Abstract
The outstanding problem of systematically developing rigorous bounds on the complex effective conductivity tensor σ* ofd-dimensional,n-component composites withn>2 is solved. The bounds incorporate information contained in successively higher order correlation functions which reflect the composite geometry. Explicit expressions are given for many of the bounds and some, but not all of them, are represented by nested sequences of circles in the complex plane that enclose, and in fact converge to, each diagonal element of σ*. They are derived from the fractional linear matrix transformations found in Part I that recursively link σ* with a hierarchy of complex effective tensors Ω(j),j=0, 1, 2, ..., of increasing dimension,d(n−1)j. Elementary bounds on Ω(j) confining the diagonal elements of Ω(j) or its inverse to half-plane, wedge or open polygon regions of the complex plane, imply narrow bounds on σ* which converge to the exact value of σ* in the limit asj → ∞. When the component conductivities are real these bounds are more restrictive than the corresponding variational bounds. Besides applying to the effective conductivity σ*, the bounds extend to a wide class of matrix-valued multivariate functions called Ω-functions, and thereby to conduction in polycrystalline media, viscoelasticity in composites, and conduction in multi-component, multiterminal, linear electrical networks. The analytic and invariance properties of Ω-functions are explored and within this class of function most of the bounds are found to be optimal or at least attainable. The bounds obtained here are essentially a generalization to matrix-valued, multivariate functions of the nested sequence of lens-shaped bounds in the complex plane derived by Gragg and Baker for single variable Stieltjes functions.
Similar content being viewed by others
References
Milton, G.W.: Multicomponent composites, electrical networks and new types of continued fraction I. Commun. Math. Phys.111, 281–327 (1987)
Dell'Antonio, G.F., Figari, R., Orlandi, E.: An approach through orthogonal projections to the study of inhomogeneous or random media with linear response. Ann. Inst. Henri Poincaré44, 1 (1986)
Wiener, O.: Abhandlungen der Mathematisch-Physischen Klasse der Königlichen Sächsischen Gesellschaft der Wissenschaften32, 509 (1912)
Hashin, Z., Shtrikman, S.: A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys.33, 3125 (1962)
Beran, M.J.: Use of the variational approach to determine bounds for the effective permittivity in random media. Nuovo Cimento38, 771 (1965)
Kröner, E.: Bounds for effective elastic moduli of disordered materials. J. Mech. Phys. Solids25, 137 (1977)
Phan-Thien, N., Milton, G.W.: New bounds on the effective thermal conductivity of N-phase materials. Proc. R. Soc. Lond A380, 333 (1982)
Prager, S.: Improved variational bounds on some bulk properties of a two-phase random medium. J. Chem. Phys.50, 4305 (1969)
Willis, J.R.: Bounds and self-consistent estimates for the overall moduli of anisotropic composites. J. Mech. Phys. Solids25, 185 (1977)
Murat, F., Tartar, L.: Calcul des variations et homogenisation. In: Les méthodes d'homogénéisation: théorie et applications en physique, Coll. de la Dir. des Etudes et Recherches d'Electricite dé France, pp. 319–370. Paris: Eyrolles 1985
Lurie, K.A., Cherkaev, A.V.: Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion. Proc. R. Soc. Edinb.99A, 71 (1984)
Kohn, R.V., Milton, G.W.: On bounding the effective conductivity of anisotropic composites. In: Homogenization and effective moduli of materials and media. Ericksen, J., Kinderlehrer, D., Kohn, R., Lions, J.L. (eds.). Berlin, Heidelberg, New York: Springer 1986
Milton, G.W., Kohn, R.V.: Variational bounds on the effective moduli of anisotropic composites (in preparation)
Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems. Commun. Pure Appl. Math. (to appear)
Bendsoe, M.P.: Optimization of plates. Thesis, Math. Inst. Tech. Univ. Denmark, Lyngby, Denmark 1983
Bergman, D.J.: The dielectric constant of a composite material — a problem in classical physics. Phys. Rep. C43, 377 (1978)
Korringa, J.: The influence of pore geometry on the dielectric dispersion of clean sandstones. Geophysics49, 1760 (1984)
Golden, K., Papanicolaou, G.: Bounds for effective parameters of heterogeneous media by analytic continuation. Commun. Math. Phys.90, 473 (1983)
Bergman, D.J.: Rigorous bounds for the complex dielectric constant of a two-component composite. Ann. Phys.138, 78 (1982)
Milton, G.W.: Bounds on the complex permittivity of a two-component composite material. J. Appl. Phys.52, 5286 (1981); see also Theoretical studies of the transport properties of inhomogeneous media, unpublished report TP/79/1: University of Sydney 1979
Schulgasser, K., Hashin, Z.: Bounds for effective permittivities of lossy dielectric composites. J. Appl. Phys.47, 424 (1975)
Milton, G.W.: Bounds on the transport and optical properties of a two-component composite. J. Appl. Phys.52, 5294 (1981)
McPhedran, R.C., Milton, G.W.: Bounds and exact theories for the transport properties of inhomogeneous media. Appl. Phys. A26, 207 (1981)
McPhedran, R.C., McKenzie, D.R., Milton, G.W.: Extraction of structural information from measured transport properties of composites. Appl. Phys. A29, 19 (1982)
Gajdardziska-Josifovska, M.: Optical properties and microstructure of cermets. Unpublished M. Sc. thesis: University of Sydney 1986
Felderhof, B.U.: Bounds for the complex dielectric constant of a two-phase composite. Physica126A, 430 (1984)
Milton, G.W., Golden, K.: Thermal conduction in composites. In: Thermal conductivity 18. Ashworth, T., Smith, D.R. (eds.). New York: Plenum Press 1985
Golden, K.: Bounds on the complex permittivity of a multicomponent material. J. Mech. Phys. Solids34, 333 (1986)
Milton, G.W., McPhedran, R.C.: A comparison of two methods for deriving bounds on the effective conductivity of composites. In: Macroscopic properties of disordered media. Burridge, R. et al. (eds.). Lecture Notes in Physics, Vol. 154, p. 183. Berlin, Heidelberg, New York: Springer 1982
Keller, J.B.: A theorem on the conductivity of a composite medium. J. Math. Phys.5, 548 (1964)
Torquato, S., Beasley, J.D.: Effective properties of fibre-reinforced materials. I. Bounds on the effective thermal conductivity of dispersions of fully penetrable cylinders. Int. J. Engng. Sci.24, 415 (1986)
Stell, G., Joslin, C.: To be published
Schulgasser, K.: On a phase interchange relationship for composite materials. J. Math. Phys.17, 378 (1976)
Korringa, J., LaTorraca, G.A.: Application of the Bergman-Milton theory of bounds to the permittivity of rocks. J. Appl. Phys.60, 2966 (1986)
Hashin, J.: Analysis of composite materials. J. Appl. Mech. Trans. ASME50, 481 (1983)
McPhedran, R.C., McKenzie, D.R., Phan-Thien, N.: Transport properties of two-phase composite materials. In: Advances in the mechanics and flow of granular materials. Shahinpoor, M., Wohlbier, R. (eds.). New York: McGraw-Hill 1983
Willis, J.R.: Variational and related methods for the overall properties of composite materials. In: Advances in applied mechanics. Yih, C.-S. (ed.) New York: Academic Press21, 2–78 (1981)
McCoy, J.J.: Macroscopic response of continua with random microstructures. In: Mechanics today, Nemat-Nasser, S. (ed.), pp. 1–40. Oxford, New York: Pergamon Press 1981
Christensen, R.M.: Mechanics of composite materials. New York: Wiley 1979
Watt, J.P., Davies, G.F., O'Connell, R.J.: The elastic properties of composite materials. Rev. Geophy. Space Phys.14, 541 (1976)
Hale, D.K.: The physical properties of composite materials. J. Mater. Sci.11, 2105 (1976)
Beran, M.J.: Statistical continuum theories, pp. 181–256. New York: Interscience 1968
Niklasson, G.A., Granquist, C.G.: Optical properties and solar selectivity of coevaporated Co-Al2O3 composite films. J. Appl. Phys.55, 3382 (1984)
Torquato, S., Stell, G.: Macroscopic approach to transport in two-phase random media. CEAS report # 352 1980
Nevanlinna, R.: Asymptotische Entwickelungen beschrankter Funktionen und das Stieltjessche Momenten problem. Ann. Acad. Sci. Fenn. A18, 1 (1922)
Shohat, J.A., Tamarkin, J.D.: The problem of moments. Baltimore: Waverly Press 1943
Henrici, P., Pfluger, P.: Truncation error estimates for Stieltjes fractions. Numer. Math.9, 129 (1966)
Gragg, W.B.: Truncation error bounds for g-fractions. Numer. Math.11, 370 (1968)
Common, A.K.: Páde approximants and bounds to series of Stieltjes. J. Math. Phys.9, 32 (1967)
Baker, G.A., Jr.: Best error bounds for Padé approximants to convergent series of Stieltjes. J. Math. Phys.10, 814 (1969)
Baker, G.A., Jr., Graves-Morris, P.R.: Encyclopedia of mathematics and its applications, Vols. 13 and 14. Rota, G.-C. (ed.). London: Addison-Wesley 1981
Jones, W.B., Thron, W.J.: Encyclopedia of mathematics and its applications, Vol. 11. Rota, G.-C. (ed.). London: Addison-Wesley 1980
Golden, K., Papanicolaou, G.: Bounds for effective parameters of multicomponent media by analytic continuation. J. Stat. Phys.40, 655 (1985)
Bergman, D.J., Milton, G.W.: Unpublished
Golden, K.: Bounds for effective parameters of multicomponent media by analytic continuation. Ph. D. thesis: New York University 1984
Author information
Authors and Affiliations
Additional information
Communicated by M. E. Fisher
Rights and permissions
About this article
Cite this article
Milton, G.W. Multicomponent composites, electrical networks and new types of continued fraction II. Commun.Math. Phys. 111, 329–372 (1987). https://doi.org/10.1007/BF01238903
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01238903