A LOCAL FRAME FORMULATION OF DUAL STRESS-STRAIN PAIRS AND TIME DERIVATIVES

This work is concerned with a formulation of the ''dual'' stress-strai n pair concept of Haupt and Tsakmakis in terms of the (spatially) loca l form of the differential geometric concept of a frame field. Among o ther things, such a formulation yields additional insight into the mat hematical structure and physical interpretation underlying the concept of ''dual'', which turns out to be based intimately on that of a time -dependent frame. To facilitate the approach taken in this work, the ( usual) referential kinematic setting (i.e., relative to a reference co nfiguration of the material body) used by Haupt and Tsakmakis is first generalized to a material setting (i.e., relative to the material bod y itself) with the help of Noll's concept of a body element. From this point of view, the formulation of dual stress-strain pairs relies ess entially upon an equivalence class of body element placements, or equi valently, material frames. In particular, the notion of a material str ain tenser is dependent on the metric tenser induced by such an equiva lence class. Euclidean representations (e.g., referential or spatial) of the basic material deformation, stress and strain tensors are induc ed by any motion of the body element in Euclidean (vector) space with respect to the corresponding induced time-dependent Euclidean frame, y ielding in particular the two families of strain and stress tensors di scussed by Haupt and Tsakmakis. The last part of this work investigate s the consequences of extending the notion of a Euclidean representati on of the basic material deformation, strain and stress tensors (induc ed by any motion of the body element) to certain physical quantities d epending on these tensors and their time derivatives, i.e., the materi al stress power density and its incremental forms. In the work of Haup t and Tsakmakis, such a representation is achieved by requiring each s tress-strain pair in each family to be conjugate with respect to the s tress power density, as well as its appropriate incremental form. This is possible in general iff the tenser time derivatives involved trans form in the same fashion as the tensors themselves, yielding the notio n of dual time derivatives of these tensors. As shown in the last part of this work, such derivatives represent a type of covariant derivati ve based on the connection of a time-dependent frame. From this point of view, one can show, among other things, that the well-known problem with ''oscillating stresses'' arising for certain hypoelastic constit utive relations (e.g., for a Maxwell fluid) can most likely be ascribe d to an extra time-dependent frame-dependence in the Jaumann form of t he constitutive relation, something that is avoided in the Oldroyd for m.