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Journal of Statistical Physics

, Volume 61, Issue 1–2, pp 365–386 | Cite as

Interchangeability and bounds on the effective conductivity of the square lattice

  • O. Bruno
  • K. Golden
Articles

Abstract

The effective conductivityσ* of an infinitely interchangeable two-component random medium is considered. This class of media includes cell materials in the continuum and the bond lattice on ℤ d , where the cells or bonds are randomly assigned the conductivitiesσ1 andσ2 (σ1,σ2ne0) with probabilitiesp1 andp2=1−p1. A rigorous basis for the very old and widely used low volume fraction expansion ofσ* is established, by proving thatσ* is an analytic function ofp2 in a suitable domain containing [0, 1]. In the case of the bond lattice ind=2, rigorous fourth-order upper and lower bounds onσ* valid for allp2,σ1, andσ2 are derived. The four perturbation coefficients entering into the bounds are obtained from the first-order volume fraction coefficient using the method of infinite interchangeability.

Key words

Effective conductivity random resistor network composites cell materials perturbation expansions bounds 

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Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • O. Bruno
    • 1
  • K. Golden
    • 2
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolis
  2. 2.Department of MathematicsPrinceton UniversityPrinceton

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