Exact relations for effective tensors of polycrystals. II. Applications to elasticity and piezoelectricity

An important necessary condition for an exact relation for effective moduli of polycrystals to hold is stability of that relation under lamination. This requirement is so restrictive that it is possible (if not always feasible) to find all such relations explicitly. In order to do this one needs to combine the results developed in Part I of this paper and the representation theory of the rotation groups SO(2) and SO(3). More precisely, one needs to know all rotationally invariant subspaces of the space of material moduli. This paper presents an algorithm for finding all such subspaces. We illustrate the workings of the algorithm on the examples of 3-dimensional elasticity, where we get all the exact relations, and the examples of 2-dimensional and 3-dimensional piezoelectricity, where we get some (possibly all) of them.