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論文名稱(外文):Boundary Element Analysis for Anisotropic Magneto-Electro-Elastic Solids
指導教授(外文):Chyanbin HwuYui-Chuin Shiah
外文關鍵詞:Anisotropic thermos-elasticity3D generally anisotropic elasticitymagneto-electro-elastic materialsStroh formalismboundary element methodRadon-Stroh formalismmulti-valued functionBranch cut
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In this dissertation, we develop both two- and three-dimensional boundary element method (BEM) for generally anisotropic magneto-electro-elastic (MEE) solids. The system of algebraic equations to calculate the nodal displacements and nodal tractions at the boundary of two-dimensional (2D) and three-dimensional (3D) solids are formulated. The relations for calculating strains and stresses at the boundary nodes as well as at the interior points are also derived. The Green’s functions in our computer codes are derived by using Stroh formalism. In Stroh formalism, the solutions for the solids with these different material types, such as elastic, piezoelectric and MEE, all bear exactly the same mathematical forms distinguished by the contents and dimensions of the related matrices and vectors. This extension is usually called expanded Stroh formalism, and it provides a big advantage for computer programming.
In 2D analysis, we consider not only general mechanical problems but also the problems with thermal effects. The available Green’s functions for an infinite plate and for an infinite plate with a central hole/crack are extended to MEE plate. To use the Green’s function containing a hole/crack in BEM, a suitable branch cut setting is provided for the multi-valued complex logarithmic functions with the mapped variable to get the correct principal value from MATLAB. For 2D thermo-MEE, the complex variable Green’s function for an infinite plate is used to derive the thermal fundamental solution for 2D thermo-MEE analysis. By using the identities of Stroh formalism, the complex form solution can be converted into real form. With the real form fundamental solution, the trouble induced by the multi-valued complex logarithmic function is circumvented and the extra line integral appeared in the thermal analysis of BEM can be eliminated. Thus, the temperature information inside the domain required by the extra line integral can be avoided, and a truly boundary element method for thermo-MEE analysis is achieved. The influence of heat source represented by the heat generation rate is also considered in our formulation.
The 3D Green’s functions of displacements and tractions in both complex and real forms, together with their first derivatives, are derived completely by Radon-Stroh formalism, which combines expanded Stroh formalism and 2D Radon transform. The calculation processes of two types of three-dimensional Green’s functions are listed step by step for comparison of efficiency. We found that the 3D BEM with complex form solution is more efficient than that with real form solution, although the opposite performance was observed for a single point calculation of Green’s function.
To verify the correctness of our solutions, some numerical examples with simple geometries involving MEE, piezoelectric, piezomagnetic and elastic material properties are demonstrated.
摘要 i
2.1 Three dimensional constitutive laws 6
2.1.1 Isothermal condition 6
2.1.2 Thermal effect 9
2.2 Two dimensional constitutive laws 11
2.2.1 Isothermal condition 11
2.2.2 Thermal effect 14
2.3 Specialization 17
2.3.1 Anisotropic piezoelectric/piezomagnetic solids 17
2.3.2 Anisotropic elastic solids 20
3.1 Stroh formalism for 2D anisotropic elasticity 22
3.2 Expanded Stroh formalism 23
3.3 Radon-Stroh formalism for 3D anisotropic elasticity 26
4.1 Two-dimensional deformation 32
4.1.1 An infinite plate 32
4.1.2 An infinite plate containing a hole or crack 33
4.1.3 Evaluation of multi-valued logarithmic functions 34
4.2 Three-dimensional deformation 38
4.2.1 An infinite solid 38
4.2.2 Evaluations of Green's function and its derivatives 39
5.1 Two-dimensional deformation 44
5.1.1 Isothermal condition 44
5.1.2 Thermal effect 50
5.2 Three-dimensional deformation 57
5.2.1 Anisotropic elastic solids 57
5.2.2 Anisotropic MEE solids 62
6.1 Two-dimensional analyses 63
6.1.1 Isothermal condition 63
6.1.2 Thermal environment 64
6.2 Three-dimensional analyses 67
6.2.1 3D Green’s function 67
6.2.2 Examples for 3D-BEM analysis 69
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