摘要:
从Hellinger-Reissner变分原理出发,通过引入适当的变换可以将两种材料组成的弹性楔问题导入极坐标哈密顿体系,从而可以在由原变量和其对偶变量组成的辛几何空间,利用分离变量法和辛本征向量展开法求解该问题的解。在极坐标哈密顿体系下的所有辛本征值中,本征值-1是一个特殊的本征值。一般情况下本征值-1为单本征值,求解其对应的基本本征函数向量就直接给出了顶端受有集中力偶的经典弹性力学解。但当两种材料的顶角和弹性模量满足特殊关系时,本征值-1成为重本征值,同时经典弹性力学解的应力分量变成无穷大,即出现佯谬。此时重本征值-1存在约当型本征解,通过对该特殊约当型本征解的直接求解就给出了两种材料组成的弹性楔顶端受有集中力偶的佯谬问题的解。结果进一步表明经典弹性力学中弹性楔的佯谬解对应的就是极坐标哈密顿体系的约当型解。
关键词:
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佯谬 /
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辛几何 /
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约当型 /
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弹性楔
Abstract:
According to the Hellinger Reissner variational principle and introducing proper transformation of variables,the problem on elastic wedge dissimilar materials can be led to Hamiltonian system,so the solution of the problem can be got by employing the separation of variables method and symplectic eigenfunction expansion under symplectic space,which consists of original variables and their dual variables.The eigenvalue -1 is a special one of all symplectic eigenvalue for Hamiltonian system in polar coordinate.In general,the eigenvalue 1 is a single eigenvalue,and the classical solution ofan elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is got directly by solving the eigenfunction vector for eigenvalue 1.But the eigenvalue 1 becomes a double eigenvalue when the vertex angles and modulus of the materials satisfy certain definite relationships and the classical solution for the stress distribution becomes infinite at this moment,that is,the paradox should occur.Here the Jordan form eigenfunction vector for eigenvalue 1 exists,and solution of the paradox on elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is obtained directly by solving this special Jordan form eigenfunction.The result shows again that the solutions of the special paradox on elastic wedge in the classical theory of elasticity are just Jordan form solutions in symplectic space under Hamiltonian system.
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