Abstract
Material heterogeneity induced by a surface or interface may be neglected at macroscale since the surface-to-volume ratio is usually small. However, its effect can become significant for structures at nanoscale with a large surface-to-volume ratio. In this paper, we incorporate such surface material heterogeneity into wave propagation analysis of a nanosized transversely isotropic cylinder. This is achieved by using the concept of surface elasticity. Instead of directly using the well-known Gurtin–Murdoch (GM) surface elasticity, we develop here another general framework based on a thin layer model. A novel approach based on state-space formalism is used to derive the approximate governing equations. Three different sources of surface effect can be identified in the first-order surface elasticity, i.e., the elasticity effect, the inertia effect and the thickness effect. It is found that the derived theory becomes identical to the GM surface elasticity if the thickness effect is dropped and when the material is isotropic. The axisymmetric wave propagation in a transversely isotropic cylinder with surface effect is then studied, and an exact solution is presented. Numerical results are finally given to show that the surface effect will play a very pronounced role in wave propagation in cylinders at nanoscale.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Kis A., Mihailovic D., Remskar M., Mrzel A., Jesih A., Piwonski I., Kulik A.J., Benoît W., Forró L.: Shear and Young’s moduli of MoS2 nanotube ropes. Adv. Mater. 15, 733–736 (2003)
Li M., Tang H.X., Roukes M.L.: Ultra-sensitive NEMS-based cantilevers for sensing, scanned probe and very high-frequency applications. Nat. Nanotechnol. 2, 114–120 (2007)
Lieber C.M., Wang Z.L.: Functional nanowires. MRS Bull. 32, 99–104 (2007)
Gao R.P., Wang Z.L., Bai Z.G., de Heer W.A., Dai L.M., Gao M.: Nanomechanics of individual carbon nanotubes from pyrolytically grown arrays. Phys. Rev. Lett. 85, 622–625 (2000)
Cuenot S., Fretigny C., Demoustier C.S., Nysten B.: Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B. 69, 165410 (2004)
Chen C.Q., Shi Y., Zhang Y.S., Zhu J., Yan Y.J.: Size dependence of Young’s modulus in ZnO nanowires. Phys. Rev. Lett. 96, 075505 (2006)
Shenoy V.B.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B. 71, 094104 (2005)
Cao G.X., Chen X.: Size dependence and orientation dependence of elastic properties of ZnO nanofilms. Int. J. Solids Struct. 45, 1730–1753 (2008)
Wang J., Lu C.S., Wang Q., Xiao P., Ke F.J., Bai Y.L., Shen Y.G., Liao X.Z., Gao H.J.: Influence of microstructures on mechanical behaviours of SiC nanowires: a molecular dynamics study. Nanotechnology 23, 025703 (2012)
Cammarata R.C.: Surface and interface stress effects on interfacial and nanostructured materials. Mater. Sci. Eng. A 237, 180–184 (1997)
Miller R.E., Shenoy V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139 (2000)
Dingreville R., Qu J.M., 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)
Gurtin M.E., Murdoch A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)
Chen T., 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)
Sharma P., Ganti S., Bhate N.: Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82, 535–537 (2003)
Wang G.F., Feng X.Q., Yu S.W.: Surface buckling of a bending microbeam due to surface elasticity. Europhys. Lett. 77, 44002 (2007)
Duan H.L., Wang J., Karihaloo B.L.: Theory of elasticity at the nanoscale. Adv. Appl. Mech. 42, 1–68 (2008)
Lu P., He L.H., Lee H.P., Lu C.: Thin plate theory including surface effects. Int. J. Solids Struct. 43, 4631–4647 (2006)
Chen W.Q., Zhang Ch.: Anti-plane shear Green’s functions for an isotropic elastic half-space. Int. J. Solids Struct. 47, 1641–1650 (2010)
Lü C.F., Wu D.Z., Chen W.Q.: Surface effects on the jump-in instability of nanomechanical structures. IEEE Trans. Nanotechnol. 10, 962–967 (2011)
Liu C., Rajapakse R.K.N.D.: A size-dependent continuum model for nanoscale circular plates. IEEE Trans. Nanotechnol. 12, 13–20 (2013)
Huang Z.P., Wang J.: A theory of hyperelasticity of multi-phase media with surface/interface energy effect. Acta Mech. 182, 195–210 (2006)
Mindlin, R.D.: High frequency vibrations of plated, crystal plates. In: Progress in Applied Mechanics, pp. 73–84. MacMillan, New York (1963)
Gurtin M.E., Murdoch A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)
Tiersten H.F.: Elastic surface waves guided by thin films. J. Appl. Phys. 40, 770–789 (1969)
Rokhlin S.I., Wang Y.J.: Analysis of boundary conditions for elastic wave interaction with an interface between two solids. J. Acoust. Soc. Am. 89, 503–515 (1991)
Bövik P.: A comparison between the Tiersten model and O(h) boundary conditions for elastic surface waves guided by thin layers. J. Appl. Mech. 63, 162–167 (1996)
Bövik P.: On the modelling of thin interface layers in elastic and acoustic scattering problems. Q. J. Mech. Appl. Math. 47, 17–42 (1994)
Benveniste Y.: A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media. J. Mech. Phys. Solids. 54, 708–734 (2006)
Ting T.C.T.: Steady waves in an anisotropic elastic layer attached to a half-space or between two half-spaces—a generalization of love waves and Stoneley waves. Math. Mech. Solids. 14, 52–71 (2009)
Chen W.Q.: Surface effect on Bleustein–Gulyaev wave in a piezoelectric half-space. Theor. Appl. Mech. Lett. 1, 041001 (2011)
Chen, W.Q.: Wave propagation in a piezoelectric plate with surface effect. In: Analysis of Piezoelectric Structures and Devices, pp. 285–312. Higher Education Press, Beijing (2013)
Wang L.F., Hu H.Y.: Flexural wave propagation in single-walled carbon nanotubes. Phys. Rev. B 71, 195412 (2005)
Wu X.F., Dzenis Y.A.: Wave propagation in nanofibers. J. Appl. Phys. 100, 124318 (2006)
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)
Assadi A., Farshi B.: Size-dependent longitudinal and transverse wave propagation in embedded nanotubes with consideration of surface effects. Acta Mech. 222, 27–39 (2011)
Huang G.Y., Kang Y.L.: Acoustic vibrations of a circular nanowire by considering the effect of surface. J. Appl. Phys. 110, 023526 (2011)
Ding H.J., Chen W.Q., Zhang L.C.: Elasticity of Transversely Isotropic Materials. Springer, Dordrecht (2006)
Timoshenko S.P., Goodier J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)
Chen, W.Q., Ding, H.J.: The state-space method and its application in analyses of FGM structures. In: Mechanics of Functionally Graded Materials and Structures, pp. 139–178. Nova Science Publishers, New York (2012)
Tarn J.Q.: A state space formalism for anisotropic elasticity. Part II: cylindrical anisotropy. Int. J. Solids Struct. 39, 5157–5172 (2002)
Ding H.J., Chen W.Q.: Three Dimensional Problems of Piezoelasticity. Nova Science Publishers, New York (2001)
Lur’e A.I.: Three-Dimensional Problems of the Theory of Elasticity. Interscience Publishers, New York (1964)
Mindlin R.D., McNiven H.D.: Axially symmetric waves in elastic rods. J. Appl. Mech. 27, 145–151 (1960)
Zhang C.L., Liu N., Yang J.S., Chen W.Q.: Thickness-shear vibration of AT-cut quartz plates carrying finite-size particles with rotational degree of freedom and rotatory inertia. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58, 666–670 (2011)
Hirth J.P., Lothe J.: Theory of Dislocations, 2nd edn. Wiley, New York (1982)
Wang G.F., Li X.D.: Predicting Young’s modulus of nanowires from first-principles calculations on their surface and bulk materials. J. Appl. Phys. 104, 113517 (2008)
Mcniven H.D., Mengi Y.: Dispersion of waves in transversely isotropic rods. J. Acoust. Soc. Am. 49, 229–236 (1971)
Hu Y.G., Liew K.M., Wang Q., He X.Q., Yakobson B.I.: Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes. J. Mech. Phys. Solids 56, 3475–3485 (2008)
Wang Q.: Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J. Appl. Phys. 98, 124301 (2005)
Hu Y.G., Liew K.M., Wang Q.: Nonlocal elastic beam models for flexural wave propagation in double-walled carbon nanotubes. J. Appl. Phys. 106, 044301 (2009)
Wang Q., Liew K.M.: Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys. Lett. A 363, 236–242 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, W.Q., Wu, B., Zhang, C.L. et al. On wave propagation in anisotropic elastic cylinders at nanoscale: surface elasticity and its effect. Acta Mech 225, 2743–2760 (2014). https://doi.org/10.1007/s00707-014-1211-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-014-1211-4