Representations of m-linear functions on tensor spaces: Duals and transpositions with applications in continuum mechanics

Abstract

The main goal of the present paper is to introduce certain duals and transpositions of a high-order tensor playing an important role in continuum mechanics. Emphasis is also placed on the comparison of the duals and the transpositions. In contrast to the duals, the transpositions depend on a metric of an underlying metric space. A high-order tensor is itself a representation of a multilinear function on a tensor space, obtained by means of the multilinear extension. The duals and the transpositions of a high-order tensor are identified as multilinear maps defined by the generalized scalar and the inner product, respectively. Consequently, the duals and the transpositions distinguish and define symmetries and symmetry-preserving transformation rules of a co-, contra- and mixed-variant tensor, respectively. As an application in continuum mechanics, the duals and the transpositions of a usual fourth-order tensor are defined and are employed to the determination of symmetries involved.