臺灣博碩士論文加值系統

在探討微觀結構時，材料固有的尺度效應與微觀效應往往不可忽略。以力學的角度來說，材料的應力已經不僅僅是內部質點的相對位移所致，更包含了旋轉的自由度，所以傳統古典彈性力學範疇裡已無法精確地描述該材料之特性，須使用更廣義的力學架構分析，也就是微極彈性理論。本文首先將比較Cosserat 彈性理論與古典彈性力學的不同，且由於微極彈性理論多考慮了旋轉角的自由度，因此，幾何方程式、材料組成律、力與力矩平衡方程式皆必須做出相對應地修正，其次本文將Cosserat 彈性理論導入複合材料中平均場量的概念，推導出微極彈性理論下，三維以及二維的平均應力、平均偶應力、平均應變以及平均曲率，利用此平均場量以及高斯散度定理，推導出等效材料係數，之後並以一週期性微結構排列的複合材料為例，使用有限元素軟體計算其等效材料係數，其中在利用代表性體積元素(RVE)分析複合材料行為時，不只選取之單位晶格需適當，且單位晶格在承受不同載重形式時也必須給予適切的邊界條件，而本文將針對用有限元素軟體模擬二維RVE 產生Cosserat 效應之模型，提出一適當之週期性邊界，並與其他文獻做比較，而本文亦探討材料微結構之幾何形狀對單位晶格產生Cosserat 效應之影響。藉由本研究方法，可合理評估週期性微結構複合材料之等效材料係數，並期許未來能依據本文之方法進行相關實驗，進而將其成果延伸至工程應用領域。

In the framework of micromechanics, under certain lengthscales, the microstructure effect of materials cannot be ignored, thus classical continuum mechanics could not be adequate for characterizing the media with complex microstructure effects. The Cosserat theory is one of the approaches we can choose to deal with this level of complexity. In Cosserat elasticity the stresses are not only characterized by the translational motion u but also rotational degrees of freedom. In this thesis, we first recall the basic formulations of Cosserat elasticity, and compare the differences between classical elasticity and Cosserat elasticity. Second, we use the average-field theory to calculate the average stresses, average couple, average strain and average curvature for a Cosserat medium. lastly, we employ the proposed average theory to determine the effective moduli of two dimensional model with periodic microstructures. Numerical simulation is also performed by a finite element method. We design a medium with periodic microstructure that is composed of an elastic material to make it effectively resemble an effect of Cosserat elasticity. It is found that the more asymmetric the microstructural configuration is, the more Cosserat effect it behaves with. Using the methodology proposed in this thesis, we can reasonably predict the overall mechanical behavior of periodic micropolar composites.

摘要.................................................... i
Abstract .............................................. ii
誌謝................................................... viii
目錄.................................................... ix
表目錄............................................... x
圖目錄............................................ xi
第一章 緒論.................................................... 1
1.1 研究動機.............................................. 1
1.2 Cosserat 彈性理論背景與文獻回顧 .................... 1
1.3 等效係數之應用與文獻回顧................................ 3
1.4 論文簡介.......................................... 5
第二章 Cosserat 彈性理論 ................................. 7
2.1 Cosserat 彈性理論的自由度 ....................... 7
2.2 Cosserat 彈性理論的組成律 ............................. 10
2.3 Cosserat 彈性理論的平衡方程式 ......................... 11
2.4 Cosserat 彈性理論的基礎方程式整理 .................. 15
第三章 Cosserat 彈性理論的平均值 ........................... 16
3.1 三維 Cosserat 彈性理論的平均值 ....................... 16
3.2 二維 Cosserat 彈性理論的平均值 ....................... 29
第四章 等效材料係數之有限元素軟體模擬.................... 38
4.1 概要.................................. 38
4.2 數值模擬之參數設定.................................... 39
4.3 等效材料係數........................................... 43
4.4 等效材料參數求解方法................................... 49
4.5 結果討論.................................... 54
4.6 數值模擬結果........................... 56
第五章 結論與未來展望.......................... 61
5.1 結論........................................... 61
5.2 未來展望..................................... 62
參考文獻............................................... 63
附錄A…………………………………………………………….….………..69

1.Aboudi, J., Linear Thermoelastic Higher-Order Theory for Periodic Multiphase Materials, Journal of Applied Mechanics, vol 68, 697-707, 2001.
2.Aboudi, J., Mechanics of Composite Materials, A Uniform Micromechanical Approach. Elsevier Science Publishers, Amsterdam, 1991.
3.Aboudi, J., Micromechanical analysis of composites by the method of cells. Appl. Mech. Rev., Vol. 42, No. 7, 193-221, 1989.
4.Addessi, D., Bellis, M. L. D. and Sacco, E., Micromechanical analysis of heterogeneous materials subjected to overall Cosserat strains. Mechanics Research Communications, 54, 27-34, 2013.
5.Anthoine, A., Derivation of the in-plane elastic characteristics of masonry through homogenization theory. Int. J. Solids Sturctures, Vol. 32, No. 2, pp. 137-163, 1995.
6.Bazant, Z. P. and Christensen, M., Analogy between micropolar continuum and grid frameworks under initial stress. Int. J. Solids Structures, Vol. 8, pp. 327-346, 1972.
7.Bigoni, D. and Drugan, W. J., Analytical Derivation of Cosserat Moduli via Homogenization of Heterogeneous Elastic Materials. Journal of Applied Mechanics, Vol. 74, 741-753, 2007.
8.Brockenbrough, J. R., Suresh, S. and Wienecke, H. A., Deformation of metal-matrix composites with continuous fibers: Geometrical effects of fiber distribution and shape. Acta Metall. Mater., 5, 735-52, 1991.
9.Buryachenko, V.A., Micromechanics of Heterogeneous Materials. 2007.
10.Caruso, J. J., Application of finite element substructuring to composite micromechanics. NASA TM-83729, NASA Lewis Research Center, Cleveland, OH, 1984.
11.Casolo, S., Macroscopic modelling of structured materials: Relationship between orthotropic Cosserat continuum and rigid elements, International Journal of Solids and Structures, 43, 475-496, 2006.
12.Chang, C. S. and Liao, C. L., Constitutive Relation for a Particulate Medium with the Effect of Particle Rotation. Int. J. Solids Structures Vol. 26, No. 4, pp. 437-435, 1990.
13.Cherkaev, A., Variational Methods for Structural Optimization. Springer. 2000.
14.Christensen, R.M. Mechanics of Composite Materials. Wiley 1979.
15.Eringen, A. C., Microcontinuum Field Theories I: Foundations and Solids. Springer, New York, 1999.
16.Forest, S. and Sab, K., Cosserat overall modelling of heterogeneous materials. Mechanics Research Communications, Vol. 25, No. 4, pp. 449-454, 1998.
17.Forest, S., Dendievel, R. and Canova, G. R., Estimating the Overall Properties of Heterogeneous Cosserat materials. Modelling Simul. Mater. Sci. Eng., 7, 829-840, 1999.
18.Forest, S., Pradel, F. and Sab, K., Asymptotic analysis of heterogeneous Cosserat media. International Journal of Solids and Structures, 38, 4585-4608, 2001.
19.Gluge, R., Generalized boundary conditions on representative volume elements and their use in determining the effective materials properties. Computational Materials Science, 79, 408-416, 2013.
20.Hashin, Z., Analysis of composite materials – A survey, J. Appl. Mech., Vol. 50, 481-505, 1983.
21.Hashin, Z., Theory of mechanical behaviour of heterogeneous media, Appl. Mech. Rev., Vol. 17, 1-9, 1964.
22.Hill, R., Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids, Vol. 11, 357-372, 1963.
23.Ieşan, D. and Chirita, S., Saint-Venant’s problem for composite micropolar elastic cylinders, Int. J. Engng. Sci., 17, 573-586, 1979.
24.Ilcewicz, L. B., Narasimhan, M. N. L. and Wilson, J. B., Micro and Macro Materials Symmetries in Generalized Continua. Int. J. Engng. Sci. Voi., 1, pp. 97-109, 1986.
25.Joumma, H. and Ostoja-Starzewski, M., Stress and Couple-stress invariance in non-centrosymmetric micropolar planar elasticity. Proceedings of the Royal Society A: Mathemmatical, Physical and Engineering Sciences, 467, 2896-2911, 2011.
26.Li, X. and Liu, Q., A version of Hill’s lemma for Cosserat continuum. Acta Mechanica Sinica, 25, 499-506, 2009.
27.Li, X., Liu, Q. and Zhang, J., A micro-macro homogenization approach for discrete particle assembly Cosserat continuum modeling of granular materials. International Journal of Solids and Structures, 47, 291-303, 2010.
28.Liu, Q., Hill’s lemma for the average-field theory of Cosserat continuum. Acta Mechanica Sinica, 224, 851-866, 2013.
29.Liu, Q., Liu X., Li, X. and Li, S., Micro-macro homogenization of granular materials based on the average-field theory of Cosserat continuum. Advanced Power Technology, 25, 436-449, 2014.
30.Liu, X. N., Huang, G. L. and Hu, G. K., Chiral effect in plane isotropic micropolar elasticity and its application to chiral lattices, J. Mech. Phys. Solids 60, 1907-1921, 2012.
31.Michel, J. C., Moulinec, H. and Suquet, P., Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Methods Appl. Mech. Engrg., 172, 109-143, 1998.
32.Milton, G.W., The Theory of Composites. Cambridge University Press, 2002.
33.Minagawa S., Arakawa K. and Yamada M, Dispersion Curves for Waves in a Cubic Micropolar Medium with Reference to Estimations of the Material Constants for Diamond, Bulletin of the JSME., Vol. 24, No. 187, 1981.
34.Naik, R. A. and Crews Jr, J. H., Micromechanical analysis of fiber-matrix interface stresses under thermomechanical loadings. Composites Materials: Testing and Design, Vol. 11, ASTM STP 1206, ed. E. T. Camponeschi Jr. American Society for Testing and Materials, Philadelphia, PA, pp. 205-19, 1993.
35.Nemat-Nasser, S. and Hori, S., Micromechanics: Overall Proerties of Heterogeneous Materials. Elsevier, North-Holland, 1993.
36.Nye, J. F., Physical properties of crystals: their representation by tensors and matrices. Oxford university press, New York, 1985.
37.Okereke, M. I. and Akpoyomare, A. I., A virtual framework for prediction of full-field elastic response of unidirectional composites, Computational Materials Science, 70, 82-99, 2013.
38.Ostoja-Starzewski, M. and Jasiuk, I., Stress invariance in planar Crosserat elasticity, Proc. R. Soc. A. 451, 453, 1995.
39.Qu, J. and Cherkaoui, M., Fundamentals of Micromechanics of Solids, Wiley, New Jersey, 2006.
40.Rodriguez- Ramos, R., Medeiros, R. D., Guinovart-Diaz, R., Bravo-Castillero J., Otero, J. A. and Tita, V., Different approaches for calculating the effective elastic properties in composite materials under imperfect contact adherence. Composite Structures, 99, 264-275, 2013.
41.Ryvkin M. and Aboudi J., Analysis of Local Thermomechanical Effects in Fiber- Reinforced Periodic Composites, Int. J. Fract, 145, 229-236, 2007.
42.Ryvkin, M. and Aboudi, J., The effect of a fiber loss in periodic composites. International Journal of Solids and Structures, 44, 3497-3513, 2007.
43.Suiker, A. S. J., Borst, R. de and Chang, C. S., Micro-mechanical modelling of granular material. Part 1: Derivation of a second-gradient micro-polar constitutive theory. Acta Mechanica, 149, 161-180, 2001.
44.Sun, C.T. and Vaidya R. S., Prediction of Composite Properties from a Representative Volume Element. Composites Science and Technology, 56, 171-179, 1996.
45.Torquato, S., Random Heterogeneous Materials. Springer 2002.
46.Vardoulakis, I., Cosserat Continuum Mechanics, , National Meeting on Generalized Continuum Theories and Applications, 2009.
47.Waseem, A., Beveridge, A. J., Wheel, M. A. and Nash, D. H., The influence of void size on the micropolar constitutive properties of model heterogeneous materials. European Journal of Mechanics A/Solids, 40, 148-157, 2013.
48.Willis, J. R., Variational and related methods for the overall properties of composites, Advances in Applied Mechanics, Vol. 21, 1-78, 1981.
49.Wu, T. H., Chen, T. and Weng, C. N., Green’s functions and Eshelby tensor for and ellipsoidal inclusion in non-centrosymmetric and anisotropic micropolar medium, 2014.
50.Xia, Z., Zhang, Y. and Ellyin, F., A unified periodical boundary conditions for representative volume elements of composites and applications. International Journal of Solids and Structures 40, 1907-1921, 2002.
51.Yuan, X. and Tomita, Y., A Homogenization Method for Analysis of Heterogeneous Cosserat Materials. Key Engineering Materials, Vols.177-180, pp 53-58, 2000.
52.Yuan, X. and Tomita, Y., Effective properties of Cosserat composites with periodic Microstructure. Mechanics Research Communications, 28, 265-270, 2001.
53.吳宗憲，Cosserat彈性介質之格林函數及內含物問題，國立成功大學土木工程研究所碩士論文，2014。
54.張博威，具多重負頻帶之彈性超材料數值模擬，國立成功大學土木工程研究所碩士論文，2013。
55.劉祐翔，圓柱中性扭轉問題於Cosserat彈性材料之探討，國立成功大學土木工程研究所碩士論文，2013。