  Chen has recently shown how the response matrix of a two-phase composite can
be written as linear combinations of products of the component matrices. We
elaborate on Chen's expansions by deriving them in a different way, which a.
shows them in a different light, and b. permits us to generalize them further.
As an application of our results we find exact microstructure-independent
relations between the moduli of the two components and those of any composite.
The body of these relations is equivalent to the compatibility relations of
Milgrom and Shtrikman (1989), but they are cast in a rather different form,
which has certain advantages. As an example, we show how any modulus of an
arbitrary two-phase composite can be written in closed form as a linear
combination of any other $n$ of its moduli, with coefficients that depend only
on the component moduli, but not on the volume fractions, or the
microstructure.
