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

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 [4, 10, 12] on polycrystals. In this paper we describe a general method for finding such relations for effective moduli of laminates. The method is applicable to any physical setting that can be put into the Hilbert space framework developed by Milton [13]. The idea is to use the W -function of Milton [13] that transforms 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. At present, each candidate relation has to be examined separately in order to confirm the stability under homogenization. We apply our general theory to the settings of conductivity and two-dimensional elasticity.