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The microscopic conduction fields in the multi-coated sphere composites under the imposed macroscopic gradient and flux fields

  • Duc Chinh Pham
  • Trung Kien Nguyen
Article
  • 16 Downloads

Abstract

Explicit analytical expressions of the microscopic conduction (gradient or flux) fields in the d-dimensional (\(d=2,3\)) multi-coated sphere assemblages under the imposed macroscopic gradient and flux fields are presented. Limiting procedures are developed to derive the results for most important specific composites, which include the inclusion composites with highly conducting imperfect interface, lowly conducting (with Kapitza resistance) imperfect interface, general imperfect interface, and those with anisotropic coatings. When the volume proportion of the outermost shells of the assemblages approaches 1, the simplified results for the dilute suspensions of the complex spherically symmetric inclusions in a major matrix are deduced.

Keywords

Conduction (gradient or flux) fields Multi-coated sphere assemblage Imperfect interfaces Anisotropic coating 

Mathematics Subject Classification

74E30 74Q05 

Notes

References

  1. 1.
    Andrianov, I.V., Bolshakov, V.I., Danishevskyy, V.V., Weichert, D.: Asymptotic study of imperfect interfaces in conduction through a granular composite material. Proc. R. Soc. Lond. A 466, 2707–2725 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benveniste, Y.: Two models of three-dimensional thin interphases with variable conductivity and their fulfilment of the reciprocal theorem. J. Mech. Phys. Solids 60, 1740–1752 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Benveniste, Y.: Models of thin interphases and the effective medium approximation in composite media with curvilinearly anisotropic coated inclusions. Int. J. Eng. Sci. 72, 140–154 (2013)CrossRefGoogle Scholar
  4. 4.
    Chen, T.: Thermoelastic properties and conductivity of composites reinforced by spherically anisotropic particles. Mech. Mater. 14, 257–268 (1993)CrossRefGoogle Scholar
  5. 5.
    Cheng, H., Torquato, S.: Effective conductivity of dispersion of spheres with a superconducting interface. Proc. R. Soc. Lond. A 453, 1331–1344 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Christensen, R.M.: Mechanics of Composite Materials. Wiley, New York (1979)Google Scholar
  7. 7.
    Hashin, Z.: Thin interphase/imperfect interface in conduction. J. Appl. Phys. 84, 2261–2267 (2001)CrossRefGoogle Scholar
  8. 8.
    Hashin, Z., Shtrikman, S.: A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 33, 3125–3131 (1962)CrossRefGoogle Scholar
  9. 9.
    Herve, E.: Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites. Int. J. Solids Struct. 39, 1041–1058 (2002)CrossRefGoogle Scholar
  10. 10.
    Kapitza, P.L.: The study of heat transfer in helium II. J. Phys. (USSR) 4, 181–210 (1941)Google Scholar
  11. 11.
    Le-Quang, H., Bonnet, G., He, Q.-C.: Size-dependent Eshelby tensor fields and effective conductivity of composites made of anisotropic phases with highly conducting imperfect interfaces. Phys. Rev. B 81, 064203 (2010)CrossRefGoogle Scholar
  12. 12.
    Le Quang, H., He, Q.-C., Bonnet, G.: Eshelbys tensor fields and effective conductivity of composites made of anisotropic phases with Kapitza’s interface thermal resistance. Philos. Mag. 91, 3358–3392 (2011)CrossRefGoogle Scholar
  13. 13.
    Le-Quang, H., Pham, D.C., Bonnet, G., He, Q.C.: Estimations of the effective conductivity of anisotropic multiphase composites with imperfect interfaces. Int. J. Heat Mass Transf. 58, 175–187 (2013)CrossRefGoogle Scholar
  14. 14.
    Lipton, R., Vernescu, B.: Composites with imperfect interface. Proc. Phys. Soc. A 452, 329–358 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Milton, G.W.: The Theory of Composites. Cambridge University Press, Cambridge (2001)Google Scholar
  16. 16.
    Pavanello, F., Manca, F., Palla, P.L., Giordano, S.: Generalized interface models for transport phenomena: unusual scale effects in composite nanomaterials. J. Appl. Phys. 112, 084306 (2012)CrossRefGoogle Scholar
  17. 17.
    Pham, D.C.: Estimations for the overall properties of some isotropic locally-ordered composites. Acta Mech. 121, 177–190 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pham, D.C.: Bounds on the effective conductivity of statistically isotropic multicomponent materials and random cell polycrystals. J. Mech. Phys. Solids 59, 497–510 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pham, D.C.: Solutions for the conductivity of multi-coated spheres and spherically-symmetric inclusion problems. Z. Angew. Math. Phys. 69, 13 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pham, D.C., Nguyen, T.K.: Polarization approximations for macroscopic conductivity of isotropic multicomponent materials. Int. J. Eng. Sci. 97, 26–39 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Pham, D.C., Tran, B.V.: Equivalent-inclusion approach and effective medium approximations for conductivity of coated-inclusion composites. Eur. J. Mech. A/Solids 47, 341–348 (2014)CrossRefGoogle Scholar
  22. 22.
    Pham, D.C., Vu, L.D., Nguyen, V.L.: Bounds on the ranges of the conductive and elastic properties of randomly inhomogeneous materials. Philos. Mag. 93, 2229–2249 (2013)CrossRefGoogle Scholar
  23. 23.
    Stratton, J.A.: Electromagnetic Theory. McGraw-Hill, New York (1941)zbMATHGoogle Scholar
  24. 24.
    Torquato, S.: Random Heterogeneous Media. Springer, New York (2002)CrossRefGoogle Scholar
  25. 25.
    Wu, L.: Bounds on the effective thermal conductivity of composites with imperfect interface. Int. J. Eng. Sci. 48, 783–794 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.VAST - Institute of MechanicsHanoiVietnam
  2. 2.Division of Structures, Department of Civil Engineering, Research and Application Center for Technology in Civil EngineeringUniversity of Transport and CommunicationsHanoiVietnam

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