The Field Equation Recursion Method

  • Graeme W. Milton
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 5)


Effective tensors, such as the effective conductivity tensor σ* and the effective elasticity tensor C* govern the macroscopic response of composites to applied fields. These tensors are strongly influenced by the details of the microgeometry and considerable emphasis has been placed on deriving microstructure independent equalities or inequalities (bounds) on effective tensors. This paper briefly reviews some of the various methods for bounding effective tensors of two-component composites, and reviews the significance of an associated tensor Y*, obtained from σ* or C* via a fractional linear tensor transformation. We also draw attention to the recent work of Cherkaev and Gibiansky [1] who independently introduced the tensor Y* as an aid in analyzing the bounds of the translation method. Their work adds substantial weight to the growing body of evidence which suggests that the tensor Y* is fundamentally important. The field equation recursion method, discussed in Section 5, gives the tensor Y* a physical interpretation and thus links together the various bounding methods.


Effective Conductivity Effective Medium Theory Translation Method Driving Field Effective Medium Approximation 
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  1. [1]
    L.V. Gibiansky and A.V. Cherkaev, private communication, Dijon, 1989.Google Scholar
  2. [2]
    M.J. Beran, “Use of the variational approach to determine bounds for the effective permittivity in random media,” Nuovo Cimento 38 (1965), 771–782.CrossRefGoogle Scholar
  3. [3]
    G.W. Milton, “Bounds on the electromagnetic, elastic, and other properties of two-component composites,” Phys. Rev. Lett. 46 (1981) 542–545.CrossRefGoogle Scholar
  4. [4]
    S. Torquato and G. Stell, “Bounds on the effective thermal conductivity of a dispersion of fully penetrable spheres,” Lett. Appl. Eng. Sci., 23 (1985), 375–384.Google Scholar
  5. [5]
    Z. Hashin and S. Shtrikman, “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” J. Appl. Phys. 33 (1962) 1514–1517.CrossRefGoogle Scholar
  6. [6]
    D.J. Bergman, “The dielectric constant of a composite material-a problem in classical physics,” Phys. Rep. C43 (1978), 377–407.MathSciNetCrossRefGoogle Scholar
  7. [7]
    D.J. Bergman, “Rigorous bounds for the complex dielectric constant of a two-component composite,” Ann. Phys. 138 (1982), 78–114.MathSciNetCrossRefGoogle Scholar
  8. [8]
    G.W. Milton, “Bounds on the complex permittivity of a two-component composite material,” J. Appl. Phys. 52 (1981), 5286–5293CrossRefGoogle Scholar
  9. [8a]
    G.W. Milton, ** “Bounds on the transport and optical properties of a two-component composite material,” J. Appl. Phys. 52 (1981), 5294–5304.CrossRefGoogle Scholar
  10. [9]
    R.C. McPhedran and G.W. Milton, “Bounds and exact theories for the transport properties of inhomogeneous media,” Appl. Phys. A26 (1981), 207–220.Google Scholar
  11. [10]
    K. Golden and G. Papanicolaou, “Bounds for effective parameters of heterogeneous media by analytic continuation,” Commun. Math. Phys. 90 (1983) 473.MathSciNetCrossRefGoogle Scholar
  12. [11]
    K. Golden and G. Papanicolaou, “Bounds for effective parameters of multicomponent media by analytic continuation,” J. Stat. Phys. 40 (1985) 655.MathSciNetCrossRefGoogle Scholar
  13. [12]
    G.F. Dell’ Antonio, R. Figari, and E. Orlandi, “An approach through orthogonal projections to the study of inhomogeneous or random media with linear response,” Ann. Inst. Henri Poincare 44 (1986) 1–28.MATHGoogle Scholar
  14. [13]
    G.F. Dell’ Antonio and V. Nesi, “A general representation for the effective dielectric constant of a composite,” J. Math. Phys. 29 (1988) 2688–2694.MathSciNetMATHCrossRefGoogle Scholar
  15. [14]
    W.F. Brown, “Solid mixture permittivities,” J. Chem. Phys. 23 (1955), 1514–1517.CrossRefGoogle Scholar
  16. [15]
    G.W. Milton and K. Golden, “Thermal conduction in composites,” in Thermal Conductivity 18, pp. 571–582, ed. by T. Ashworth and D.R. Smith, Plenum, 1985.Google Scholar
  17. [16]
    J.B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. 5 (1964) 548–549.MATHCrossRefGoogle Scholar
  18. [17]
    A.M. Dykhne, “Conductivity of a two-dimensional, two-phase system,” Zh. Eksp. Teor. Fiz 59 (1970) 110–115Google Scholar
  19. [17a]
    A.M. Dykhne, “Conductivity of a two-dimensional, two-phase system,” **[*Soviet Phys JETP 32 (1971) 63–65].Google Scholar
  20. [18]
    L. Tartar, “Estimations fines des coefficients homogeneises,” in Ennio De Giorgi’s Colloquium, ed. P. Kree, Research notes in mathematics 125, pp. 168–187, Pitman Press, London, 1985.Google Scholar
  21. [19]
    F. Murat and L. Tartar, “Calcul des variations et homogeneisation,” in Les Methodes d’Homogeneisation:Theorie et Applications en Physique, Coll. de la Dir. des Etudes et Recherches d’Electricite de France, pp. 319–369 Eyrolles, Paris, 1985.Google Scholar
  22. [20]
    K.A. Lurie and A.V. Cherkaev, “Exact estimates of the conductivity of a binary mixture of isotropic components,” Proc. Roy. Soc. Edinburgh 104A (1986) 21–38.MathSciNetCrossRefGoogle Scholar
  23. [21]
    K.A. Lurie and A.V. Cherkaev, “The effective properties of composites and problems of optimal designs of constructions” (in Russian), Uspekhi Mekaniki (Advances in Mechanics) 2 (1987) 3–81.Google Scholar
  24. [22]
    R.V. Kohn and G.W. Milton, “On bounding the effective conductivity of anisotropic composites,” in Homogenization and effective moduli of materials and media, pp. 97–125, ed. J.L. Ericksen, D. Kinderlehrer, R. Kohn and J.-L. Lions, Springer-Verlag, New York, 1986.CrossRefGoogle Scholar
  25. [23]
    G.W. Milton, “A brief review of the translation method for bounding effective elastic tensors of composites,” to appear in the proceedings of the 6th Symposium on Continuum Models and Discrete Systems, ed. G.A. Maugin, Longman, 1990.Google Scholar
  26. [24]
    G.A. Francfort and F. Murat, “Homogenization and optimal bounds in linear elasticity,” Archives Rat. Mech. and Analysis 94 (1986) 307–334.MathSciNetMATHCrossRefGoogle Scholar
  27. [25]
    Z. Hashin and S. Shtrikman, “A variational approach to the theory of the elastic behavior of multiphase materials,” J. Mech. Phys. Solids 11, (1963) 127–140.MathSciNetMATHCrossRefGoogle Scholar
  28. [26]
    G.W. Milton, “On characterizing the set of possible effective tensors of composites: the variational method and the translation method,” Commun. Pure. Appl. Math 43 (1990) 63–125.MathSciNetMATHCrossRefGoogle Scholar
  29. [27]
    G.W. Milton, “The coherent potential approximation is a realizable effective medium scheme,” Commun. Math. Phys. 99 (1985) 463–500.MathSciNetCrossRefGoogle Scholar
  30. [28]
    M. Avellaneda, “Iterated homogenization, differential effective medium theory and applications,” Commun. Pure Appl. Math. 40 (1987) 527.MathSciNetMATHCrossRefGoogle Scholar
  31. [29]
    J. Berryman, “Effective medium theory for elastic composites,” in Elastic wave scattering and propagation, p. 111, ed. V.K. Varadan and V.V. Varadan, Ann Arbor, MI, 1982.Google Scholar
  32. [30]
    G.W. Milton, “Multicomponent composites, electrical networks and new types of continued fractions I and II,” Commun. Math. Phys. 111 (1987), 281–327; 329–372.MathSciNetMATHCrossRefGoogle Scholar
  33. [31]
    G. Allaire and R.V. Kohn, private communication, New York, 1990.Google Scholar

Copyright information

© Birkhäuser Boston 1991

Authors and Affiliations

  • Graeme W. Milton
    • 1
  1. 1.Courant InstituteNew York UniversityNew YorkUSA

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