Exact Relations for Effective Tensors of Polycrystals. II. Applications to Elasticity and Piezoelectricity

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 [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.

Racah coefficients, subrepresentation semirings, and composite materials

“…When T is relatively simple, it is possible to find all solutions to (1) by brute force calculations. For example, this approach succeeded in finding all exact relations for threedimensional elasticity [10]. However, these naive methods are no longer feasible even in the next simplest case of piezoelectricity.…”

Exact relations for Green’s functions in linear PDE and boundary field equalities: a generalization of conservation laws

“…Then the associated manifold M = W −1 n (K 0 ) consists of 2 × 2 symmetric matrices with determinant σ 2 0 , and this is the manifold corresponding to the Dykhne [11] exact relation. Grabovsky's pioneering work, developed further with Sage in [19], provided essential clues that led to the breakthrough result [18] establishing conditions that guarantee an exact relation holds for all composites, and not just laminates. Using carefully devised perturbation expansions that had their basis in [32] Sect.…”

A generalization of the Chandler Davis convexity theorem

“…Physically Our idea is that it may be more advantageous to work in the K = W (L) space of variables than in the physical space of L variables. For example, it was the key idea for the recently developed theory of exact relations for composites [6][7][8][9]. In the K-space we have a problem of finding, for a given subset U of…”