基于原子势的碳纳米管有限元模拟
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摘要
碳纳米管具有非常优异的力学、热学、电学和光学性能,应用前景广阔。自从碳纳米管被发现以来,对其力学性能的理论预测一直是一个热门研究领域。本文基于原子势建立了碳纳米管的有限元模型,基于该模型开展了碳纳米管的若干关键力学问题研究,主要包括以下内容:
(1)碳纳米管有限元模型构建。提出一种新的能够完备地替代谐和势描述C-C共价键的等效梁单元,该单元能够区分C-C键的σ-σ键角和σ-π键角的不同弯曲刚度。通过分析石墨层的本构关系确定了等效梁单元的参数。基于该等效梁单元建立了单壁碳纳米管的有限元模型,并用非线性弹簧等效Lennard-Jones势描述多壁碳纳米管层间范德华作用从而建立了多壁碳纳米管的有限元模型。本文模型能够直接在标准商业有限元软件中实现,从而可以利用商业有限元软件的强大功能开展碳纳米管力学问题的模拟。
(2)碳纳米管弹性模量计算。将单壁碳纳米管看作横观各向同性材料,系统地计算了其独立的5个弹性模量随直径和手性的变化。发现其直径越小,弹性模量对直径和手性的依赖性越强,且各向异性越明显;随着直径的增大,弹性模量的各向异性逐渐消失,当直径足够大时候,碳纳米管退化成各向同性材料,弹性模量趋于石墨层的模量。这些结果拓展了前人的文献报道,提供了一个对单壁碳纳米管各向异性性质的完整理解。
(3)碳纳米管屈曲行为模拟。应用特征值屈曲计算方法系统地分析了单壁和多壁碳纳米管临界压缩屈曲模态和临界压缩屈曲应变随其长径比的变化,发现相同长度的单壁碳纳米管承受临界压缩屈曲应变的能力有一个最优直径,处在该直径的单壁碳纳米管承受压缩屈曲应变的能力最强。还用一种附加阻尼的方法对单壁碳纳米管的弯曲屈曲行为进行了初步探讨,结果为相关分子动力学计算所验证。
(4)碳纳米管振动模态分析。应用特征值振动方法计算了单壁和多壁碳纳米管的各种振动模态,着重分析了多壁碳纳米管的“刚性”同轴振动模态(仅由范德华力决定的模态)。本文计算范围内碳纳米管的振动频率均在GHz以上,表明碳纳米管有望成为设计高频纳米器件的理想材料。
(5)碳纳米管拉曼活性模态计算。用特征值振动方法计算了单壁碳纳米管的拉曼模态,计算表明低阶拉曼模态与相关实验和理论报道相当吻合,尤其径向呼吸模态与前人第一原理计算结果高度一致,说明本文模型在拉曼模态计算方面具有一定的适用性。本文还对多壁碳纳米管的径向呼吸模态进行了详细预测,结果表明多壁碳纳米管存在与其层数相同阶数的径向呼吸模态,且其振动频率分别高于各层单壁碳纳米管径向呼吸模态的频率。
(6) Casimir力和静电力驱动下的GHz振动探讨。从多壁碳纳米管得到启发,抽象出双层同轴圆柱管模型,探讨了该模型在Casimir力和静电力作用下的振动特性,结果显示在纳米尺度该模型振动频率即可达到GHz以上,表明在纳米尺度GHz高频振动的普遍存在性。
Carbon nanotubes (CNTs) have very amazing mechanical, electronic, thermal and optical properties, which enable them many potential applications. Since the discovery of CNTs, great attention has been paid to the prediction of their mechanical behaviors. In this dissertation, a finite element model of CNTs is proposed basing on their atomic potentials, and several key mechanical problems of CNTs are studied by the model. The major contents are following:
(1) Finite element modeling of CNTs. In order to capture the distinguishing in-planeσ-σand out-of-planeσ-πbond angle bending rigidities of C-C bonds in CNTs, we propose a novel equivalent beam element, which can perfectly replace the harmonic potentials to describe the bond stretching, bond angle variance, inversion angle variance and torsion of the C-C bonds. The parameters of the equivalent beam element are extracted from the constitutive relations of a graphite sheet under different load conditions. Using the equivalent beam elements, the frame structure finite element model of single-walled carbon nanotubes (SWNTs) is developed. Then nonlinear spring elements are used to replace the Lennard–Jones potentials to represent the interlayer van der Waals interactions of multi-walled carbon nanotubes (MWNTs) and the corresponding finite element model of MWNTs is constructed. The present model can be incorporated into any standard commercial finite element software. Since the commercial finite element softwares are comprehensively developed and have very powerful calculation abilities, many mechanical problems of CNTs can be solved by the models in prospect.
(2) Calculations of the elastics moduli of SWNTs. The five independent elastic moduli of SWNTs with arbitrary chirality and diameter are evaluated systematically. It is found that the elastic properties of SWNTs are transversely isotropic when the tube diameter is small. The smaller the tube diameter, the stronger the dependence of the elastic properties on the tube size and chirality, while when the tube diameter is large enough, the SWNTs degenerate from transversely isotropic to isotropic and the elastic moduli tend to that of a graphite sheet. This work extends the previously reported results and provides a full understanding of the anisotropic elastic properties of SWNTs
(3) Simulation of the elastic buckling of CNTs. The compression buckling behavior of CNTs is predicted by using the eigenvalue buckling analysis. It is found that the critical buckling modes and the critical buckling strains are varied with the aspect of the CNTs. The results also indicate that there exists an optimum diameter for SWNTs with the same lengths at which the critical compression buckling strain reaches its maximum value. The bending buckling of SWNTs is also been studied elementarily through the automatic addition of artificial damping to the model. The results show good agreement with previous molecular dynamics calculations.
(4) Analysis of the vibration modes of CNTs. By using the eigenvalue extraction method, the vibration modes of SWNTs and MWNTs are studied, and the rigid coaxial vibration modes, which are mainly determined by the interlayer van der Waals interations, are especially concerned. It is shown that in our calculation range, all the vibration frequencies of CNTs are above the order of GHz, indicating that CNTs may be an ideal material for high frequency nano devices.
(5) Calculation of the Raman-active modes of CNTs. The Raman modes of SWNTs are extracted by the eigenvalue extraction method. It is shown that the present calculated frequencies of the low-order Raman modes agree well with the related experimental and theoretical results, especially that of the radial breathing modes (RBMs) are greatly consistent with existing ab initio results. This indicates that the present method is applicable for the calculation of Raman-active modes of CNTs, especially for the calculation of RBMs. So we conduct detailed prediction of the RBMs of MWNTs. Results show that RBM number of a MWNT equals to its layer number, and the frequency of each RBM is higher than the RBM frequency of the corresponding SWNT of each MWNT layer.
(6) Investigation on the GHz oscillation driven by Casimir or electrostatic force. A mechanical oscillator model of double-walled coaxial cylindrical tubes, which is analogous to double-walled carbon nanotubes, is proposed and the oscillation behavior of the model driven by Casimir or electrostatic force is studied by energy approach It is shown that when the model is at nano-scale, its oscillation frequency can reach or even exceed the order of GHz, indicating that GHz high frequency oscillation may be quite common phenomenon at nano-scale.
(1)碳纳米管有限元模型构建。提出一种新的能够完备地替代谐和势描述C-C共价键的等效梁单元,该单元能够区分C-C键的σ-σ键角和σ-π键角的不同弯曲刚度。通过分析石墨层的本构关系确定了等效梁单元的参数。基于该等效梁单元建立了单壁碳纳米管的有限元模型,并用非线性弹簧等效Lennard-Jones势描述多壁碳纳米管层间范德华作用从而建立了多壁碳纳米管的有限元模型。本文模型能够直接在标准商业有限元软件中实现,从而可以利用商业有限元软件的强大功能开展碳纳米管力学问题的模拟。
(2)碳纳米管弹性模量计算。将单壁碳纳米管看作横观各向同性材料,系统地计算了其独立的5个弹性模量随直径和手性的变化。发现其直径越小,弹性模量对直径和手性的依赖性越强,且各向异性越明显;随着直径的增大,弹性模量的各向异性逐渐消失,当直径足够大时候,碳纳米管退化成各向同性材料,弹性模量趋于石墨层的模量。这些结果拓展了前人的文献报道,提供了一个对单壁碳纳米管各向异性性质的完整理解。
(3)碳纳米管屈曲行为模拟。应用特征值屈曲计算方法系统地分析了单壁和多壁碳纳米管临界压缩屈曲模态和临界压缩屈曲应变随其长径比的变化,发现相同长度的单壁碳纳米管承受临界压缩屈曲应变的能力有一个最优直径,处在该直径的单壁碳纳米管承受压缩屈曲应变的能力最强。还用一种附加阻尼的方法对单壁碳纳米管的弯曲屈曲行为进行了初步探讨,结果为相关分子动力学计算所验证。
(4)碳纳米管振动模态分析。应用特征值振动方法计算了单壁和多壁碳纳米管的各种振动模态,着重分析了多壁碳纳米管的“刚性”同轴振动模态(仅由范德华力决定的模态)。本文计算范围内碳纳米管的振动频率均在GHz以上,表明碳纳米管有望成为设计高频纳米器件的理想材料。
(5)碳纳米管拉曼活性模态计算。用特征值振动方法计算了单壁碳纳米管的拉曼模态,计算表明低阶拉曼模态与相关实验和理论报道相当吻合,尤其径向呼吸模态与前人第一原理计算结果高度一致,说明本文模型在拉曼模态计算方面具有一定的适用性。本文还对多壁碳纳米管的径向呼吸模态进行了详细预测,结果表明多壁碳纳米管存在与其层数相同阶数的径向呼吸模态,且其振动频率分别高于各层单壁碳纳米管径向呼吸模态的频率。
(6) Casimir力和静电力驱动下的GHz振动探讨。从多壁碳纳米管得到启发,抽象出双层同轴圆柱管模型,探讨了该模型在Casimir力和静电力作用下的振动特性,结果显示在纳米尺度该模型振动频率即可达到GHz以上,表明在纳米尺度GHz高频振动的普遍存在性。
Carbon nanotubes (CNTs) have very amazing mechanical, electronic, thermal and optical properties, which enable them many potential applications. Since the discovery of CNTs, great attention has been paid to the prediction of their mechanical behaviors. In this dissertation, a finite element model of CNTs is proposed basing on their atomic potentials, and several key mechanical problems of CNTs are studied by the model. The major contents are following:
(1) Finite element modeling of CNTs. In order to capture the distinguishing in-planeσ-σand out-of-planeσ-πbond angle bending rigidities of C-C bonds in CNTs, we propose a novel equivalent beam element, which can perfectly replace the harmonic potentials to describe the bond stretching, bond angle variance, inversion angle variance and torsion of the C-C bonds. The parameters of the equivalent beam element are extracted from the constitutive relations of a graphite sheet under different load conditions. Using the equivalent beam elements, the frame structure finite element model of single-walled carbon nanotubes (SWNTs) is developed. Then nonlinear spring elements are used to replace the Lennard–Jones potentials to represent the interlayer van der Waals interactions of multi-walled carbon nanotubes (MWNTs) and the corresponding finite element model of MWNTs is constructed. The present model can be incorporated into any standard commercial finite element software. Since the commercial finite element softwares are comprehensively developed and have very powerful calculation abilities, many mechanical problems of CNTs can be solved by the models in prospect.
(2) Calculations of the elastics moduli of SWNTs. The five independent elastic moduli of SWNTs with arbitrary chirality and diameter are evaluated systematically. It is found that the elastic properties of SWNTs are transversely isotropic when the tube diameter is small. The smaller the tube diameter, the stronger the dependence of the elastic properties on the tube size and chirality, while when the tube diameter is large enough, the SWNTs degenerate from transversely isotropic to isotropic and the elastic moduli tend to that of a graphite sheet. This work extends the previously reported results and provides a full understanding of the anisotropic elastic properties of SWNTs
(3) Simulation of the elastic buckling of CNTs. The compression buckling behavior of CNTs is predicted by using the eigenvalue buckling analysis. It is found that the critical buckling modes and the critical buckling strains are varied with the aspect of the CNTs. The results also indicate that there exists an optimum diameter for SWNTs with the same lengths at which the critical compression buckling strain reaches its maximum value. The bending buckling of SWNTs is also been studied elementarily through the automatic addition of artificial damping to the model. The results show good agreement with previous molecular dynamics calculations.
(4) Analysis of the vibration modes of CNTs. By using the eigenvalue extraction method, the vibration modes of SWNTs and MWNTs are studied, and the rigid coaxial vibration modes, which are mainly determined by the interlayer van der Waals interations, are especially concerned. It is shown that in our calculation range, all the vibration frequencies of CNTs are above the order of GHz, indicating that CNTs may be an ideal material for high frequency nano devices.
(5) Calculation of the Raman-active modes of CNTs. The Raman modes of SWNTs are extracted by the eigenvalue extraction method. It is shown that the present calculated frequencies of the low-order Raman modes agree well with the related experimental and theoretical results, especially that of the radial breathing modes (RBMs) are greatly consistent with existing ab initio results. This indicates that the present method is applicable for the calculation of Raman-active modes of CNTs, especially for the calculation of RBMs. So we conduct detailed prediction of the RBMs of MWNTs. Results show that RBM number of a MWNT equals to its layer number, and the frequency of each RBM is higher than the RBM frequency of the corresponding SWNT of each MWNT layer.
(6) Investigation on the GHz oscillation driven by Casimir or electrostatic force. A mechanical oscillator model of double-walled coaxial cylindrical tubes, which is analogous to double-walled carbon nanotubes, is proposed and the oscillation behavior of the model driven by Casimir or electrostatic force is studied by energy approach It is shown that when the model is at nano-scale, its oscillation frequency can reach or even exceed the order of GHz, indicating that GHz high frequency oscillation may be quite common phenomenon at nano-scale.
引文
[1] 刘吉平, 孙洪强. 碳纳米材料. 北京: 科学出版社, 2004
[2] 张立德, 牟季美. 纳米材料和纳米结构. 北京: 科学出版社, 2001
[3] 朱宏伟, 吴德海, 徐才录. 碳纳米管. 北京: 机械工业出版社, 2003
[4] 成会明. 纳米碳纳米管: 制备、结构、物性及应用. 北京: 化学工业出版社, 2002
[5] Kroto HW, Heath JR, O'Brien SC, Curl RF, Smalley RE. C60: Buckminsterfullerene. Nature, 1985, 318: 162~163
[6] Smalley RE. Discovering the fullerenes. Reviews of Modern Physics, 1997, 69: 723~730
[7] Iijima S. Helical microtubules of graphitic carbon. Nature, 1991, 354: 56~58
[8] Iijima S, Ichihashi T. Single-shell carbon nanotubes of 1-nm diameter. Nature, 1993, 363: 603~605
[9] Bethune DS, Kiang CH, DeVries MS, Gorman G, Savoy R, Beyers R. Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls. Nature, 1993, 363: 605~607
[10] Smith BW, Monthioux M, Luzzi DE. Encapsulated C60 in carbon nanotubes. Nature, 1998, 396: 323~324
[11] Li J, Papadopoulos C, Xu J. Nanoelectronics: Growing Y-junction carbon nanotubes. Nature, 1999, 402: 253~254
[12] Zhang G, Jiang X, Wang E. Tubular Graphite Cones. Science, 2003, 300: 472~474
[13] Barros EB, Jorio A, Samsonidze GG, et al. Review on the symmetry-related properties of carbon nanotubes. Physics Reports, 2006, 431: 261~302
[14] Dresselhaus MS, Dresselhaus G, Saito R. Physics of carbon nanotubes. Carbon, 1995, 33(7): 883~891
[15] Thostensona ET, Ren Z, Chou TW. Advances in the science and technology of carbon nanotubes and their composites: a review. Comput. Sci. Techol., 2001, 61: 1899~1912
[16] Jiang H, Zhang P, Liu B, Huang Y, Geubelle PH, Gao H, Hwang KC. The effect of nanotube radius on the constitutive model for carbon nanotubes. Computational Materials Science, 2003, 28: 429~442
[17] Baughman RH, Zakhidov AA, de Heer WA. Carbon Nanotubes—the Route Toward Applications. Science, 2002, 297(2): 787~792
[18] Chico L, Crespi VH, Benedict LX, Louie SG, Cohen ML. Pure Carbon Nanoscale Devices: Nanotube Heterojunctions. Phys. Rev. Lett., 1996, 76: 971~974
[19] Kociak M, Kasumov AY, Guéron1 S, Reulet B, Khodos II, Gorbatov YB, Volkov VT, Vaccarini L, Bouchiat H. Superconductivity in Ropes of Single-Walled Carbon Nanotubes.Phys. Rev. Lett., 2001, 86: 2416~2419
[20] 刘哲. 碳纳米管若干力学问题的研究[博士学位论文]. 北京: 清华大学, 2002
[21] Yu MF, Lourie O, Dyer MJ, Moloni K, Kelly TF, Ruoff RS. Strength and breaking mechanism of multiwalled Carbon Nanotubes Under Tensile Load. Science, 2000, 287: 637~640
[22] Lourie O, Cox DM, Wagner HD. Buckling and Collapse of Embedded Carbon Nanotubes. Phys. Rev. Lett., 1998, 81: 1638~1641
[23] Shen W, Jiang B, Han BS, Xie S. Investigation of the Radial Compression of Carbon Nanotubes with a Scanning Probe Microscope. Phys. Rev. Lett., 2000 , 84: 3634~3637
[24] Dai H, Hafner JH, Rinzler AG, Colbert DT, Smalley RE. Nanotubes as nanoprobes in scanning probe microscopy. Nature, 1996, 384: 147~150
[25] Wang ZL, Poncharal WA, de Heer WA. Measuring physical and mechanical properties of individual carbon nanotubes by in situ TEM. J. Phys. Chem. Solids, 2000, 61: 1025~1030
[26] Fennimore AM, Yuzvinsky TD, Han WQ, Fuhrer MS, Cumings J, Zettl A. Rotational actuators based on carbon nanotubes. Nature, 2003, 424: 408~410
[27] Baughman RH, Cui CX, Zakhidov AA, et al. Carbon nanotube actuators. Science, 1999, 284: 1340~1344
[28] Guo W, Guo Y. Giant Axial Electrostrictive Deformation in Carbon Nanotubes. Phys. Rev. Lett., 2003, 91: 115501
[29] Zheng Q, Jiang Q. Multiwalled Carbon Nanotubes as Gigahertz Oscillators. Physical Review Letters, 2002, 88(4): 045503
[30] Zheng Q, Liu JZ, Jiang Q. Excess van der Waals interaction energy of a multiwalled carbon nanotube with an extruded core and the induced core oscillation. Physical Review B, 2002, 65: 245409
[31] Guo W, Guo Y, Gao H, Zheng Q, Zhong W. Energy dissipation in gigahertz oscillators from multiwalled carbon nanotubes. Phys Rev Lett, 2003, 91: 125501
[32] Berber S, Kwon YK, Tománek D. Unusually High Thermal Conductivity of Carbon Nanotubes. Phys. Rev. Lett., 2000, 84: 4613~4616
[33] Che J, Cagin T, Goddard, WA. Thermal conductivity of carbon nanotubes. Nanotechnology, 2000, 11: 65~69
[34] Rinzler AG, Hafner JH, Nikolaev P, Lou L, Kim SG, Tománek D, Nordlander P, Colbert DT, Smalley RE. Unraveling Nanotubes: Field Emission from an Atomic Wire. Science, 1995, 269: 1550~1553
[35] Dillon AC, Jones KM, Bekkendahl TA, Kiang CH, Bethune DS, Heben MJ. Storage of hydrogen in single-walled carbon nanotubes. Nature, 1997, 386: 377~379
[36] Li YH, Wang S, Cao A, Zhao D, Zhang X, Xu C, Luan Z, Ruan D, Liang J, Wu D, Wei B.Adsorption of fluoride from water by amorphous alumina supported on carbon nanotubes. Chem. Phys. Lett., 2001, 350: 412~416
[37] Qian D, Wagner GJ, Liu WK, Yu MF, Ruoff RS. Mechanics of carbon nanotubes. Appl. Mech. Rev., 2002, 55 (2): 495~533
[38] 张田忠, 郭万林. 纳米力学的数值模拟方法. 力学进展, 2002, 32(2): 175~187
[39] 文玉华, 朱如曾, 周富信, 王崇愚. 分子动力学模拟的主要技术. 力学进展, 2003, 33(1): 65~73
[40] Rafii-Tabar H. Computational modelling of thermo-mechanical and transport properties of carbon nanotubes. Physics Reports, 2004, 390: 235~452
[41] 杨卫, 马新玲, 王洪涛, 洪伟. 纳米力学进展. 力学进展, 2002, 32(2): 161~174
[42] 杨卫, 马新玲, 王洪涛, 洪伟. 纳米力学进展(续). 力学进展, 2003, 33(2): 175~186
[43] Ohno K, Esfarjani K, Kawazoe Y. Computational Material Science: From ab initio to Monte Carlo Methods, Springer, 1999
[44] Haile JM. Molecular Dynamics Simulation: Elementary Methods. New York: Wiley, 1992
[45] Yakobson BI, Brabec CJ, Bernholc J. Nanomechanics of carbon tubes: instabilities beyond linear Response. Physical Review Letters, 1996, 76: 2511~2514.
[46] Ru CQ. Effective Bending Stiffness of Carbon Nanotubes. Phys. Rev. B, 2000, 62: 9973~9976
[47] Ru CQ. Degraded Axial Buckling Strain of Multiwalled Carbon Nanotubes Due to Interlayer Slips. J. Appl. Phys., 2001, 89: 3426~3433
[48] Ru CQ. Axially compressed buckling of a doublewalled nanotube embedded within an elastic medium. Journal of the Mechanics and Physics of Solids, 2001, 49: 1265~1279
[49] Yoon J, Ru CQ, Mioduchowski A. Sound wave propagation in multiwall carbon nanotubes. Journal of Applied Physics, 2003, 93: 4801~4806
[50] Falvo MR, Clary GJ, Taylor RM, Chi V, Brooks F P, Washburn S, Superfine R. Bending and Buckling of Carbon Nanotubes Under Large Strain. Nature, 1997, 389: 582~584
[51] Yoon J, Ru CQ, Mioduchowski A. Non-coaxial resonance of an isolated multiwall carbon nanotube. Physical Review B, 2002, 66: 233402
[52] Liu JZ, Zheng Q, Jiang Q. Effect of a rippling mode on resonances of carbon nanotubes. Physical Review Letters, 2001, 86: 4843~4846
[53] Yoon J, Ru, CQ, Mioduchowski A. Vibration of an embedded multiwall carbon nanotube. Compos. Sci. Technol., 2003, 63: 1533~1542
[54] Xu KY, Guo XN, Ru CQ. Vibration of a double-walled carbon nanotube aroused by nonlinear intertube van der Waals forces. J. Appl. Phys., 2006, 99: 064303
[55] Chang T, Gao H. Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model nanotubes. Journal of the Mechanics and Physics of Solids,2003, 51: 1059~1074
[56] Shen L, Li J. Transversely isotropic elastic properties of single-walled carbon nanotubes. Physical Review B, 2004, 69: 045414
[57] Xiao JR, Gama BA, Gillespie Jr. An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes. International Journal of Solids and Structures, 2005, 42: 3075~3092
[58] Zhang P, Huang Y, Geubelle PH, Klein PA, Hwang KC. The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials. International Journal of Solids and Structures, 2002, 39: 3893~3906
[59] 郭旭, 王晋宝, 张洪武, 基于高阶 Cauchy-Born 准则的单壁碳纳米管本构模型. 计算力学学报, 22(2): 135~140
[60] Zhang HW, Wang JB, Guo X. Predicting the elastic properties of single-walled carbon nanotubes. Journal of the Mechanics and Physics of Solids, 2005, 53: 1929~1950
[61] Wang JB, Guo X, Zhang HW, Wang L, Liao JB. Energy and mechanical properties of single-walled carbon nanotubes predicted using the higher order Cauchy-Born rule. Physical Review B, 2006, 73: 115428
[62] Zhang P, Huang Y, Gao H, Hwang KC. Fracture nucleation in single-wall carbon nanotubes under tension: a continuum analysis incorporating interatomic potentials. Journal of Applied Mechanics-Transactions of the ASME, 2002, 69: 454~458
[63] Li CY, Chou TW. A structural mechanics approach for the analysis of carbon nanotubes. International Journal of Solids and Structures, 2003, 40: 2487~2499
[64] Li C, Chou TW. Elastic moduli of multi-walled carbon nanotubes and the effect of van der Waals forces. Composites Science and Technology, 2003, 63: 1517~1524
[65] Li CY, Chou TW. Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators. Phys. Rev. B, 2003, 68: 073405
[66] Li CY, Chou TW. Elastic properties of single-walled carbon nanotubes in transverse directions. Phys. Rev. B, 2004, 69: 073401
[67] Li CY, Chou TW. Modeling of elastic buckling of carbon nanotubes by molecular structural mechanics approach. Mechanics of Materials, 2004, 36(11): 1047~1055
[68] Li CY, Chou TW. Vibrational behaviors of multiwalled-carbon-nanotube-based nanomechanical resonators. Applied Physics Letters, 2004, 84(1): 121~123
[69] Tadmor EB, Ortiz M, Phillips R. Quasicontinuum analysis of defects in solids. Philos. Mag. A, 1996, 73 (6): 1529~1563
[70] Liu B, Huang Y, Jiang H, Qu S, Hwang KC. The atomic-scale finite element method. Comput. Methods Appl. Mech. Engrg., 2004, 193: 1849~1864
[71] Liu B, Jiang H, Huang Y, Qu S, Yu MF, Hwang KC. Atomic-scale finite element method inmultiscale computation with applications to carbon nanotubes. Phys. Rev. B, 2005, 72: 035435
[72] Qian D, Wagner GJ, Liu WK. A multiscale projection method for the analysis of carbon nanotubes.Comput. Methods Appl. Mech. Engrg., 2004, 193: 1603~1632
[73] 张田忠. 单壁碳纳米管屈曲的原子/连续介质混合模型. 力学学报, 2004, 36(6): 744~748
[74] Popov VN, Van Doren VE. Elastic properties of single-walled carbon nanotubes. Physical Review B, 61(4): 3078~3084
[75] Zienkiewicz OC. The finite element method. London: McGraw-Hill,1991
[76] Rappe AK, Casewit CJ. Molecular Mechanics Across Chemistry. Sausalito, California: University Science Books, 1997
[77] Rappe AK, Casewit CJ, Colwell KS, Goddard WA. UFF, A full periodic-table force-field for molecular mechanics and molecular dynamics simulations. Journal of American Chemical Society, 1992, 114: 10024~10035
[78] Odegard GM, Gates TS, Nicholson LM, Wise KE. Equivalent-continuum modeling with application to carbon nanotubes. NASA/TM-2002-211454
[79] Timoshenko, S, Strength of materials: part 1: Elementary theory and problems (third edition). New York: Van Nostrand Reinhold Company, 1955
[80] Weaver Jr., W., Gere, J.M. Matrix analysis of framed structures, second ed. New York: D. Van Nostrand Company, 1980
[81] Lu, JP. Elastic properties of carbon nanotubes and nanoropes. Physical Review Letters, 1997, 79: 1297~1300
[82] Gülseren, O, Yildirim, T, Ciraci, S. Systematic ab initio study of curvature effects in carbon nanotubes. Physical Review B, 2002, 65: 153405
[83] Blakslee OL, Proctor DG, Seldin EJ, Spence GB, Weng T. Elastic constants of compression-annealed pyrolytic graphite. J. Appl. Phys., 1970, 41: 3373~3382
[84] Kelly BT. Physics of Graphite. London: Applied Science, 1981
[85] Timoshenko, S, Woinowsky-Krieger, S. Theory of plates and shells (second edition). New York : McGraw-Hill, 1959
[86] Cornell WD, Cieplak P, Bayly CI, et al. A second generation force-field for the simulation of proteins, nucleic-acids, and organic-molecules. Journal of American Chemical Society, 1995, 117: 5179~5197
[87] Jorgensen WL, Severance DL. Aromatic aromatic interactions––free-energy profiles for the benzene dimer in water, chloroform, and liquid benzene. Journal of American Chemical Society, 1990, 112: 4768~4774
[88] Van Lier G, Van Alsenoy C, Van Doren V, Geerlings P. Ab initio study of the elasticproperties of single-walled carbon nanotubes and graphene. Chemical Physics Letters, 2000, 326: 181~185
[89] Hernández E, Goze C, Bernier P, Rubio A. Elastic properties of C and BxCyNz composite nanotubes. Physical Review Letters, 1998, 80: 4502~4505
[90] Wang L, Zheng Q, Liu JZ, Jiang Q. Size dependence of the thin-shell model for carbon nanotubes. Physical Review Letters, 2005, 95: 105501
[91] Odegard GM, Gates TS, Nicholson LM, Wise KE. Equivalent-continuum modeling of nano-structured materials. Composites Science and Technology, 2002, 62: 1869~1880
[92] Liew KM, Wong CH, He XQ, Tan MJ, Meguid SA. Nanomechanics of single and multiwalled carbon nanotubes. Phys. Rev. B, 2004, 69: 115429
[93] Treacy MMJ, Ebbesen TW, Gibson JM. Exceptionally high Young's modulus observed for individual carbon nanotubes. Nature, 1996, 381: 678~680
[94] Lourie O, Wagner HD. Evaluation of Young’s modulus of carbon nanotubes by micro-Raman spectroscopy. J. Mater. Res., 1998, 13: 2418~2422
[95] Wong EW, Sheehan PE, Lieber CM. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science, 1997, 277: 1971~1975
[96] Tombler TW, Zhou C, Alexseyev L, Kong J, Dai H, Liu L, Jayanthi CS, Tang M, Wu SY. Reversible electromechanical characteristics of carbon nanotubes under local-probe manipulation. Nature, 2000, 405: 769~772
[97] Sánchez-Portal D, Artacho E, Soler JM, Rubio A, Ordejon P. Ab initio structural, elastic, and vibrational properties of carbon nanotubes. Phys. Rev. B, 1999, 59: 12678~12688
[98] Zhou G, Duan W, Gu B. First-principles study on morphology and mechanical properties of single-walled carbon nanotube. Chemical Physics Letters, 2001, 333: 344~349
[99] Goze C, Vaccarini L, Henrard L, Bernier P, Hernández E, Rubio A. Elastic and mechanical properties of carbon nanotubes. Synthetic Metals, 1999, 103: 2500~2501
[100] Vaccarini L, Goze C, Henrard L, Hernández E, Bernier P, Rubio A. Mechanical and electronic properties of carbon and boron-nitride nanotubes. Carbon, 2000, 38: 1681~1690
[101] Cai J, Bie RF, Tan XM, Lu C. Application of the tight-binding method to the elastic modulus of C60 and carbon nanotube. Physica B, 2004, 344: 99~102
[102] Robertson DH, Brenner DW, Mintmire JW. Energetics of nanoscale graphitic tubules. Physical Review B, 1992, 45: 12592~12595
[103] Bao WX, Zhu CC, Cui WZ. Simulation of Young’s modulus of single-walled carbon nanotubes by molecular dynamics. Physica B, 2004, 352: 156~163
[104] Popov VN, Van Doren VE, Balkanski M. Lattice dynamics of single-walled carbon nanotubes. Physical Review B, 1999, 59: 8355~8358
[105] Brenner DW. Empirical potential for hydrocarbons for use in simulation the chemicalvapor deposition of diamond films. Physical Review B, 1990, 42: 9458~9471
[106] 李海军, 郭万林. 单壁碳纳米管的等效梁单元有限元模型. 力学学报, 2006, 38(4): 488~495
[107] Chang T, Geng J, Guo X. Chirality- and size- dependent elastic properties of single-walled carbon nanotubes. Appl. Phys. Lett., 2005, 87: 251929
[108] Chang T, Geng J, Guo X. Prediction of chirality- and size-dependent elastic properties of single-walled carbon nanotubes via a molecular mechanics model. Proc. R. Soc. A, 2006, 462: 2523~2540
[109] 丁锡洪. 结构力学. 北京: 航空工业出版社, 1991
[110] Hall AR, An L, Liu J, Vicci L, Falvo MR, Superfine R, Washburn S. Experimental measurement of single-wall carbon nanotube torsional properties. Physical Review Letters, 2006, 96: 256102
[111] Leung AYT, Wu Y, Zhong W. Computation of Young’s moduli for chiral single-walled carbon nanotubes. Applied Physics Letters, 2006, 88: 251908
[112] Brenner DW, Shenderova OA, Harrison JA, Stuart SJ, Ni B, Sinnott SB. A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J. Phys.: Condens. Matter, 2002, 14: 783~802
[113] Allinger NL. Conformational analysis. 130. MM2. A hydrocarbon force field utilizing V1 and V2 torsional terms. J. Am. Chem. Soc., 1977, 99(25): 8127~8134
[114] Allinger NL, Yuh YH, Lii JH. Molecular mechanics. The MM3 force field for hydrocarbons. 1. J. Am. Chem. Soc., 1989, 111(23): 8551~8566
[115] Tersoff J. New empirical-model for the structural-properties of silicon. Phys. Rev. Lett., 1986, 56(6): 632~635
[116] Tersoff J. New empirical-approach for the structure and energy of covalent systems. Phys. Rev. B, 1988, 37(12): 6991~7000
[117] Tersoff J. Empirical interatomic potential for carbon, with applications to amorphous-carbon. Phys. Rev. Lett., 1988, 61(25): 2879~2882
[118] Tersoff J. Modeling solid-state chemistry-interatomic potentials for multicomponent systems. Phys. Rev. B, 1989, 39(8): 5566~5568
[119] Jones JE. On the Determination of Molecular Fields. I. From the Variation of the Viscosity of a Gas with temperature. Proc. Roy. Soc., 1924, 106: 441~462
[120] Jones JE. On the Determination of Molecular Fields. II. From the Equation of State of a Gas. Proc. Roy. Soc., 1924, 106: 463~477
[121] Ercolessi F. A molecular dynamics primer. http://www.fisica.uniud.it/~ercolessi/md/md/
[122] Wang Y, Tomanek D, and Bertsch GF. Stiffness of a solid composed of c60 clusters. Phys. Rev. B, 1991, 44(12), 6562~6565
[123] Kolmogorov AN, Crespi VH. Smoothest bearings: Interlayer sliding in multiwalled carbon nanotubes. Phys. Rev. Lett., 2000, 85(22), 4727~4730
[124] Kolmogorov AN, Crespi VH, Schleier-Smith MH, Ellenbogen JC. Nanotube-Substrate Interactions: Distinguishing Carbon Nanotubes by the Helical Angle. Phys. Rev. Lett., 2004, 92: 085503
[125] Popov VN, Henrard L. Breathinglike phonon modes of multiwalled carbon nanotubes. Phys. Rev. B, 2002, 65: 235415
[126] He XQ, Kitipornchai S, Wang CM, Liew KM. Modeling of van der Waals force for infinitesimal deformation of multi-walled carbon nanotubes treated as cylindrical shells. International Journal of Solids and Structures, 2005, 42: 6032~6047
[127] Chang T, Guo W, Guo X. Buckling of multiwalled carbon nanotubes under axial compression and bending via a molecular mechanics model. Phys. Rev. B, 2005, 72: 064101
[128] Lu JP, Yang W. The shape of large single- and multiple-shell fullerenes. Phys. Rev. B, 1994, 49: 11421
[129] Girifalco LA, Lad RA. Energy of cohesion, compressibility and the potential energy functions of the graphite system. J. Chem. Phys., 1956, 25(4), 693~697
[130] Vodenitcharova L, Zhang LC. Mechanism of bending with kinking of a single-walled carbon nanotube. Phys. Rev. B, 2004, 69: 115410
[131] 张田忠, 郭兴明. 径向压缩单壁碳纳米管的力学行为. 力学学报, 2005, 37(4): 408~412
[132] Shen L, Li J.Transversely isotropic elastic properties of multiwalled carbon nanotubes. Phys. Rev. B, 2005, 71: 035412
[133] Iijima S, Brabec C, Maiti A, Bernholc J. Structural flexibility of carbon nanotubes. J. Chem. Phys. 1996, 104: 2089~2092
[134] Poncharal P, Wang ZL, Ugarte D, der Herr WA. Electrostatic Deflections and Electromechanical Resonances of Carbon Nanotubes. Science, 1999, 283: 1513~1516
[135] Postma HWC, der Jonge M, Yao Z, Dekker C. Electrical transport through carbon nanotube junctions created by mechanical manipulation. Phys. Rev. B, 2000, 62: R10653~R10656
[136] Postma HWC, Teepen T, Yao Z, Grifoni M, Dekker C. Carbon Nanotube Single-Electron Transistors at Room Temperature. Science, 2001, 293: 76~79
[137] Ru CQ. Elastic buckling of single-walled carbon nanotube ropes under high pressure. Phys. Rev. B, 2000, 62: 10405~10408
[138] Ru CQ. Column buckling of multiwalled carbon nanotubes with interlayer radial displacements. Phys. Rev. B, 2000, 62: 16962~16967
[139] Ru CQ. Effect of van der Waals forces on axial buckling of a double-walled carbon nanotube. J. Appl. Phys. 2000, 87: 7227~7231
[140] Chang T, Li G, Guo X. Elastic axial buckling of carbon nanotubes via a molecular mechanics model. Carbon, 2005, 43: 287~294
[141] Arroyo M, Belytschko T. Large deofrmation atomistic-based continuum analysis of carbon nanotubes. AIAA-2002-1317, 1~10
[142] Li C, Guo W. Continuum mechanics simulation of post-buckling of single-walled nanotubes. Int. J. Nonlinear Sci. Numer. Simul, 2003, 4: 387~393
[143] Leung AYT, Guo X, He XQ, Jiang H, Huang Y. Postbuckling of carbon nanotubes by atomic-scale finite element. J. Appl. Phys., 2006, 99: 124308
[144] Buehler MJ, Kong Y, Gao H. Deformation Mechanisms of Very Long Single-Wall Carbon Nanotubes Subject to Compressive Loading. Journal of Engineering Materials and Technology-Transactions of the ASME, 2004, 126: 245~249
[145] Cao G, Chen X. Buckling of single-walled carbon nanotubes upon bending: Molecular dynamics simulations and finite element method. Phys. Rev. B, 2006, 73: 155435
[146] ABAQUS Analysis User's Manual. ABAQUS Version 6.5 Documentation. ABAQUS, Inc
[147] ABAQUS Theory Manual. ABAQUS Version 6.5 Documentation. ABAQUS, Inc
[148] 朱伯芳. 有限单元法原理与应用. 北京: 中国水利水电出版社, 1979
[149] Gibson RF, Ayorinde EO, Wen YF. Vibrations of carbon nanotubes and their composites: A review. Composites Science and Technology, 2006 (in press)
[150] Rao AM, Richter E, Bandow S, et al. Diameter-selective Raman scattering from vibrational modes in carbon nanotubes. Science, 1997, 275: 187~191
[151] Popov VN, Lambin P. Radius and chirality dependence of the radial breathing mode and the G-band phonon modes of single-walled carbon nanotubes. Phys. Rev. B, 2006, 73: 085407
[152] Kahn D, Lu JP. Vibrational modes of carbon nanotubes and nanoropes. Phys. Rev. B, 1999, 60: 6535~6540
[153] Legoas SB, Coluci VR, Braga SF, et al. Molecular-Dynamics Simulations of Carbon Nanotubes as Gigahertz Oscillators. Physical Review Letters , 2003, 90: 055504
[154] Yoon J, Ru CQ, Mioduchowski A. Terahertz vibration of short carbon nanotubes modeled as Timoshenko beams. J. Appl. Mech., 2005, 72: 10
[155] Wang CY, Ru CQ, Mioduchowski A. Free vibration of multiwall carbon nanotubes. J. Appl. Phys., 2005, 97: 114323
[156] Wang CY, Ru CQ, Mioduchowski A. Pressure effect on radial breathing modes of multiwall carbon nanotubes. J. Appl. Phys., 2005, 97: 024310
[157] Li C, Chou TW. Mass detection using carbon nanotube-based nanomechanical resonators.Appl. Phys. Lett., 2004, 84: 5246~5248
[158] Li C, Chou TW. Vibrational behaviors of multiwalled-carbon-nanotube-based nanomechanical resonators. Appl. Phys. Lett., 2004, 84: 121~123
[159] Jorio A, Saito R, Hafner JH, et al. Structural (n, m) Determination of Isolated Single-Wall Carbon Nanotubes by Resonant Raman Scattering. Phys. Rev. Lett., 2001, 86: 1118~1121
[160] Zhang YY, Zhang J, Son H, Kong J, Liu Z. Substrate-Induced Raman Frequency Variation for Single-Walled Carbon Nanotubes. J. Am. Chem. Soc., 2005, 127: 17156~17157
[161] Jishi RA, Venkataraman L, Dresselhaus MS, Dresselhaus G. Phonon modes in carbon nanotubules. Chem. Phys. Lett., 1993, 209: 77~82
[162] Kürti J, Kresse G, Kuzmany H. First-principles calculations of the radial breathing mode of single-wall carbon nanotubes. Phys. Rev. B, 1998, 58: R8869~R8872
[163] Lawler HM, Areshkin D, Mintmire JW, White CT. Radial-breathing mode frequencies for single-walled carbon nanotubes of arbitrary chirality: First-principles calculations. Phys. Rev. B, 2005, 72: 233403
[164] Dresselhaus MS, Eklund PC. Phonons in carbon nanotubes. Advances In Physics, 2000, 49: 705~814
[165] Dresselhaus MS, Dresselhaus G, Saito R, Jorio A. Raman spectroscopy of carbon nanotubes. Physics Reports, 2005, 409: 47~99
[166] Alon OE. Number of Raman- and infrared-active vibrations in single-walled carbon nanotubes. Phys. Rev. B, 2001, 63: 201403
[167] Zhao X, Ando Y, Qin LC, Kataura H, Maniwa Y, Satio R. Radial breathing modes of multiwalled carbon nanotubes. Chem. Phys. Lett., 2002, 361: 169~174
[168] Craighead HG. Nanoelectromechanical Systems. Science, 2000, 290: 1532~1535
[169] Roukes ML. Nanoelectromechanical systems face the future. Phys. World, 2001, 14: 25~31
[170] Ekinci KL, Roukes ML. Nanoelectromechanical systems. Rev. Sci. Instrum., 2005, 76: 061101
[171] Huang XMH, Zorman CA, Mehregany M, Roukes ML. Nanoelectromechanical systems:Nanodevice motion at microwave frequencies. Nature, 2003, 421: 496
[172] Cumings J, Zettl A. Low-Friction Nanoscale Linear Bearing Realized from Multiwall Carbon Nanotubes. Science, 2000, 289: 602~605
[173] Forró L. Nanotechnology: Beyond Gedanken Experiments. Science, 2000, 289: 560~561
[174] Casimir HBG. On the attraction between two perfectly conducting plates. Proc. Kon. Ned. Akad. Wet., 1948, 51: 793~796
[175] Smythe WR. Static and dynamic electricity. New York: McGraw-Hill, 1968
[176] Bordag M, Mohideen U, Mostepanenko VM. New developments in the Casimir effect.Physics Reports, 2001, 353: 1~205
[177] Mazzitelli FD, Sánchez MJ, Scoccola NN. Casimir interaction between two concentric cylinders: Exact versus semiclassical results. Physical Review A, 2003, 67: 013807
[178] Jiang WZ, Wang ZX, Fu DJ, et al. Casimir energy of concentric and infinite cylindrical shells of perfect conductor. Physics Letters A, 2003, 315: 273~279
[2] 张立德, 牟季美. 纳米材料和纳米结构. 北京: 科学出版社, 2001
[3] 朱宏伟, 吴德海, 徐才录. 碳纳米管. 北京: 机械工业出版社, 2003
[4] 成会明. 纳米碳纳米管: 制备、结构、物性及应用. 北京: 化学工业出版社, 2002
[5] Kroto HW, Heath JR, O'Brien SC, Curl RF, Smalley RE. C60: Buckminsterfullerene. Nature, 1985, 318: 162~163
[6] Smalley RE. Discovering the fullerenes. Reviews of Modern Physics, 1997, 69: 723~730
[7] Iijima S. Helical microtubules of graphitic carbon. Nature, 1991, 354: 56~58
[8] Iijima S, Ichihashi T. Single-shell carbon nanotubes of 1-nm diameter. Nature, 1993, 363: 603~605
[9] Bethune DS, Kiang CH, DeVries MS, Gorman G, Savoy R, Beyers R. Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls. Nature, 1993, 363: 605~607
[10] Smith BW, Monthioux M, Luzzi DE. Encapsulated C60 in carbon nanotubes. Nature, 1998, 396: 323~324
[11] Li J, Papadopoulos C, Xu J. Nanoelectronics: Growing Y-junction carbon nanotubes. Nature, 1999, 402: 253~254
[12] Zhang G, Jiang X, Wang E. Tubular Graphite Cones. Science, 2003, 300: 472~474
[13] Barros EB, Jorio A, Samsonidze GG, et al. Review on the symmetry-related properties of carbon nanotubes. Physics Reports, 2006, 431: 261~302
[14] Dresselhaus MS, Dresselhaus G, Saito R. Physics of carbon nanotubes. Carbon, 1995, 33(7): 883~891
[15] Thostensona ET, Ren Z, Chou TW. Advances in the science and technology of carbon nanotubes and their composites: a review. Comput. Sci. Techol., 2001, 61: 1899~1912
[16] Jiang H, Zhang P, Liu B, Huang Y, Geubelle PH, Gao H, Hwang KC. The effect of nanotube radius on the constitutive model for carbon nanotubes. Computational Materials Science, 2003, 28: 429~442
[17] Baughman RH, Zakhidov AA, de Heer WA. Carbon Nanotubes—the Route Toward Applications. Science, 2002, 297(2): 787~792
[18] Chico L, Crespi VH, Benedict LX, Louie SG, Cohen ML. Pure Carbon Nanoscale Devices: Nanotube Heterojunctions. Phys. Rev. Lett., 1996, 76: 971~974
[19] Kociak M, Kasumov AY, Guéron1 S, Reulet B, Khodos II, Gorbatov YB, Volkov VT, Vaccarini L, Bouchiat H. Superconductivity in Ropes of Single-Walled Carbon Nanotubes.Phys. Rev. Lett., 2001, 86: 2416~2419
[20] 刘哲. 碳纳米管若干力学问题的研究[博士学位论文]. 北京: 清华大学, 2002
[21] Yu MF, Lourie O, Dyer MJ, Moloni K, Kelly TF, Ruoff RS. Strength and breaking mechanism of multiwalled Carbon Nanotubes Under Tensile Load. Science, 2000, 287: 637~640
[22] Lourie O, Cox DM, Wagner HD. Buckling and Collapse of Embedded Carbon Nanotubes. Phys. Rev. Lett., 1998, 81: 1638~1641
[23] Shen W, Jiang B, Han BS, Xie S. Investigation of the Radial Compression of Carbon Nanotubes with a Scanning Probe Microscope. Phys. Rev. Lett., 2000 , 84: 3634~3637
[24] Dai H, Hafner JH, Rinzler AG, Colbert DT, Smalley RE. Nanotubes as nanoprobes in scanning probe microscopy. Nature, 1996, 384: 147~150
[25] Wang ZL, Poncharal WA, de Heer WA. Measuring physical and mechanical properties of individual carbon nanotubes by in situ TEM. J. Phys. Chem. Solids, 2000, 61: 1025~1030
[26] Fennimore AM, Yuzvinsky TD, Han WQ, Fuhrer MS, Cumings J, Zettl A. Rotational actuators based on carbon nanotubes. Nature, 2003, 424: 408~410
[27] Baughman RH, Cui CX, Zakhidov AA, et al. Carbon nanotube actuators. Science, 1999, 284: 1340~1344
[28] Guo W, Guo Y. Giant Axial Electrostrictive Deformation in Carbon Nanotubes. Phys. Rev. Lett., 2003, 91: 115501
[29] Zheng Q, Jiang Q. Multiwalled Carbon Nanotubes as Gigahertz Oscillators. Physical Review Letters, 2002, 88(4): 045503
[30] Zheng Q, Liu JZ, Jiang Q. Excess van der Waals interaction energy of a multiwalled carbon nanotube with an extruded core and the induced core oscillation. Physical Review B, 2002, 65: 245409
[31] Guo W, Guo Y, Gao H, Zheng Q, Zhong W. Energy dissipation in gigahertz oscillators from multiwalled carbon nanotubes. Phys Rev Lett, 2003, 91: 125501
[32] Berber S, Kwon YK, Tománek D. Unusually High Thermal Conductivity of Carbon Nanotubes. Phys. Rev. Lett., 2000, 84: 4613~4616
[33] Che J, Cagin T, Goddard, WA. Thermal conductivity of carbon nanotubes. Nanotechnology, 2000, 11: 65~69
[34] Rinzler AG, Hafner JH, Nikolaev P, Lou L, Kim SG, Tománek D, Nordlander P, Colbert DT, Smalley RE. Unraveling Nanotubes: Field Emission from an Atomic Wire. Science, 1995, 269: 1550~1553
[35] Dillon AC, Jones KM, Bekkendahl TA, Kiang CH, Bethune DS, Heben MJ. Storage of hydrogen in single-walled carbon nanotubes. Nature, 1997, 386: 377~379
[36] Li YH, Wang S, Cao A, Zhao D, Zhang X, Xu C, Luan Z, Ruan D, Liang J, Wu D, Wei B.Adsorption of fluoride from water by amorphous alumina supported on carbon nanotubes. Chem. Phys. Lett., 2001, 350: 412~416
[37] Qian D, Wagner GJ, Liu WK, Yu MF, Ruoff RS. Mechanics of carbon nanotubes. Appl. Mech. Rev., 2002, 55 (2): 495~533
[38] 张田忠, 郭万林. 纳米力学的数值模拟方法. 力学进展, 2002, 32(2): 175~187
[39] 文玉华, 朱如曾, 周富信, 王崇愚. 分子动力学模拟的主要技术. 力学进展, 2003, 33(1): 65~73
[40] Rafii-Tabar H. Computational modelling of thermo-mechanical and transport properties of carbon nanotubes. Physics Reports, 2004, 390: 235~452
[41] 杨卫, 马新玲, 王洪涛, 洪伟. 纳米力学进展. 力学进展, 2002, 32(2): 161~174
[42] 杨卫, 马新玲, 王洪涛, 洪伟. 纳米力学进展(续). 力学进展, 2003, 33(2): 175~186
[43] Ohno K, Esfarjani K, Kawazoe Y. Computational Material Science: From ab initio to Monte Carlo Methods, Springer, 1999
[44] Haile JM. Molecular Dynamics Simulation: Elementary Methods. New York: Wiley, 1992
[45] Yakobson BI, Brabec CJ, Bernholc J. Nanomechanics of carbon tubes: instabilities beyond linear Response. Physical Review Letters, 1996, 76: 2511~2514.
[46] Ru CQ. Effective Bending Stiffness of Carbon Nanotubes. Phys. Rev. B, 2000, 62: 9973~9976
[47] Ru CQ. Degraded Axial Buckling Strain of Multiwalled Carbon Nanotubes Due to Interlayer Slips. J. Appl. Phys., 2001, 89: 3426~3433
[48] Ru CQ. Axially compressed buckling of a doublewalled nanotube embedded within an elastic medium. Journal of the Mechanics and Physics of Solids, 2001, 49: 1265~1279
[49] Yoon J, Ru CQ, Mioduchowski A. Sound wave propagation in multiwall carbon nanotubes. Journal of Applied Physics, 2003, 93: 4801~4806
[50] Falvo MR, Clary GJ, Taylor RM, Chi V, Brooks F P, Washburn S, Superfine R. Bending and Buckling of Carbon Nanotubes Under Large Strain. Nature, 1997, 389: 582~584
[51] Yoon J, Ru CQ, Mioduchowski A. Non-coaxial resonance of an isolated multiwall carbon nanotube. Physical Review B, 2002, 66: 233402
[52] Liu JZ, Zheng Q, Jiang Q. Effect of a rippling mode on resonances of carbon nanotubes. Physical Review Letters, 2001, 86: 4843~4846
[53] Yoon J, Ru, CQ, Mioduchowski A. Vibration of an embedded multiwall carbon nanotube. Compos. Sci. Technol., 2003, 63: 1533~1542
[54] Xu KY, Guo XN, Ru CQ. Vibration of a double-walled carbon nanotube aroused by nonlinear intertube van der Waals forces. J. Appl. Phys., 2006, 99: 064303
[55] Chang T, Gao H. Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model nanotubes. Journal of the Mechanics and Physics of Solids,2003, 51: 1059~1074
[56] Shen L, Li J. Transversely isotropic elastic properties of single-walled carbon nanotubes. Physical Review B, 2004, 69: 045414
[57] Xiao JR, Gama BA, Gillespie Jr. An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes. International Journal of Solids and Structures, 2005, 42: 3075~3092
[58] Zhang P, Huang Y, Geubelle PH, Klein PA, Hwang KC. The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials. International Journal of Solids and Structures, 2002, 39: 3893~3906
[59] 郭旭, 王晋宝, 张洪武, 基于高阶 Cauchy-Born 准则的单壁碳纳米管本构模型. 计算力学学报, 22(2): 135~140
[60] Zhang HW, Wang JB, Guo X. Predicting the elastic properties of single-walled carbon nanotubes. Journal of the Mechanics and Physics of Solids, 2005, 53: 1929~1950
[61] Wang JB, Guo X, Zhang HW, Wang L, Liao JB. Energy and mechanical properties of single-walled carbon nanotubes predicted using the higher order Cauchy-Born rule. Physical Review B, 2006, 73: 115428
[62] Zhang P, Huang Y, Gao H, Hwang KC. Fracture nucleation in single-wall carbon nanotubes under tension: a continuum analysis incorporating interatomic potentials. Journal of Applied Mechanics-Transactions of the ASME, 2002, 69: 454~458
[63] Li CY, Chou TW. A structural mechanics approach for the analysis of carbon nanotubes. International Journal of Solids and Structures, 2003, 40: 2487~2499
[64] Li C, Chou TW. Elastic moduli of multi-walled carbon nanotubes and the effect of van der Waals forces. Composites Science and Technology, 2003, 63: 1517~1524
[65] Li CY, Chou TW. Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators. Phys. Rev. B, 2003, 68: 073405
[66] Li CY, Chou TW. Elastic properties of single-walled carbon nanotubes in transverse directions. Phys. Rev. B, 2004, 69: 073401
[67] Li CY, Chou TW. Modeling of elastic buckling of carbon nanotubes by molecular structural mechanics approach. Mechanics of Materials, 2004, 36(11): 1047~1055
[68] Li CY, Chou TW. Vibrational behaviors of multiwalled-carbon-nanotube-based nanomechanical resonators. Applied Physics Letters, 2004, 84(1): 121~123
[69] Tadmor EB, Ortiz M, Phillips R. Quasicontinuum analysis of defects in solids. Philos. Mag. A, 1996, 73 (6): 1529~1563
[70] Liu B, Huang Y, Jiang H, Qu S, Hwang KC. The atomic-scale finite element method. Comput. Methods Appl. Mech. Engrg., 2004, 193: 1849~1864
[71] Liu B, Jiang H, Huang Y, Qu S, Yu MF, Hwang KC. Atomic-scale finite element method inmultiscale computation with applications to carbon nanotubes. Phys. Rev. B, 2005, 72: 035435
[72] Qian D, Wagner GJ, Liu WK. A multiscale projection method for the analysis of carbon nanotubes.Comput. Methods Appl. Mech. Engrg., 2004, 193: 1603~1632
[73] 张田忠. 单壁碳纳米管屈曲的原子/连续介质混合模型. 力学学报, 2004, 36(6): 744~748
[74] Popov VN, Van Doren VE. Elastic properties of single-walled carbon nanotubes. Physical Review B, 61(4): 3078~3084
[75] Zienkiewicz OC. The finite element method. London: McGraw-Hill,1991
[76] Rappe AK, Casewit CJ. Molecular Mechanics Across Chemistry. Sausalito, California: University Science Books, 1997
[77] Rappe AK, Casewit CJ, Colwell KS, Goddard WA. UFF, A full periodic-table force-field for molecular mechanics and molecular dynamics simulations. Journal of American Chemical Society, 1992, 114: 10024~10035
[78] Odegard GM, Gates TS, Nicholson LM, Wise KE. Equivalent-continuum modeling with application to carbon nanotubes. NASA/TM-2002-211454
[79] Timoshenko, S, Strength of materials: part 1: Elementary theory and problems (third edition). New York: Van Nostrand Reinhold Company, 1955
[80] Weaver Jr., W., Gere, J.M. Matrix analysis of framed structures, second ed. New York: D. Van Nostrand Company, 1980
[81] Lu, JP. Elastic properties of carbon nanotubes and nanoropes. Physical Review Letters, 1997, 79: 1297~1300
[82] Gülseren, O, Yildirim, T, Ciraci, S. Systematic ab initio study of curvature effects in carbon nanotubes. Physical Review B, 2002, 65: 153405
[83] Blakslee OL, Proctor DG, Seldin EJ, Spence GB, Weng T. Elastic constants of compression-annealed pyrolytic graphite. J. Appl. Phys., 1970, 41: 3373~3382
[84] Kelly BT. Physics of Graphite. London: Applied Science, 1981
[85] Timoshenko, S, Woinowsky-Krieger, S. Theory of plates and shells (second edition). New York : McGraw-Hill, 1959
[86] Cornell WD, Cieplak P, Bayly CI, et al. A second generation force-field for the simulation of proteins, nucleic-acids, and organic-molecules. Journal of American Chemical Society, 1995, 117: 5179~5197
[87] Jorgensen WL, Severance DL. Aromatic aromatic interactions––free-energy profiles for the benzene dimer in water, chloroform, and liquid benzene. Journal of American Chemical Society, 1990, 112: 4768~4774
[88] Van Lier G, Van Alsenoy C, Van Doren V, Geerlings P. Ab initio study of the elasticproperties of single-walled carbon nanotubes and graphene. Chemical Physics Letters, 2000, 326: 181~185
[89] Hernández E, Goze C, Bernier P, Rubio A. Elastic properties of C and BxCyNz composite nanotubes. Physical Review Letters, 1998, 80: 4502~4505
[90] Wang L, Zheng Q, Liu JZ, Jiang Q. Size dependence of the thin-shell model for carbon nanotubes. Physical Review Letters, 2005, 95: 105501
[91] Odegard GM, Gates TS, Nicholson LM, Wise KE. Equivalent-continuum modeling of nano-structured materials. Composites Science and Technology, 2002, 62: 1869~1880
[92] Liew KM, Wong CH, He XQ, Tan MJ, Meguid SA. Nanomechanics of single and multiwalled carbon nanotubes. Phys. Rev. B, 2004, 69: 115429
[93] Treacy MMJ, Ebbesen TW, Gibson JM. Exceptionally high Young's modulus observed for individual carbon nanotubes. Nature, 1996, 381: 678~680
[94] Lourie O, Wagner HD. Evaluation of Young’s modulus of carbon nanotubes by micro-Raman spectroscopy. J. Mater. Res., 1998, 13: 2418~2422
[95] Wong EW, Sheehan PE, Lieber CM. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science, 1997, 277: 1971~1975
[96] Tombler TW, Zhou C, Alexseyev L, Kong J, Dai H, Liu L, Jayanthi CS, Tang M, Wu SY. Reversible electromechanical characteristics of carbon nanotubes under local-probe manipulation. Nature, 2000, 405: 769~772
[97] Sánchez-Portal D, Artacho E, Soler JM, Rubio A, Ordejon P. Ab initio structural, elastic, and vibrational properties of carbon nanotubes. Phys. Rev. B, 1999, 59: 12678~12688
[98] Zhou G, Duan W, Gu B. First-principles study on morphology and mechanical properties of single-walled carbon nanotube. Chemical Physics Letters, 2001, 333: 344~349
[99] Goze C, Vaccarini L, Henrard L, Bernier P, Hernández E, Rubio A. Elastic and mechanical properties of carbon nanotubes. Synthetic Metals, 1999, 103: 2500~2501
[100] Vaccarini L, Goze C, Henrard L, Hernández E, Bernier P, Rubio A. Mechanical and electronic properties of carbon and boron-nitride nanotubes. Carbon, 2000, 38: 1681~1690
[101] Cai J, Bie RF, Tan XM, Lu C. Application of the tight-binding method to the elastic modulus of C60 and carbon nanotube. Physica B, 2004, 344: 99~102
[102] Robertson DH, Brenner DW, Mintmire JW. Energetics of nanoscale graphitic tubules. Physical Review B, 1992, 45: 12592~12595
[103] Bao WX, Zhu CC, Cui WZ. Simulation of Young’s modulus of single-walled carbon nanotubes by molecular dynamics. Physica B, 2004, 352: 156~163
[104] Popov VN, Van Doren VE, Balkanski M. Lattice dynamics of single-walled carbon nanotubes. Physical Review B, 1999, 59: 8355~8358
[105] Brenner DW. Empirical potential for hydrocarbons for use in simulation the chemicalvapor deposition of diamond films. Physical Review B, 1990, 42: 9458~9471
[106] 李海军, 郭万林. 单壁碳纳米管的等效梁单元有限元模型. 力学学报, 2006, 38(4): 488~495
[107] Chang T, Geng J, Guo X. Chirality- and size- dependent elastic properties of single-walled carbon nanotubes. Appl. Phys. Lett., 2005, 87: 251929
[108] Chang T, Geng J, Guo X. Prediction of chirality- and size-dependent elastic properties of single-walled carbon nanotubes via a molecular mechanics model. Proc. R. Soc. A, 2006, 462: 2523~2540
[109] 丁锡洪. 结构力学. 北京: 航空工业出版社, 1991
[110] Hall AR, An L, Liu J, Vicci L, Falvo MR, Superfine R, Washburn S. Experimental measurement of single-wall carbon nanotube torsional properties. Physical Review Letters, 2006, 96: 256102
[111] Leung AYT, Wu Y, Zhong W. Computation of Young’s moduli for chiral single-walled carbon nanotubes. Applied Physics Letters, 2006, 88: 251908
[112] Brenner DW, Shenderova OA, Harrison JA, Stuart SJ, Ni B, Sinnott SB. A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J. Phys.: Condens. Matter, 2002, 14: 783~802
[113] Allinger NL. Conformational analysis. 130. MM2. A hydrocarbon force field utilizing V1 and V2 torsional terms. J. Am. Chem. Soc., 1977, 99(25): 8127~8134
[114] Allinger NL, Yuh YH, Lii JH. Molecular mechanics. The MM3 force field for hydrocarbons. 1. J. Am. Chem. Soc., 1989, 111(23): 8551~8566
[115] Tersoff J. New empirical-model for the structural-properties of silicon. Phys. Rev. Lett., 1986, 56(6): 632~635
[116] Tersoff J. New empirical-approach for the structure and energy of covalent systems. Phys. Rev. B, 1988, 37(12): 6991~7000
[117] Tersoff J. Empirical interatomic potential for carbon, with applications to amorphous-carbon. Phys. Rev. Lett., 1988, 61(25): 2879~2882
[118] Tersoff J. Modeling solid-state chemistry-interatomic potentials for multicomponent systems. Phys. Rev. B, 1989, 39(8): 5566~5568
[119] Jones JE. On the Determination of Molecular Fields. I. From the Variation of the Viscosity of a Gas with temperature. Proc. Roy. Soc., 1924, 106: 441~462
[120] Jones JE. On the Determination of Molecular Fields. II. From the Equation of State of a Gas. Proc. Roy. Soc., 1924, 106: 463~477
[121] Ercolessi F. A molecular dynamics primer. http://www.fisica.uniud.it/~ercolessi/md/md/
[122] Wang Y, Tomanek D, and Bertsch GF. Stiffness of a solid composed of c60 clusters. Phys. Rev. B, 1991, 44(12), 6562~6565
[123] Kolmogorov AN, Crespi VH. Smoothest bearings: Interlayer sliding in multiwalled carbon nanotubes. Phys. Rev. Lett., 2000, 85(22), 4727~4730
[124] Kolmogorov AN, Crespi VH, Schleier-Smith MH, Ellenbogen JC. Nanotube-Substrate Interactions: Distinguishing Carbon Nanotubes by the Helical Angle. Phys. Rev. Lett., 2004, 92: 085503
[125] Popov VN, Henrard L. Breathinglike phonon modes of multiwalled carbon nanotubes. Phys. Rev. B, 2002, 65: 235415
[126] He XQ, Kitipornchai S, Wang CM, Liew KM. Modeling of van der Waals force for infinitesimal deformation of multi-walled carbon nanotubes treated as cylindrical shells. International Journal of Solids and Structures, 2005, 42: 6032~6047
[127] Chang T, Guo W, Guo X. Buckling of multiwalled carbon nanotubes under axial compression and bending via a molecular mechanics model. Phys. Rev. B, 2005, 72: 064101
[128] Lu JP, Yang W. The shape of large single- and multiple-shell fullerenes. Phys. Rev. B, 1994, 49: 11421
[129] Girifalco LA, Lad RA. Energy of cohesion, compressibility and the potential energy functions of the graphite system. J. Chem. Phys., 1956, 25(4), 693~697
[130] Vodenitcharova L, Zhang LC. Mechanism of bending with kinking of a single-walled carbon nanotube. Phys. Rev. B, 2004, 69: 115410
[131] 张田忠, 郭兴明. 径向压缩单壁碳纳米管的力学行为. 力学学报, 2005, 37(4): 408~412
[132] Shen L, Li J.Transversely isotropic elastic properties of multiwalled carbon nanotubes. Phys. Rev. B, 2005, 71: 035412
[133] Iijima S, Brabec C, Maiti A, Bernholc J. Structural flexibility of carbon nanotubes. J. Chem. Phys. 1996, 104: 2089~2092
[134] Poncharal P, Wang ZL, Ugarte D, der Herr WA. Electrostatic Deflections and Electromechanical Resonances of Carbon Nanotubes. Science, 1999, 283: 1513~1516
[135] Postma HWC, der Jonge M, Yao Z, Dekker C. Electrical transport through carbon nanotube junctions created by mechanical manipulation. Phys. Rev. B, 2000, 62: R10653~R10656
[136] Postma HWC, Teepen T, Yao Z, Grifoni M, Dekker C. Carbon Nanotube Single-Electron Transistors at Room Temperature. Science, 2001, 293: 76~79
[137] Ru CQ. Elastic buckling of single-walled carbon nanotube ropes under high pressure. Phys. Rev. B, 2000, 62: 10405~10408
[138] Ru CQ. Column buckling of multiwalled carbon nanotubes with interlayer radial displacements. Phys. Rev. B, 2000, 62: 16962~16967
[139] Ru CQ. Effect of van der Waals forces on axial buckling of a double-walled carbon nanotube. J. Appl. Phys. 2000, 87: 7227~7231
[140] Chang T, Li G, Guo X. Elastic axial buckling of carbon nanotubes via a molecular mechanics model. Carbon, 2005, 43: 287~294
[141] Arroyo M, Belytschko T. Large deofrmation atomistic-based continuum analysis of carbon nanotubes. AIAA-2002-1317, 1~10
[142] Li C, Guo W. Continuum mechanics simulation of post-buckling of single-walled nanotubes. Int. J. Nonlinear Sci. Numer. Simul, 2003, 4: 387~393
[143] Leung AYT, Guo X, He XQ, Jiang H, Huang Y. Postbuckling of carbon nanotubes by atomic-scale finite element. J. Appl. Phys., 2006, 99: 124308
[144] Buehler MJ, Kong Y, Gao H. Deformation Mechanisms of Very Long Single-Wall Carbon Nanotubes Subject to Compressive Loading. Journal of Engineering Materials and Technology-Transactions of the ASME, 2004, 126: 245~249
[145] Cao G, Chen X. Buckling of single-walled carbon nanotubes upon bending: Molecular dynamics simulations and finite element method. Phys. Rev. B, 2006, 73: 155435
[146] ABAQUS Analysis User's Manual. ABAQUS Version 6.5 Documentation. ABAQUS, Inc
[147] ABAQUS Theory Manual. ABAQUS Version 6.5 Documentation. ABAQUS, Inc
[148] 朱伯芳. 有限单元法原理与应用. 北京: 中国水利水电出版社, 1979
[149] Gibson RF, Ayorinde EO, Wen YF. Vibrations of carbon nanotubes and their composites: A review. Composites Science and Technology, 2006 (in press)
[150] Rao AM, Richter E, Bandow S, et al. Diameter-selective Raman scattering from vibrational modes in carbon nanotubes. Science, 1997, 275: 187~191
[151] Popov VN, Lambin P. Radius and chirality dependence of the radial breathing mode and the G-band phonon modes of single-walled carbon nanotubes. Phys. Rev. B, 2006, 73: 085407
[152] Kahn D, Lu JP. Vibrational modes of carbon nanotubes and nanoropes. Phys. Rev. B, 1999, 60: 6535~6540
[153] Legoas SB, Coluci VR, Braga SF, et al. Molecular-Dynamics Simulations of Carbon Nanotubes as Gigahertz Oscillators. Physical Review Letters , 2003, 90: 055504
[154] Yoon J, Ru CQ, Mioduchowski A. Terahertz vibration of short carbon nanotubes modeled as Timoshenko beams. J. Appl. Mech., 2005, 72: 10
[155] Wang CY, Ru CQ, Mioduchowski A. Free vibration of multiwall carbon nanotubes. J. Appl. Phys., 2005, 97: 114323
[156] Wang CY, Ru CQ, Mioduchowski A. Pressure effect on radial breathing modes of multiwall carbon nanotubes. J. Appl. Phys., 2005, 97: 024310
[157] Li C, Chou TW. Mass detection using carbon nanotube-based nanomechanical resonators.Appl. Phys. Lett., 2004, 84: 5246~5248
[158] Li C, Chou TW. Vibrational behaviors of multiwalled-carbon-nanotube-based nanomechanical resonators. Appl. Phys. Lett., 2004, 84: 121~123
[159] Jorio A, Saito R, Hafner JH, et al. Structural (n, m) Determination of Isolated Single-Wall Carbon Nanotubes by Resonant Raman Scattering. Phys. Rev. Lett., 2001, 86: 1118~1121
[160] Zhang YY, Zhang J, Son H, Kong J, Liu Z. Substrate-Induced Raman Frequency Variation for Single-Walled Carbon Nanotubes. J. Am. Chem. Soc., 2005, 127: 17156~17157
[161] Jishi RA, Venkataraman L, Dresselhaus MS, Dresselhaus G. Phonon modes in carbon nanotubules. Chem. Phys. Lett., 1993, 209: 77~82
[162] Kürti J, Kresse G, Kuzmany H. First-principles calculations of the radial breathing mode of single-wall carbon nanotubes. Phys. Rev. B, 1998, 58: R8869~R8872
[163] Lawler HM, Areshkin D, Mintmire JW, White CT. Radial-breathing mode frequencies for single-walled carbon nanotubes of arbitrary chirality: First-principles calculations. Phys. Rev. B, 2005, 72: 233403
[164] Dresselhaus MS, Eklund PC. Phonons in carbon nanotubes. Advances In Physics, 2000, 49: 705~814
[165] Dresselhaus MS, Dresselhaus G, Saito R, Jorio A. Raman spectroscopy of carbon nanotubes. Physics Reports, 2005, 409: 47~99
[166] Alon OE. Number of Raman- and infrared-active vibrations in single-walled carbon nanotubes. Phys. Rev. B, 2001, 63: 201403
[167] Zhao X, Ando Y, Qin LC, Kataura H, Maniwa Y, Satio R. Radial breathing modes of multiwalled carbon nanotubes. Chem. Phys. Lett., 2002, 361: 169~174
[168] Craighead HG. Nanoelectromechanical Systems. Science, 2000, 290: 1532~1535
[169] Roukes ML. Nanoelectromechanical systems face the future. Phys. World, 2001, 14: 25~31
[170] Ekinci KL, Roukes ML. Nanoelectromechanical systems. Rev. Sci. Instrum., 2005, 76: 061101
[171] Huang XMH, Zorman CA, Mehregany M, Roukes ML. Nanoelectromechanical systems:Nanodevice motion at microwave frequencies. Nature, 2003, 421: 496
[172] Cumings J, Zettl A. Low-Friction Nanoscale Linear Bearing Realized from Multiwall Carbon Nanotubes. Science, 2000, 289: 602~605
[173] Forró L. Nanotechnology: Beyond Gedanken Experiments. Science, 2000, 289: 560~561
[174] Casimir HBG. On the attraction between two perfectly conducting plates. Proc. Kon. Ned. Akad. Wet., 1948, 51: 793~796
[175] Smythe WR. Static and dynamic electricity. New York: McGraw-Hill, 1968
[176] Bordag M, Mohideen U, Mostepanenko VM. New developments in the Casimir effect.Physics Reports, 2001, 353: 1~205
[177] Mazzitelli FD, Sánchez MJ, Scoccola NN. Casimir interaction between two concentric cylinders: Exact versus semiclassical results. Physical Review A, 2003, 67: 013807
[178] Jiang WZ, Wang ZX, Fu DJ, et al. Casimir energy of concentric and infinite cylindrical shells of perfect conductor. Physics Letters A, 2003, 315: 273~279