## Abstract

This paper studies the determination of third- and fourth-order bounds on the effective conductivity σ_{e} of a composite material composed of aligned, infinitely long, identical, partially penetrable, circular cylinders of conductivity σ_{2} randomly distributed throughout a matrix of conductivity σ_{1}. Both bounds involve the microstructural parameter ζ_{2} which is a multifold integral that depends upon S_{3}, the three-point probability function of the composite. This key integral ζ_{2} is computed (for the possible range of cylinder volume fraction φ_{2}) 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=σ_{2}/σ_{1} (conductivity ratio) and φ_{2}, the third-order bounds on σ_{e} significantly improve upon second-order bounds which just depend upon φ_{2}. 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.

Original language | English (US) |
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Pages (from-to) | 893-900 |

Number of pages | 8 |

Journal | Journal of Applied Physics |

Volume | 65 |

Issue number | 3 |

DOIs | |

State | Published - 1989 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)