based on the coupled Donnell theory and the Eringen nonlocal elasticity. The following findings are summarized:
• The significance of the small size effects on the natural frequencies of double-walled carbon nanotubes is shown to be dependent on the geometric sizes of CNT and the elastic surrounding medium.
• The small-scale effects in the nonlocal continuum model make small-size CNTs more flexible. In other words, the classical continuum model tends to overestimate the natural frequencies of small size nanotubes, and one must recourse to the nonlocal version to reduce the relative error. As the small-scale parameter increases, the frequencies obtained for the nonlocal shell become smaller than those for its local counterpart.
• The natural frequencies corresponding to higher vibration modes are more sensitive to the small length scale.
• The existence of the elastic medium significantly enhances the values of natural frequencies particularly for lower ones.
• The small size effect becomes more pronounced when the modulus of the elastic medium becomes smaller.
References
[1] Kroto H W, Heath J R, O‘Brien S C, Curl R F 1985 Smalley RE, C60:
buckminsterfullerene Nature. 318 162-163.
[2] Iijima S 1991 Helical microtubes of graphitic carbon Nature 8 354-356.
[3] Kong X Y, Ding Y, Yang R, Wang Z L 2004 Single-Crystal Nanorings Formed by Epitaxial Self-Coiling of Polar Nanobelts. Science. 303 1348-1351.
[4] Iijima S, Brabec C, Maiti A and Bernholc J 1996 Chem. Phys.104 2089
[5] Yakobson B I, Campbell M P, Brabec C J and Bernholc J 1997 Comput. Mater. Sci. 8 241
[6] Hernandez E, Goze C, Bernier P and Rubio A 1998 Phys. Rev. Lett. 80 4502 [7] Sanchez-Portal D et al. 1999 Phys. Rev. B 59 12678
[8] Qian D, Wagner J G, Liu W K, Yu M F and Ruoff R S 2002 Appl. Mech. Rev. 55 495 [9] Yakobson B I, Brabec C J, Bernholc J 1996 Nanomechanics of carbon tubes: instability
beyond linear response Phys. Rev. Lett. 76 2511.
[10] Ru C Q 2001 Axially compressed buckling of a double-walled carbon nanotube embedded in an elastic medium J. Mech. Phys. Solids 49 1265-1279.
[11] Peng J, Wu J, Hwang K C, Song J, Huang Y 2008 Can a single-wall carbon nanotube be modeled as a thin shell? J. Mech. Phys. Solids 56 2213–2224.
[12] Belytschko, T., Xiao, S.P., Schatz, G.C., Ruoff, R.S., 2002. Atomistic simulations of nanotube fracture. Phys. Rev. B 65, 235430.
[13] Zhang, P., Huang, Y., Gao, H., Hwang, K.C., 2002a. Fracture nucleation in single-wall carbon nanotubes under tension: a ontinuum analysis incorporating interatomic potentials. J. Appl. Mech. 69, 454–458.
[14] Zhang, P., Huang, Y.G., Geubelle, P.H., Hwang, K.C., 2002b. On the continuum modeling of carbon nanotubes. Acta Mech. Sin. 18, 528–536.
[15] Zhang, P., Huang, Y., Geubelle, P.H., Klein, P.A., Hwang, K.C., 2002c. The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials. Int. J. Solids Struct. 39, 3893–3906.
[16] Wu, J., Hwang, K.C., Huang, Y., 2008. An atomistic-based finite-deformation shell theory for single-wall carbon nanotubes. J. Mech. Phys. Solids. 56, 279–292.
[17] Eringen A C 1983 On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves J. Appl. Phys. 54 4703–4710.
[18] Eringen A C 2002 Nonlocal Continuum Field Theories. (Springer NewYork).
[19] Peddieson J, Buchanan G R, McNitt R P 2003 Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41 305–312.
[20] Yoon J, Ru C Q, Mioduchowski A 2003 Vibration of an embedded multiwall carbon nanotube. Compos. Sci. Tech. 63 1533–1542.
[21] Yoon J, Ru C Q, Mioduchowski A 2005 Vibration and instability of carbon nanotubes conveying fluid. Compos. Sci. Tech. 65 1326–1336.
[22] Zhang Y, Liu G, Han X 2005 Transverse vibrations of double-walled carbon nanotubes under compressive axial load. Phys. Lett. A 340 258–266.
[23] Fu Y M, Hong J W, Wang X Q 2006 Analysis of nonlinear vibration for embedded carbon nanotubes. J. Sound Vib. 296 746–756.
[24] Wang C M, Tan V B C, Zhang Y Y 2006 Timoshenko beam model for vibration analysis of multiwalled carbon nanotubes. J. Sound Vib. 294 1060–1072.
[25] Wang Q, Varadan V K, 2006 Wave characteristics of carbon nanotubes. Int. J. Solids Struct. 43 254–265.
[26] Wang L, Ni Q, Li M, Qian Q 2008 The thermal effect on vibration and instability of carbon nanotubes conveying fluid. Physica E. 40 3179-3182.
[27] Aydogdu M 2008 Vibration of multiwalled carbon nanotubes by generalized shear deformation theory. Int. J. Mech. Sci. 50 837–844.
[28] Xu K Y, Aifantis E C, Ya Xu n Y H 2008 Vibrations of Double-Walled Carbon Nanotubes With Different Boundary Conditions Between Inner and Outer Tubes. J.
Appl. Mech. 75 021013.
[29] Kuang Y D, He X Q, Chen C Y, Li G Q 2009 Analysis of nonlinear vibrations of double-walled carbon nanotubes conveying fluid. Comput. Mat. Sci. 45 875–880.
[30] Ansari R., Hemmatnezhad M., Ramezannezhad H. 2009 Application of HPM to the Nonlinear Vibrations of Multiwalled Carbon Nanotubes. Numerical Methods for Partial Differential Equations Journal DOI 10.1002/num.20499.
[31] Elishakoff I, Pentaras D 2009 Fundamental natural frequencies of double-walled carbon nanotubes. J. Sound Vib. 322 652–664.
[32] Chang W J, Lee H L 2009 Free vibration of a single-walled carbon nanotube containing a fluid flow using the Timoshenko beam model. Phys. Lett. A 373 982–985.
[33] Natsuki T, Endo M 2006 Vibration analysis of embedded carbon nanotubes using wave propagation approach. J. Appl. Phys. 99 034311.
[34] Liew K M, Wang Q 2007 Analysis of wave propagation in carbon nanotubes via elastic shell theories. International Journal of Engineering Science 45 227–241.
[35] Sun C, Liu K 2007 Vibration of multiwalled carbon nanotubes with initial axial loading. Solid State Commun. 143 202–207.
[36] Yan Y, Wang W Q, Zhang L X 2009 Noncoaxial vibration of fluid-filled multiwalled carbon nanotubes. Appl. Math. Modell. doi:10.1016/j.apm.2009.03.031
[37] Wang Q, Zhou G Y, Lin K C 2006 Scale effect on wave propagation of double-walled carbon nanotubes. Int. J. Solids Struct. 43 6071–6084.
[38] Wang Q, Varadan V K 2006 Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Mater. Struct. 15 659–666.
[39] Lu P, Lee H P, Lu C, Zhang P Q 2007 Application of nonlocal beam models for carbon nanotubes. Int. J. Solids Struct. 44 5289–5300.
[40] Reddy J N 2007 Nonlocal theories for bending, buckling and vibration of beams.
International Journal of Engineering Science 45 288–307.
[41] Wang Q, Wang C M 2007 The constitutive relation and small-scale parameter of nonlocal continuum mechanics for modeling carbon nanotubes Nanotechnology 18 075702.
[42] Wang C M, Zhang Y Y, He X Q 2007 Vibration of nonlocal Timoshenko beams.
Nanotechnology. 18 105401 (9pp)
[43] Khosravian N, Rafii-Tabar H 2008 Computational modelling of a non-viscous fluid flow in a multiwalled carbon nanotube modelled as a Timoshenko beam.
Nanotechnology. 19 275703.
[44] Heireche H, Tounsi A, Benzair A 2008 Scale effect on wave propagation of double- walled carbon nanotubes with initial axial loading. Nanotechnology. 19 185703 (11pp) [45] Heireche H, Tounsi A, Benzair A, Maachou M, Adda Bedia E A 2008 Sound wave
propagation in single-walled carbon nanotubes using nonlocal elasticity. Physica. E 40 2791–2799.
[46] Aydogdu M 2009 Axial vibration of the nanorods with the nonlocal continuum rod model. Physica. E 41 861–864.
[47] Wang L 2009 Dynamical behaviors of double-walled carbon nanotubes conveying fluid accounting for the role of small length scale. Computational Materials Science. 45 584–
588.
[48] Murmu T, Pradhan S C, 2009 Small-Scale Effect on the Vibration of Non-uniform Nanocantilever based on Nonlocal Elasticity Theory Physica E. doi:10.1016/
j.physe.2009.04.015
[49] Murmu T, Pradhan S C 2009 Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory.
Computational Materials Science.
[50] Lee H L, Chang W J 2009 Vibration analysis of a viscous-fluid-conveying single- walled carbon nanotube embedded in an elastic medium. Physica. E 41 529–532.
[51] Wang Q, Varadan V K 2007 Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes. Smart Mater. Struct. 16 178–190.
[52] Li R, Kardomateas G A 2007 Vibration Characteristics of Multiwalled Carbon Nanotubes Embedded in Elastic Media by a Nonlocal Elastic Shell Model. J. Appl.
Mech. 74 1087-1094.
[53] Hu Y G, Liew K M, Wang Q, He X Q, Yakobson B I 2008 Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes. J. Mech. Phys.
Solids 56 3475–3485.
[54] Eringen A C 1983 On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54 4703–4710.
[55] Eringen A C 2002 Nonlocal Continuum Field Theories. (Springer NewYork).
[56] He X Q, Kitipornchaia S, Liew K M 2005 Buckling analysis of multiwalled carbon nanotubes: a continuum model accounting for van der Waals interaction J. Mech. Phys.
solids 53 303-326.
Editors: A. K. Haghi and G. E. Zaikov © 2013 Nova Science Publishers, Inc.