Exact relations for effective tensors of polycrystals.
                   I:  Necessary Conditions.

                             by

                       Yury Grabovsky

           JTB 120 3:20pm, Monday, 28 April, 1997

                          Abstract

The set of all effective moduli of a polycrystal usually has
a nonempty interior. When it does not, we say that there is
an exact relation for effective moduli. This can indeed
happen as evidenced by recent results on polycrystals. In
this paper we describe a general method for finding such
relations without establishing any bounds on the effective
moduli. The method is applicable to any physical setting
that can be put into the Hilbert space framework developed
by Milton. The idea is to look for exact relations for
effective moduli of laminates and use the W-function of
Milton that transforms a lamination formula into a convex
combination. The method reduces the problem of finding exact
relations to a problem from representation theory of SO(d)
(d=2 or 3) corresponding to a particular physical setting.
When this last problem is solved there is a finite amount of
calculation required to be done in order to answer the
question completely.


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