本研究以微極彈性理論為基礎，利用平面線性三角形元素，推導出二維微極彈性理論之有限元素方程式，分析受力及環境溫度變化影響變形後微極彈性蜂巢式結構波桑比值的變化。藉由不同的結構肋寬、肋長與內凹角度等幾何參數，並改變微極材料常數及微極彈性常數，以探討結構幾何參數及微極常數改變，與受力變形及溫度變化影響後內凹型蜂巢式結構波桑比之間的相互關係。
由本研究之結果可以發現，雖然微極彈性材料本身固有之波桑比均為正值，但由微極彈性材料所組成之內凹型蜂巢式結構，若在適當的內凹角度、結構肋寬與肋長等幾何條件下，適當地改變微極常數，則此結構受力及溫度影響變形後可具有可觀的負波桑比值。

A two-dimensional triangular finite element formulation including extra degree of freedom was derived on the basis of the Eringen's micropolar elasticity theory using linear triangular element. We analyze the variation of the structural Poisson’s ratio for a deformed micropolar elastic honeycomb structure.
By varying the structural cell rib width and length, the re-entrant angle of the honeycomb structure, the micropolar material constants and the micropolar elastic constants in accordance with the micropolar elastic restrictions, we can obtain the effects on the structural Poisson’s ratio of a deformed re-entrant honeycomb structure.
According to our numerical results, with appropriate re-entrant angle, cell rib length and width of the honeycomb structure, by changing the micropolar material and elastic constants, the honeycomb structure after normal distributed loading can exhibit amazing negative Poisson’s ratio.

目錄

誌謝 ii
摘要 iii
ABSTRACT iv
目錄 v
表目錄 viii
圖目錄 ix
符號說明 xii
1. 緒論 1
1.1 研究動機 1
1.2 研究目的與方法 1
1.3 文獻回顧 2
1.3.1 微極彈性理論之發展 2
1.3.2 負波桑比之沿革 3
2. 微極彈性理論簡介 5
2.1 前言 5
2.2 微極彈性基本理論 6
2.2.1 平衡方程式 6
2.2.2 組成方程式 7
2.2.3 應變與位移之關係 8
2.2.4 邊界條件 8
2.2.5 協調方程式 9
2.2.6 微極彈性常數的限制 9
2.3 二維微極彈性理論 9
3. 二維微極彈性理論之有限元素法 14
3.1 前言 14
3.2 線性三角形元素 14
3.3 應變計算 18
3.4 應力計算 19
3.5 能量法 20
3.6 數值積分 22
3.7 溫度效應 23
4. 內凹型蜂巢式結構波桑比效應之研究 25
4.1 前言 25
4.2 收斂測試 25
4.3 微極彈性蜂巢式結構─溫度變化及幾何形狀對結構波桑比值之影響 29
4.3.1 溫度變化對結構波桑比值之影響 32
4.3.2 幾何形狀─肋寬對結構波桑比值之影響 34
4.3.3 幾何形狀─肋長與內凹角度對結構波桑比值之影響 36
4.3.4 小結 48
4.4 微極常數對結構波桑比值之影響 49
4.4.1 微極楊氏模數 對結構波桑比值之影響 50
4.4.2 微極波桑比 對結構波桑比值之影響 53
4.4.3 特徵長度 l 對結構波桑比值之影響 55
4.4.4 力偶因子N對結構波桑比值之影響 57
4.4.5 微極彈性常數 對結構波桑比值之影響 59
4.4.6 微極彈性常數 對結構波桑比值之影響 61
4.4.7 微極彈性常數 對結構波桑比值之影響 63
4.4.8 微極彈性常數 對結構波桑比值之影響 65
4.4.9 小結 67
5. 結論 68
參考文獻 70
自傳 73

[1]A. C. Eringen, “Linear theory of micropolar elasticity”, Journal of Mathematics and Mechanics, Vol. 15, No. 6, pp. 909-923, 1966.
[2]S. C. Cowin, “An incorrect inequality in micropolar elasticity theory”, Zeitschrift fur Angewandte Mathematik und Physik: ZAMP (Journal of Applied Mathematics and Physics), Vol.21, pp. 494-497, 1970.
[3]R. D. Gauthier and W. E. Jahsman, “A quest for micropolar elastic constants”, Journal of Applied Mechanics, Transactions of ASME, Vol. 42, pp. 369-374, 1975.
[4]S. Nakamura, R. Benedict and R. Lakes, “Finite element method for orthotropic micropolar elasticity”, International Journal of Engineering Science, Vol. 22, No. 3, pp. 319-330, 1984.
[5]F. Y. Huang and K. Z. Liang, ”Torsional analysis of micropolar elasticity using the finite element method”, International Journal of Engineering Science, Vol. 32, No. 2, pp. 347-358, 1994.
[6]K. Z. Liang and F. Y. Huang, “Boundary element method for micropolar elasticity”, International Journal of Engineering Science, Vol. 34, No. 5, pp. 509-521, 1996.
[7]F. Y. Huang and K. Z. Liang, “Boundary element analysis of stress concentration in micropolar elastic plate”, International Journal for Numerical Methods in Engineering, Vol. 40, pp. 1611-1622, 1997.
[8]F. Y. Huang, B. H. Yan, J. L. Yan and D. U. Yang, “Bending analysis of micropolar elastic beam using 3-D finite element method”, International Journal of Engineering Science, Vol. 38, pp. 275-286, 2000.
[9]R. F. Almgren, “An isotropic three-dimensional structure with Poisson’s ratio = -1”, Journal of Elasticity, Vol. 15, pp. 427-430, 1985.
[10]K. E. Evans, “Tensile network microstructures exhibiting negative Poisson’s ratios”, Journal of Physics D: Applied Physics, Vol. 22, pp. 1870-1876, 1989.
[11]B. D. Caddock and K. E. Evans, “Microporous materials with negative Poisson’s ratios: I. Microstructure and mechanical properties”, Journal of Physics D: Applied Physics, Vol. 22, pp. 1877-1882, 1989.
[12]K. E. Evans and B. D. Caddock, “Microporous materials with negative Poisson’s ratios: II. Mechanisms and interpretation”, Journal of Physics D: Applied Physics, Vol. 22, pp. 1883-1887, 1989.
[13]J. B. Choi and R. S. Lakes, “Non-linear properties of polymer cellular materials with a negative Poisson’s ratio”, Journal of Materials Science, Vol. 27, pp. 4678-4684, 1992.
[14]J. B. Choi and R. S. Lakes, “Design of a fastener based on negative Poisson’s ratio foam”, Cellular Polymers, Vol. 10, No. 3, pp. 205-212, 1991.
[15]J. Lee, J. B. Choi and K. Choi, “Application of homogenization FEM analysis to regular and re-entrant honeycomb structures”, Journal of Materials Science, Vol. 31, pp. 4105-4110, 1996.
[16]D. U. Yang and F. Y. Huang, “Analysis of poisson’s ratio for a micropolar elastic rectangular plate using the finite element method”, Engineering Computation, Vol. 18, No. 7, pp. 1012-1030, 2001.
[17]D. U. Yang, B. H. Yan and F. Y. Huang, “The effects of material constants on the micropolar elastic honeycomb structure with negative Poisson’s ratio using the finite element method”, Engineering Computation, Vol. 19, No. 7, pp. 742-763, 2002.
[18]D. U. Yang, S Lee and F. Y. Huang, “Geometric effects on micropolar elastic honeycomb structure with negative Poisson’s ratio using the finite element method”, Finite Element in Analysis and Design, Vol. 39, pp. 187-205, 2003.
[19]S. Nakamura and R. Lakes, “Finite element analysis of Saint-Venant end effects in micropolar elastic solids”, Engineering Computations. Vol. 12, pp. 571-587, 1995.
[20]T. R. Chandrupatla and A. D. Belegundu, Introduction to Finite Element in Engineering, 3rd. Prentice-Hall International, Inc, pp. 130-151, 2002.
[21]T. Y. Yang, Finite Element Structural Analysis, Prentice-Hall, Inc. Englewood Cliffs, N. J., pp. 368-369, 1986.
[22]K. Z. Liang and F. Y. Huang, “Boundary element method for micropolar thermoelasticity”, Engineering Analysis with Boundary Elements, Vol. 17, pp. 19-26, 1996.