Some of the rigorous bounds that are expressible in terms of the
canonical functions for *J* = 2 are listed in TABLE 1. Functions and
averages required as definitions for some of the more complex terms in
TABLE 1 are:

(4) |

(5) |

(6) |

Alternative bounds that are at least as tight as the McCoy-Silnutzer (MS) bounds for any choice of microstructure were given by Milton and Phan-Thien (1982) as

(7) |

(8) |

(9) |

Note that ``improved bounds'' are not necessarily improved for every choice of volume fraction, constituent moduli, and microgeometry. It is possible in some cases that ``improved bounds'' will actually be less restrictive, than say the Hashin-Shtrikman bounds, for some range of the parameters. In such cases we obviously prefer to use the more restrictive bounds when our parameters happen to fall in this range.

Milton (1987; 2002) has shown that, for the commonly
discussed case of two-component composites, the canonical functionals
can be viewed as fractional linear transforms with the arguments
and of the canonical functionals as the transform
variables. In light of the monotonicity properties of the functionals,
this point of view is very useful because the problem of determining
estimates of the moduli can then be reduced to that of finding
estimates of the parameters and . Furthermore,
properties of the canonical functions also imply that excellent
estimates of the moduli can be obtained from fairly crude estimates of the
transformation parameters and . (Recall, for example,
that estimates of zero and infinity for these parameters result in
Reuss and Voigt bounds on the moduli.) Milton calls this
transformation procedure the *Y*-transform, where *Y* stands for one
of these transform parameters (*i.e.*, and in
elasticity, or another combination when electrical conductivity
and/or other mathematically analogous properties are being
considered).

5/3/2005