上三角算子矩阵的本征函数展开法在应力形式的二维弹性问题中的应用

Eigenfunction Expansion Method of Upper Triangular Operator Matrix and Application to Two-Dimensional Elasticity Problems Based on Stress Formulation

深入研究了求解基于应力形式的二维弹性问题的本征函数展开法.根据已有的研究结果,将基于应力形式的二维弹性问题的基本偏微分方程组等价地转化为上三角微分系统,并导出了相应的上三角算子矩阵.通过深入研究,分别获得了该算子矩阵的两个对角块算子更为简洁的正交本征函数系,并证明了它们在相应空间中的完备性,进而应用本征函数展开法给出了该二维弹性问题的更为简洁实用的一般解.此外,对该二维弹性问题,还指出了什么样的边界条件可以应用此方法求解.最后应用具体的算例验证了所得结论的合理性.

The eigenfunction expansion method to solve two-dimensional （2D） elasticity prob- lems based on stress formulation was studied. The fundamental system of partial differential e- quations of the 2D problems was rewritten as an upper triangular differential system based on the known results, and then the associated upper triangular operator matrix was obtained. By further researching, the two simpler complete orthogonal systems of eigenftmctions in some space were obtained, which belong to the two block operators arising in the operator matrix. Then a more simple and convenient general solution for the 2D problem was given by the eigen- function expansion method. Furthermore, it was indicated what boundary conditions for the 2D problem can be solved by this method. Finally, the validity of the obtained results was verified by a specific example.