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A Spherical Inclusion with Inhomogeneous Interface in Conduction

  • T. Chen (a1), C. H. Hsieh (a1) and P. C. Chuang (a1)


A series solution is presented for a spherical inclusion embedded in an infinite matrix under a remotely applied uniform intensity. Particularly, the interface between the inclusion and the matrix is considered to be inhomegeneously bonded. We examine the axisymmetric case in which the interface parameter varies with the cone angle θ. Two kinds of imperfect interfaces are considered: an imperfect interface which models a thin interphase of low conductivity and an imperfect interface which models a thin interphase of high conductivity. We show that, by expanding the solutions of terms of Legendre polynomials, the field solution is governed by a linear set of algebraic equations with an infinite number of unknowns. The key step of the formulation relies on algebraic identities between coefficients of products of Legendre series. Some numerical illustrations are presented to show the correctness of the presented procedures. Further, solutions of the boundary-value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of spherical inclusions with equal size. The effective conductivity solely depends on one particular constant among an infinite number of unknowns.


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1Sanchez-Palencia, E., “Comportement limite d'um probleme de transmission a traverse une plaque faiblement conductrice,” Comp. Rend. Acad. Sci. Paris, A270, pp. 10261028 (1970).
2Beneniste, Y. and Miloh, T., “The Effective Conductivity of Composite with Imperfect Contract at Constitutent Interfaces,” Int. J. Engng. Sci., 24, pp. 15371552 (1986).
3Hashin, Z., “Thin Interphase/Imperfect Interface in Conduction,” J. Appl. Phys., 89, pp. 22612267 (2001).
4Pham Huy, H., and Sanchez-Palencia, E., “Phenomenes de transmission a travers des couches minces de conductivite elevee,” J. Math. Analysis Appl., 47, pp. 284309 (1974).
5Torquato, S. and Rintoul, M. D., “Effect of the Interface on the Properties of Composite Media,” Phys. Rev. Lett., 75, pp. 40674070 (1995).
6Lipton, R., “Reciprocal Relations, Bounds, and Size Effects for Composites with Highly Conducting Inter-face,” SIAM J. Appl. Math., 57, pp. 347363 (1997).
7Miloh, T. and Beneniste, Y., “On the Effective Conductivity of Composites with Ellipsoidal Inhome-geneities and Highly Conducting Interfaces,” Proc. R. Soc. Lond., A455, pp. 26872706 (1999).
8Hasselman, D. P. H., Donaldson, K. Y., Thomas, J. R. Jr., and Brennan, J. J., “Thermal Conductivity of Vapor-Liquid-Solid and Vapor-Solid Silicon Carbide Whisker-Reinforced Lithium Alumino-silcate Glass-Ceramic Composites,” J. Am. Ceram. Soc., 79, pp. 742748 (1996).
9Ru, C. Q., “Interface Design of Neutral Elastic Inclusion,” Int. J. Solids Structure, 35,pp. 557572 (1998).
10Benveniste, Y. and Miloh, T., “Neutral Inhomogeneities in Conduction Phenomena,” J. Mech. Phys. Solids, 47, pp. 18731892 (1999).
11Benveniste, Y. and Chen, T., “On the Saint-Venant Torsion of Composite Bars with Imperfect,” Proc. Roy. Soc. Lond., A457, pp. 231255 (2001).
12Nan, C. W., Li, X.-P. and Birringer, R., “Inverse Problem for Composites with Imperfect Interface: Determination of Interfacial Thermal Resistance, Thermal Conductivity of Constituents and Micro- structural Parameters,” J. Am. Ceram. Soc., 83, pp. 848854 (2000).
13Chen, T.Thermal Conduction of a Circular Inclusion with Variable Interface Parameter,” Int. J. Solids Structures, 38, pp. 30813097 (2001).
14Jackson, J. D., Classical Electrodynamics, 2nd ed., John Wiley and Sons, New York (1975).
15Ru, C. Q. and Schiavone, P., “A Circular Inclusion with Circumferentially Inhomogeneous Interface in Anti-plane Shear,” Proc. R. Soc. Lond., A453, pp. 25512572 (1997).
16Sudak, L. J., Ru, C. Q., Schiavone, P. and Mioduchowski, A., “A Circular Inclusion with Inhomogeneously Imperfect Interface in Plane Elasticity,” J. Elasticity, 55, pp. 1941 (1999).
17Van Bladel, J., Electromagnetic Fields, McGraw- Hill, New York (1964).
18Byerly, W. E., An Elementary Treatise on Fourier Series and Spherical, Cylindircal and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics, Gin and Company, Boston (1893).
19MacRobert, T. M., Spherical Harmonics:an Elementary Treatise on Harmonic Functions with Applications, Methuen and Co., London (1927).
20Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics, Chelesea, New York (1965).
21Arfken, G. B. and Weber, H. J., Mathematical Methods for Physicists, 4th ed., Academia Press, San Diego (1995).



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