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Isotropic discrete orientation distributions on the 3D special orthogonal group

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

Series: 2000-26, Preprints

62K99 None of the above, but in this section

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.

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