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Interchangeability and bounds on the effective conductivity of the square lattice

Bruno, O. and Golden, K. (1990) Interchangeability and bounds on the effective conductivity of the square lattice. Journal of Statistical Physics, 61 (1-2). pp. 365-386. ISSN 0022-4715. https://resolver.caltech.edu/CaltechAUTHORS:20181107-080002180

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Abstract

The effective conductivity σ* 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 conductivities σ_1 and σ_2 (σ_1, σ_2 ne0) with probabilities p_1 and p_2 = 1−p_1. A rigorous basis for the very old and widely used low volume fraction expansion of σ* is established, by proving that σ* is an analytic function of p_2 in a suitable domain containing [0, 1]. In the case of the bond lattice in d = 2, rigorous fourth-order upper and lower bounds on σ* valid for all p_2, σ_1, and σ_2 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/BF01013970DOIArticle
https://rdcu.be/baU5sPublisherFree ReadCube access
ORCID:
AuthorORCID
Bruno, O.0000-0001-8369-3014
Additional Information:© 1990 Plenum Publishing Corporation. Received July 13, 1989; final May 11, 1990. The work of O.B. was supported in part by ARO through grant DAAL 03-88-F-0110, and the work of K.G. by AFOSR through grant AFOSR-90-0203.
Funders:
Funding AgencyGrant Number
Army Research Office (ARO)DAAL 03-88-F-0110
Air Force Office of Scientific Research (AFOSR)AFOSR-90-0203
Subject Keywords:Effective conductivity; random resistor network; composites; cell materials; perturbation expansions; bounds
Issue or Number:1-2
Record Number:CaltechAUTHORS:20181107-080002180
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20181107-080002180
Official Citation:Bruno, O. & Golden, K. J Stat Phys (1990) 61: 365. https://doi.org/10.1007/BF01013970
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:90686
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:07 Nov 2018 17:52
Last Modified:03 Oct 2019 20:27

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