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For random polycrystals (see the earlier discussion of the basic
model in the second section), it is most convenient to define a new
canonical function:
| |
(36) |

where the mean and harmonic mean
of the layer constituents are the pertinent conductivities (off-axis
and on-axis of symmetry, respectively) in each layered grain. Then, the
Hashin-Shtrikman bounds for the conductivity of the random polycrystal are
| |
(37) |

where and .These bounds are known not to be the most general ones since they rely
on an implicit assumption that the grains are equiaxed.
A more general lower bound that is known to be optimal is due to Schulgasser
(1983) and Avellaneda *et al.* (1988):
| |
(38) |

Helsing and Helte (1991) have reviewed the state of the art for
conductivity bounds for polycrystals, and in particular have noted
that the self-consistent
[or CPA (*i.e.*, coherent potential approximation)]
for the random polycrystal conductivity is given by
| |
(39) |

It is easy to show (39) always lies between the two rigorous bounds
and , and also between
and . Note that and cross when , with
becoming the superior lower bound for mean/harmonic-mean
contrast ratios greater than 10.

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Stanford Exploration Project

5/3/2005