解这个边值问题,便得到原问题的半离散近似解。值得指出的是,这个用分离变量形式给出的半离散近似解自然地具有原问题的奇性。数值例子显示,用该方法可以很方便地计算出在裂纹顶端的应力强度因子的近似值。

THE DISCRETE METHOD OF SEPARATION OF VARIABLES FOR COMPUTATION OF STRESS INTENSITY FACTORS

A semi-discrete method is proposed for finding the numerical solution of the boundary value problem(BVP) of Navier's equations on the polygon with a single edge-crack. After a suitable transformation of the coordinates, the BVP is reduced to a discontinuous coefficients problem on a semi-infinite strip, and the semi-discrete approximation of the problem is obtained, equivalent to a BVP of a system of O.D.E's with constant coefficients. Furthermore, the semi-discrete approximation of the BVP can be acquired by a direct method. It is worthwhile to note that, the semi-discrete approximation in the form of separable variables naturally possesses the singularity of the given problem. Finally numerical examples show the effectiveness of the method to calculate the approximation of the stress intensity factors.