### Abstract

We evaluate the third-order Beran-Molyneux bounds on the effective bulk modulus K_{e} and the third-order McCoy bounds on the effective shear modulus μ_{e} of a model of a two-phase composite in which one of the phases consists of spherical inclusions (or voids), with bulk and shear moduli, K_{2} and μ_{2}, respectively, and volume fraction φ_{2}, dispersed randomly throughout a matrix phase, with bulk and shear moduli, K_{1} and μ_{1}, respectively, and volume fraction φ_{1}. We tabulate the two fundamental microstructural parameters I_{1} and L_{1} required to evaluate the bounds, which depend upon the three-point matrix probability function of the model, for the aforementioned fully-penetrablesphere model. We compare the third-order bounds on K_{e} and μ_{e} to the second-order bounds due to Hashin and Shtrikman and to Walpole. We find that the third-order bounds for our model are always more restrictive than the corresponding second-order bounds. When the moduli of the phases differ by an order of magnitude, the third-order bounds are sharp enough to provide quantitatively useful estimates of K_{e} and μ_{e} for all φ_{2}. The third-order bounds are very restrictive at low φ_{2} values (e.g., φ_{2} = 0.1) where they remain useful for cases in which the moduli of the phases differ by two orders of magnitude. Experimental values of μ_{e} measured by Corson for a tungstenlead composite are found to lie within the McCoy bounds for our model, with the lower bound giving a good estimate of the data.

Original language | English (US) |
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Pages (from-to) | 385-392 |

Number of pages | 8 |

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

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## Cite this

*International Journal of Engineering Science*,

*23*(3), 385-392. https://doi.org/10.1016/0020-7225(85)90058-8