Research on an Orthogonal Relationship for Orthotropic Elasticity
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摘要: 新的正交关系被推广到正交各向异性三维弹性力学．将弹性力学新正交关系中构造对偶向量的思路推广到正交各向异性问题．将弹性力学求解辛体系的对偶向量重新排序后，提出了一种新的对偶向量．由混合变量求解法直接得到对偶微分方程．所导出的对偶微分矩阵具有主对角子矩阵为零矩阵的特点．由于对偶微分矩阵的这一特点,对于正交各向异性三维弹性力学发现了2个独立的、对称的正交关系．采用分离变量法求解对偶微分方程．从正交各向异性弹性力学求解体系的积分形式出发，利用一些恒等式证明了新的正交关系．新的正交关系不但包含原有的辛正交关系，而且比原有的关系简洁．新正交关系的物理意义是对偶方程的解关于z坐标的对称性的体现．辛正交关系是一个广义关系，但辛正交关系可以在一定的条件下以狭义的强形式出现．新的研究成果将为研究正交各向异性三维弹性力学的解析解和有限元解提供新的有效工具．Abstract: The new orthogonal relationship is generalized for orthotropic elasticity of three-dimensions. The thought of how dual vectors are constructed in a new orthogonal relationship for theory of elasticity is generalized into orthotropic problems. A new dual vector is presented by the dual vector of the symplectic systematic methodology for elasticity that is over again sorted. A dual differential equation is directly obtained by using a mixed variables method. A dual differential matrix to be derived possesses a peculiarity of which principal diagonal sub-matrixes are zero matrixes. As a result of the peculiarity of the dual differential matrix, two independently and symmetrically orthogonal sub-relationships are discovered for orthotropic elasticity of three-dimensions. The dual differential equation is solved by a method of separation of variable. Based on the integral form of orthotropic elasticity a new orthogonal relationship is proved by using some identical equations. The new orthogonal relationship not only includes the symplectic orthogonal relationship but is also simpler. The physical significance of the new orthogonal relationship is the symmetry representation about an axis z for solutions of the dual equation. The symplectic orthogonal relationship is a generalized relationship but it may be appeared in a strong form with narrow sense in certain condition. This theoretical achievement will provide new effective tools for the research on analytical and finite element solutions to orthotropic elasticity of three-dimensions.
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Key words:
- elasticity /
- dual vector /
- orthogonal relationship
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