Skip to main content
SpringerLink
Log in

Size effect in the bending of a Timoshenko nanobeam

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

The size effect should be considered due to the large ratio of surface area to volume when the characteristic length of a beam lies in the nanoscale. The size effect in the bending of a Timoshenko nanobeam is investigated in this paper based on a recently developed elastic theory for nanomaterials, in which only the bulk surface energy density and the surface relaxation parameter are involved as independent parameters to characterize the surface property of nanomaterials. In contrast to the Euler nanobeams and the classical Timoshenko beam, not only the size effect but also the shear deformation effect in Timoshenko nanobeams is included. Closed-form solutions of the deflection and the effective elastic modulus for both a fixed–fixed Timoshenko nanobeam and a cantilevered one are achieved. Comparing to the classical solution of Timoshenko beams, the size effect is obviously significant in Timoshenko nanobeams. The shear deformation effect in nanobeams cannot be neglected in contrast to the solution of Euler–Bernoulli nanobeams when the aspect ratio of a nanobeam is relatively small. Furthermore, the size effect exhibits different influences on the bending behavior of nanobeams with different boundary conditions. A nanobeam with a fixed–fixed boundary would be stiffened, while a cantilevered one is softened by the size effect, compared to the classical solution. All the findings are consistent with the existing experimental measurement. The results in this paper should be very useful for the precision design of nanobeam-based devices.

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

Access this article

Log in via an institution

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. Rogers, J.A., Someya, T., Huang, Y.G.: Materials and mechanics for stretchable electronics. Science 327, 1603–1607 (2010)

    Article  Google Scholar 

  2. Cui, Y., Wei, Q.Q., Park, H., Lieber, C.M.: Nanowire nanosensors for highly sensitive and selective detection of biological and chemical species. Science 293, 1289–1292 (2001)

    Article  Google Scholar 

  3. Xie, P., Xiong, Q.H., Fang, Y., Qing, Q., Lieber, C.M.: Local electrical potential detection of DNA by nanowire-nanopore sensors. Nat. Nanotechnol. 7, 119–125 (2012)

    Article  Google Scholar 

  4. Gong, C., Liang, J., Hu, W., Niu, X., Ma, S., Hahn, H.T., Pei, Q.: A healable, semitransparent silver nanowire-polymer composite conductor. Adv. Mater. 25, 4186–4191 (2013)

    Article  Google Scholar 

  5. Liang, H., Upmanyu, M., Huang, H.: Size-dependent elasticity of nanowires: nonlinear effects. Phys. Rev. B 71, 241403 (2005)

    Article  Google Scholar 

  6. Cuenot, S., Frétigny, C., Demoustier-Champagne, S., Nysten, B.: Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B 69, 165410 (2004)

    Article  Google Scholar 

  7. Chen, T.Y., Chiu, M.S., Weng, C.N.: Derivation of the generalized Young-Laplace equation of curved interfaces in nanoscaled solids. J. Appl. Phys. 100, 074308 (2006)

    Article  Google Scholar 

  8. Jing, G.Y., Duan, H.L., Sun, X.M., Zhang, Z.S., Xu, J., Li, Y.D., Wang, J.X., Yu, D.P.: Surface effects on elastic properties of silver nanowires: contact atomic-force microscopy. Phys. Rev. B 73, 235409 (2006)

    Article  Google Scholar 

  9. Nam, C.Y., Jaroenapibal, P., Tham, D., Luzzi, D.E., Evoy, S., Fischer, J.E.: Diameter-dependent electromechanical properties of GaN nanowires. Nano Lett. 6, 153–158 (2006)

    Article  Google Scholar 

  10. Gavan, K.B., Westra, H.J., van der Drift, E.W., Venstra, W.J., van der Zant, H.S.: Size-dependent effective Young’s modulus of silicon nitride cantilevers. Appl. Phys. Lett. 94, 233108 (2009)

    Article  Google Scholar 

  11. Sadeghian, H., Yang, C.K., Goosen, J.F., Bossche, A., Staufer, U., French, P.J., van Keulen, F.: Effects of size and defects on the elasticity of silicon nanocantilevers. J. Micromech. Microeng. 20, 064012 (2010)

    Article  Google Scholar 

  12. Celik, E., Guven, I., Madenci, E.: Mechanical characterization of nickel nanowires by using a customized atomic force microscope. Nanotechnology 22, 155702 (2011)

    Article  Google Scholar 

  13. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)

    Article  MATH  Google Scholar 

  15. Dingreville, R., Qu, J., Cherkaoui, M.: Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. J. Mech. Phys. Solids 53, 1827–1854 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L.: Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J. Mech. Phys. Solids 53, 1574–1596 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, G.F., Feng, X.Q.: Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl. Phys. Lett. 90, 231904 (2007)

    Article  Google Scholar 

  18. He, J., Lilley, C.M.: Surface effect on the elastic behavior of static bending nanowires. Nano Lett. 8, 1798–1802 (2008)

    Article  Google Scholar 

  19. Chhapadia, P., Mohammadi, P., Sharma, P.: Curvature-dependent surface energy and implications for nanostructures. J. Mech. Phys. Solids 59, 2103–2115 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chiu, M.S., Chen, T.: Effects of high-order surface stress on static bending behavior of nanowires. Phys. E 44, 714–718 (2011)

    Article  Google Scholar 

  21. Song, F., Huang, G., Park, H., Liu, X.: A continuum model for the mechanical behavior of nanowires including surface and surface-induced initial stresses. Int. J. Solids Struct. 48, 2154–2163 (2011)

    Article  Google Scholar 

  22. Liu, J., Mei, Y., Xia, R., Zhu, W.: Large displacement of a static bending nanowire with surface effects. Phys. E 44, 2050–2055 (2012)

    Article  Google Scholar 

  23. Park, S., Kim, J., Park, J., Lee, J., Choi, Y., Kwon, O.: Molecular dynamics study on size-dependent elastic properties of silicon nanocantilevers. Thin Solid Films 492, 285–289 (2005)

    Article  Google Scholar 

  24. Mohammadi, P., Sharma, P.: Atomistic elucidation of the effect of surface roughness on curvature-dependent surface energy, surface stress, and elasticity. Appl. Phys. Lett. 100, 133110 (2012)

    Article  Google Scholar 

  25. Georgakaki, D., Ziogos, O., Polatoglou, H.: Vibrational and mechanical properties of Si/Ge nanowires as resonators: a molecular dynamics study. Phys. Status Solidi Appl. Mater. 211, 267–276 (2014)

    Article  Google Scholar 

  26. Wang, G.F., Feng, X.Q.: Timoshenko beam model for buckling and vibration of nanowires with surface effects. J. Phys. D: Appl. Phys. 42, 155411 (2009)

    Article  Google Scholar 

  27. Jiang, L.Y., Yan, Z.: Timoshenko beam model for static bending of nanowires with surface effects. Phys. E 42, 2274–2279 (2010)

    Article  Google Scholar 

  28. Li, X.F., Zhang, H., Lee, K.Y.: Dependence of Young’s modulus of nanowires on surface effect. Int. J. Mech. Sci. 81, 120–125 (2014)

    Article  Google Scholar 

  29. Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139 (2000)

    Article  Google Scholar 

  30. Shenoy, V.B.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B 71, 094104 (2005)

    Article  Google Scholar 

  31. Mi, C., Jun, S., Kouris, D.A., Kim, S.Y.: Atomistic calculations of interface elastic properties in noncoherent metallic bilayers. Phys. Rev. B 77, 075425 (2008)

    Article  Google Scholar 

  32. Chen, S.H., Yao, Y.: Elastic theory of nanomaterials based on surface-energy density. J. Appl. Mech. 81, 121002 (2014)

    Article  Google Scholar 

  33. Yao, Y., Chen, S.H.: Surface effect in the bending of nanowires. Mech. Mater. 100, 12–21 (2016)

    Article  Google Scholar 

  34. Yao, Y., Chen, S.H.: Surface effect on resonant properties of nanowires predicted by an elastic theory for nanomaterials. J. Appl. Phys. 118, 044303 (2015)

    Article  Google Scholar 

  35. Yao, Y., Chen, S.: Buckling behavior of nanowires predicted by a new surface energy density model. Acta Mech. 227, 1799–1811 (2016)

  36. Huang, Z.P., Sun, L.: Size-dependent effective properties of a heterogeneous material with interface energy effect: from finite deformation theory to infinitesimal strain analysis. Acta Mech. 190, 151–163 (2007)

    Article  MATH  Google Scholar 

  37. Nix, W.D., Gao, H.J.: An atomistic interpretation of interface stress. Scr. Mater. 39, 1653–1661 (1998)

    Article  Google Scholar 

  38. Huang, Z.P., Wang, J.: A theory of hyperelasticity of multi-phase media with surface/interface energy effect. Acta Mech. 182, 195–210 (2006)

    Article  MATH  Google Scholar 

  39. Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)

  40. Chen, Y., Dorgan Jr., B.L., Mcllroy, D.N., Aston, D.E.: On the importance of boundary conditions on nanomechanical bending behavior and elastic modulus determination of silver nanowires. J. Appl. Phys. 100, 104301 (2006)

    Article  Google Scholar 

  41. Diao, J., Gall, K., Dunn, M.L.: Atomistic simulation of the structure and elastic properties of gold nanowires. J. Mech. Phys. Solids 52, 1935–1962 (2004)

    Article  MATH  Google Scholar 

  42. Ouyang, G., Li, X., Tan, X., Yang, G.: Surface energy of nanowires. Nanotechnology 19, 045709 (2008)

    Article  Google Scholar 

  43. Olsson, P.A., Park, H.S.: On the importance of surface elastic contributions to the flexural rigidity of nanowires. J. Mech. Phys. Solids 60, 2064–2083 (2012)

    Article  MathSciNet  Google Scholar 

  44. Jaccodine, R.: Surface energy of germanium and silicon. J. Electrochem. Soc. 110, 524–527 (1963)

    Article  Google Scholar 

  45. Sheng, H., Kramer, M., Cadien, A., Fujita, T., Chen, M.: Highly optimized embedded-atom-method potentials for fourteen fcc metals. Phys. Rev. B 83, 134118 (2011)

    Article  Google Scholar 

  46. Moshtaghin, A.F., Naghdabadi, R., Asghari, M.: Effects of surface residual stress and surface elasticity on the overall yield surfaces of nanoporous materials with cylindrical nanovoids. Mech. Mater. 51, 74–87 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

The work reported here is supported by NSFC through Grants #11532013, #11372317, #11402270 and the CAS/SAFEA International Partnership Program for Creative Research Teams.

Author information

Authors and Affiliations

  1. LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

    Ning Jia

  2. Institute of Advanced Structure Technology and Beijing Key Laboratory of Lightweight Multi-functional Composite Materials and Structures, Beijing Institute of Technology, Beijing, 100081, China

    Yin Yao, Yazheng Yang & Shaohua Chen

Authors
  1. Ning Jia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yin Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yazheng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shaohua Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shaohua Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jia, N., Yao, Y., Yang, Y. et al. Size effect in the bending of a Timoshenko nanobeam. Acta Mech 228, 2363–2375 (2017). https://doi.org/10.1007/s00707-017-1835-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-017-1835-2

Search

Navigation