Averaging material tensors of any rank in textured polycrystalline materials: Extending the scope beyond crystallographic proper point groups

Abstract

In many modern micromechanical applications, there is usually a need to perform the orientational averaging of certain material tensors weighted by an orientation distribution function (ODF). The computation of these averages is seen to be very simplified by means of the so-called generalized spherical harmonic method (GSHM), which is based on the classical assumption that the ODFs are defined in the rotation group SO(3). A priori, this makes the averaging strictly applicable only to polycrystals with crystallite symmetry defined by one of the 11 proper point groups. Despite the interest in the study of materials belonging to any of the other 21 point groups, few studies have properly considered such cases. This is crucial for physical properties represented by odd-order tensors such as the third-order linear piezoelectric tensor. The goal of this work is to extend the applicability of the GSHM to crystallites with symmetry that belongs to the orthogonal group O(3). Thus, a simple formula is provided for the averaging of material tensors of any rank in textured polycrystals containing crystallites with symmetry defined by any of the 32 crystallographic point groups. Our work presents a simple yet rigorous method for indirect averaging on the orthogonal group, using averaging on SO(3). We demonstrate the full equivalence between our proposed method and averaging properly on O(3) through a detailed proof provided in this paper. This proof represents a significant contribution to the field, providing a practical and reliable approach for researchers working with crystalline materials. The results reported here confirm the validity of closed-form expressions previously derived by the authors for piezoelectric materials.