Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. (English) Zbl 1120.74718

Summary: The fundamental framework of micromechanical procedure is generalized to take into account the surface/interface stress effect at the nano-scale. This framework is applied to the derivation of the effective moduli of solids containing nano-inhomogeneities in conjunction with the composite spheres assemblage model, the Mori-Tanaka method and the generalized self-consistent method. Closed-form expressions are given for the bulk and shear moduli, which are shown to be functions of the interface properties and the size of the inhomogeneities. The dependence of the elastic moduli on the size of the inhomogeneities highlights the importance of the surface/interface in analysing the deformation of nano-scale structures. The present results are applicable to analysis of the properties of nano-composites and foam structures.


74Q15 Effective constitutive equations in solid mechanics
74E30 Composite and mixture properties
Full Text: DOI


[1] Aboudi, J., Mechanics of composite materials: A unified micromechanical approach, (1991), Elsevier Amsterdam · Zbl 0837.73003
[2] Benveniste, Y., The effective mechanical behaviour of composite materials with imperfect contact between the constituents, Mech. mater., 4, 197-208, (1985)
[3] Benveniste, Y., A new approach to the application of mori – tanaka’s theory in composite materials, Mech. mater., 6, 147-157, (1987)
[4] Benveniste, Y.; Miloh, T., Imperfect soft and stiff interfaces in two-dimensional elasticity, Mech. mater., 33, 309-323, (2001)
[5] Bottomley, D.J., Ogino, T., 2001. Alternative to the Shuttleworth formulation of solid surface stress. Phys. Rev. B 63, 165412-1-165412-5.
[6] Cahn, J.W., Thermodynamics of solid and fluid surfaces, (), 3-23
[7] Cammarata, R.C., Surface and interface stress effects in thin films, Prog. surf. sci., 46, 1-38, (1994)
[8] Christensen, R.M.; Lo, K.H., Solutions for effective shear properties in three phase sphere and cylinder models, J. mech. phys. solids, 27, 315-330, (1979) · Zbl 0419.73007
[9] Cuenot, S., Frétigny, 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, 165410-1-165410-5.
[10] 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) · Zbl 1115.74303
[11] Duan, H.L., Wang, J., Huang, Z.P., Luo, Z.Y., 2005. Stress concentration tensors of inhomogeneities with interface effects. Mech. Mater. 37, 723-736.
[12] Ghoniem, N.M.; Busso, E.P.; Kioussis, N.; Huang, H.C., Multiscale modelling of nanomechanics and micromechanics: an overview, Philos. mag., 83, 3475-3528, (2003)
[13] Gibbs, J.W., 1906. The Scientific Papers of J. Willard Gibbs. vol 1. Longmans-Green, London. · JFM 33.0708.01
[14] Gu, P.; Miao, H.; Liu, Z.T.; Wu, X.P.; Zhao, J.H., Investigation of elastic modulus of nanoporous alumina membrane, J. mater. sci., 39, 3369-3373, (2004)
[15] Gurtin, M.E.; Murdoch, A.I., A continuum theory of elastic material surfaces, Arch. rat. mech. anal., 57, 291-323, (1975) · Zbl 0326.73001
[16] Gurtin, M.E.; Weissmüller, J.; Larché, F., A general theory of curved deformable interfaces in solids at equilibrium, Philos. mag. A, 78, 1093-1109, (1998)
[17] Hashin, Z., The elastic moduli of heterogeneous materials, J. appl. mech., 29, 143-150, (1962) · Zbl 0102.17401
[18] Hashin, Z., Thermoelastic properties of particulate composites with imperfect interface, J. mech. phys. solids, 39, 745-762, (1991)
[19] Herring, C., The use of classical macroscopic concepts in surface energy problems, (), 5-81
[20] Huo, B.; Zheng, Q.-S.; Huang, Y., A note on the effect of surface energy and void size to void growth, Eur. J. mech. A/solids, 18, 987-994, (1999) · Zbl 0954.74053
[21] Ibach, H., The role of surface stress in reconstruction, epitaxial growth and stabilization of mesoscopic structures, Surf. sci. rep., 29, 195-263, (1997)
[22] Lur’e, A.I., Three-dimensional problems of theory of elasticity, (1964), Interscience New York · Zbl 0122.19003
[23] Martin, C.R.; Siwy, Z., Molecular filters-pores within pores, Nat. mater., 3, 284-285, (2004)
[24] Masuda, H.; Fukuda, K., Ordered metal nanohole arrays made by a two-step replication of honeycomb structures of anodic alumina, Science, 268, 1466-1468, (1995)
[25] Miller, R.E.; Shenoy, V.B., Size-dependent elastic properties of nanosized structural elements, Nanotechnology, 11, 139-147, (2000)
[26] Milton, G.W., The theory of composites, (2002), Cambridge University Press Cambridge · Zbl 0631.73011
[27] Mori, T.; Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta metall., 21, 571-574, (1973)
[28] Müller, P.; Saúl, A., Elastic effects on surface physics, Surf. sci. rep., 54, 157-258, (2004)
[29] Murr, L.E., Interfacial phenomena in metals and alloys, (1975), Addison-Wesley London
[30] Nemat-Nasser, S.; Hori, M., Micromechanics: overall properties of heterogeneous materials, second ed. elsevier, (1999), Amsterdam · Zbl 0924.73006
[31] Nix, W.D.; Gao, H.J., An atomistic interpretation of interface stress, Scr. mater., 39, 1653-1661, (1998)
[32] Orowan, E., Surface energy and surface tension in solids and liquids, Proc. R. soc. London A, 316, 473-491, (1970)
[33] Pokropivnyi, V.V., Two-dimensional nanocomposites: photonic crystals and nanomembranes (review). I. types and preparation, Powder metall. met. ceram., 41, 264-272, (2002)
[34] Ponte Castañeda, P.; Suquet, P., Nonlinear composites, Adv. appl. mech., 34, 171-302, (1998) · Zbl 0889.73049
[35] Povstenko, Y.Z., Theoretical investigation of phenomena caused by heterogeneous surface tension in solids, J. mech. phys. solids, 41, 1499-1514, (1993) · Zbl 0784.73072
[36] Rottman, C., Landau theory of coherent interphase interfaces, Phys. rev. B, 38, 12031-12034, (1988)
[37] Sharma, P., Dasgupta, A., 2002. Average elastic fields and scale-dependent overall properties of heterogeneous micropolar materials containing spherical and cylindrical inhomogeneities. Phys. Rev. B 66, 224110-1-224110-10.
[38] 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)
[39] Shuttleworth, R., The surface tension of solids, Proc. phys. soc. A, 63, 444-457, (1950)
[40] Steigmann, D.J.; Ogden, R.W., Elastic surface – substrate interactions, Proc. R. soc. London A, 455, 437-474, (1999) · Zbl 0926.74016
[41] Streitz, F.H., Cammarata, R.C., Sieradzki, K., 1994. Surface-stress effects on elastic properties. I. Thin metal films. Phys. Rev. B 49, 10699-10706.
[42] Sun, L.; Wu, Y.M.; Huang, Z.P.; Wang, J., Interface effect on the effective bulk modulus of a particle-reinforced composite, Acta mech. sin., 20, 676-679, (2004)
[43] Sun, C.T.; Zhang, H.T., Size-dependent elastic moduli of platelike nanomaterials, J. appl. phys., 93, 1212-1218, (2003)
[44] Suo, Z., Evolving material structures of small feature sizes, Int. J. solids struct., 37, 367-378, (2000) · Zbl 1075.74008
[45] Torquato, S., Random heterogeneous materials: microstructure and macroscopic properties, (2002), Springer New York · Zbl 0988.74001
[46] Walpole, L.J., A coated inclusion in an elastic medium, Math. proc. Cambridge philos. soc., 83, 495-506, (1978) · Zbl 0378.73019
[47] Wang, J., Duan, H.L., Zhang, Z., Huang, Z.P., 2005. An anti-interpenetration model and connections between interphase and interface models in particle-reinforced composites. Int. J. Mech. Sci., in press. · Zbl 1192.74084
[48] Weissmüller, J.; Cahn, J.W., Mean stresses in microstructures due to interface stresses: A generalization of a capillary equation for solids, Acta mater., 45, 1899-1906, (1997)
[49] Willis, J.R., The overall response of nonlinear composite media, Eur. J. mech. A/solids, 19, S165-S184, (2000)
[50] Wu, C.H.; Hsu, J.; Chen, C.-H., The effect of surface stress on the stability of surfaces of stressed solids, Acta mater., 46, 3755-3760, (1998)
[51] Wu, H.A.; Liu, G.R.; Wang, J.S., Atomistic and continuum simulation on extension behaviour of single crystal with nano-holes, Modelling simul. mater. sci. eng., 12, 225-233, (2004)
[52] Xun, F.; Hu, G.K.; Huang, Z.P., Effective in plane moduli of composites with a micropolar matrix and coated fibers, Int. J. solids struct., 41, 247-265, (2004) · Zbl 1069.74044
[53] Yang, F.Q., Size-dependent effective modulus of elastic composite materials: spherical nanocavities at dilute concentrations, J. appl. phys., 95, 3516-3520, (2004)
[54] Zhou, L.G.; Huang, H.C., Are surfaces elastically softer or stiffer?, Appl. phys. lett., 84, 1940-1942, (2004)
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