## 00-26

#### Isotropic discrete orientation distributions on the 3D special orthogonal group

by Gaffke, N.; Heiligers, B.; Offinger, R.

**Preprint series:** 00-26, Preprints

**The paper is published:** Linear Algebra and its Applications 354 (2002) 119-139

**MSC****:**- 15A90 Applications of matrix theory to physics
- 62K99 None of the above but in this section
- 73C02 Classical linear elasticity

**Abstract:** In modeling the linear elastic behavior of a polycrystalline material on the microscopic level, a special problem is to determine so-called discrete orientations (DODs) which satisfy a certain 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 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 modelled by stiffness tensors having a nontrivial symmetry group. There are six possible nontrivial 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.

**Keywords:** Linear elasticity; stiffness tensor; Voigt average; discrete orientation distribution; weakly and strongly invariant designs; invariant subspace; symmetry group.

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