Journal of Statistical Physics

, Volume 40, Issue 5–6, pp 655–667 | Cite as

Bounds for effective parameters of multicomponent media by analytic continuation

  • K. Golden
  • G. Papanicolaou
Articles

Abstract

Recently D. Bergman introduced a method for obtaining bounds for the effective dielectric constant (or conductivity) of a two-component medium. This method does not rely on a variational principle but instead exploits the properties of the effective parameter as an analytic function of the ratio of the component parameters. We extend the method to multicomponent media using techniques of several complex variables.

Key words

Multicomponent stationary random media bounds for the effective dielectric constant integral representations several complex variables 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. Wiener,Abhandl. Math.-Phys. Kl. Königl. Sachsischen Ges. 32:509 (1912).Google Scholar
  2. 2.
    Z. Hashin and S. Shtrikman, A variational approach to the theory of effective magnetic permeability of multiphase materials,J. Appl. Phys. 33:3125 (1962).Google Scholar
  3. 3.
    D. J. Bergman, The dielectric constant of a composite material-a problem in classical physics,Phys. Rep. C43:377 (1978).Google Scholar
  4. 4.
    D. J. Bergman, The dielectric constant of a single cubic array of identical spheres,J. Phys. C12:4947 (1979).Google Scholar
  5. 5.
    D. J. Bergman and D. Stroud, Theory of resonances in electromagnetic scattering by macroscopic bodies,Phys. Rev. B 22:3527 (1980).Google Scholar
  6. 6.
    D. J. Bergman, Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,Phys. Rev. Lett. 44:1285 (1980).Google Scholar
  7. 7.
    D. J. Bergman, Bounds for the complex dielectric constant of a two-component composite material,Phys. Rev. B 23:3058 (1981).Google Scholar
  8. 8.
    D. J. Bergman, Resonances in the bulk properties of composite media-Theory and applications,Lecture Notes in Physics, Vol. 154 (Springer, New York, 1982), pp. 10–37.Google Scholar
  9. 9.
    D. J. Bergman, Bulk physical properties of composite media, Lecture Notes, École d'Été d'Analyse Numérique, 1983.Google Scholar
  10. 10.
    D. J. Bergman, Rigorous bounds for the complex dielectric constant of a two-component composite,Ann. Phys. (Leipzig) 138:78 (1982).Google Scholar
  11. 11.
    G. W. Milton, Bounds on the transport and optical properties of a two-component composite,J. Appl. Phys. 52:5294 (1981).Google Scholar
  12. 12.
    G. W. Milton, Bounds on the complex permittivity of a two-component composite material,J. Appl. Phys. 52:5286 (1981).Google Scholar
  13. 13.
    G. W. Milton, Bounds on the electromagnetic, elastic, and other properties of two-component composites,Phys. Rev. Lett. 46:542 (1981).Google Scholar
  14. 14.
    G. W. Milton and R. C. McPhedran, A comparison of two methods for deriving bounds on the effective conductivity of composites,Lecture Notes in Physics, Vol. 154 (Springer, New York, 1982), pp. 183–193.Google Scholar
  15. 15.
    G. W. Milton, Bounds on the complex dielectric constant of a composite material,Appl. Phys. Lett. 37:300 (1980).Google Scholar
  16. 16.
    G. W. Milton, Bounds on the elastic and transport properties of two-component composites,J. Mech. Phys. Solids 30:177 (1982).Google Scholar
  17. 17.
    K. Golden and G. Papanicolaou, Bounds for effective parameters of heterogeneous media by analytic continuation,Commun. Math. Phys. 90:473 (1983).Google Scholar
  18. 18.
    K. Golden, Bounds on the complex permittivity of a multicomponent material, to appear inJ. Mech. Phys. Solids. Google Scholar
  19. 19.
    G. Papanicolaou and S. Varadhan, Boundary value problems with rapidly oscilating random coefficients, inColloquia Mathematica Societatis János Bolyai 27, Random Fields, Esztergom, Hungary, 1979 (North-Holland, Amsterdam, 1982), pp. 835–873.Google Scholar
  20. 20.
    K. Golden, Bounds for effective parameters of multicomponent media by analytic continuation, Ph.D. thesis, New York University, 1984.Google Scholar
  21. 21.
    D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order (Springer, New York, 1977).Google Scholar
  22. 22.
    N. I. Akhiezer and I. M. Glazman, The Theory of Linear Operators in Hilbert Space (F. Ungar Publ. Co., New York, 1966).Google Scholar
  23. 23.
    N. I. Akhiezer,The Classical Moment Problem (Hafner, New York, 1965).Google Scholar
  24. 24.
    J. B. Keller, A theorem on the conductivity of a composite medium,J. Math. Phys. 5:548 (1964).Google Scholar
  25. 25.
    K. Schulgasser, On a phase interchange relationship for composite materials,J. Math. Phys. 17:378 (1976).Google Scholar
  26. 26.
    G. A. Baker, Jr., Best error bounds for Padé approximants to convergent series of Stieltjes,J. Math. Phys. 10:814 (1969).Google Scholar
  27. 27.
    G. A. Baker, Jr.,Essentials of Padé Approximants (Academic Press, New York, 1975).Google Scholar
  28. 28.
    B. U. Felderhof, Bounds for the complex dielectric constant of a two-phase composite,Physica 126A:430 (1984).Google Scholar
  29. 29.
    G. W. Milton and K. Golden, Thermal conduction in composites, inProceedings of the 18th International Thermal Conductivity Congress, Rapid City, S.D., 1983 (in press); T. Ashworth and David R. Smith; eds. Thermal Conductivity 18 (Plenum Press, New York, 1985).Google Scholar
  30. 30.
    Y. Kantor and D. J. Bergman, Improved rigorous bounds on the effective elastic moduli of a composite material,J. Mech Phys. Solids 32:41 (1984).Google Scholar
  31. 31.
    W. Rudin,Function Theory in Polydiscs (W. A. Benjamin, New York, 1969).Google Scholar
  32. 32.
    A. Korányi and L. Pukánszky, Holomorphic function with positive real part on polycylinders,Am. Math. Soc. Trans. 108:449 (1963).Google Scholar
  33. 33.
    V. S. Vladimirov and Yu. N. Drozhzhinov, Holonomic functions in a polycircle with non-negative imaginary part,Matematicheskie Zametki 15:55 (1974).Google Scholar
  34. 34.
    W. Rudin, Harmonic analysis in polydiscs,Actes, Congrès Int. Math., 489 (1970).Google Scholar
  35. 35.
    W. Rudin, private communication, 1983.Google Scholar
  36. 36.
    J. N. McDonald, Measures on the torus which are real parts of holomorphic functions (preprint).Google Scholar
  37. 37.
    G. W. Milton, private communication, 1984.Google Scholar

Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • K. Golden
    • 1
  • G. Papanicolaou
    • 1
  1. 1.Courant InstituteNew York UniversityNew York

Personalised recommendations