On Mathematical Aspects of Dual Variables in Continuum Mechanics. Part 1: Mathematical Principles

In this paper consisting of two parts we consider mathematical aspects of dual variables appearing in continuum mechanics. Tensor calculus on manifolds as introduced into continuum mechanics is used as a point of departure. This mathematical formalism leads to additional structure of continuum mechanical theories. Specifically invariance of certain bilinear forms renders unambiguous transformation rules for tensors between the reference and the current configuration. These transformation rules are determined by push-forwards and pull-backs, respectively. - In Part 1 we consider the basic mathematical features of our theory. The key aspect of our approach is that, contrary to the usual considerations in this field, we distinguish carefully between inner products and scalar products. This discrimination is motivated by physical considerations and is subsequently given a firm mathematical basis. Inner products can only be formed with objects living in one and the same vector space. Scalar products, on the other hand, are formed between objects living in different spaces. The distinction, between inner and scalar products leads to a distinction between transposes and duals of tensors. Therefore, we distinguish between symmetry and self-duality. An important result of this approach are new formulae for the computation of push-forwards and pull-backs, respectively, of second-order tensors, which are derived from invariance requirements of inner and scalar products, respectively. In contrast to prior approaches these new formulae preserve symmetry of symmetric mixed tensors.