尺寸效應在奈米尺度材料或結構的力學性質中甚為重要，本研究以界面區間(interphase)來導入奈米尺度固體中界面/表面的力學行為，在正交曲線座標架構下經由力平衡及力偶平衡推導界面間曳引力不連續的條件。本文研究的重點在於高階界面應力效應，其源自於沿著界面區間厚度方向非均勻之應力；當此界面區間的勁度達到足夠量級(order)，將使得此非均勻應力沿厚度方向積分後產生等效彎矩的作用，即為本研究所謂的高階界面應力效應。在二維問題中我們探討此界面區間的勁度和尺度在量級上的差異進而推導出四種型式的界面條件；並考慮一基材內嵌圓形內含物，遠端承受側向剪變形的邊界值問題。假設內含物為孔洞時，我們解出應力集中因子的解析解；同時亦考慮內含物為異質，經由廣義自洽法求解等效剪力模數。我們闡述在此四種界面條件下，應力集中因子和等效剪力模數的連續性過渡力學行為。在二維界面應力條件的應用上，本文亦在古典結構力學樑理論架構下探討奈米線(nanowires)於高階界面應力效應下不同的力學行為，包含靜態撓曲、挫曲和振動。我們將高階界面應力條件導入Euler-Bernoulli以及Timoshenko樑理論數學模式，求解臨界挫曲力量、撓度、特徵頻率等，並呈現高階與一階(generalized Young-Laplace equation)界面應力效應的差異。同時也推導等效楊氏模數的解析解而與現存實驗數據作比較；可以發現相較於其它理論，此處的理論架構較為簡單且對於實驗數據的趨勢能提供適當的預測。最後，我們提出高階界面應力條件在三維空間下的推導，並且基於古典薄板理論(Kirchhoff-Love plate theory)來檢視奈米板(nanoplates)在高階界面應力效應下的尺寸相依行為。

A theoretical framework which accounts for in-plane membrane surface stresses and surface moments is developed to simulate the interfacial behavior between two different nanoscaled solids. The interface behavior is derived from a thin interphase of constant thickness. Depending on the order difference in magnitudes of stiffness and length scales of the interphase, we find that four different types of interface conditions in two dimensions can be deduced by proper limit analysis. A boundary value problem of a circular inclusion in an infinite matrix is studied as an illustration. To highlight the high-order effects, we derive analytically the stress concentration factor of an infinite plate containing a circular cavity with interface stresses of different orders subjected to a remote transverse shear loading. The closed-form expressions show how the orders of interface stress will influence the concentration factor in a successive manner. In addition, we assess the effective shear modulus of a medium containing circular nano inclusions with high-order interface stress effects. The effective transverse shear modulus is derived based on the generalized self-consistent method. We also apply the high-order interface stress model using the Euler-Bernoulli beam and Timoshenko beam theories to predict the mechanical behavior of nanowires (NWs). Size-dependent effective Young’s modulus is characterized versus the diameter of the NWs, which agrees well with the existing experimental data. We conclude that the present continuum mechanics approach could be served as one of the feasible tools to analyze the mechanical behavior of nanostructures. Lastly, the high-order interface stress in three dimensions and the size-dependent behavior of nanoplates are examined.

References
Allara, D., 2005. A perspective on surfaces and interfaces. Nature 437, 638-639.
Andreussi, F., Gurtin, M.E., 1977. On the wrinkling of a free surface. J. Appl. Phys. 48, 3798.
Ansari, R., Sahmani, S., 2011. Surface stress effects on the free vibration behavior of nanoplates. Int. J. Eng. Sci. 49, 1204-1215.
Asghari, M., Kahrobaiyan, M.H., Rahaeifard, M., Ahmadian, M.T., 2011. Investigation of the size effects in Timoshenko beams based on the couple stress theory. Arch. Appl. Mech. 81, 863-874.
Asghari, M., 2012. Geometrically nonlinear micro-plate formulation based on the modified couple stress theory. Int. J. Eng. Sci. 51, 292-309.
Assadi, A., Farshi, B., Alinia-Ziazi A., 2010. Size dependent dynamic analysis of nanoplates. J. Appl. Phys. 107, 124310.
Assadi, A. and Farshi, B., 2010. Vibration characteristics of circular nanoplates. J. Appl. Phys. 108, 074312.
Finn, R., 1986. Equilibrium Capillary Surfaces. Springer-Verlag, New York Inc.
Benveniste, Y., Miloh, T., 2001. Imperfect soft and stiff interfaces in two-dimensional elasticity. Mech. Mater. 33, 309-323.
Benveniste, Y., 2006a. A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media. J. Mech. Phys. Solids 54, 708-734.
Benveniste, Y., 2006b. An interface model of a three-dimensional curved interphase in conduction phenomena. Proc. R. Soc. A. 462, 1593-1617.
Benveniste, Y., Baum, G., 2007. An interface model of a graded three-dimensional anisotropic curved interphase. Proc. R. Soc. A. 463, 419-434.
Benveniste, Y., Miloh, T., 2007. Soft neutral elastic inhomogeneities with membrane-type interface conditions. J. Elasticity 88, 87-111.
Benveniste, Y., Berdichevsky, O., 2010. On two models of arbitrarily curved three-dimensional thin interphases in elasticity. Int. J. Solids Struct. 47, 1899-1915.
Bovik, P., 1994. On the modeling of thin interface layers in elastic and acoustic scattering problems. Q. J. Mech. Appl. Math. 47, 17-42.
Bowden, N., Brittain, S., Evans, A.G., Hutchinson, J.W., Whitesides, G.M., 1998. Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer. Nature 393, 146-149.
Brooks, H., 1952. Metal Surface, pp30-31. American Society for Metals, Ohio.
Cammarata, R.C., 1994. Surface and interface stress effects in thin films. Prog. Surf. Sci. 46, 1-38.
Cammarata, R.C., Trimble, T.M., Srolovitz, D.J., 2000a. Surface stress model for intrinsic stresses in thin films. J. Mater. Res. 15, 2468-2474.
Cammarata, R.C., Sieradzki, K., Spaepen, F., 2000b. Simple model for interface stresses with application to misfit dislocation generation in epitaxial thin films. J. Appl. Phys. 87, 1227.
Cao, Y.P., Li, B., Feng, X.Q., 2012. Surface wrinkling and folding of core-shell soft cylinders. Soft Matter 8, 556-562.
Chakraborty, A., 2010. The effect of surface stress on the propagation of Lamb waves. Ultrasonics 50, 645-649.
Chang, G.E., Chang, C.O., Cheng, H.H., 2012. Strain analysis of a wrinkled SiGe bilayer thin film. J. Appl. Phys. 111, 034314.
Chen, X., Hutchinson, J.W., 2004. Herringbone buckling patterns of compressed thin films on compliant substrates. J. Appl. Mech. (ASME) 71, 597-603.
Chen, T., 1993. Thermoelastic properties and conductivity of composites reinforced by spherically anisotropic particles. Mech. Mater. 14, 257-268.
Chen, T., Dvorak, G.J., 2006. Fibrous nanocomposites with interface stress: Hill's and Levin's connections for effective moduli. Appl. Phys. Lett. 88, 211912.
Chen, T., Chiu, M.S., Weng, C.N., 2006a. Derivation of the generalized Young-Laplace equation of curved interface in nanoscaled solids. J. Appl. Phys. 100, 074308.
Chen, C.Q., Shi, Y., Zhang, Y.S., Zhu, J., Yan, Y.J., 2006b. Size dependence of Young’s modulus in ZnO nanowires. Phys. Rev. Lett. 96, 075505.
Chen, T., Dvorak, G.J., Yu, C.C., 2007a. Size-dependent elastic properties of unidirectional nano-composites with interface stresses. Acta Mech. 188, 39-54.
Chen, T., Dvorak, G.J., Yu, C.C., 2007b. Solids containing spherical nano-inclusions with interface stresses: Effective properties and thermal-mechanical connections. Int. J. Solids Struct. 44, 941-955.
Chen, H.A., Hu, G.K., Huang, Z.P., 2007c. Effective moduli for micropolar composite with interface effect. Int. J. Solids Struct. 44, 8106-8118.
Chen, T., 2008. Exact size-dependent connections between effective moduli of fibrous piezoelectric nanocomposites with interface effects. Acta Mech. 196, 205-217.
Chen, H.A., Liu, X., Hu, G.K., 2008. Overall plasticity of micropolar composites with interface effect. Mech. Mater. 40, 721-728.
Chen, A.L., Wang, Y.S., 2011. Size-effect on band structures of nanoscale phononic crystals. Physica E 44, 317-321.
Chen, T., Chiu, M.S., 2011. Effects of higher-order interface stresses on the elastic states of two-dimensional composites. Mech. Mater. 43, 212-221.
Cheng, Z.Q., He, L.H., 1995. Micropolar elastic fields due to a spherical inclusion. Int. J. Eng. Sci. 33, 389-397.
Chiu, M.S., Chen, T., 2011. Effects of high-order surface stress on static bending behavior of nanowires. Physica E 44, 714-718.
Chiu, M.S., Chen, T., 2011. Effects of high-order surface stress on buckling and resonance behavior of nanowires. Acta Mech. 223, 1473-1484.
Chiu, M.S., Chen, T., 2011. Higher-order surface stress effects on buckling of nanowires under uniaxial compression. Procedia Engineering 10, 397-402. (11th International Conference on the Mechanical Behavior of Materials, ICM11).
Cho, J., Joshi, M.S., Sun, C.T., 2006. Effect of inclusion size on mechanical properties of polymeric composites with micro and nano particles. Compos. Sci. Technology. 66, 1941-1952.
Christensen, R. M., Lo, K.H., 1979. Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids 27, 315-330.
Christensen, R. M., Lo, K.H., 1986. Erratum to Christensen and Lo (1979), J. Mech. Phys. Solids 34, 639.
Chung, J.Y., Chastek, T.Q., Fasolka, M.J., Ro, H.W., Stafford, C.M., 2009. Quantifying residual stress in nanoscale thin polymer films via surface wrinkling. ACS NANO 3, 844-852.
Cicco, S., Ieşan, D., 2012. A theory of chiral cosserat elastic plates, J. Elasticity, 1-19. DOI: 10.1007/s10659-012-9400-7
Cuenot, S., Fretigny, C., Demoustier-Champagne, S., Nysten, B., 2004. Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B. 69, 165410.
Dai, S., Park, H.S., 2013. Surface effects on the piezoelectricity of ZnO nanowires. J. Mech. Phys. Solids 61, 385-397.
Dingreville, R., Qu, J., Cherkaoui, M., 2005. Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. J. Mech. Phys. Solids 53, 1827-1854.
Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L., 2005a. Eshelby formalism for nano-inhomogeneities. Proc. R. Soc. A. 461, 3335-3353.
Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L., 2005b. Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J. Mech. Phys. Solids 53, 1574-1596.
Duan, H.L., Wang, J., Huang, Z.P., Zhong, Y., 2005c. Stress fields of a spheroidal inhomogeneity with an interphase in an infinite medium under remote loadings. Proc. R. Soc. A. 461, 1055-1080.
Dyszlewicz, J., 2004. Micropolar Theory of Elasticity. Springer, Berlin, New York.
Eshelby, J.D., 1961. Elastic inclusions and inhomogeneities. Progress in Solid Mechanics 2, Edited by I.N. Sneddon and R. Hill, Amsterdam: North Holland, pp.89-140.
Fang, X.Q., Wang, X.H., Zhang, L.L., 2010. Interface effect on the dynamic stress around an elliptical nano-inhomogeneity subjected to anti-plane shear waves. CMC Comput. Mater. Contin. 16, 229-246.
Feng, X.Q., Xia, R., Li, X., Li, B., 2009. Surface effects on the elastic modulus of nanoporous materials. Appl. Phys. Lett. 94, 011916.
Freund, L.B., Suresh, S., 2003. Thin Film Materials: Stress, Defect Formation and Surface Evolution. Cambridge Univ. Press, Cambridge.
Gibbs, J.W., 1928. The collected works of J. W. Gibbs Vol. 1. Longman, New York.
Goudarzi, T., Avazmohammadi, R., Naghdabadi, R., 2010. Surface energy effects on the yield strength of nanoporous materials containing nanoscale cylindrical voids. Mech. Mater. 42, 852-862.
Groenewold, J., 2001. Wrinkling of plates coupled with soft elastic media. Physica A 298, 32-45.
Gurtin, M.E., Murdoch, A.I., 1975. A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291-323.
Gurtin, M.E., Murdoch, A.I., 1976. Effect of surface stress on wave propagation in solids. J. Appl. Phys. 47, 4414.
Gurtin, M.E., Murdoch, A.I., 1978. Surface stress in solids. Int. J. Solids Struct. 14, 431-440.
Gurtin, M. E., Weissmüller, J. and Larché, F., 1998. A general theory of curved deformable interfaces in solids at equilibrium, Philo. Mag. A 78, 1093-1109.
Hadjesfandiari, A.R., 2013. Size-dependent piezoelectricity. Int. J. Solids Struct. 50, 2781-2791.
Haftbaradaran, H., Shodja, H.M., 2009. Elliptic inhomogeneities and inclusions in anti-plane couple stress elasticity with application to nano-composites. Int. J. Solids Struct. 46, 2978-2987.
He, Q.C., Benveniste, Y., 2004. Exactly solvable spherically anisotropic thermoelastic microstructures. J. Mech. Phys. Solids. 52, 2661-2682.
He, L.H., Li, Z.R., 2006. Impact of surface stress on stress concentration. Int. J. Solids Struct. 43, 6208-6219.
He, J., Lilley, C.M., 2008a. Surface effect on the elastic behavior of static bending nanowires. Nano Letters 8, 1798-1802.
He, J., Lilley, C.M., 2008b. Surface stress effect on bending resonance of nanowires with different boundary conditions. Appl. Phys. Lett. 93, 263108.
Hill, R., 1964. Theory of mechanical properties of fibre-strengthened materials. I. Elastic behavior. J. Mech. Phys. Solids. 12, 199-212.
Hildebrand, F. B., 1965. Methods of Applied Mathematics. Prentice-Hall, Englewood Cliffs.
Huang, R., Suo, Z., 2002. Wrinkling of a compressed elastic film on a viscous layer. J. Appl. Phys. 91, 1135.
Huang, R., 2005a. Electrically induced surface instability of a conductive thin film on a dielectric substrate. Appl. Phys. Lett. 87, 151911.
Huang, R., 2005b. Kinetic wrinkling of an elastic film on a viscoelastic substrate. J. Mech. Phys. Solids. 53, 63-89.
Huang, Y., Rosakis, A.J., 2005. Extension of Stoney's formula to non-uniform temperature distributions in thin film/substrate systems. The case of radial symmetry. J. Mech. Phys. Solids. 53, 2483-2500.
Huang, Y., Ngo, D., Rosakis, A.J., 2005a. Non-uniform, axisymmetric misfit strain: in thin films bonded on plate substrates/substrate systems: the relation between non-uniform film stresses and system curvatures. Acta Mechanica Sinica 21, 362-370.
Huang, Z.Y., Hong, W., Suo, Z., 2005b. Nonlinear analyses of wrinkles in a film bonded to a compliant substrate. J. Mech. Phys. Solids. 53, 2101-2118.
Huang, Y., Rosakis, A.J., 2007. Extension of Stoney's formula to arbitrary temperature distributions in thin film/substrate systems. J. Appl. Mech. (Transactions of the ASME) 74, 1225-1233.
Huang, D.W., 2008. Size-dependent response of ultra-thin films with surface effects. Int. J. Solids Struct. 45, 568-579.
Huang, G.Y. Yu, S.W., 2006. Effect of surface piezoelectricity on the electromechanical behaviour of a piezoelectric ring. Phys. Stat. Solidi (b) 243, R22-R24.
Huck, W.T.S., Bowden, N., Onck, P., Pardoen, T., Hutchinson, J.W., Whitesides, G.M., 2000. Ordering of spontaneously formed buckles on planar surfaces. Langmuir 16, 3497-3501.
Ieşan, D., 1971. Torsion of micropolar elastic beams. Int. J. Eng. Sci. 9, 1047-1060.
Ieşan, D., 2010a. Chiral effects in uniformly loaded rods. J. Mech. Phys. Solids 58, 1272-1285.
Ieşan, D., 2010b. Torsion of chiral Cosserat elastic rods. Eur. J. Mech. A/Solids 29, 990-997.
Ieşan, D., 2010c. Thermal effects in chiral elastic rods. Int. J. Thermal Sci. 49, 1593-1599.
Ji, L.W., Young, S.J., Fang, T.H., Liu, C.H., 2007. Buckling characterization of vertical ZnO nanowires using nanoindentation. Appl. Phys. Lett. 90, 033109.
Jiang, L.Y., Yan, Z., 2010. Timoshenko beam model for static bending of nanowires with surface effects. Physica E 42, 2274-2279.
Jing, G.Y., Duan, H.L., Sun, X.M., Zhang, Z.S., Xu, J., Li, Y.D., Wang, X.J., Yu, D.P., 2006. Surface effects on elastic properties of silver nanowires: Contact atomic-force microscopy. Phys. Rev. B. 73, 235409.
Jing, Y., Meng, Q., Gao, Y., 2009. Molecular dynamics simulation on the buckling behavior of silicon nanowires under uniaxial compression. Comput. Mater. Sci. 45, 321-326.
Jomehzadeh, E., Noori, H.R., Saidi, A.R., 2011. The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Physica E 43, 877-883.
Kim, T.Y., Puntel, E., Fried, E., 2012. Numerical study of the wrinkling of a stretched thin sheet. Int. J. Solids Struct. 49, 771-782.
Kornev, K.G., Srolovitz, D.J., 2004. Surface stress-driven instabilities of a free film. Appl. Phys. Let. 85, 2487-2489.
Kushwaha, M.S., Halevi, P., Dobrzynski, L., Djafari-Rouhani, B., 1993. Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71, 2022–2025.
Kwon, S.J., Lee, H.H., 2005. Theoretical analysis of two-dimensional buckling patterns of thin metal-polymer bilayer on the substrate. J. Appl. Phys. 98, 063526.
Lachut, M.J., Sader, J.E., 2007. Effect of Surface Stress on the Stiffness of Cantilever Plates. Phys. Rev. Lett. 99, 206102.
Lachut, M.J., Sader, J.E., 2009. Effect of surface stress on the stiffness of cantilever plates: Influence of cantilever geometry. Appl. Phys. Let. 95, 193505.
Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P., 2003. Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477-1508.
Laplace, P. S., 1806. Traite de mechanique celeste; supplements au Livre X. Euvres Complete Vol. 4, Gauthier-Villars, Paris.
Landau, L. D. and Lifshitz, E. M., 1987. Fluid Mechanic, 2nd Ed., Pergamon Press, Oxford.
Li, B., Huang, S.Q., Feng, X.Q., 2010. Buckling and postbuckling of a compressed thin film bonded on a soft elastic layer: a three-dimensional analysis. Arch. Appl. Mech. 80, 175-188.
Li, Y., Fang, B., Zhang, J., Song J., 2011. Surface effects on the wrinkling of piezoelectric films on compliant substrates. J. Appl. Phys. 110, 114303.
Li, Y.D., Lee, K.Y., 2010. Size-dependent behavior of Love wave propagation in a nanocoating. Modern Physics Letters B 24, 3015-3023.
Li, B., Cao, Y.P., Feng, X.Q., Gao, H., 2012a. Mechanics of morphological instabilities and surface wrinkling in soft materials: a review. Soft Matter 8, 5728-5745.
Li, Y., Fang, B., Zhang, J., Song, J., 2012b. Surface effects on the wrinkles in a stiff thin film bonded to a compliant substrate. Thin Solid Films 520, 2077-2079.
Liu, F., Rugheimer, P., Mateeva, E., Savage, D.E., Lagally, M.G., 2002. Nanomechanics: Response of a strained semiconductor structure. Nature 416, 498.
Liu, W., Chen, J.W., Liu, Y.Q., Su, X.Y., 2012. Effect of interface/surface stress on the elastic wave band structure of two-dimensional phononic crystals. Phys. Lett. A 376, 605-609.
Liu, H., Liu, H., Yang, J., 2013. Surface effects on the propagation of shear horizontal waves in thin films with nano-scalethickness. Physica E 49, 13-17.
Lu, T.Q., Zhang, W.X., Wang, T.J., 2011. The surface effect on the strain energy release rate of buckling delamination in thin film–substrate systems. Int. J. Eng. Sci. 49, 967-975.
Luo, J., Wang, X., 2009. On the anti-plane shear of an elliptic nano inhomogeneity. European J. Mech. A-Solids 28, 926-934.
Ma, H.M., Gao, X.L., Reddy, J.N., 2008. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379-3391.
Ma, H.M., Gao, X.L., Reddy, J.N., 2011. A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220, 217-235.
MacLaurin, J., Chapman, J., Jones, G.W., Roose, T., 2013. The study of asymptotically fine wrinkling in nonlinear elasticity using a boundary layer analysis. J. Mech. Phys. Solids 61, 1691-1711.
Mayes, A.M., 2005. Nanocomposites: Softer at the boundary. Nature Materials 4, 651-652.
Miller, R.E., Shenoy, V.B., 2000. Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139-147.
Mogilevskaya, S.G., Crouch, S.L., Stolarski, H.K., 2008. Multiple interacting circular nano-inhomogeneities with surface/interface effects. J. Mech. Phys. Solids 56, 2298-2327.
Mogilevskaya, S.G., Crouch, S.L., La Grotta, A., Stolarski, H.K., 2010a. The effect of surface elasticity and surface tension on the transverse overall elastic behavior of unidirectional nano-composites. Comp. Sci. Technology 70, 427-434..
Mogilevskaya, S.G., Crouch, S.L., Stolarski, H.K., Benusiglio, A., 2010b. Equivalent inhomogeneity method for evaluating the effective elastic properties of unidirectional multi-phase composites with surface/interface effects. Int. J. Solids Struct. 47, 407-418.
Murdoch, A.I., 1976. The propagation of surface waves in bodies with material boundaries. J. Mech. Phys. Solids 24, 137-146.
Nemat-Nasser, S., Hori, M., 1999. Micromechanics: Overall Properties of Heterogeneous Materials, second ed. Elsevier, North-Holland.
Ngo, D., Huang, Y., Rosakis, A.J., Feng, X., 2006. Spatially non-uniform, isotropic misfit strain in thin films bonded on plate substrates: The relation between non-uniform film stresses and system curvatures. Thin Solid Films 515, 2220-2229.
Ni, H., Li, X., 2006. Young's modulus of ZnO nanobelts measured using atomic force microscopy and nanoindentation techniques. Nanotechnology 17, 3591-3597.
Nix, W.D., Gao, H., 1998. An atomistic interpretation of interface stress. Scr. Mater. 39, 1653-1661.
Nowacki, W., 1986. Theory of Asymmetric Elasticity (translated by Zorski, H.). Oxford, Pergamon Press, New York.
Olssona, P.A.T., Park, H.S., 2012. On the importance of surface elastic contributions to the flexural rigidity of nanowires, J. Mech. Phys. Solids 60, 2064-2083.
Ou, Z.Y., Wang, G.F., Wang, T.J., 2008. Effect of residual surface tension on the stress concentration around a nanosized spheroidal cavity. Int. J. Eng. Sci. 46, 475-485.
Ou, Z.Y., Wang, G.F., Wang, T.J., 2009. Elastic fields around a nanosized spheroidal cavity under arbitrary uniform remote loadings. European J. Mech. A-Solids 28, 110-120.
Ou, Z.Y., Lee, D.W., 2012. Effects of interface energy on scattering of plane elastic wave by a nano-sized coated fiber. J. Sound and Vibration 331, 5623-5643.
Park, S.K., Gao, X.L., 2006. Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355-2359.
Park, H.S., 2008. Surface stress effects on the resonant properties of silicon nanowires. J. Appl. Phys. 103, 123504.
Park, H.S., 2012. Surface stress effects on the critical buckling strains of silicon nanowires. Comput. Mater. Sci. 51, 396-401.
Park, H.S., Klein, P.A., 2008. 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.
Povstenko, Y. Z., 1993. Theoretical investigation of phenomena caused by heterogeneous surface tension in solids. J. Mech. Phys. Solids. 41, 1499-1514.
Quang, H. L., He, Q.C., 2007. Size-dependent effective thermoelastic properties of nanocomposites with spherically anisotropic phases. J. Mech. Phys. Solids. 55, 1889-1921.
Quang, H. L., He, Q.C., 2008. Variational principles and bounds for elastic inhomogeneous materials with coherent imperfect interfaces. Mech. Mater. 40, 865-884.
Quang, H. L., He, Q.C., 2009. Estimation of the effective thermoelastic moduli of fibrous nanocomposites with cylindrically anisotropic phases. Arch. Appl. Mech. 79, 225-248.
Reddy, J.N., 2007. Theory and analysis of elastic plates and shells. Taylor & Francis, Philadelphia, P.A., London.
Riaz, M., Nur, O., Willander, M., Klason, P., 2008. Buckling of ZnO nanowires under uniaxial compression. Appl. Phys. Lett. 92, 103118.
Ru, C.Q., 2010. Simple geometrical explanation of Gurtin-Murdoch model of surface elasticity with clarification of its related versions. Science China 53, 536-544.
Ru, Y., Wang, G.F., Su, L.C., Wang, T.J., 2013. Scattering of vertical shear waves by a cluster of nanosized cylindrical holes with surface effect. Acta Mech. 224, 935-944.
Saada, A. S., 2009. Elasticity: Theory and Applications, 2nd ed, J. Ross Pub., Ft. Lauderdale.
Samaei, A.T., Bakhtiari, M., Wang, G.F., 2012. Timoshenko beam model for buckling of piezoelectric nanowires with surface effects. Nanoscale Res. Lett. 7, 201.
Sharma, P., Ganti, S., Bhate, N., 2003. Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82, 535-537.
Sharma, P., Ganti, S., 2004. Size-dependent Eshelby's tensor for embedded nano-inclusions incorporating surface/interface energies. J. Appl. Mech. 71, 663-671.
Shenoy, V.B., 2005. Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B. 71, 094104.
Shuttleworth, R., 1950. The surface tension of solids. Proc. Phys. Soc. A. 63, 444-457.
Song, J., Wang, X., Riedo, E., Wang, Z.L., 2005. Elastic property of vertically aligned nanowires. Nano Lett. 5, 1954-1958.
Song, J., Jiang, H., Liu, Z.J. Khang, D.Y., Huang, Y., Rogers, J.A., Lu, C., Koh, C.G.., 2008. Buckling of a stiff thin film on a compliant substrate in large deformation. Int. J. Solids Struct. 45, 3107-3121.
Song, F., Huang, G.L., Park, H.S., Liu, X.N., 2011. A continuum model for the mechanical behavior of nanowires including surface and surface-induced initial stresses. Int. J. Solids Struct. 48, 2154-2163.
Spaepen, F., 2000. Interfaces and stresses in thin films. Acta Mater. 48, 31-42.
Steigmann, D.J., Ogden, R.W., 1999. Elastic surface-substrate interactions. Proc. R. Soc. A. 455, 437-474.
Stoney, G.G., 1909. The tension of metallic films deposited by electrolysis. Proc. R. Soc. A. 82, 172-175.
Sun, J.Y., Xia, S., Moon, M.W., Oh, K.H., Kim, K.S., 2012. Folding wrinkles of a thin stiff layer on a soft substrate. Proc. R. Soc. A. 468, 932-953.
Tai, C.T., 1997. Generalized Vector and Dyadic Analysis: Applied Mathematics in Field Theory, IEEE Press, New York.
Ting, T.C.T., 2007. Mechanics of a thin anisotropic elastic layer and a layer that is bonded to an anisotropic elastic body or bodies. Proc. R. Soc. A. 463, 2223-2239.
Timoshenko, S.P., Gere, J.M., 1961. Theory of Elastic Stability. McGraw-Hill, New York.
Timoshenko, S.P., Goodier, J.N., 1970. Theory of Elasticity, McGraw-Hill, New York.
Timoshenko, S.P., Gere, J.M., 1972. Mechanics of Materials, Van Nostrand Reinhold Co., New York.
Thomson, R., Chuang, T.J., Lin, I.H., 1986. The role of surface stress in fracture. Acat Metall. 34, 1133-1143.
Tsiatas, G.C., 2009. A new Kirchhoff plate model based on a modified couple stress theory, Int. J. Solids Struct. 46, 2757-2764.
Ugural, A.C., 1999. Stresses in plates and shells. WCB/McGraw Hill, Boston.
Vella, D., Ajdari, A., Vaziri, A., Boudaoud, A., 2011. Wrinkling of Pressurized Elastic Shells. Phys. Rev. Lett. 107, 174301.
Wan, F.Y.M., Weinitschke, H.J., 1988. On shells of revolution with the Love-Kirchhoff hypothesis. J. Eng. Math. 22, 285-334.
Wang, G.F., Feng, X.Q., 2007. Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl. Phys. Lett. 90, 231904.
Wang, G.F., Feng, X.Q., 2009a. Surface effects on buckling of nanowires under uniaxial compression. Appl. Phys. Lett. 94, 141913.
Wang, G.F., Feng, X.Q., 2009b. Timoshenko beam model for buckling and vibration of nanowires with surface effects. J. Phys. D: Appl. Phys. 42, 155411.
Wang, G.F., Feng, X.Q., 2010. Effect of surface stresses on the vibration and buckling of piezoelectric nanowires. Europhysics Letters 91, 56007. Doi: 10.1209/0295-5075/91/56007.
Wang, G.F., Wang, T.J., 2006. Deformation around a nanosized elliptical hole with surface effect. Appl. Phys. Lett. 89, 161901.
Wang, D.H., Wang, G.F., 2011. Surface Effects on the Vibration and Buckling of Double-Nanobeam-Systems. J. Nanomaterials 2011, 518706
Wang, G.F., Yang, F., 2011. Postbuckling analysis of nanowires with surface effects. J. Appl. Phys. 109, 063535.
Wang, Z.Q., Zhao, Y.P., Huang, Z.P., 2010. The effects of surface tension on the elastic properties of nano structures. Int. J. Eng. Sci. 48, 140-150.
Weaver, W., Timoshenko, S.P., Young, D.H., 1990. Vibration problems in engineering. Wiley, New York.
Xia, R., Li, X.D., Qin, Q.H., Liu, J.L., Feng, X.Q., 2011. Surface effects on the mechanical properties of nanoporous materials. Nanotechnology 22, 265714.
Xiao, J.H., Xu, Y.L., Zhang, F.C., 2011. Size-dependent effective electroelastic moduli of piezoelectric nanocomposites with interface effect. Acta Mech. 222, 59-67.
Xiao, J.H., Xu, Y.L., Zhang, F.C., 2013. Evaluation of effective electroelastic properties of piezoelectric coated nano-inclusion composites with interface effect under antiplane shear. Int. J. Eng. Sci. 69, 61-68.
Yan, Z., Jiang, L.Y., 2011a. The vibrational and buckling behaviors of piezoelectric nanobeams with surface effects. Nanotechnology 22, 245703.
Yan, Z., Jiang, L.Y., 2011b. Surface effects on the electromechanical coupling and bending behaviours of piezoelectric nanowires. J. Phys. D: Appl. Phys. 44, 075404.
Yang, F.Q., 2004. Size-dependent effective modulus of elastic composite materials: spherical nanocavities at dilute concentrations. J. Appl. Phys. 95, 3516-3520.
Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P., 2002. Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731-2743.
Young, T. Phil., 1805. An essay on the cohesion of fluid, Philosophical Transactions of the Royal Society of London 95, 65-87.
Young, S.J., Ji, L.W., Chang, S.J., Fang, T.H., Hsueh, T.J., Meen, T.H., Chen, I.C., 2007. Nanoscale mechanical characteristics of vertical ZnO nanowires grown on ZnO:Ga/glass templates. Nanotechnology 18, 225603.
Zhang, Y., Ren, Q., Zhao, Y., 2004. Modeling analysis of surface stress on a rectangular cantilever beam. J. Phys. D: Appl. Phys. 37, 2140-2145.
Zhen, N., Wang, Y.S., Zhang, C.Z., 2012. Surface/interface effect on band structures of nanosized phononic crystals. Mech. Res. Communications 46, 81-89.
Zhou, L.G., Huang, H.C., 2004. Are surfaces elastically softer or stiffer? Appl. Phys. Lett. 84, 1940-1942.
Zhu, Y., Xu, F., Qin, Q., Fung, W.Y., Lu, W., 2009. Mechanical properties of vapor-liquid-solid synthesized silicon nanowires. Nano Lett. 9, 3934-3939.