A closure relation for the higher order alignment tensors in liquid crystal theory and the alignment-fabric tensors in damage mechanics from a statistical background

In several different systems there occurs an internal orientational degree of freedom and possibly an orientational order connected to it. In these cases an orientation distribution function can be introduced, and orientational moments of this distribution function are relevant. Practical examples of this kind are the orientational order in liquid crystals, the order of dipoles in paramagnetic materials and the orientation distribution of micro-cracks in damaged material. In all cases higher order moments of the orientation distribution function occur in the theory, which have to be eliminated by some closure relation. A derivation of such a closure relation is discussed in the present work.