The preliminary study for complex-valued general solution of anisotropic elasticity
Date Issued
2010
Date
2010
Author(s)
Huang, Yu-Tang
Abstract
For the two dimentional problem of static anisotropic elasticity, we use a viewpoint of coordinate transformation. As long as we take appropriate coordinate transformation. The general solution will be appear, and not only a coordinate transformation can let general solution appear. In generality, there exist three coordinate transformation which can let general solution appear. Each coordinae transformation get a pair conjugate vector field. Therefore, the general solution of the anisotropic elastic governing equation will be summation of three pair conjugate vector feild. And then we compare above result and Stroh formalism. It is the same. Besides, we can prove the quadratic eigenvalue problem of two method. it is the same. But the method is restricted by two dimentional problem. For the static anisotropic elastic problem, we propose other method. We do twice eigenvalue problem for the fourth order tensor of elastic modulus. And then we can find the orthogonal property in second eigenvalue problem. According to the orthogonal property, we can choose appropriate the form of general solution. The advantage of second method is not restricted by two dimentional problem.
For the problem of dynamic anisotropic elasticity or static anisotropic magneto-electroelasticity, we only rewrite the governing equation. And then, we do singular value decomposition for new fourth order tensor. And then, we do eigenvalue decomposition again. The orthogonal property will be appear. Therefore, we can choose appropriate the form of general solution of dynamic anisotropic elasticity or static anisotropic magneto-electro-elasticity according to the orthogonal property.
Subjects
anisotropic elasticity
anisotropic magneto-electro-elasticity
clifford analysis
complex analysis
quaternion analysis
Stroh formalism
quadratic eigenvalue problem
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