## Abstract

We evaluate upper and lower bounds on the effective thermal conductivity K_{e} of a model of two-phase composite materials in which one of the phases consists of spherical inclusions (or voids) of conductivity K_{2} and volume fraction φ_{2}, dispersed randomly throughout a matrix phase of conductivity K_{1} and volume fraction φ_{1}. Our evaluations compare third-order bounds of Beran and of Brown, which utilize the three-point matrix probability function of the model, with bounds of De Vera and Strieder (which apply only to the aforementioned "fully penetrable sphere" model) and of Hashin and Shtrikman. The comparisons are made over extended ranges of values of both φ_{2} and α = K_{2} K_{1} and reveal that the best bounds among those we have tested (generally those of Beran) are sharp enough to give quantitatively useful estimates of K_{e} for 0.1 ≤ K_{2} K_{1} ≤ 10 over a wide range of φ_{2} values. They are sharp at high φ_{2} values (i.e., φ_{2} = 0.9) and very sharp at low φ_{2} values (e.g. φ_{2} = 0.1) where they remain useful for K_{2} K_{1} ≈ 100. They are less sharp at intermediate values (e.g. φ_{2} = 0.5). As is well known, such results immediately translate into equivalent results for the electrical conductivity, dielectric constant, or magnetic permeability of composites.

Original language | English (US) |
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Pages (from-to) | 375-383 |

Number of pages | 9 |

Journal | International Journal of Engineering Science |

Volume | 23 |

Issue number | 3 |

DOIs | |

State | Published - 1985 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Materials Science(all)
- Engineering(all)
- Mechanics of Materials
- Mechanical Engineering