Acta Mechanica Solida Sinica

, Volume 27, Issue 2, pp 202–209 | Cite as

Transverse Vibration of a Hanging Nonuniform Nanoscale Tube Based on Nonlocal Elasticity Theory with Surface Effects

  • Hossein Roostai
  • Mohammad Haghpanahi


The aim of this paper is to study the free transverse vibration of a hanging nonuniform nanoscale tube. The analysis procedure is based on nonlocal elasticity theory with surface effects. The nonlocal elasticity theory states that the stress at a point is a function of strains at all points in the continuum. This theory becomes significant for small-length scale objects such as micro- and nanostructures. The effects of nonlocality, surface energy and axial force on the natural frequencies of the nanotube are investigated. In this study, analytical solutions are formulated for a clamped-free Euler-Bernoulli beam to study the free vibration of nanoscale tubes.

Key Words

nonlocal elasticity theory vibration surface effects nanoscale tube 


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  1. 1.
    Liu, C. and Rajapakse, R.K.N.D., Continuum models incorporating surface energy for static and dynamic response of nanoscale beams. IEEE Transactions on Nanotechnology, 2010, 9(4): 422–431.CrossRefGoogle Scholar
  2. 2.
    Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 1983, 54(9): 4703.CrossRefGoogle Scholar
  3. 3.
    Eringen, A.C., Speziale, C.G. and Kim, B.S., Crack-tip problem in non-local elasticity. Journal of the Mechanics and Physics of Solids, 1977, 25(5): 339–355.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Wang, Q., Wave propagation in carbon nanotubes via nonlocal continuum mechanics. Journal of Applied Physics, 2005, 98(12): 124301.CrossRefGoogle Scholar
  5. 5.
    Wang, G.F., Wang, T.J. and Feng, X.Q., Surface effects on the diffraction of plane compressional waves by a nanosized circular hole. Applied Physics Letters, 2006, 89(23): 231923.CrossRefGoogle Scholar
  6. 6.
    Wang, Q. and Varadan, V.K., Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Materials and Structures, 2006, 15(2): 659–666.CrossRefGoogle Scholar
  7. 7.
    Aydogdu, M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E, 2009, 41(9): 1651–1655.CrossRefGoogle Scholar
  8. 8.
    Chowdhury, R., Adhikari, S., Wang, C.Y. and Scapra, F., A molecular mechanics approach for the vibration of single-walled carbon nanotubes. Computational Materials Science, 2010, 48(4): 730–735.CrossRefGoogle Scholar
  9. 9.
    Murmu, T. and Adhikari, S., Nonlocal effects in the longitudinal vibration of double-nanorod systems. Physica E, 2010, 43(1): 415–422.CrossRefGoogle Scholar
  10. 10.
    Murmu, T. and Adhikari, S., Nonlocal transverse vibration of double-nanobeam-systems. Journal of Applied Physics, 2010, 108(8): 083514.CrossRefGoogle Scholar
  11. 11.
    Hasheminejad, S.M., Gheshlaghi, B., Mirzaei, Y. and Abbasion, S., Free transverse vibrations of cracked nanobeams with surface effects. Thin Solid Films, 2011, 519(8): 2477–2482.CrossRefGoogle Scholar
  12. 12.
    Wang, G.F. and Feng, X.Q., Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Applied Physics Letters, 2007, 90(23): 231904.CrossRefGoogle Scholar
  13. 13.
    Abbasion, S., Rafsanjani, A., Avazmohammadi, R. and Farshidianfar, A., Free vibration of microscaled Timoshenko beams. Applied Physics Letters, 2009, 95(14): 143122.CrossRefGoogle Scholar
  14. 14.
    Farshi, B., Assadi, A. and Alinia-ziazi, A., Frequency analysis of nanotubes with consideration of surface effects. Applied Physics Letters, 2010, 96(9): 093105.CrossRefGoogle Scholar
  15. 15.
    Lee, H.L. and Chang, W.J., Surface and small-scale effects on vibration analysis of a nonuniform nanocantilever beam. Physica E, 2010, 43(1): 466–469.CrossRefGoogle Scholar
  16. 16.
    Murmu, T. and Adhikari, S., Scale-dependent vibration analysis of prestressed carbon nanotubes undergoing rotation. Journal of Applied Physics, 2010, 108(12):123507.CrossRefGoogle Scholar
  17. 17.
    Murmu, T. and Pradhan, S.C., Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory. Computational Materials Science, 2009, 46(4): 854–859.CrossRefGoogle Scholar
  18. 18.
    Wang, G.F. and Feng, X.Q., Timoshenko beam model for buckling and vibration of nanowires with surface effects. Journal of Physics D:Applied Physics, 2009, 42(15): 155411.CrossRefGoogle Scholar
  19. 19.
    Meirovitch, L., Fundamentals of Vibrations. New York: McGraw-Hill, 2001.Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2014

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIran University of Science & Technology (IUST)TehranIslamic Republic of Iran

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