Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-21T03:02:33.489Z Has data issue: false hasContentIssue false
  • English
  • Français

A Spherical Inclusion with Inhomogeneous Interface in Conduction

Published online by Cambridge University Press:  05 May 2011

T. Chen ,
C. H. Hsieh  and
P. C. Chuang
T. Chen*
Affiliation:
Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
C. H. Hsieh*
Affiliation:
Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
P. C. Chuang*
Affiliation:
Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
*
*Professor
**Graduate student
**Graduate student
Get access

Abstract

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.

Keywords

Inclusion problemsImperfect interfaceConduction
Type
Articles
Information
Journal of Mechanics , Volume 19 , Issue 1 , March 2003 , pp. 1 - 8
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1Sanchez-Palencia, E., “Comportement limite d'um probleme de transmission a traverse une plaque faiblement conductrice,” Comp. Rend. Acad. Sci. Paris, A270, pp. 1026–1028 (1970).Google Scholar
2Beneniste, Y. and Miloh, T., “The Effective Conductivity of Composite with Imperfect Contract at Constitutent Interfaces,” Int. J. Engng. Sci., 24, pp. 1537–1552 (1986).CrossRefGoogle Scholar
3Hashin, Z., “Thin Interphase/Imperfect Interface in Conduction,” J. Appl. Phys., 89, pp. 2261–2267 (2001).CrossRefGoogle Scholar
4Pham Huy, H., and Sanchez-Palencia, E., “Phenomenes de transmission a travers des couches minces de conductivite elevee,” J. Math. Analysis Appl., 47, pp. 284–309 (1974).Google Scholar
5Torquato, S. and Rintoul, M. D., “Effect of the Interface on the Properties of Composite Media,” Phys. Rev. Lett., 75, pp. 4067–4070 (1995).CrossRefGoogle ScholarPubMed
6Lipton, R., “Reciprocal Relations, Bounds, and Size Effects for Composites with Highly Conducting Inter-face,” SIAM J. Appl. Math., 57, pp. 347–363 (1997).CrossRefGoogle Scholar
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. 2687–2706 (1999).CrossRefGoogle Scholar
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. 742–748 (1996).CrossRefGoogle Scholar
9Ru, C. Q., “Interface Design of Neutral Elastic Inclusion,” Int. J. Solids Structure, 35,pp. 557–572 (1998).CrossRefGoogle Scholar
10Benveniste, Y. and Miloh, T., “Neutral Inhomogeneities in Conduction Phenomena,” J. Mech. Phys. Solids, 47, pp. 1873–1892 (1999).CrossRefGoogle Scholar
11Benveniste, Y. and Chen, T., “On the Saint-Venant Torsion of Composite Bars with Imperfect,” Proc. Roy. Soc. Lond., A457, pp. 231–255 (2001).CrossRefGoogle Scholar
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. 848–854 (2000).CrossRefGoogle Scholar
13Chen, T. “Thermal Conduction of a Circular Inclusion with Variable Interface Parameter,” Int. J. Solids Structures, 38, pp. 3081–3097 (2001).CrossRefGoogle Scholar
14Jackson, J. D., Classical Electrodynamics, 2nd ed., John Wiley and Sons, New York (1975).Google Scholar
15Ru, C. Q. and Schiavone, P., “A Circular Inclusion with Circumferentially Inhomogeneous Interface in Anti-plane Shear,” Proc. R. Soc. Lond., A453, pp. 2551–2572 (1997).CrossRefGoogle Scholar
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. 19–41 (1999).CrossRefGoogle Scholar
17Van Bladel, J., Electromagnetic Fields, McGraw- Hill, New York (1964).Google Scholar
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).Google Scholar
19MacRobert, T. M., Spherical Harmonics:an Elementary Treatise on Harmonic Functions with Applications, Methuen and Co., London (1927).Google Scholar
20Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics, Chelesea, New York (1965).Google Scholar
21Arfken, G. B. and Weber, H. J., Mathematical Methods for Physicists, 4th ed., Academia Press, San Diego (1995).Google Scholar