圆柱型正交各向异性弹性楔的佯谬解

SOLUTION OF PARADOX IN CYLINDRICAL ORTHOGONAL ANISOTROPIC ELASTIC WEDGE

圆柱型正交各向异性弹性楔体顶端受有集中力偶的经典解 ,当顶角满足一定关系时 ,其应力成为无穷大 ,这是个佯谬 .该文在哈密顿体系下将该问题进行重新求解 ,即利用极坐标各向异性弹性力学哈密顿体系 ,在原变量和其对偶变量组成的辛几何空间求解特殊本征值的约当型本征解 ,从而直接给出该佯谬问题的解析解 .结果再次表明经典力学中的弹性楔佯谬解对应的是哈密顿体系下辛几何的约当型解 .

Classical solution of a cylindrical orthogonal anisotropic elastic wedge subjected to a concentrated couple at the vertex become infinite when the vertex angle satisfies certain definite relationships, this is a paradox. In this paper, the paradox is restudied under Hamiltonian system. The polar coordinate Hamiltonian system of anisotropic elasticity is used to solve the Jordan canonical form eigen solution for the special eigenvalue in symplectic space which consists of the original variables and their dual variables. In this way, solution of the paradox can be obtained directly. It shows further that solution of the special paradox in classical elasticity is just Jordan canonical form solutions in symplectic space under Hamiltonian system.