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Surface effect investigation for static bending of nanowiresresting on elastic substrate using Timoshenko beam theory in tandem with the Laplace-Youngequation

Published online by Cambridge University Press:  16 November 2012

Amin Khajeansari  and
Gholam Hosein Baradaran
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Abstract

In the present study, an enriched continuum mechanics framework is employed to study thesurface effects on bending behavior of silver nanowires (NWs) resting on elasticsubstrate. The Timoshenko beam theory and the Laplace-Young equation are employed toinvestigate static behavior of silver NWs lying on Winkler-Pasternak elastic substrate.Three types of boundary conditions are considered as doubly simply supported (S-S), doublyclamped (C-C) and cantilevered (C-F). Analytical solutions are obtained for NWs withsurface crystallographic orientation of [001] subjected to a concentrated external force.By defining different normalized contact stiffness, extensive numerical results arecarried out to study the influence of effective parameters such as substrate, surface,aspect ratio (L/D) and diameter onthe stiffness of NWs. According to the obtained results, the effect of surface and itsrate of variation on stiffness of NWs lying on Winkler and Winkler-Pasternak elasticfoundation models are more significant in (C-F) type of boundary condition compared to theNWs without foundation. By increasing the modulus of elastic substrate, the effect ofshear deformation increases which it is more considerable in (C-C) and (S-S) NWs restingon the Winkler-Pasternak and Winkler substrate models, respectively.

Keywords

Nanowireelastic substratesurface effectsize dependencytimoshenko beam
Type
Research Article
Information
Mechanics & Industry , Volume 13 , Issue 3 , 2012 , pp. 163 - 174
Copyright
© AFM, EDP Sciences 2012

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References

Références

Lieber, C.M., Nanoscale science and technology : building a big future from small things, MRS Bull 28 (2003) 486–491 CrossRefGoogle Scholar
Xia, Y., Yang, P., Sun, Y., Wu, Y., Mayers, B., Gates, B., Yin, Y., Kim, F., Yan, H., One-dimensional nanostructures : synthesis, characterization, and applications, Adv. Mater. 15 (2003) 353–389 CrossRefGoogle Scholar
Lieber, C.M., Wang, Z.L., Functional Nanowires, MRS Bull. 32 (2007) 99–108 CrossRefGoogle Scholar
Canham, L.T., Silicon quantum wire array fabricated by electrochemical and chemical dissolution of wafers, Appl. Phys. Lett. 57 (1990) 1046–48 CrossRefGoogle Scholar
Wong, E.W., Sheehan, P.E., Lieber, C.M., Nanobeam mechanics : elasticity, strength, and toughness of nanorods and nanotubes, Science 277 (1997) 1971–75 CrossRefGoogle Scholar
Kacem, N., Baguet, S., Hentz, S., Dufour, R., Nonlinear phenomena in nanomechanical resonators : mechanical behaviors and physical limitations, Mécanique & Industries 11 (2010) 521–529 CrossRefGoogle Scholar
Cui, Y., Zhong, Z., Wang, D., Wang, W.U., Lieber, C.M., High performance silicon nanowire field effect transistors, Nano. Lett. 3 (2003) 149–152 CrossRefGoogle Scholar
M. Cahay, J.P. Leburton, D.J. Lockwood, S. Bondyopadhyay, J.S. Harris, Quantum confinement VI : nanostructured materials and devices, Electrochemical Society, inc. USA, 2001
H. Haug, S.W. Koch, Quantum theory of the optical and electronic properties of semiconductors, World Scientific, Singapore, 2004
Shenoy, V.B., Atomistic calculations of elastic properties of metallic fcc crystal surfaces, Phys. Rev. B. 71 (2005) 094104–11 CrossRefGoogle Scholar
Craighead, H.G., Nanoelectromechanical systems, Science 290 (2000) 1532–35 CrossRefGoogle ScholarPubMed
Ekinci, K.L., M.L. Roukes, Nanoelectromechanical systems. Rev. Sci. Instrum. 76 (2005) 061101–12 CrossRefGoogle Scholar
R. Michael, S.C. Wolfram, Handbook of theoretical and computational nanotechnology, American Scientific, 2005, Vol. 1
Diao, J., Gall, K., Dunn, M.L., Atomistic simulation of the structure and elastic properties of gold nanowires, J. Mech. Phys. Sol. 52 (2004) 1935–1962 CrossRefGoogle Scholar
Wu, H.A., Molecular dynamics study on mechanics of metal nanowire, Mech. Res. Commun. 33 (2006) 9–16 CrossRefGoogle Scholar
Z.L. Wang, Mechanical properties of nanowires and nanobelts, Dekker Encyclopedia of Nanoscience and Nanotechnology (2004) 1773–1786, DOI : 10.1081/E-ENN.120013387
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 (2006) 235409–6 CrossRefGoogle Scholar
Chen, Y.X., Dorgan, B.L., McIlroy, 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 (2006) 104301–7 CrossRefGoogle Scholar
Chen, C.Q., Zhang, Y.S., Zhu, J., Yan, Y.J., Size dependence of Young’s modulus in ZnO nanowires, Phys. Rev. Lett. 96 (2006) 075505–4 CrossRefGoogle ScholarPubMed
Miller, R.E., Shenoy, V.B., Size-dependent elastic properties of nanosized structural elements, Nanotechnology 11 (2000) 139–147 CrossRefGoogle Scholar
Guo, J.-G., Zhao, Y.-P., The size-dependent bending elastic properties of nanobeams with surface effects, Nanotechnology 18 (2007) 295701–6 CrossRefGoogle Scholar
Li, X.F., Wang, B.L., Lee, K.Y., Size effects of the bending stiffness of nanowires, J. Appl. Phys. 105 (2009) 074306–6 CrossRefGoogle Scholar
Gurtin, M.E., Murdoch, A.I., A continuum theory of elastic material surfaces, Arch. Rational. Mech. Anal. 57 (1975) 291–323 CrossRefGoogle Scholar
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 (2005) 1827–1854 CrossRefGoogle Scholar
Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L., Eshelby formalism for nano-inhomogeneities, Proc. R. Soc. A 461 (2005) 3335–3353 CrossRefGoogle Scholar
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 (2005) 1574–1596 CrossRefGoogle Scholar
Duan, H.L., Yi, X., Huang, Z.P., Wang, J., A unified scheme for prediction of effective moduli of multiphase composites with interface effects, Part I : Theoretical framework, Mech. Mater. 39 (2007) 81–93 CrossRefGoogle Scholar
Sharma, P., Ganti, S., Bhate, N., Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities, Appl. Phys. Lett. 82 (2003) 535–537 CrossRefGoogle Scholar
Sharma, P., Ganti, S., Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies, ASME J. Appl. Mech. 71 (2004) 663–71 CrossRefGoogle Scholar
Yvonnet, J., Quang, H.L., He, Q.-C., An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites, Comput. Mech. 42 (2008) 119–131 CrossRefGoogle Scholar
Yvonnet, J., Mitrushchenkov, A., Chambaud, G., He, Q.-C., Finite element model of ionic nanowires with size-dependent mechanical properties determined by ab initio calculations, Comput. Methods. Appl. Mech. Eng. 200 (2011) 614–625 CrossRefGoogle Scholar
Wang, G.F., Feng, X.Q., Effects of surface elasticity and residual surface tension on the natural frequency of microbeams, Appl. Phys. Lett. 90 (2007) 231904 CrossRefGoogle Scholar
Wang, G.F., Feng, X.Q., Effects of surface stresses on contact problems at nanoscale, J. Appl. Phys. 101 (2007) 013510–6 CrossRefGoogle Scholar
Wang, G.F., Feng, X.Q., Surface effects on buckling of nanowires under uniaxial compression, Appl. Phys. Lett. 94 (2009) 141913 CrossRefGoogle Scholar
Wang, G.F., Feng, X.Q., Timoshenko beam model for buckling and vibration of nanowires with surface effects, J. Phys. 42 (2009) 155411 Google Scholar
He, J., Lilley, C.M., Surface effect on the elastic behavior of static bending nanowires, Nano Lett. 8 (2008) 1798–1802 CrossRefGoogle ScholarPubMed
Jiang, L.Y., Yan, Z., Timoshenko beam model for static bending of nanowires with surface effects, Physica E : Low-dimens. Syst. Nanostruct. 42 (2010) 2274 CrossRefGoogle Scholar
Wang, M.C.P., Gates, B.D., Directed assembly of nanowires, Mater. Today 12 (2009) 34–43 CrossRefGoogle Scholar
Lu, W., Lieber, C.M., Nanoelectronics from the bottom up, Nat. Mater. 6 (2007) 541–850 CrossRefGoogle ScholarPubMed
Ru, C.Q., Axially compressed buckling of a doublewalled carbon nanotube embedded in an elastic medium, J. Mech. Phys. Solids 49 (2001) 1265–79 CrossRefGoogle Scholar
Pradhan, S.C., Kumar, A., Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method, Comput. Mater. Sci. 50 (2010) 239–245 CrossRefGoogle Scholar
Mohammadimehr, M., Saidi, A.R., A. Ghorbanpour Arani, A. Arefmanesh, Q. Han, Torsional buckling of a DWCNT embedded on Winkler and Pasternak foundations using nonlocal theory, J. Mech. Sci. Technol. 24 (2010) 1289–1299 CrossRefGoogle Scholar
R.F. Scott, Foundation Analysis, Prentice-Hall : Englewood Cliffs N.J, 1981
A.C. Ugural, S.K. Fenster, Advanced strength and applied elasticity, Printice Hall : USA, 2003
P.L. Pasternak, On a new method of analysis of an elastic foundation by means of two foundation constants, Goz Izd Lip Po Strait i Arkh : Moscow (in Russian), 1954
Cammarata, R.C., Surface and interface stress effects in thin films, Prog. Surf. Sci. 46 (1994) 1–38 CrossRefGoogle Scholar
Hutchinson, J.R., Shear coefficients for timoshenko beam theory, J. Appl. Mech. Trans : ASME 68 (2001) 87–92 CrossRefGoogle Scholar
Ghani razaqpur, A., Shah, K.R., Exact analysis of beams on two-parameter elastic foundations, Int. J. Solids Struct. 27 (1991) 435–54 CrossRefGoogle Scholar
Liew, K.M., He, X.Q., Kitipornchai, S., Predicting nanovibration of multi-layered graphene sheets embedded in an elastic matrix, Acta. Mater. 54 (2006) 4229–4236 CrossRefGoogle Scholar
Murmu, T., Pradhan, S.C., Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM, Physica E : Low-dimens. Syst. Nanostruct. 41 (2009) 1232–1239 CrossRefGoogle Scholar
Li, X.F., Fei, G.T., Zhou, W.F., Zhang, L.D., A convenient method to determine the bulk modulus of nanowires and its temperature dependence based on X-ray diffraction measurement, Solid. State. Commun. 150 (2010) 1117–1119 CrossRefGoogle Scholar