Designing Textured Polycrystals with Specific Isotropic Material Tensors: the Odf Method

Herein we study the following problem: Suppose we are given a supply of grains, which are of the same material and have equal volume. Given a finite set of material tensors (i), can we find an arrangement of grains in an aggregate so that all the tensors (i) pertaining to this aggregate are isotropic? In this paper we examine the preceding problem within the special context of physical theories where material anisotropy of polycrystalline aggregates is determined by crystallographic texture, and we restrict our attention to tensors whose anisotropic part is linear in the texture coefficients. A method is developed by which the preceding problem is answered positively for tensors of various orders and grains of various crystal symmetries. Our method uses the machinery developed in quantitative texture analysis. It is based on the symmetry properties of the orientation distribution function (ODF) and appeals to some recent findings on how crystallographic texture affects material tensors of weakly textured polycrystals. As illustration, explicit solutions are worked out for the fourth-order elasticity tensor and for the sixth-order acoustoelastic tensor.