Isotropic discrete orientation distributions on the 3D special orthogonal group

Abstract

In modeling the linear elastic behavior of a polycrystalline material on the microscopic level, a special problem is to determine a so-called discrete orientation distributions (DODs) which satisfy the isotropy condition. A DOD is a probability measure with finite support on SO(3), the special orthogonal group in three dimensions. Isotropy of a DOD can be viewed as an invariance property of a certain moment matrix of the DOD. So the problem of finding isotropic DODs resembles that of finding weakly invariant linear regression designs. In fact, methods from matrix and group theory which have been successfully applied in linear regression design can also be utilized here to construct various isotropic DODs. Of particular interest are isotropic DODs with small support. Crystal classes with additional symmetry properties are modeled by stiffness tensors having a non-trivial symmetry group. There are six possible non-trivial symmetry groups, up to conjugation. In either cases we find isotropic DODs with fairly small support, in particular for the cubic and the transversal symmetry groups.