二維彈性平面問題中任意邊界條件下應力分布的封閉解

Abstract

應用辛方法研究了正交各向異性二維平面（x，z）彈性問題，在任意邊界和不考慮梁假設條件下的解析應力分布解．辛方法通過將位移和應力作為對偶量推導得到一組辛的偏微分方程組，並且應用變量分離法對方程組進行了求解．同動力學中的問題比較，將彈性問題中的x軸模擬成時間軸，這樣z軸成為唯一一個獨立的坐標軸．問題中的Hamilton矩陣的指數展開具有辛的特征．在齊次問題求解中，通過邊界條件和邊界上的積分求得級數中的未知數．齊次解中包括減階的零特征值的特征向量（零本征向量）和完好的非零本征值的特征向量（非零本征向量）．零本征值的Jordan鏈給出了經典的Saint Venant解，反映了平均的整體行為像剛體位移、剛體旋轉和彎曲等．另外，非零本征向量反映的是指數衰減的局部解，它們通常在Saint Venant原理下被忽略．文中給出了完整的算例，並且和已有結果進行了對比．

A Hamiltonian method was applied to study analytically the stress distributions of orthotropic two-dimensional elasticity in (x, z) plane for arbitrary boundary conditions without beam assumptions. It is a method of separable variables for partial differential equations using displacements and their conjugate stresses as unknowns. Since coordinates (x, z) cannot be easily separated, an alternative symplectic expansion was used. Similar to the Hamiltonian formulation in classical dynamics, the x coordinate as time variable so that z becomes the only independent coordinate in the Hamiltonian matrix differential operator. The exponential of the Hamiltonian matrix is symplectic. There are homogenous solutions with constants to be determined by the boundary conditions and particular integrals satisfying the loading conditions. The homogenous solutions consist of the eigen-solutions of the derogatory zero eigenvalues (zero eigen-solutions) and that of the wellbehaved nonzero eigenvalues (nonzero eigen-solutions). The Jordan chains at zero eigenvalues give the classical Saint Venant solutions associated with averaged global behaviors such as rigid-body translation, rigid-body rotation or bending. On the other hand, the nonzero eigen-solutions describe the exponentially decaying localized solutions usually ignored by Saint-Venant's principle. Completed numerical examples were newly given to compare with established results.