A new method for determination of natural frequency in bending vibration mode of single-walled carbon nanotubes

Document Type : English


Department of Mechanical Engineering, Shahryar Branch, Islamic Azad University, Shahryar, Iran


This paper investigates the bending vibration of single-walled carbon nanotubes based on a new theory called doublet mechanics with a scale parameter. A sixth order partial differential equation that governs the bending vibration for such nanotubes is derived. Euler-Bernoulli beam model is used in this paper. Using doublet mechanics, the relation between natural frequency and scale parameter is derived in the bending vibration mode. It is proven that scale parameter plays significant role in the vibration behavior of such nanotubes in lateral direction. Such effect decreases the natural frequency compared to the predictions of the classical continuum mechanics models. However, with increasing the tube length, the scale effect on the natural frequency decreases. To validate this method, the results obtained herein are compared with the existing nonlocal and molecular dynamics results and good agreement is observed. It is the first time that DM is used to model the bending vibration of carbon nanotube.


[1] I. Elishakoff, D. Pentaras, Fundamental natural frequencies of double-walled carbon nanotubes, Journal of Sound and Vibration, 322 (2009) 652–664
[2] A. C. Eringen, Nonlocal polar elastic, International Journal of Engineering Science, 1972, 10, l-16
[3] M. Ferrari, V. T. Granik, A. Imam, J. Nadeau, Advances in Doublet Mechanics, 1997, Springer, Berlin
[4] V. T. Granik, 1978, Microstructural mechanics of granular media, Technique Report IM/MGU 78-241, Institute of Mechanics of Moscow State University, in Russian.
[5] V. T. Granik , M. Ferrari, Microstructural mechanics of granular media, Mechanics of Materials 15 (1993) 301-322
[6] M. Kojic, I. Vlastelica, P. Decuzzi, V. T. Granik, M. Ferrari, A finite element formulation for the doublet mechanics modeling of microstructural materials, Comput. Methods Appl. Mech. Engrg. 200 (2011) 1446–1454
[7] J. Xin, L. X. Zhou, W. J. Ru, Ultrasound Attenuation in Biological Tissue Predicted by the Modified Doublet Mechanics Model, Chin. Phys. Lett. 26(7) (2009) 074301.1-074301.4
[8] M. Ferrari, Nanomechanics, and Biomedical Nanomechanics: Eshelby's Inclusion and Inhomogeneity Problems at the Discrete Continuum Interface, Biomedical Microdevices 2(4), 273-281, 2000
[9] F. Gentile, J. Sakamoto, R. Righetti, P. Decuzzi, M. Ferrari, A doublet mechanics model for the ultrasound characterization of malignant tissues, J. Biomedical Science and Engineering, 2011, 4, 362-374
[10] S. S. Lin, Y. C. Shen, Stress fields of a half-plane caused by moving loads-resolved using doublet mechanics, Soil Dynamics and Earthquake Engineering 25 (2005) 893–904
[11] M. H. Sadd, Q. Dai, A comparison of micro-mechanical modeling of asphalt materials using finite elements and doublet mechanics, Mechanics of Materials 37 (2005) 641–662
[12] J. Y. Fang, Z. Jue, F .Jing, M. Ferrari, Dispersion analysis of wave propagation in cubic-Tetrahedral assembly by doublet mechanics, Chin.Phys.Lett, 21(8) (2004) 15621565
[13] M. H. Sadd, Elasticity theory, applications, and numeric, 2005, Elsevier, New York
[14] M. Ferrari, BioMEMS and Biomedical Nanotechnology, Volume I Biological and Biomedical Nanotechnology, 2006, Springer, New York
[15] Fatahi-Vajari, A. and Imam, A.: Axial vibration of single-walled carbon nanotubes using doublet mechanics. Indian Journal of Physics. 90(4), 447–455 (2016)
[16] A. Fatahi-Vajari and A. Imam, Analysis of radial breathing mode of vibration of single-walled carbon nanotubes via doublet mechanics, ZAMM, 96(9) 2016
[17] Iijima S., 1991, Helical microtubes of graphitic carbon, Nature (London) 354, 56–58
[20] P. Ghosh, T. Soga, R. A. Afre, T. Jimbo, Simplified synthesis of single-walled carbon nanotubes from a botanical hydrocarbon, Turpentine oil, Journal of Alloys and Compounds 462 (2008) 289–293
[21] S. M. Bachilo, M. S. Strano, C. Kittrell, R. H. Hauge, R. E. Smalley, R. B. Weisman, Structure-Assigned Optical Spectra of Single-Walled Carbon Nanotubes, SCIENCE, 298, 2002, 2361-2366
[21] D. Sanchez-Portal, E. Artacho, J. M. Soler, A. Rubio, P. Ordejon, "Ab initio structural, elastic, and vibrational properties of carbon nanotubes", Physical Review B, 59(19), (1999), 12678.
[22] S. S. Gupta, F. G. Bosco, R. C. Batra, Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and in extensional modes of vibration, Computational Materials Science 47 (2010) 1049–1059
[23] S.S. Gupta, R.C. Batra, Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes, Computational Materials Science 43 (2008) 715–723
[24] A. Szabó, C. Perri, A. Csató, G. Giordano, D. Vuono  and J. B. Nagy, Synthesis Methods of Carbon Nanotubes and Related Materials, Materials, 2010, 3, 3092-3140
[25] F. Taleshi, A. A. Hosseini, M. Mohammadi, M. Pashaee, Effect of hydrocarbon gas on synthesis and diameter of carbon nanotubes, Indian Journal of Physics, 2013 , 87(9) 873-877
[26] HONGJIE DAI, Carbon Nanotubes: Synthesis, Integration, and Properties, American Chemical Society, 2002, 35, 1035-1044
[27] I. Elishakoff, D. Pentaras, Fundamental natural frequencies of double-walled carbon nanotubes, Journal of Sound and Vibration, 2009, 322(4) 652-664
[28] R. Ansari, R. Gholami, H. Rouhi, Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories, Composites: Part B 43 (2012) 2985–2989
[29] F Sohani, H.R Eipakchi, Response Determination of a Beam with Moderately Large Deflection Under Transverse Dynamic Load Using First Order Shear Deformation Theory, Journal of Solid Mechanics, 5(4) 2013, 391-401
[30] Sh Hosseini – Hashemi, M Fakher, R Nazemnezhad, Surface Effects on Free Vibration Analysis of Nanobeams Using Nonlocal Elasticity: A Comparison Between Euler-Bernoulli and Timoshenko, Journal of Solid Mechanics, 5(3) 2013, 290-304
[31] A Ghorbanpour Arani, H Rabbani, S Amir, Z Khoddami Maraghi, M Mohammadimehr, E Haghparast, Analysis of Nonlinear Vibrations for Multi-walled Carbon Nanotubes Embedded in an Elastic Medium, Journal of Solid Mechanics, 3( 3) 2011, 258-270
[32] J. Bocko, P. Lengvarský, Bending Vibrations of Carbon Nanotubes by using Nonlocal Theory, Procedia Engineering 96 ( 2014 ) 21 – 27
[33] R Ranjan, Nonlinear Finite Element Analysis of Bending of Straight Beams Using hp-Spectral Approximations, Journal of Solid Mechanics, 3(1) 2011, 96-113
[34] Fatahi-Vajari, A.: A new method for evaluating the natural frequency in radial breathing like mode vibration of double-walled carbon nanotubes. ZAMM. Doi: 10.1002/zamm.201600234
[35] Fatahi-Vajari, A. and Imam, A. Torsional vibration of single-walled carbon nanotubes using doublet mechanics. ZAMP. Doi: 10.1007/s00033-016-0675-6
[36] Fatahi-Vajari, A. and Imam, A.: Lateral Vibrations of Single-Layered Graphene Sheets Using Doublet Mechanics. Journal of Solid Mechanics. 8(4), 875-894 (2016)