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

**Series:** 2000-26, Preprints

- MSC:
- 62K99 None of the above, but in this section

**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.

**This paper was published in:**

Linear Algebra and its Applications 354 (2002) 119-139