Clips operation between type II and type III O(3)-subgroups with application to piezoelectricity

TL;DR

This paper completes the classification of subgroup interactions in O(3), specifically between type II and III subgroups, enabling better analysis of 3D piezoelectric materials and their symmetry properties.

Contribution

It introduces the missing clips between type II and III O(3) subgroups, enhancing the understanding of isotropy classes in 3D coupled constitutive laws.

Findings

Computed 25 isotropy classes for 3D piezoelectric law

Filled gaps in the clips tables for O(3) subgroups

Applied results to elasticity, piezoelectricity, and permittivity tensors

Abstract

Clips is a binary operation between conjugacy classes of closed subgroups of a compact group, which has been introduced to obtain the isotropy classes of a direct sum of representations E1 $\oplus$ E2, if the isotropy classes are known for each factor E1, E2. This allows, in particular, to compute the isotropy classes of any reducible representation, once one knows the decomposition into irreducible factors and the symmetry classes for the irreducible factors. In the specific case of the three dimensional orthogonal group O(3), clips between some types of subgroups of O(3) have already been calculated. However, until now, clips between type II and type III subgroups were missing. Those are encountered in 3D coupled constitutive laws. In this paper, we complete the clips tables by computing the missing clips. As an application, we obtain 25 isotropy classes for the standard O(3) representation on the full 3D Piezoelectric law, which involves the three Elasticity, Piezoelectricity and Permittivity constitutive tensors.