ON MATHEMATICAL ASPECTS OF DUAL VARIABLES IN CONTINUUM-MECHANICS .1. MATHEMATICAL PRINCIPLES

In this paper consisting of two parts we consider mathematical aspects of dual variables appearing in continuum mechanics. Tensor calculus o n manifolds as introduced into continuum mechanics by MARSDEN and HUGH ES [1] is used as a point of departure. This mathematical formalism le ads to additional structure of continuum mechanical theories. Specific ally invariance of certain bilinear forms renders unambiguous transfor mation rules for tensors between the reference and the current configu ration These transformation rules are determined by push-forwards and pull-backs, respectively. - In Part 1 We consider the basic mathematic al features of our theory. The key aspect of our approach is that, con trary to the usual considerations in this field, we distinguish carefu lly 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 l iving in one and the same vector space. Scalar products. on the other hand, are formed between objects living in different spaces. The disti nction between inner and scalar products leads to a distinction betwee n transposes and duals of tensors. Therefore, we distinguish between s ymmetry and self-duality. An important result of this approach are new formulae for the computation of push-forwards and pull-backs, respect ively, of second-order tensors, which are derived from invariance requ irements of inner and scalar products, respectively. In contrast to pr ior approaches these new formulae preserve symmetry of symmetric mixed tensors.