臺灣博碩士論文加值系統

本論文探討冪函數漸變纖維複合材料之彈性力學反平面問題，功能梯度材料(Functionally graded material)之材料性質沿著徑向或厚度方向變化，其漸變性質可使其材料之間無明顯交界面並減少應力集中的現象。本文探討材料性質隨著纖維半徑r呈冪函數 漸變( 為漸變參數)。
本研究將內含物以方形陣列(Square array)及六邊形陣列(Hexagonal array)排列，利用週期性排列時的Rayleigh法求得等效彈性模數 ，並透過聚合複合圓柱模型(Composite cylinder assemblage model)與有限元素法(COMSOL Multiphysics 5.0)驗證其結果之正確性。此外，透過收斂性分析檢核位移係數與位移之收斂性，以及有限元素法分析中所適用的網格，並確認位移函數在交界面上之連續性。
本文分為有無殼層兩種形式，數值結果顯示在無殼層纖維複合材料中，當漸變參數 為-2時，對等效彈性模數 有較顯著的提升，但其誤差也會隨著漸變參數 變大而遞增。後續在改變材料性質比例的部分發現，當 時，對於等效彈性模數 之影響較大。而在具殼層纖維複合材料中，當漸變參數 為+25且c (殼層與核層厚度比例)為0.1時，對等效彈性模數 的影響較為顯著，其誤差在不同內含物體積百分率下皆有良好一致性。最後藉由改變材料性質比例與殼層厚度探討對等效彈性模數 的影響，材料性質比例 對等效彈性模數 變化之影響較大，並發現在不同漸變參數 的情況下，等效彈性模數 皆會在特定的厚度達到極值，而非與厚度成正比。

This research studies the anti-plane problem of functionally graded fibrous composites in power law of elasticity theory, in which the functionally graded material’s gradient properties express without a clear interface between the materials. We investigate the changes in material property with the fiber radius r as a power function ( is a gradient parameter).
This experiment consists of square array and hexagonal array, the equivalent elastic modulus of the composite material that is obtained by the Rayleigh method; finite element analysis (COMSOL Multiphysics 5.0) and composite cylinder assemblages model are used to compare the results predicated by the Rayleigh’s method. In addition, the convergence analysis is used to understand the convergence of the displacement coefficient and the displacement and choose the mesh applied in the finite element analysis. Furthermore, we confirm the continuity of the displacement equation at the interface.
Numerical calculations are divided into two cases. First, in the fiber-matrix case, when the gradient parameter is -2, there is a significant improvement on the equivalent elastic modulus , but the error will increase as the gradient parameter becomes larger. In the section that changes the material property ratio, it is found that has a greater influence on the equivalent elastic modulus . Next, in the fiber-shell-matrix case, when the gradient parameter is +25 and c is 0.1 (where c is the shell thickness to core thickness ratio), the effect on the equivalent elastic modulus is more significant, and the error has good consistency in different volume fractions. Additionally, we discuss the impact on equivalent elastic modulus by changing the material property ratios and shell thickness. Numerical results demonstrated that the material property ratio has a greater influence on the equivalent elastic modulus . In the case of different gradual parameters, the equivalent elastic modulus will reach an extreme value at a particular thickness rather than being proportional to the thickness.

摘要 I
Abstract II
目錄 IV
圖目錄 VII
表目錄 XIV
符號表 XV
第一章 導論 1
1-1 研究背景與目的 1
1-2 功能梯度材料 2
1-2-1 纖維複合材料 2
1-2-2 非等向性材料 3
1-3 文獻回顧 3
1-3-1 功能梯度材料 3
1-3-2 複合材料 5
1-3-3 反平面問題 6
1-4 本文架構 7
第二章 無殼層纖維複合材料 8
2-1 彈性力學基本方程與反平面問題 9
2-1-1 彈性力學基本方程 9
2-1-2 反平面問題 11
2-2 等效彈性模數 14
2-2-1 內含物位移函數 14
2-2-2 內含物為任意排列之位移係數關係 16
2-2-3 內含物為週期性排列之位移係數關係 20
2-2-4 等效彈性模數 22
2-2-5 收斂性分析 25
2-3 聚合複合圓柱模型 47
2-4 有限元素法之收斂性分析 49
2-5 數值結果與分析 54
2-5-1 材料選擇 54
2-5-2 位移圖 54
2-5-3 等效彈性模數 61
2-5-4 數值結果分析 70
第三章 具殼層纖維複合材料 75
3-1 材料性質與反平面問題 76
3-1-1 材料性質 76
3-1-2 反平面問題 77
3-2 位移函數 78
3-2-1 殼層位移函數 79
3-2-2 內含物為週期性排列之位移係數關係 83
3-2-3 收斂性分析 87
3-3 聚合複合圓柱模型 111
3-4 有限元素法之收斂性分析 114
3-5 數值結果與分析 117
3-5-1 材料選擇 118
3-5-2 位移圖 118
3-5-3 等效彈性模數 123
3-5-4 數值結果分析 130
第四章 結論與未來展望 136
4-1 結論 136
4-2 未來展望 138
參考文獻 139
附錄A COMSOL Multiphysics 5.0操作流程 144
A-1 研究領域選擇 144
A-2 全域參數定義 145
A-3 元件定義 145
A-4 元件幾何 148
A-5 元件材料 149
A-6 元件固體力學 150
A-7 元件網格 152
A-8 研究求解器設定 153
A-9 讀取結果設定 154
附錄B 收斂性誤差分析 155
B-1 無殼層纖維複合材料之收斂性誤差 155
B-2 具殼層纖維複合材料之收斂性誤差 158

Avellaneda, M., Harshe, G., 1994. "Magnetoelectric effect in piezoelectric /magnetostrictive multilayer (2-2) composites," Journal of Intelligent Material Systems and Structures, 5, 501-513.
Arfken, G. B., Weber. H. J., 2001. Mathematical Methods for Physicists, San Diago : Academic Press.
Artioli, E., Bisegna. P., Maceri, F., 2010. "Effective longitudinal shear moduli of periodic fibre-reinforced composites with radially-graded fibres," International Journal of Solid and Structures, 47, 383-397.
Benveniste, Y., 1995. "Magnetoelectric effect in fibrous composite with piezoelectric and piezomagnetic phases," Physical Review B, 51, 16424.
Chen, T., Kuo, H. Y., 2005. “Transport properties of composites consisting of periodic arrays of exponentially graded cylinders with cylindrically orthotropic materials,” Journal of Applied Physics, 98, 033716.
Coddington, E. A., 1961. An introduction to ordinary differential equations. New York : Dover Publications.
Elliott, H. A., 1948. “Three-dimensional stress distributions in hexagonal aelotropic crystals,” Mathematical Proceedings of the Cambridge Philosophical Society, 44, 522-533.
Giannakopoulos, A. E., Suresh, S., 1997. "Indentation of solids with gradients in
elastic properties: part I. point force," International Journal of Solids and
Structure, 34, 2357-2392.

Gu, G., Yu, K. W., 2003. “Conductivities of dilute suspensions of graded fibers,
Journal of Applied Physics, 94, 3376-3383.
Harshe, G., Dougherty, J. P., Newnham, R. E., 1993. "Theoretical modeling of 3-0/0-3 magnetoelectric," International Journal of Applied Electromagnetics in Materials, 4, 161-171.
Hashin, Z., Rosen, B. W., 1964. "The elastic moduli of fiber-reinforced materials," Journal of Applied Mechanics, 31, 223-232.
Hashin, Z., Hemphreys, E. A., Goering, J., 1987. "Analysis of thermoviscoelastic behavior of unidirectional fiber composites," Composites Science and Technology, 29, 103-131.
Hori, M., Nemat-Nasser, S., 1999. "On two micromechanics theories for determining micro-macro relations in heterogeneous solids," Mechanics of Materials, 31, 667-682.
Horgan, C. O., 1995. "Anti-plane shear deformations in linear and nonlinear solid mechanics," Society for Industrial and Applied Mathematics, 37, 53-81.
Iwahori, Y., Ishiwata, S., Sumizawa, T., Ishikawa, T., 2005. "Mechanical properties improvements in two-phase and three-phase composites using carbon nano-fiber dispersed resin," Composites Part A : Applied Science and Manufacturing, 36, 1430-1439.
Ju, J. W., Tseng, K. H., 1996. "Effective elastoplastic behavior of two-phase ductile matrix composites: A micromechanical framework," International Journal of Solids and Structures, 33, 4267-4291.
Kumlutas, D., Tavman, I. H., Coban, M. T., 2003. "Thermal conductivity of particle filled polyethylene composite materials," Composites Science and Technology, 63, 113-117.

Kuo, H. Y., Chen, T., 2008. "Electrostatic fields of an infinite medium containing arbitrarily
positioned coated cylinders," International Journal of Engineering Science, 46,
1157-1172.
Kuo, H. Y., Bhattacharya, K., 2013. "Fibrous composites of piezoelectric and piezomagnetic
phases," Mechanics of Materials, 60, 159-170.
Liu, N., Yang, J., Qian, Z. H., Hirose, S., 2010. "Interface wave in functionally graded piezoelectric materials," International Journal of Engineering Science, 48, 151-159.
Lutz , M. P., Zimmerman, R.W, 1996. "Thermal stresses and effective thermal expansion coefficient of a functionally gradient sphere," Journal of Thermal Stresses, 19, 39-54.
Lutz , M. P., Zimmerman, R.W, 2016. "Effect of the interphase zone on the conductivity or diffusivity of a particulate composite using Maxwell’s homogenization method," International Journal of Engineering Science, 98, 51-59.
Movchan, B. A., 2002. "Functionally graded EB PVD coating," Surface and Coatings Technology, 149, 252-262.
Nan, C. W., 1994. "Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases," Physical Review B, 50, 6082-6088.
Niino, M., Suzuki, A., Hirai, T., Watanabe, R., Hirano, T., Kuroishi, N., 1988. “Method of producing a functionally gradient material,” U.S. Patent No. 4,751,099, Washington, DC : U.S. Patent and Trademark Office.
Perrins, W. T., Mckenzie, D. R., Mcphedran, R. C., 1979. "Transport properties of regular arrays of cylinders," Proceedings of The Royal Society A, 369, 207-225.
Peng, X. L., Li, X. F., 2010. "Thermoelastic analysis of a cylindrical vessel of functionally graded materials," International Journal of Pressure Vessels and Piping, 87, 203-210.
Roos, E., Maile, K., Berreth, K., Lyutovich, A., Weiss, R., Perova, T., Moore, A., 2004. “Chemical vapour deposition of multilayer with graded C…SiC transition,” Surface and Coatings Technology, 180, 465-469.
Schapery, R. A.,1968. "Thermal expansion coefficients of composite materials based on energy principles," Journal of Composite Materials, 2, 380-404.
Shindo, Y., Narita, F., Tanaka, K., 1996. "Electroelastic intensification near anti-plane shear crack in orthotropic piezoelectric ceramic strip," Theoretical and Applied Fracture Mechamics, 25, 65-71.
Tsang, C. H., Chau, K. H., Wong, C. K., Wong, Y. W., Shin, F. G., 2008. "Modeling of the magnetoelectric effect of three-phase multiferroic particulate composites," Integrated Ferroelectrics, 100, 177-197.
Wei, E. B., Tian, J. W., Song, J. B., 2003. "Dielectric response of composites with graded cylindrical particles," Journal of Physics Condensed, 15, 8907-8915.
Wei, E. B., Poon, Y. M., 2005. “Dielectric response of graded composites having general power-law-graded cylindrical inclusions,” Journal of Applied Physics, 98, 014104.
Wang, B. L., Mai, Y. W., 2004. "Fracture of piezoelectromagnetic materials," Mechanics Research Communications, 31, 65-73.
Yang, X. M., Hu, Y. T., Yang, J. S.,2004. "Electric field gradient effects in anti-plane problems of polarized ceramics," International Journal of Solid and Structures, 41, 6801-6811.
Zheng, J. L., Fang, Q. H., Liu, Y. W., 2007. "A generalized screw dislocation interacting with interfacial cracks along a circular inhomogeneity in magnetoelectroelastic solids," Theoretical and Applied Fracture Mechanics, 47, 205-218.
朱世杰。2003。先進複合材料。北京市:機械工業出版社。
馮小明、張崇才。2007。複合材料。重慶市:重慶大學出版社。
黃麗。2012。聚合物複合材料。北京市:中國輕工業出版社。
陳靖元。2014。具非完美交界面多鐵性纖維複合材料之等效材料性質。碩士論文。新竹市:交通大學土木工程學系。