本研究以原子-連體力學法(Atomistic-Continuum Mechanics, ACM)，輔以有限元素法(Finite Element Method, FEM)為模擬基礎，建立奈米碳管之等效連體結構模型，並以第二代貝氏潛勢能(Brenner’s Reactive Empirical Bond Order Potential, REBO)為碳原子間鍵結力的描述方式來進行奈米碳管的力學行為模擬分析。在傳統連體力學中，材料的楊氏係數除了可由傳統的拉伸試驗得知，亦可藉由對材料振動特性的量測反推得到材料的楊氏係數。將上述概念應用到奈米碳管的楊氏係數預估上，理論上應可得到相同的對應關係。本研究首先針對使用線性鍵結力或第二代貝氏潛勢能的原子連體轉換元素，進行兩者所得模擬結果的比較與探討；並參考已知奈米碳管的力學性質實驗結果，討論本研究之等效連體結構在實際物理行為上的合理性。而藉由適當的潛勢能函數取用，本研究亦針對奈米碳管之應變與應力關係進行分析。最後將針對此原子-連體結構進行振動模態分析，並建立相關連體殼層結構。藉由分析兩者間的振動與力學性質相互關連性，探討奈米碳管有效的管壁厚度。
針對奈米材料之數值模擬分析，有鑑於原子等級多粒子系統模擬計算，如量子理論計算、分子動力學計算等等，其龐大的計算量使得所能分析的系統侷限於數奈米的範圍，本研究採用之原子-連體力學分析法成功的將模擬之單壁奈米碳管尺寸增加至數十奈米。此外，本方法於電腦運算速度上比分子動力學更具有時間上的效率。於實驗模擬過程中，拉伸負載計算的最長單壁奈米碳管可超過100 nm ，於振動模態分析上計算超過90 nm 的單壁奈米碳管，以上電腦運算時間在使用個人電腦環境下均不超過一個小時的時間單位。
研究結果顯示單壁奈米碳管的楊氏係數和管徑與管長的變化無顯著關連性，且楊氏係數平均值為1050GPa，符合目前文獻中公認的單壁奈米碳管楊氏係數強度。在振動模態部分，單壁奈米碳管的共振頻率在同一個模態之下，與管徑有正比關係，而與管長為反比關係。經由中空管狀樑之解析解與等效殼層連體模型的模擬分析解之比較，可得知利用管壁厚度為0.34 nm所得到的楊氏係數，與利用此楊氏係數與管壁厚度進行振動模態分析可得到五個百分比以內的差異量。此結果顯示利用管壁厚度為0.34 nm進行單壁奈米碳管的楊氏係數計算為合理的假設值。

The atomistic-continuum mechanics (ACM) and finite element method (FEM) were applied to construct an equivalent-continuum model to investigate the mechanical properties of carbon nanotube (CNT). In addition, the interatomic potential function between carbon atoms were described by two kinds of potential function, one was the Cornell’s potential function which was a linear function and the other was the Brenner’s Reactive Empirical Bond Order Potential (REBO) which was a non-linear one. Several models using continuum mechanics had been published including the equivalent-beam and the equivalent-truss models. By using the later model with linear interatomic force constant, the results of the linear analysis could be validated with the experiments. As a result, the authenticity of applying continuum mechanics and finite element method into the analysis of atomic-level material has been verified. Moreover, the modal analysis of carbon nanotubes by using finite element method was also validated.
In classical continuum mechanics, the Young’s modulus could be directly measured by tensile test, or be calculated from the results of the corresponding vibration test. Applying the above concept to the prediction of Young’s modulus of carbon nanotubes theoretically could result in the same corresponding relation. In this research, both linear interatomic force constant and non-linear potential function REBO were chosen as the material properties of the atomistic-continuum transformation element. The discussion about the rationality of the equivalent-continuum model was proceeded by referring to the available experiments. Moreover, by selecting the appropriate potential function, the stress-strain relation could be carried out. At last, the modal analyses of the equivalent atomistic-continuum SWCNT model were proceeded and the corresponding continuum models were constructed. By analyzing the relation between the above two models, the reasonable wall-thickness of SWCNT could be acquired.
The results showed that the Young’s modulus wouldn’t change obviously with the variation of tube radius and length, and the average number is about 1,050GPa which agreed with the most experimental and analytical results. In the modal analyses, the resonant frequency of SWCNT with the same mode shape was proportional to the tube radius, and was disproportional to the tube length. Finally, by comparing the analytical solution of the tube structure with the simulation results of equivalent-continuum shell model, the reasonable wall-thickness of SWCNT was acquired as 0.34nm which was chosen in the most publications.

[1] Iijima S., “Helical microtubules of graphitic carbon.”, Nature, vol. 354, 56-58, 1991.
[2] Iijima S., Ichlhasshi T., “Single-shell carbon nanotubes of 1-nm diameter.”, Nature, vol. 363, 603-605, 1993.
[3] Bethune D. S., Kiang C. H., Devries M. S., Gorman G., Savoy R., Vazquez J., et al., “Cobaltcatalyzed growth of carbon nanotubes with single-atomic layer walls.”, Nature, vol. 363, 605-607, 1993.
[4] Dresselhaus M. S., Dresselhaus G., Saito R., “Physics of carbon nanotubes.”, Carbon, vol. 33, 883-891, 1995.
[5] Dresselhaus M. S., Dresselhaus G., Avouris P.(Eds.), Carbon Nanotubes: Synthesis, Structure, Properties, and Applications., Springer, Topics in Appl. Phys., vol. 80, 3-9,2001.
[6] Treacy M. M. J., Ebbesen T. W., Gibson T. M., “Exceptionally high Young’s modulus observed for individual carbon nanotubes.”, Nature, vol. 381, 680-687, 1996.
[7] Salvetat J. P., Bonard J. M., Thomson N. H., Kulik A. J., Forró L., Benoit W., Zuppiroli L., “Mechanical properties of carbon nanotubes.”, Appl. Phys. A, vol. 69, 255-260, 1999.
[8] Falvo M. R., Clary G. J., Taylor R. M., Chi V., Brooks F. P., Washburn S., et al., “Bending and bulking of carbon nanotubes under large strain.”, Nature, vol. 389, 582-584, 1997.
[9] Bower C., Rosen R., Jin L., Han J., Zhou O., “Deformation of carbon nanotubes in nanotube-polymer composites.”, Appl. Phys. Let., vol. 74, 3317-3319, 1999.
[10] Therrones M., Hsu W. K., Kroto H. W., Walton D. R. M., Nanotubes: a revolution in materials science and electronics., Topic in Current Chem., vol. 199, 190-234, 1999.
[11] Yakobson B. I., Campbell M. P., Brabec C. J., Bernholc J., “High strain rate fracture and C-chain unraveling in carbon nanotubes.”, Comp. Mat. Sci., vol. 8, 341-348, 1997.
[12] Liew K. M., Wong C. H., Tan M. J., “Buckling properties of carbon nanotube bundles.”, Appl. Phys. Let., vol. 87, 041901-041903, 2005.
[13] Tersoff J., “New empirical approach for the structure and energy of covalent systems.”, Phys. Rev. B, vol. 37, 6991-7000, 1988.
[14] Brenner D. W., “Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films.”, Phys. Rev. B, vol. 42, 9458-9471, 1990.
[15] Brenner D. W., Shenderova O. A., Harrison J. A., Stuart S. J., Ni B., Sinnott S. B., “A second-generation reactive empirical bond order (REBO)potential energy expression for hydrocarbons.”, J. Phys. Condens. Matter, vol. 14, 783-802, 2002.
[16] Jones J. E., “On the Determination of Molecular Fields. I. From the Variation of the Viscosity of a Gas with Temperature.”, Proc. R. Soc. London, Ser. A, vol. 106, 441-462, 1924.
[17] Jones J. E., “On the Determination of Molecular Fields. II. From the Equation of State of a Gas.”, Proc. R. Soc. London, Ser. A, vol. 106, 463-477, 1924.
[18] Hernandez E., Goze C., Bernier P., Rubio A., “Elastic properties of C and BxCyNz composite nanotubes.”, Phys. Rev. Let., vol. 80, 4502-4505, 1998.
[19] Sanchez-Portal D., et al., “ab initio structural, elastic, and vibrational properties of carbon nanotubes.”, Phys. Rev. B, vol. 59, 12678-12688, 1999.
[20] Tersoff J., “Energies of fullerenes.”, Phys. Rev. B, vol. 46, 15546-15549, 1992.
[21] Yakobson B. I., Brabec C. J., Bernholc J., “Nanomechanics of carbon tubes: instabilities beyond linear range.”, Phys. Rev. Let., vol. 76, 2511-2514, 1996.
[22] Li C. Y., Chou T. W., “A structural mechanics approach for the analysis of carbon nanotubes.”, Int. J. Sol. Struc., vol. 40, 2487-2499, 2003.
[23] Li C. Y., Chou T. W., “Elastic moduli of multi-walled carbon nanotubes and the effect of van der Waals forces.”, Comp. Sci. Tech., vol. 63, 1517-1524, 2003.
[24] Li C. Y., Chou T. W., “Vibrational behaviors of multiwalled-carbon-nanobute- based nanomechanical resonators.”, Appl. Phys. Let., vol. 84, 121-123, 2004.
[25] Odegard G. M., Gates T. S., Nicholson L. M., Wise K. E., “Equivalent-continuum modeling of nano-structured materials.”, Comp. Sci. Tech., vol. 62, 1869-1880, 2002.
[26] Leung A. Y. T, Guo X., He Q., Kitipornchai S., “A continuum model for zigzag single-walled carbon nanotubes.”, Appl. Phys. Let., vol. 86, 083110-3, 2005.
[27] Cornell W. D., Cieplak P., Bayly C. I., Gould I. R., Merz K. M., Ferguson D. M., Spellmeyer D. C., et al., “A second generation force field for the simulation of proteins, nucleic acids, and organic molecules.”, J. Ame. Chem. Soc., vol. 117, 5179-5197, 1995.
[28] Zhang P., Huang Y., Geubelle P.H., Klein P.A., Hwang K.C., “The elastic modulus of single-wall carbon nanotubes:a continuum analysis incorporating interatomic potentials.”, Int. J. Sol. Struc., vol. 39, 3893-3903, 2002.
[29] Chang T. and Gao H., “Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model.”, J. Mech. Phys. Sol., vol. 51, 1059-1074, 2003.
[30] Chang T. and Gao H., “Chirality- and size-dependent elastic properties of single-walled carbon nanotubes.”, Appl. Phys. Let., vol. 87, 251929-1-251929-3, 2005.
[31] C. A. Yuan and K. N. Chiang, "Investigation of dsDNA stretching meso-mechanics using finite Element Method”, NSTI Nanotech Conference 2004, March 7-11, 2004, Boston, Massachusetts, U.S.A.
[32] C. A .Yuan, C. N. Han and K. N. Chiang, “Investigation of the Sequence- Dependent dsDNA Mechanical Behavior using Clustered Atomistic-Continuum Method,” NSTI Nanotechnology Conference 2005, May 8-12, 2005, Anaheim, California, U.S.A.
[33] K. N. Chiang , C.A. Yuan, C. N. Han, C. Y. Chou and Yujia Cui, “Mechanical Characteristic of ssDNA /dsDNA Molecule Under External Loading”, Appl. Phys. Let., vol. 88, 023902-1-023902-3, 2006.
[34] C. N. Han, C. Y. Chou, C. J. Wu and K. N. Chiang, “Investigation of ssDNA Backbone Molecule Mechanical Behavior Using Atomistic-Continuum Mechanics Method”, NSTI Nanotech Conference 2006, May 7-11, 2006, Boston, Massachusetts, U.S.A.
[35] Chan-Yen Chou, Cadmus Yuan, Chung-Jung Wu and Kuo -Ning Chiang, “Numerical Simulation of the Mechanical Properties of Nanoscale Metal Clusters Using the Atomistic-Continuum Mechanics Method.”, NSTI EuroNanoSystem 2005, December 14-16 2005, Paris, France.
[36] Pettersson I., Liljefors T., ”Molecular Mechanics Calculates Conformation Energies of Organic Molecules: A Comparison of Force Field.”, Rev. Comp. Chem., vol. 9, 167-189, 1996.
[37] Abell G. C., “Empirical chemical pseudopotential theory of molecular and metallic bonding.”, Phys. Rev. B, vol. 31, 6184-6196, 1985.
[38] Tersoff J., “New empirical approach for the structure and energy of covalent systems.”, Phys. Rev. B, vol. 37, 6991-7000, 1988.
[39] Beiser A., Concepts of Modern Physics, McGraw Hill, 6th Ed., 2003.
[40] Murray G. T., Murray G. T., Introduction to Engineering Meterials, Marcel Deeker, 2nd Ed., 1993.
[41] Bathe K. J., Finite Element Procedures in Engineering Analysis, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1982.
[42] Faires J. D., Burden R., Numerical Methods, Thomson Learning Inc., 2nd Ed., 1998.
[43] Maiti A., Brabec C. J., Bernholc J., “Kinetics of metal-catalyzed growth of single-walled carbon nanotubes.”, Phys. Rev. B, vol. 55, R6097-R6100, 1997.
[44] ANSYS User’s Manual, 00049 Updo ANSYS Revision 5.2, vol. 3, Elements. SAS IP Inc, 4-217,1995.
[45] Krane K. S., Introductory Nuclear Physics, John Wiley New York, 3rd Ed., 1987.
[46] Chiang K. N., Chou C. Y., Wu C. J., and Yuan C. A., "Prediction of The Bulk Elastic Constant of Metals Using Atomic-Level Single-Lattice Analytical Method" Appl. Phys. Let., vol. 88, 171904-1- 171904-3, 2006.
[47] Blakslee O. L., Proctor D. G., Selden E. J., Spence G. B., and Weng T., “Elastic Constants of Compression-Annealed Pyrolytic Graphite.”, J. Appl. Phys., vol. 41, 3373-3382, 1970.
[48] Jacobsen R. L., Tritt T. M., Guth J. R., Ehrlich A. C., and Gillespie D. J., “Mechanical properties of vapor-grown carbon fiber.”, Carbon, vol. 33, 1217-1221, 1995
[49] Yu M. F., Files B. S, Arepalli S., Ruoff R. S., “Tensile Loading of Ropes of Single Wall Nanotubes and Their Mechanical Properties.”, Phy. Rev. Let., vol. 84, 5552-5555, 2000.
[50] Krishnan A., Dujardin E., Ebbesen T. W., Yianilos P. N., Treacy M. M. J., “Young’s Modulus of Single-Walled Nanotubes.”, Phy. Rev. B, vol. 58, 14013-14019, 1998.
[51] R. R. Craig, Jr., Structural Dynamics: An Introduction to Computer Methods, John Wiley New York, 1981.