TY - JOUR
T1 - Computer simulation results for bounds on the effective conductivity of composite media
AU - Smith, P. A.
AU - Torquato, S.
PY - 1989
Y1 - 1989
N2 - This paper studies the determination of third- and fourth-order bounds on the effective conductivity σe of a composite material composed of aligned, infinitely long, identical, partially penetrable, circular cylinders of conductivity σ2 randomly distributed throughout a matrix of conductivity σ1. Both bounds involve the microstructural parameter ζ2 which is a multifold integral that depends upon S3, the three-point probability function of the composite. This key integral ζ2 is computed (for the possible range of cylinder volume fraction φ2) using a Monte Carlo simulation technique for the penetrable-concentric-shell model in which cylinders are distributed with an arbitrary degree of impenetrability λ, 0≤λ≤1. Results for the limiting cases λ=0 ("fully penetrable" or randomly centered cylinders) and λ=1 ("totally impenetrable" cylinders) compare very favorably with theoretical predictions made by Torquato and Beasley [Int. J. Eng. Sci. 24, 415 (1986)] and by Torquato and Lado [Proc. R. Soc. London Ser. A 417, 59 (1988)], respectively. Results are also reported for intermediate values of λ: cases which heretofore have not been examined. For a wide range of α=σ2/σ1 (conductivity ratio) and φ2, the third-order bounds on σe significantly improve upon second-order bounds which just depend upon φ2. The fourth-order bounds are, in turn, narrower than the third-order bounds. Moreover, when the cylinders are highly conducting (α≫1), the fourth-order lower bound provides an excellent estimate of the effective conductivity for a wide range of volume fractions.
AB - This paper studies the determination of third- and fourth-order bounds on the effective conductivity σe of a composite material composed of aligned, infinitely long, identical, partially penetrable, circular cylinders of conductivity σ2 randomly distributed throughout a matrix of conductivity σ1. Both bounds involve the microstructural parameter ζ2 which is a multifold integral that depends upon S3, the three-point probability function of the composite. This key integral ζ2 is computed (for the possible range of cylinder volume fraction φ2) using a Monte Carlo simulation technique for the penetrable-concentric-shell model in which cylinders are distributed with an arbitrary degree of impenetrability λ, 0≤λ≤1. Results for the limiting cases λ=0 ("fully penetrable" or randomly centered cylinders) and λ=1 ("totally impenetrable" cylinders) compare very favorably with theoretical predictions made by Torquato and Beasley [Int. J. Eng. Sci. 24, 415 (1986)] and by Torquato and Lado [Proc. R. Soc. London Ser. A 417, 59 (1988)], respectively. Results are also reported for intermediate values of λ: cases which heretofore have not been examined. For a wide range of α=σ2/σ1 (conductivity ratio) and φ2, the third-order bounds on σe significantly improve upon second-order bounds which just depend upon φ2. The fourth-order bounds are, in turn, narrower than the third-order bounds. Moreover, when the cylinders are highly conducting (α≫1), the fourth-order lower bound provides an excellent estimate of the effective conductivity for a wide range of volume fractions.
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U2 - 10.1063/1.343422
DO - 10.1063/1.343422
M3 - Article
AN - SCOPUS:0000567716
VL - 65
SP - 893
EP - 900
JO - Journal of Applied Physics
JF - Journal of Applied Physics
SN - 0021-8979
IS - 3
ER -