The Field Equation Recursion Method
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  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.
KeywordsEffective Conductivity Effective Medium Theory Translation Method Driving Field Effective Medium Approximation
Unable to display preview. Download preview PDF.
- L.V. Gibiansky and A.V. Cherkaev, private communication, Dijon, 1989.Google Scholar
- 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
- 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
- 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
- A.M. Dykhne, “Conductivity of a two-dimensional, two-phase system,” Zh. Eksp. Teor. Fiz 59 (1970) 110–115Google Scholar
- [17a]A.M. Dykhne, “Conductivity of a two-dimensional, two-phase system,” **[*Soviet Phys JETP 32 (1971) 63–65].Google Scholar
- 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
- 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
- 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
- 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
- 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
- G. Allaire and R.V. Kohn, private communication, New York, 1990.Google Scholar