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

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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.

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References

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  4. Chen, T.: Thermoelastic properties and conductivity of composites reinforced by spherically anisotropic particles. Mech. Mater. 14, 257–268 (1993)

    Article  Google Scholar 

  5. Cheng, H., Torquato, S.: Effective conductivity of dispersion of spheres with a superconducting interface. Proc. R. Soc. Lond. A 453, 1331–1344 (1997)

    Article  MathSciNet  Google Scholar 

  6. Christensen, R.M.: Mechanics of Composite Materials. Wiley, New York (1979)

    Google Scholar 

  7. Hashin, Z.: Thin interphase/imperfect interface in conduction. J. Appl. Phys. 84, 2261–2267 (2001)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  9. Herve, E.: Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites. Int. J. Solids Struct. 39, 1041–1058 (2002)

    Article  Google Scholar 

  10. Kapitza, P.L.: The study of heat transfer in helium II. J. Phys. (USSR) 4, 181–210 (1941)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  14. Lipton, R., Vernescu, B.: Composites with imperfect interface. Proc. Phys. Soc. A 452, 329–358 (1996)

    Article  MathSciNet  Google Scholar 

  15. Milton, G.W.: The Theory of Composites. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  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)

    Article  Google Scholar 

  17. Pham, D.C.: Estimations for the overall properties of some isotropic locally-ordered composites. Acta Mech. 121, 177–190 (1997)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  19. Pham, D.C.: Solutions for the conductivity of multi-coated spheres and spherically-symmetric inclusion problems. Z. Angew. Math. Phys. 69, 13 (2018)

    Article  MathSciNet  Google Scholar 

  20. Pham, D.C., Nguyen, T.K.: Polarization approximations for macroscopic conductivity of isotropic multicomponent materials. Int. J. Eng. Sci. 97, 26–39 (2015)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  23. Stratton, J.A.: Electromagnetic Theory. McGraw-Hill, New York (1941)

    MATH  Google Scholar 

  24. Torquato, S.: Random Heterogeneous Media. Springer, New York (2002)

    Book  Google Scholar 

  25. Wu, L.: Bounds on the effective thermal conductivity of composites with imperfect interface. Int. J. Eng. Sci. 48, 783–794 (2010)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. VAST - Institute of Mechanics, 264 Doi Can, Hanoi, Vietnam

    Duc Chinh Pham

  2. Division of Structures, Department of Civil Engineering, Research and Application Center for Technology in Civil Engineering, University of Transport and Communications, 3 Cau Giay, Dong Da, Hanoi, Vietnam

    Trung Kien Nguyen

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  1. Duc Chinh Pham
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  2. Trung Kien Nguyen
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Correspondence to Duc Chinh Pham.

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Pham, D.C., Nguyen, T.K. The microscopic conduction fields in the multi-coated sphere composites under the imposed macroscopic gradient and flux fields. Z. Angew. Math. Phys. 70, 24 (2019). https://doi.org/10.1007/s00033-018-1062-2

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  • DOI: https://doi.org/10.1007/s00033-018-1062-2

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