Skip to main content
Log in

On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams: (i) why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i.e., increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise? (ii) the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam with a point load at its tip; and (iii) the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, domain governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary conditions and the true basic static responses for bending of nanobeams. It is also concluded that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Iijima, S. Helical microtubules of graphitic carbon. Nature 354, 56–58 (1991)

    Article  Google Scholar 

  2. Treacy, M. M. J., Ebbesen, T. W., and Gibson, T. M. Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 381, 680–687 (1996)

    Article  Google Scholar 

  3. Ball, P. Roll up for the revolution. Nature 414, 142–144 (2001)

    Article  Google Scholar 

  4. Iijima, S., Brabec, C., Maiti, A., and Bernhole, J. Structural flexibility of carbon nanotubes. J. Chem. Phys. 104, 2089–2092 (1996)

    Article  Google Scholar 

  5. Yakobson, B. I., Campbell, M. P., Brabec, C. J., and Bernholc, J. High strain rate fracture and C-chain unraveling in carbon nanotubes. Comput. Mater. Sci. 8, 341–348 (1997)

    Article  Google Scholar 

  6. 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. J. Mech. Phys. Solids 53, 303–326 (2005)

    Article  MATH  Google Scholar 

  7. Yakobson, B. I., Brabec, C. J., and Bernholc, J. Nanomechanics of carbon tubes: instabilities beyond linear range. Phys. Rev. Lett. 76, 2511–2514 (1996)

    Article  Google Scholar 

  8. Ru, C. Q. Effective bending stiffness of carbon nanotubes. Phys. Rev. B 62, 9973–9976 (2000)

    Article  Google Scholar 

  9. Ru, C. Q. Elastic buckling of single-walled carbon nanotubes ropes under high pressure. Phys. Rev. B 62, 10405–10408 (2000)

    Article  Google Scholar 

  10. Zhang, P., Huang, Y., Geubelle, P. H., Klein, P. A., and Hwang, K. C. The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials. I. J. Solids Struct. 39, 3893–3906 (2002)

    Article  MATH  Google Scholar 

  11. Gurtin, M. E. and Murdoch, A. A continuum theory of elastic material surfaces. Archives of Rational Mechanics and Analysis 57, 291–323 (1975)

    MATH  MathSciNet  Google Scholar 

  12. Gurtin, M. E. and Murdoch, A. I. Effect of surface stress on wave propagation in solids. J. Applied Physics 47, 4414–4421 (1976)

    Article  Google Scholar 

  13. He, L. H. and Lim, C. W. On the bending of unconstrained thin crystalline plates caused by change in surface stress. Surface Sci. 478(3), 203–210 (2001)

    Article  Google Scholar 

  14. He, L. H., Lim, C. W., and Wu, B. S. A continuum model for size-dependent deformation of elastic films of nano-scale thickness. I. J. Solids Struct. 41, 847–857 (2004)

    Article  MATH  Google Scholar 

  15. Lim, C. W. and He, L. H. Size-dependent nonlinear response of thin elastic films with nano-scale thickness. I. J. Mech. Sci. 46(11), 1715–1726 (2004)

    Article  MATH  Google Scholar 

  16. Lim, C. W., Li, Z. R., and He, L. H. Size dependent, nonuniform elastic field inside a nano-scale spherical inclusion due to interface stress. I. J. Solids Struct. 43, 5055–5065 (2006)

    Article  MATH  Google Scholar 

  17. Wang, Z. Q., Zhao, Y. P, and Huang, Z. P. The effects of surface tension on the elastic properties of nano structures. I. J. Engineering Science, in press (2009) DOI 10.1016/j.ijengsci.2009.07.007

  18. Eringen, A. C. Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Engng. Sci. 10(5), 425–435 (1972)

    Article  MATH  Google Scholar 

  19. Eringen, A. C. Nonlocal polar elastic continua. International Journal of Engineering Science 10(1), 1–16 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  20. Eringen, A. C. On nonlocal fluid mechanics. Int. J. Eng. Sci. 10(6), 561–575 (1972)

    Article  MATH  Google Scholar 

  21. Eringen, A. C. and Edelen, D. G. B. On nonlocal elasticity. International Journal of Engineering Science 10(3), 233–248 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  22. Eringen, A. C. Linear theory of nonlocal microelasticity and dispersion of plane waves. Lett. Appl. Eng. Sci. 1, 129–146 (1973)

    Google Scholar 

  23. Eringen, A. C. On nonlocal microfluid mechanics. Int. J. Eng. Sci. 11(2), 291–306 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  24. Eringen, A. C. Theory of nonlocal electromagnetic elastic solids. J. Math. Phys. 14(6), 733–740 (1973)

    Article  MATH  Google Scholar 

  25. Eringen, A. C. Theory of nonlocal thermoelasticity. Int. J. Eng. Sci. 12, 1063–1077 (1974)

    Article  MATH  Google Scholar 

  26. Eringen, A. C. Memory-dependent nonlocal thermoelastic solids. Lett. Appl. Eng. Sci. 2, 145–149 (1974)

    Google Scholar 

  27. Eringen, A. C., Nonlocal elasticity and waves. Continuum Mechanics Aspect of Geodynamics and Rock Fracture Mechanics (ed. Thoft-Christensen, P.), Kluwer Academic Publishers Group, Netherlands, 81–105 (1974)

    Google Scholar 

  28. Eringen, A. C. Continuum Physics, Academic Press, New York (1975)

    Google Scholar 

  29. Eringen, A. C. Nonlocal micropolar elastic moduli. Lett. Appl. Engng. Sci. 3(5), 385–393 (1975)

    Google Scholar 

  30. Eringen, A. C. Nonlocal Polar Field Theories, Academic Press, New York (1976)

    Google Scholar 

  31. Eringen, A. C. Mechanics of Continua, 2nd Ed., Krieger, Melbourne, FL (1980)

    Google Scholar 

  32. Eringen, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics 54(9), 4703–4710 (1983)

    Article  Google Scholar 

  33. Eringen, A. C. Theory of nonlocal piezoelectricity. J. Math. Phys. 25, 717–727 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  34. Eringen, A. C. Point charge, infra-red dispersion and conduction in nonlocal piezoelectricity. The Mechanical Behavior of Electromagnetic Solid Continua (ed. Maugin, G.A.), North-Holland, Elsevier Science, 187–196 (1984)

  35. Eringen, A. C. Nonlocal Continuum Field Theories, Springer, New York (2002)

    MATH  Google Scholar 

  36. Peddieson, J., Buchanan, G. R., and McNitt, R. P. Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science 41(3–5), 305–312 (2002)

    Google Scholar 

  37. Sudak, L. J. Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. Journal of Applied Physics 94, 7281–7287 (2003)

    Article  Google Scholar 

  38. Nix, W. and Gao, H. Indentation size effects in crystalline materials: A law for strain gradient plasticity. Journal of the Mechanics and Physics of Solids 46(3), 411–425 (2007)

    Article  Google Scholar 

  39. Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J., and Tong, P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids 51, 1477–1508 (2003)

    Article  MATH  Google Scholar 

  40. Li, C. Y. and Chou, T. W. Vibrational behaviors of multi-walled carbon nanotube-based nanomechancial resonators. Appl. Phys. Lett. 84, 121–123 (2004).

    Article  Google Scholar 

  41. Park, S. K. and Gao, X. L. Bernoulli-Euler beam model based on a modified couple stress theory. Journal of Micromechanics and Microengineering 16, 2355–2359 (2006)

    Article  Google Scholar 

  42. Park, S. K. and Gao, X. L. Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z. angew. Math. Phys. 59, 904–917 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  43. Ma, H. M., Gao, X. L., and Reddy, J. N. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of Solids 56(12), 3379–3391 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  44. Was, G. S. and Foecke, T. Deformation and fracture in microlaminates. Thin Solid Films 286, 1–31 (1996)

    Article  Google Scholar 

  45. McFarland, A. W., and Colton, J. S. Role of material microstructure in plate stiffness with relevance to microcantilever sensors. Journal of Micromechanics and Microengineering 15, 1060–1067 (2005)

    Article  Google Scholar 

  46. Liew, K. M., Hu, Y. G., and He, X. Q. Flexural wave propagation in single-walled carbon nanotubes. Journal of Computational and Theoretical Nanoscience 5, 581–586 (2008)

    Google Scholar 

  47. Zhang, Y. Y., Wang, C. M., Duan, W. H., Xiang, Y., and Zong, Z. Assessment of continuum mechanics models in predicting buckling strains of single-walled carbon nanotubes. Nanotechnology 20, 395707 (2009)

    Article  Google Scholar 

  48. Lim, C. W. and Wang, C. M. Exact variational nonlocal stress modeling with asymptotic higherorder strain gradients for nanobeams. Journal of Applied Physics 101, 054312 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to C. W. Lim.

Additional information

Communicated by Xing-ming GUO

Project supported by a grant from Research Grants Council of the Hong Kong Special Administrative Region (No. CityU 117406)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lim, C.W. On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection. Appl. Math. Mech.-Engl. Ed. 31, 37–54 (2010). https://doi.org/10.1007/s10483-010-0105-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-010-0105-7

Key words

Chinese Librarary Classification

2000 Mathematics Subject Classification

Navigation