Kronecker Product Operations on Tensors

Abstract

Kronecker product algebra (KPA) is widely applied in control theory, signal processing, image processing and statistics. However, it does not appear to commonly applied to continuum mechanics. The reason may be that, in its current state of development, KPA applies to matrices but not to third and fourth order tensors which are pervasive in continuum mechanics. In broad terms the goal of the current investigation is to extend Kronecker product algebra to tensors. In particular, Kronecker counterparts of quadratic products and of tensor outer products are presented. Kronecker product operators on third and fourth order tensors are introduced. The tensorial nature of Kronecker products of tensors is established. Finally, conditions for symmetry classes in fourth order tensors are stated in terms of Kronecker products. In several related studies on continuum mechanics, the KPA extensions have been shown to furnish compact expressions for elaborate quantities such as the tangent modulus tensor in thermohyperelasticity.