Skip to main content
SpringerLink
Log in

Buckling behavior of nanowires predicted by a new surface energy density model

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

Abstract

The axial buckling behavior of nanowires is investigated with a new continuum theory, in which the surface effect of nanomaterials is characterized by the surface energy density. Only the surface energy density of bulk materials and the surface relaxation parameter are involved, instead of the surface elastic constants in the classical surface elasticity theory. Two kinds of nanowires with different boundary conditions are discussed. It is demonstrated that the new continuum theory can predict the buckling behavior of nanowires very well. Similar to the prediction of the classical elasticity theory, the critical compressive load of axial buckling of nanowires predicted by the new continuum theory increases with an increasing characteristic length, such as the diameter or height of nanowires. With the same aspect ratio, a nanowire with a rectangular cross section possesses a larger critical buckling load than that with a circular one. However, the surface effect could enhance the critical buckling load not only for a fixed–fixed nanowire but also for a cantilevered one in contrast to the classical elastic model. All the results predicted by the new continuum theory agree well with predictions by the surface elasticity models. The present research not only verifies the validation of the new continuum theory, but also gives a much more convenient characterization of buckling behaviors of nanowires. This should be helpful for the design of nanodevices based on nanomaterials, for example, nanobeams in NEMS or high-precision instruments.

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. Craighead H.G.: Nanoelectromechanical systems. Science 290, 1532–1535 (2000)

    Article  Google Scholar 

  2. Wu B., Heidelberg A., Boland J.J.: Mechanical properties of ultrahigh-strength gold nanowires. Nat. Mater. 4, 525–529 (2005)

    Article  Google Scholar 

  3. Llobet J., Sansa M., Gerboles M., Mestres N., Arbiol J., Borrise X., Murano-Perez F.: Enabling electromechanical transduction in silicon nanowire mechanical resonators fabricated by focused ion beam implantation. Nanotechnology 25, 135302 (2014)

    Article  Google Scholar 

  4. McDowell M.T., Leach A.M., Gall K.: On the elastic modulus of metallic nanowires. Nano Lett. 8, 3613–3618 (2008)

    Article  Google Scholar 

  5. Wang J.X., Huang Z.P., Duan H.L., Yu S.W., Feng X.Q., Wang G.F., Zhang W.X., Wang T.J.: Surface stress effect in mechanics of nanostructured materials. Acta. Mech. Solida Sin. 24, 52–82 (2011)

    Article  Google Scholar 

  6. Cuenot S., Fretigny C., Champagne S.D., 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 Y.X., Dorgan 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 

  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. Gavan, K.B., Westra, H.J.R., Vander drift, E.W.J.M., Venstra, W.J., Vander zant, H.S.J.: Surface effects on elastic properties of silver nanowires: Contact atomic-force microscopy. Appl. Phys. Lett. 94, 233108 (2009)

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

    Google Scholar 

  11. 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 

  12. 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 

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

    Article  MATH  Google Scholar 

  14. 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 

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

    Article  Google Scholar 

  16. Song F., Huang G.L., Park H.S., Liu X.N.: 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 

  17. Chiu M.S., Chen T.Y.: Effects of high-order surface stress on buckling and resonance behavior of nanowires. Acta Mech. 223, 1473–1484 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    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. 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 

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

    Article  Google Scholar 

  22. Park H.S., Klein P.A.: Surface stress effects on the resonant properties of metal nanowires: The importance of finite deformation kinematics and the impact of the residual surface stress. J. Mech. Phys. Solids 56, 3144–3166 (2008)

    Article  MATH  Google Scholar 

  23. Feng Y.K., Liu Y.L., Wang B.: Finite element analysis of resonant properties of silicon nanowires with consideration of surface effects. Acta Mech. 217, 149–155 (2011)

    Article  MATH  Google Scholar 

  24. Dobrokhotov V.V., Yazdanpanah M.M., Pabba S., Safir A., Cohn R.W.: Visual force sensing with flexible nanowire buckling springs. Nanotechnology 19, 035502 (2008)

    Article  Google Scholar 

  25. Hsin C.L., Mai W.J., Gu Y.D., Gao Y.F., Huang C.T., Liu Y.Z., Chen L.J., Wang Z.L.: Elastic properties and buckling of silicon nanowires. Adv. Mater. 20, 3919–3923 (2008)

    Article  Google Scholar 

  26. Olsson P.A.T., Park H.S.: Atomistic study of the buckling of gold nanowires. Acta Mater. 59, 3883–3894 (2011)

    Article  Google Scholar 

  27. Ji L.W., Young S.J., Fang T.H., Liu C.H.: Buckling characterization of vertical ZnO nanowires using nanoindentation. Appl. Phys. Lett. 90, 033109 (2007)

    Article  Google Scholar 

  28. Young S.J., Ji L.W., Chang S.J., Fang T.H., Hsueh T.J., Meen T.H., Chen I.C.: Nanoscale mechanical characteristics of vertical ZnO nanowires grown on ZnO:Ga/glass templates. Nanotechnology 18, 225603 (2007)

    Article  Google Scholar 

  29. Wen Y.H., Wang Q., Liew K.M., Zhu Z.Z.: Compressive mechanical behavior of Au nanowires. Phys. Lett. A 374, 2949–2952 (2010)

    Article  Google Scholar 

  30. Wang G.F., Feng X.Q.: Surface effects on buckling of nanowires under uniaxial compression. Appl. Phys. Lett. 94, 141913 (2009)

    Article  Google Scholar 

  31. 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 

  32. Yao H.Y., Yun G.H.: The effect of nonuniform surface elasticity on buckling of ZnO nanowires. Phys. E 44, 1916–1919 (2012)

    Article  Google Scholar 

  33. Challamel N., Elishakoff E.: Surface stress effects may induce softening: Euler–Bernoulli and Timoshenko buckling solutions. Phys. E 44, 1862–1867 (2012)

    Article  Google Scholar 

  34. Juntarasaid C., Pulngern T., Chucheepsakul S.: Bending and buckling of nanowires including the effects of surface stress and nonlocal elasticity. Phys. E 46, 68–76 (2012)

    Article  Google Scholar 

  35. Wang Y., Song J.Z., Xiao J.L.: Surface effects on in-plane buckling of nanowires on elastomeric substrates. J. Phys. D: Appl. Phys. 46, 125309 (2013)

    Article  Google Scholar 

  36. Liu C., Rajapakse R.K.N.D., Phani A.S.: Finite element modeling of beams with surface energy effects. ASME J. Appl. Mech. 78, 031014 (2011)

    Article  Google Scholar 

  37. Miller R.E., Shenoy V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)

    Article  Google Scholar 

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

    Article  Google Scholar 

  39. Mi C.W., 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 

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

    Article  Google Scholar 

  41. Yao Y., Wei Y.C., Chen S.H.: Size effect of the surface energy density of nanoparticles. Surf. Sci. 636, 19–24 (2015)

    Article  Google Scholar 

  42. Yao, Y., Chen, S.H.: Surface effect in the bending of nanowires. (2015) (Under review)

  43. 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 

  44. Zhang C., Yao Y., Chen S.H.: Size-dependent surface energy density of typically fcc metallic nanomaterials. Comput. Mater. Sci. 82, 372–377 (2014)

    Article  Google Scholar 

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

    Article  Google Scholar 

  46. Sun C.Q.: Oxidation electronics: bond–band–barrier correlation and its applications. Prog. Mater. Sci. 48, 521–685 (2003)

    Article  Google Scholar 

  47. 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 

  48. Timoshenko S.P., Gere J.M.: Mechanics of Materials. Van Nostrand Reinhold Co., New York (1972)

    Google Scholar 

  49. Zhang W.X., Wang T.J., Chen X.: Effect of surface/interface stress on the plastic deformation of nanoporous materials and nanocomposites. Int. J. Plast. 26, 957–975 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  50. Chen T.Y., Chiu M.S.: Effects of higher-order interface stresses on the elastic states of two-dimensional composites. Mech. Mater. 43, 212–221 (2011)

    Article  Google Scholar 

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

    Article  Google Scholar 

  52. Diao J.K., 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 

  53. Weinberger C.R., Jennings A.T., Kang K.W., Greer J.R.: Atomistic simulations and continuum modeling of dislocation nucleation and strength in gold nanowires. J. Mech. Phys. Solids 60, 84–103 (2012)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Yin Yao & Shaohua Chen

Authors
  1. Yin Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. 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

Yao, Y., Chen, S. Buckling behavior of nanowires predicted by a new surface energy density model. Acta Mech 227, 1799–1811 (2016). https://doi.org/10.1007/s00707-016-1597-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-016-1597-2

Keywords

Search

Navigation