本文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，並依此撰寫相關之Fortran程式，作為數值分析之工具。
首先以微極彈性方形板為例，假設其受到軸向的均佈負荷作用，計算方形板的應力應變值以驗證本程式之收斂性與精確性。藉由微極材料常數與微極彈性常數之改變，來分析受力變形後微極彈性方形板其波桑比與微極材料常數或微極彈性常數之間的相互關係。
其次分析受力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，以探討結構幾何參數改變與受力變形後內凹型蜂巢式結構波桑比之間的相互關係。除此之外，改變微極材料常數如微極楊氏模數 、微極波桑比 、特徵長度 與力偶因子N，或在微極彈性限制條件下改變 、 、 與 等四個微極彈性常數，可以得到對受力變形後內凹型蜂巢式結構波桑比值之影響。
經由本文的研究結果可以發現，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，藉由微極材料常數或微極彈性常數的改變，此結構受力變形後可具有可觀的負波桑比值。但是對受力變形後之微極彈性方形板而言，微極常數改變幾乎不會造成其波桑比值的變化。

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 and a corresponding computer program was also developed.
Firstly, we investigate the relation between the Poisson’s ratio of a deformed micropolar elastic rectangular plate by changing the micropolar material constants and the micropolar elastic constants.
Secondly, 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. But for the deformed micropolar elastic rectangular plate, the variation of the micropolar constants can not alter the corresponding Poisson’s ratio.

[1] Y. C. Fung, Foundation of Solid Mechanics, Prentice-Hall, Inc. Englewood
Cliffs, N. J., 1968.
[2] R. F. Almgren, “An isotropic three-dimensional structure with Poisson’s
ratio = -1”, Journal of Elasticity, Vol. 15, pp. 427-430, 1985.
[3] K. E. Evans, “Tensile network microstructures exhibiting negative Poisson’
s ratios”, Journal of Physics D: Applied Physics, Vol. 22, pp. 1870-1876,
1989.
[4] U. D. Larsen, O. Sigmund and S. Bouwstra, “Design and fabrication of
compliant micromechanisms and structure with negative Poisson’s ratio”,
Journal of micrielectromechanical systems, Vol. 6, No. 2, pp. 99-106, 1997.
[5] R. Lakes, “Foam structures with a negative Poisson’s ratio”, Science,
Vol. 235, pp. 1038-1040, 1987.
[6] 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.
[7] 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.
[8] W. E. Warren and A. M. Kraynik, “Foam mechanics: The linear elastic
response of two-dimensional spatially periodic cellular materials”,
Mechanics of Materials, Vol. 6, pp. 27-37, 1987.
[9] T. L. Warren, “Negative Poisson’s ratio in a transversely isotropic foam
structure”, Journal of Applied Physics, Vol. 6, No. 15, pp. 7591-7594,
1990.
[10] 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.
[11] 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.
[12] 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.
[13] W. Voigt, “Theoretische studin uber die elasticitusverhultnisse der
kystall”, Ahandlungen der konigllshaft der wissenshaften zu gottingen,
Vol. 24, Gottingen, 1887.
[14] E. Cosserat and F. Cosserat, “Theorie des corps deformables”, Paris, A.
Hermann and Sons. 1909.
[15] R. D. Mindlin and H. F. Tierstn, “Effects of couple stress in linear
elasticity”, Archive for Rational Mechanics and Analysis, Vol. 11, pp.
415-448, 1962.
[16] A. E. Green and R. S. Rivlin, “Directors and multipolar displacements in
continuum mechanics”, International Journal of Engineering Science, Vol.
2, pp. 611-620, 1965.
[17] A. C. Eringen and E. S. Suhubi, “Nonlinear theory of simple micro-elastic
solids-I“, International Journal of Engineering Science, Vol. 2, pp. 189-
203, 1964.
[18] A. C. Eringen, “Linear theory of micropolar elasticity”, Journal of
Mathematics and Mechanics, Vol. 15, No. 6, pp. 909-923, 1966.
[19] A. C. Eringen, “Theory of micropolar fluids”, Journal of Mathematics and
Mechanics, Vol. 16, pp. 1-18, 1966.
[20] A. C. Eringen, Fracture, in Liebowitz, M. (Ed.), Academic Press, New York,
2, pp. 621-729, 1968.
[21] T. R. Tauchert, W. D. Claus and T. Ariman, “The linear theory of
micropolar thermoelasticity”, International Journal of Engineering
Science, Vol. 6, pp. 37-47, 1968.
[22] D. Iesan, “Existence theorems in the theory of micropolar elasticity”,
International Journal of Engineering Science, Vol. 8, pp. 777-791, 1970.
[23] P. P. Teodorescu (Editor), Actual Problems in Solid Mechanics, Vol. 1,
ed., Academiei, Bucharest, 1975.
[24] L. Dragos, “Fundamental solutions in micropolar elasticity”,
International Journal of Engineering Science, Vol. 22, pp. 265-275, 1984.
[25] 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.
[26] R. D. Gauthier, Analytical and Experimental Investigations in Linear
Isotropic Micropolar Elasticity, doctoral dissertation, University of
Colorado, 1974.
[27] 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.
[28] J. F. C. Yang and R. S. Lakes, “Transient study of couple stress effects
in compact bone: torsion”, Journal of Biomechanical Engineering,
Transactions of ASME, Vol. 103, pp. 275-279, 1981.
[29] R. S. Lakes, “Dynamical study of couple stress effects in human compact
bone”, Journal of Biomechanical Engineering, Trans-actions of ASME, Vol.
104, pp. 6-11, 1982.
[30] 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.
[31] J. T. Yeh and W. H. Chen, “Shell elements with drilling degree of
freedoms based on micropolar elasticity theory”, International Journal
for Numerical Methods in Engineering, Vol. 36, pp. 1145-1159, 1993.
[32] 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.
[33] C. A. Brebbia, J. C. F. Telles and L. C. Wrobel, Boundary Element
Techniques, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1984.
[34] C. A. Brebbia and J. Dominguez, Boundary Elements: An Introductory Course,
Computational Mechanics Publication, Southampton Boston, 1989.
[35] A. N. Das and P. K. Chaudhuri, “A note on boundary elements for
micropolar elasticity”, International Journal of Engineering Science,
Vol. 30, No. 3, pp. 397-400, 1992.
[36] 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.
[37] 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.
[38] 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.
[39] T. R. Chandrupatla and A. D. Belegundu, Introduction to Finite Element in
Engineering, 2nd ed. Prentice-Hall International, Inc, 1997.
[40] 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.
[41] E. Hinton and D. R. J. Owen, An Introduction to Finite Element
Computations, Swansea, UK: Pineridge Press, 1979; Taipei: Kai Fa Book,
1982.
[42] T. Y. Yang, Finite Element Structural Analysis, Prentice-Hall, Inc.
Englewood Cliffs, N. J., 1986.