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Acta Mechanica Solida Sinica

, Volume 24, Issue 6, pp 484–494 | Cite as

Twisting Statics and Dynamics for Circular Elastic Nanosolids by Nonlocal Elasticity Theory

  • Cheng Li
  • C. W. Lim
  • Jilin Yu
Article

Abstract

The torsional static and dynamic behaviors of circular nanosolids such as nanoshafts, nanorods and nanotubes are established based on a new nonlocal elastic stress field theory. Based on a new expression for strain energy with a nonlocal nanoscale parameter, new higher-order governing equations and the corresponding boundary conditions are first derived here via the variational principle because the classical equilibrium conditions and/or equations of motion cannot be directly applied to nonlocal nanostructures even if the stress and moment quantities are replaced by the corresponding nonlocal quantities. The static twist and torsional vibration of circular, nonlocal nanosolids are solved and discussed in detail. A comparison of the conventional and new nonlocal models is also presented for a fully fixed nanosolid, where a lower-order governing equation and reduced stiffness are found in the conventional model while the new model reports opposite solutions. Analytical solutions and numerical examples based on the new nonlocal stress theory demonstrate that nonlocal stress enhances stiffness of nanosolids, i.e. the angular displacement decreases with the increasing nonlocal nanoscale while the natural frequency increases with the increasing nonlocal nanoscale.

Key words

angular displacement nanoscale nonlocal stress torsion vibration 

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References

  1. [1]
    Eringen, A.C. and Kim, B.S., Stress concentration at the tip of the crack. Mechanics Research Communications, 1974, 1(4): 233–237.CrossRefGoogle Scholar
  2. [2]
    Eringen, A.C. and Edelen, D.G.B., On nonlocal elasticity. International Journal of Engineering Science, 1972, 10(3): 233–248.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Stoney, G.C., The tension of metallic films deposited by electrolysis. Proceeding of Royal Society of London, 1909, A82: 172–175.CrossRefGoogle Scholar
  4. [4]
    Cammarata, R.C., Surface and interface stress effects in thin films. Progress in Surface Science, 1994, 46(1): 1–38.CrossRefGoogle Scholar
  5. [5]
    Gao, H., Huang, Y., Nix, W.D. and Hutchinson, J.W., Mechanism-based strain gradient plasticity—I theory. Journal of the Mechanics and Physics of Solids, 1999, 47(6): 1239–1263.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Wang, J.X., Huang, Z.P., Duan, H.L., Yu, S.W., Feng, X.Q., Wang, G.F., Zhang, W.X. and Wang, T.J., Surface stress effect in mechanics of nanostructured materials. Acta Mechanica Solida Sinica, 2011, 24(1): 52–82.CrossRefGoogle Scholar
  7. [7]
    Maranganti, R. and Sharma, P., Length scales at which classical elasticity breaks down for various materials. Physical Review Letters, 2007, 98(19): 195504.CrossRefGoogle Scholar
  8. [8]
    Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 1983, 54(9): 4703–4710.CrossRefGoogle Scholar
  9. [9]
    Zhou, Z.G., Wang, B. and Sun, Y.G., Analysis of the dynamic behavior of a Griffith permeable crack in piezoelectric materials with the non-local theory. Acta Mechanica Solida Sinica, 2003, 16(1): 52–60.Google Scholar
  10. [10]
    Peddieson, J., Buchanan, G.G. and McNitt, R.P., Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 2003, 41(3-5): 305–312.CrossRefGoogle Scholar
  11. [11]
    He, X.Q., Kitipornchai, S. and Liew, K.M., Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. Journal of the Mechanics and Physics of Solids, 2005, 53(2): 303–326.CrossRefGoogle Scholar
  12. [12]
    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
  13. [13]
    Lu, P., Lee, H.P., Lu, C. and Zhang, P.Q., Dynamic properties of flexural beams using a nonlocal elasticity model. Journal of Applied Physics, 2006, 99(7): 073510.CrossRefGoogle Scholar
  14. [14]
    Liang, J., Scattering of harmonic anti-plane shear stress waves by a crack in functionally graded piezoelectric/piezomagnetic materials. Acta Mechanica Solida Sinica, 2007, 20(1): 75–86.CrossRefGoogle Scholar
  15. [15]
    Wang, C.M., Zhang, Y.Y. and Kitipornchai, S., Vibration of initially stressed micro- and nano-beams. International Journal of Structural Stability and Dynamics, 2007, 7(4): 555–570.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Duan, W.H. and Wang, C.M., Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory. Nanotechnology, 2007, 18(38): 385704.CrossRefGoogle Scholar
  17. [17]
    Lim, C.W., Equilibrium and static deflection for bending of a nonlocal nanobeam. Advances in Vibration Engineering, 2009, 8(4): 277–300.Google Scholar
  18. [18]
    Lim, C.W., On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection. Applied Mathmatics and Mechanics, 2010, 31(1): 37–54.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Lim, C.W., A nanorod (or nanotube) with lower Young’s modulus is stiffer? Is not Young’s modulus a stiffness indicator? Science in China Series G: Physics, Mechanics & Astronomy, 2010, 53(4): 712–724.CrossRefGoogle Scholar
  20. [20]
    Lim, C.W. and Yang, Y., New predictions of size-dependent nanoscale based on nonlocal elasticity for wave propagation in carbon nanotubes. Journal of Computational and Theoretical of Nanoscience, 2010, 7(6): 988–995.CrossRefGoogle Scholar
  21. [21]
    Lim, C.W. and Yang, Y., Wave propagation in carbon nanotubes: nonlocal elasticity-induced stiffness and velocity enhancement effects. Journal of Mechanics of Materials and Structures, 2010, 5(3): 459–476.CrossRefGoogle Scholar
  22. [22]
    Lim, C.W., Niu, J.C. and Yu, Y.M., Nonlocal stress theory for buckling instability of nanotubes: new predictions on stiffness strengthening effects of nanoscales. Journal of Computational and Theoretical of Nanoscience, 2010, 7(10): 2104–2111.CrossRefGoogle Scholar
  23. [23]
    Li, C., Lim, C.W., Yu, J.L. and Zeng, Q.C., Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force. International Journal of Structural Stability and Dynamics, 2011, 11(2): 257–271.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Li, C., Lim, C.W. and Yu, J.L., Dynamics and stability of transverse vibrations of nonlocal nanobeams with a variable axial load. Smart Materials and Structures, 2011, 20(1): 015023.CrossRefGoogle Scholar
  25. [25]
    Yang, Y. and Lim, C.W., A new nonlocal cylindrical shell model for axisymmetric wave propagation in carbon nanotubes. Advanced Science Letters, 2011, 4(1): 121–131.CrossRefGoogle Scholar
  26. [26]
    Ramamurti, V., Mechanical Vibration Practice And Noise Control. Oxford: Alpha Science, 2008.Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2011

Authors and Affiliations

  1. 1.School of Urban Rail TransportationSoochow UniversitySuzhouChina
  2. 2.Department of Building and ConstructionCity University of Hong KongHong KongChina
  3. 3.Department of Modern MechanicsUniversity of Science and Technology of ChinaHefeiChina
  4. 4.USTC-CityU Joint Advanced Research CenterSuzhouChina

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