zbMATH — the first resource for mathematics

Size-dependent effective thermoelastic properties of nanocomposites with spherically anisotropic phases. (English) Zbl 1173.74035
Summary: Composites made of semi-crystalline polymers and nanoparticles have a spherulitic microstructure which can be reasonably represented by a spherically anisotropic volume element. Due to the high surface-to-volume ratio of a nanoparticle, the particle-matrix interface stress, usually neglected in determining the effective elastic moduli of particle-reinforced composites, may have a non-negligible effect. To account for the latter in estimating the effective thermoelastic properties of a composite consisting of nanoparticles embedded in a semi-crystalline polymeric matrix, this work adopts a coherent interface model for the nanoparticle-matrix interface and proposes an extended version of the classical generalized self-consistent method. In particular, Eshelby’s formulae widely used to calculate the elastic energy change of a homogeneous medium due to the introduction of inhomogeneity are extended to the thermoelastic case. The nanoparticle size effect on the effective thermoelastic moduli of the composite is theoretically shown and numerically illustrated.

74Q15 Effective constitutive equations in solid mechanics
74E30 Composite and mixture properties
74M25 Micromechanics of solids
Full Text: DOI
[1] Bassett, D.C., Principles of polymer morphology, (1981), Cambridge University Press London
[2] Bottomley, D.J.; Ogino, T., Alternative to the shuttleworth formulation of solid surface stress, Phys. rev. B, 63, 165412-1-165412-5, (2001)
[3] Bouby, C., He, Q.-C., Gu, S.-T., Pensée, V., 2007. Coordinate-free derivation of a thermoelastic curved interface model, submitted for publication.
[4] Cahn, J.W., Surface stress and the chemical equilibrium of small crystals-I. the case of the isotropic surface, Acta metall., 28, 1333-1338, (1980)
[5] Causin, V.; Marega, C.; Marigo, A.; Ferrara, G.; Idiyatullina, G.; Fantinel, F., Morphology, structure and properties of a poly(1-butene)/montmorillonite nanocomposite, Polymer, 47, 4773-4780, (2006)
[6] Chen, T., Thermoelastic properties and conductivity of composites reinforced by spherically particles, Mech. mater., 14, 257-268, (1993)
[7] Chen, T.; Dvorak, G.J.; Yu, C.C., Solids containing spherical nano-inclusions with interface stress: effective properties and thermal-mechanical connections, Int. J. solids struct., 44, 941-955, (2007) · Zbl 1120.74045
[8] Christensen, R.M., A critical evaluation for a class of micromechanics models, J. mech. phys. solids, 38, 379-404, (1990)
[9] 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
[10] 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, 8, 1827-1854, (2005) · Zbl 1120.74683
[11] Dryden, J.R., Elastic constants of spherulitic polymer, J. mech. phys. solids, 36, 477-498, (1988)
[12] Duan, H.L., Karihaloo, B.L., 2007. Thermoelastic properties of heterogeneous materials with imperfect interfaces: generalized Levins’s formula and Hill’s connections. J. Mech. Phys. Solids, in press, doi:10.1016/j.jmps.2006.10.006. · Zbl 1170.74017
[13] Duan, H.L.; Wang, J.; Huang, Z.P.; Karihaloo, B.L., Eshelby formalism for nano-inhomogeneities, Proc. R. soc. A, 461, 3335-3353, (2005) · Zbl 1370.74068
[14] 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, 1574-1596, (2005) · Zbl 1120.74718
[15] Eshelby, J.D., The continuum theory of lattice defects, (), 79-144
[16] Gurtin, M.E.; Murdoch, A.I., A continuum theory of elastic material surfaces, Arch. ration. mech. anal., 57, 291-323, (1975), and 59, 389-390 · Zbl 0326.73001
[17] Hadal, R.; Yuan, Q.; Jog, J.P.; Misra, R.D.K., On stress whitening during surface deformation in Clay-containing polymer nanocomposites: a microstructural approach, Mater. sci. eng. A, 418, 268-281, (2006)
[18] He, L.-H.; Cheng, Z.-Q., Correspondence relations between the effective thermoelastic properties of composites reinforced by spherically particles, Int. J. eng. sci., 34, 1-8, (1996) · Zbl 0900.73449
[19] He, Q.-C.; Benveniste, Y., Exactly solvable spherically anisotropic thermoelastic microstructures, J. mech. phys. solids, 52, 2661-2682, (2004) · Zbl 1087.74022
[20] Hill, R., Theory of mechanical properties of fibre-strengthened materials: I, Elastic behaviour. J. mech. phys. solids, 12, 199-212, (1964)
[21] Kerner, E.H., The elastic and thermoelastic properties of composite media, Proc. phys. soc. B., 69, 808-813, (1956)
[22] Kim, G.; Han, C.C.; Libera, M.; Jackson, C.L., Crystallization within melt ordered semicrystalline block copolymers: exploring the coexistence of microphase-separated and spherulitic morphologies, Macromolecules, 34, 7336-7342, (2001)
[23] Le Quang, H.; He, Q.-C., Micromechanical estimation of the shear modulus of a composite consisting of anisotropic phases, J. mater. sci. technol., 20, 6-10, (2004)
[24] Levin, V.M., Thermal expansion coefficients of heterogeneous media, Mekhanika tverdogo tela., 2, 88-94, (1967), English version, Mech. Solids, 11, 58-61
[25] Liu, Z.; Chen, K.; Yan, D., Crystallization, morphology, and dynamic mechanical properties of poly(trimethylene terephthalate)/Clay nanocomposites, Eur. polym. J., 39, 2359-2366, (2003)
[26] Miller, R.E.; Shenoy, V.B., Size-dependent elastic properties of nanosized structural elements, Nanotechnology, 11, 139-147, (2000)
[27] Milton, G.W., The theory of composites, (2002), Cambridge University Press London · Zbl 0631.73011
[28] Murdoch, A.I., A thermodynamical theory of elastic-material interfaces, Q. J. mech. appl. math., 29, 245-275, (1976) · Zbl 0398.73003
[29] Murdoch, A.I., Some fundamental aspects of surface modelling, J. elasticity, 80, 33-52, (2005) · Zbl 1089.74012
[30] Nowacki, R.; Monasse, B.; Piorkowska, E.; Galeski, A.; Haudin, J.M., Spherulite nucleation in isotactic polypropylene based nanocomposites with montmorillonite under shear, Polymer, 45, 4877-4892, (2004)
[31] Pathak, S.; Shenoy, V.B., Size dependence of thermal expansion of nanostructures, Phys. rev. B, 72, 113404, (2005)
[32] 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
[33] Sharma, P.; Dasgupta, A., 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, (2002)
[34] Shuttleworth, R., The surface tension of solid, Proc. phys. soc. A, 63, 444-457, (1950)
[35] Smith, J.C., Correction and extension of Van der Poel’s method for calculating the shear modulus of a particulate composite, J. res. nat. bur. stand., 78A, 355-361, (1974)
[36] Smith, J.C., Simplification of Van der Poel’s formula for the shear modulus of a particulate composite, J. res. nat. bur. stand., 79A, 419-423, (1975)
[37] Torquato, S., Random heterogeneous materials: microstructure and macroscopic properties, (2001), Springer Berlin · Zbl 0988.74001
[38] Van der Poel, C., On the rheology of concentrated suspensions, Rheol. acta, 1, 198-205, (1958)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.