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Study of wave propagation in nanowires with surface effects by using a high-order continuum theory

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Abstract

A high-order continuum model is developed to study wave propagation in nanowires. By using the model, heterogeneous nanostructure effects can be captured especially for high wave frequency cases. Surface stress effects are also included by using the incremental deformation approach. The governing equations of motion in the nanowire are derived including both the strain-independent and strain-dependent surface stresses. For simplicity and clarity, specific attention will be paid to the effects of strain-independent surface stress in this study. The accuracy of the proposed model is validated by comparing dispersion curves of longitudinal wave propagation from the current model with those from the exact solution. By conducting a reduced formulation, the results predicted by the current model will be compared with those based on existed high-order models to show capability of the current model. Numerical simulations are then conducted to study both longitudinal and flexural wave propagation in nanowires. The surface stress effects upon both longitudinal and flexural wave propagation in nanowires are demonstrated, from which the size dependent wave information in nanowires can be observed. Some new physical wave phenomena related to the surface stress effects are discussed.

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References

  1. Zhong Z., Wang D., Cui Y., Bockrath M.W., Lieber C.M.: Nanowire crossbar arrays as address decoders for integrated nanosystems. Science 302, 1377–1379 (2003)

    Article  Google Scholar 

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

    Article  Google Scholar 

  3. Keating C.D., Natan M.J.: Striped metal nanowires as building blocks and optical tags. Adv. Mater. 15, 451–454 (2003)

    Article  Google Scholar 

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

    Article  Google Scholar 

  5. Haiss W.: Surface stress of clean and adsorbate-covered solids. Rep. Progr. Phys. 64, 591–648 (2001)

    Article  Google Scholar 

  6. Park H.S., Klein P.A.: Surface Cauchy–Born analysis of surface stress effects on metallic nanowires. Phys. Rev. B 75, 085408 (2007)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Wang G.F., Li X.D.: Predicting the Young’s modulus of nanowires from first-principles calculations on their surface and bulk materials. J. Appl. Phys. 104, 113517 (2008)

    Article  Google Scholar 

  9. Hu J., Liu X.W., Pan B.C.: A study of the size-dependent elastic properties of ZnO nanowires and nanotubes. Nanotechnology 19, 285710 (2008)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  12. Zhang W.X., Wang T.J., Chen X.: Effect of surface stress on the asymmetric yield strength of nanowires. J. Appl. Phys. 103, 123504 (2008)

    Article  Google Scholar 

  13. Koh S.J.A., Lee H.P., Lu C., Cheng Q.H.: Molecular dynamics simulation of a solid platinum nanowire under uniaxial tensile strain: temperature and strain-rate effects. Phys. Rev. B 72, 085414 (2005)

    Article  Google Scholar 

  14. 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 

  15. Wu X.F., Dzenis Y.A.: Wave propagation in nanofibers. J. Appl. Phys. 100, 124318 (2006)

    Article  Google Scholar 

  16. 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 

  17. Chen T., Dvorak G.J., Yu C.C.: Size-dependent elastic properties of unidirectional nano-composites with interface stresses. Acta Mech. 188, 39–54 (2007)

    Article  MATH  Google Scholar 

  18. Huang G.L., Sun C.T.: Higher order continuum modeling of wave propagation in elastic media with microstructures. Mech. Adv. Mat. Struct. 15, 550–557 (2008)

    Article  Google Scholar 

  19. Huang G.L., Sun C.T.: Continuum modeling of heterogeneous media with microstructures or nanostructures. Phil. Mag. 87, 3689–3707 (2007)

    Article  Google Scholar 

  20. Sun C.T., Huang G.L.: Modeling heterogeneous media with microstructures of different scales. J. Appl. Mech. (Trans. ASME) 74, 203–209 (2007)

    Article  MATH  Google Scholar 

  21. Feng X.L., He R.R., Yang P.D., Roukes M.L.: Very high frequency silicon nanowire electromechanical resonators. Nano Lett. 7, 1953–1959 (2007)

    Article  Google Scholar 

  22. Biot M.A.: Mechanics of Incremental Deformations. Wiley, New York (1965)

    Google Scholar 

  23. Sun C.T.: On the equations for a Timoshenko beam under initial stress. J. Appl. Mech. (Trans. ASME) 39, 282–285 (1972)

    Google Scholar 

  24. Achenbach J.D.: Wave Propagation in Elastic Solids. Elsevier, New York (1973)

    MATH  Google Scholar 

  25. Graff K.F.: Wave Motion in Elastic Solids. Dover, New York (1991)

    Google Scholar 

  26. Kaneko T.: On Timoshenko’s correction for shear in vibrating beams. J. Phys. D Appl. Phys. 8, 1927–1936 (1975)

    Article  Google Scholar 

  27. Askes H., Aifantis E.C.: Gradient elasticity theories in statics and dynamics—a unification of approaches. Int. J. Fract. 139, 297–304 (2006)

    Article  Google Scholar 

  28. Askes H., Bennett T., Aifantis E.C.: A new formulation and ζ0-implementation of dynamically consistent gradient elasticity. Int. J. Numer. Methods Eng. 72, 111–126 (2007)

    Article  MathSciNet  Google Scholar 

  29. Aifantis E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30, 1279–1299 (1992)

    Article  MATH  Google Scholar 

  30. Ru C.Q., Aifantis E.C.: A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech. 101, 59–68 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  31. Askes H., Suiker A.S.J., Sluys L.J.: A classification of high-order strain-gradient models-linear analysis. Arch. Appl. Mech. 72, 171–188 (2002)

    Article  MATH  Google Scholar 

  32. Maranganti R., Sharma P.: A novel atomistic approach to determine strain-gradient elasticity constants: tabulation and comparison for various metals, semiconductors, silica, polymers and the (Ir) relevance for nanotechnologies. J. Mech. Phys. Solids 55, 1823–1852 (2007)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  34. Huang D.W.: Size-dependent response of ultra-thin films with surface effects. Int. J. Solids Struct. 45, 568–579 (2008)

    Article  MATH  Google Scholar 

  35. Cuenot S., Fre’tigny 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 

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Authors and Affiliations

  1. Department of Applied Science, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA

    F. Song

  2. Department of Systems Engineering, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA

    G. L. Huang

  3. Department of Electrical Engineering, University of Arkansas, Fayetteville, AR, 72701, USA

    V. K. Varadan

Authors
  1. F. Song
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  2. G. L. Huang
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  3. V. K. Varadan
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Correspondence to G. L. Huang.

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Song, F., Huang, G.L. & Varadan, V.K. Study of wave propagation in nanowires with surface effects by using a high-order continuum theory. Acta Mech 209, 129–139 (2010). https://doi.org/10.1007/s00707-009-0156-5

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  • DOI: https://doi.org/10.1007/s00707-009-0156-5

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