应用边界积分法求圆形夹杂问题的解析解

边界元方法作为一种数值方法, 在各种科学工程问题中得到了广泛的应用.本文参考了边界元法的求解思路, 从Somigliana等式出发, 利用格林函数性质,得到了一种边界积分法, 使之可以用来寻求弹性问题的解析解.此边界积分法也可以从Betti互易定理得到. 应用此新方法, 求解了圆形夹杂问题.首先设定夹杂与基体之间完美连接, 将界面处的位移与应力按照傅里叶级数展开,根据问题的对称性与三角函数的正交性来简化假设, 减少待定系数的个数.其次选择合适的试函数(试函数满足位移单值条件以及无体力的线弹性力学问题的控制方程),应用边界积分法, 求得界面处的位移与应力的值. 然后再求解域内位移与应力.得到了问题的精确解析解, 当夹杂弹性模量为零或趋向于无穷大时,退化为圆孔或刚性夹杂问题的解析解. 求解过程表明,若问题的求解区域包含无穷远处时, 所取的试函数应满足无穷远处的边界条件.若求解区域包含坐标原点, 试函数在原点处位移与应力应是有限的.结果表明了此方法的有效性. 

EXACT SOLUTION OF CIRCULAR INCLUSION PROBLEMS BY A BOUNDARY INTEGRAL METHOD 1)

As an excellent numerical method, boundary element method (BEM) has been widely applied in various scientific and engineering problems. In this paper, a new boundary integral method is obtained based on Somigliana's equation and the properties of Green's function by referring to the idea of boundary element method. It can be used to find the analytic solution of linear elastic problems. The boundary integral method can also be obtained from Betti's reciprocity theorem. By using this new method, the classical problem of elastic circular inclusion under a uniform tensile field at infinity is solved. Firstly, the perfect bonding between inclusion and matrix is set up, and the displacement and stress at interface are expanded according to Fourier series. According to the symmetry of the problem and the orthogonality of trigonometric function, the hypothesis is simplified and the number of undetermined coefficients is reduced. Secondly, the appropriate trial functions are selected (these trial functions satisfy the condition of displacement single value and the control equation of linear elasticity without body force). And the boundary integral method is used to calculate the displacement and stress at the interface. Then the displacement and stress in the domain are solved using similar tricks. The exact analytical solution of the problem is obtained, which is exactly the same with the results in literatures. When the elastic modulus of the inclusion is zero or tends to infinity, it degenerates to the analytical solution of the problem of circular hole or rigid inclusion. The solution process shows that if the problem has boundary conditions at infinity, trial functions should meet the boundary condition at infinity. If the domain of the problem contains the coordinate origin, the displacement and stress of trial functions at the origin should be limited. The results show that the method is effective.