Closed form stress distribution in 2D elasticity for all boundary conditions  

作　　者：梁以德 郑建军 

机构地区：[1]Department of Building and Construction,City University of Hong Kong

出　　处：《Applied Mathematics and Mechanics(English Edition)》2007年第12期1629-1642,共14页应用数学和力学（英文版）

基　　金：the Research Grant Council of Hong Kong(No.CERG1157/06)

摘　　要：This paper applies a Hamiltonian method 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） can not be easily separated, an alternative symplectic expansion is used. Similar to the Hamiltonian formulation in classical dynamics, we treat 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 well-behaved nonzero eigenvalues （nonzero eigen-solutions）. The Jordan chains at zerO eigenvalues give the classical Saint-Venant solutions associated with aver- aged 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 are newly given to compare with established results.This paper applies a Hamiltonian method 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） can not be easily separated, an alternative symplectic expansion is used. Similar to the Hamiltonian formulation in classical dynamics, we treat 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 well-behaved nonzero eigenvalues （nonzero eigen-solutions）. The Jordan chains at zerO eigenvalues give the classical Saint-Venant solutions associated with aver- aged 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 are newly given to compare with established results.

关 键 词：transversely isotropy EIGENFUNCTIONS symplectic expansion 

分 类 号：O343[理学—固体力学]