### Abstract

We derive a new perturbation expansion for the effective conductivity tensor e of a macroscopically anisotropic d-dimensional two-phase composite of arbitrary microstructure. The nth-order tensor coefficients ssAn(i) of the expansion (termed n-point microstructural parameters) are given explicitly in terms of integrals over the set of n-point probability functions (associated with the ith phase) which statistically characterize the microstructure. Macroscopic anisotropy arises out of some asymmetry in the microstructure, i.e., due to statistical anisotropy (e.g., a distribution of oriented, nonspherical inclusions in a matrix, layered media, such as sandstones and laminates, etc.). General and useful properties of the n-point microstructural parameters are established, and contact is made with the formal results of Milton. We then derive rigorous nth-order bounds on e (from our perturbation expansion) that depend upon the n-point parameters ssAn(i) for n=1, 2, 3, and 4. This is the first time that such bounds (for n>1) have been explicitly given in terms of the ssAn(i).

Original language | English (US) |
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Pages (from-to) | 4504-4515 |

Number of pages | 12 |

Journal | Physical Review B |

Volume | 39 |

Issue number | 7 |

DOIs | |

State | Published - Jan 1 1989 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Condensed Matter Physics

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## Cite this

*Physical Review B*,

*39*(7), 4504-4515. https://doi.org/10.1103/PhysRevB.39.4504