### Abstract

We compare the general Beran bounds on the effective electrical conductivity of a two-phase composite to the bounds derived by Torquato for the specific model of spheres distributed throughout a matrix phase. For the case of impenetrable spheres, these bounds are shown to be identical and to depend on the microstructure through the sphere volume fraction φ_{2} and a three-point parameter ζ_{2}, which is an integral over a three-point correlation function. We evaluate ζ_{2} exactly through third order in φ_{2} for distributions of impenetrable spheres. This expansion is compared to the analogous results of Felderhof and of Torquato and Lado, all of whom employed the superposition approximation for the three-particle distribution function involved in ζ_{2}. The results indicate that the exact ζ_{2} will be greater than the value calculated under the superposition approximation. For reasons of mathematical analogy, the results obtained here apply as well to the determination of the thermal conductivity, dielectric constant, and magnetic permeability of composite media and the diffusion coefficient of porous media.

Original language | English (US) |
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Pages (from-to) | 3576-3581 |

Number of pages | 6 |

Journal | Journal of Applied Physics |

Volume | 60 |

Issue number | 10 |

DOIs | |

State | Published - Dec 1 1986 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

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## Cite this

*Journal of Applied Physics*,

*60*(10), 3576-3581. https://doi.org/10.1063/1.337614