二维弹性平面问题中任意边界条件下应力分布的封闭解

Closed Form Stress Distribution in 2D Elasticity for all Boundary Conditions

应用辛方法研究了正交各向异性二维平面(x,z)弹性问题,在任意边界和不考虑梁假设条件下的解析应力分布解.辛方法通过将位移和应力作为对偶量推导得到一组辛的偏微分方程组,并且应用变量分离法对方程组进行了求解.同动力学中的问题比较,将弹性问题中的x轴模拟成时间轴,这样z轴成为唯一一个独立的坐标轴.问题中的Hamilton矩阵的指数展开具有辛的特征.在齐次问题求解中,通过边界条件和边界上的积分求得级数中的未知数.齐次解中包括减阶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链给出了经典的Saint Venant解,反映了平均的整体行为像刚体位移、刚体旋转和弯曲等.另外,非零本征向量反映的是指数衰减的局部解,它们通常在Saint Venant原理下被忽略.文中给出了完整的算例,并且和已有结果进行了对比.

A Hamiltonian method was applied to study analytically the stress distxibutions 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 marx 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.