  Viscoelastic fluids are non-Newtonian fluids that exhibit both "viscous" and
"elastic" characteristics in virtue of mechanisms to store energy and produce
entropy. Usually the energy storage properties of such fluids are modelled
using the same concepts as in the classical theory of nonlinear solids.
Recently new models for elastic solids have been successfully developed by
appealing to implicit constitutive relations, and these new models offer a
different perspective to the old topic of elastic response of materials. In
particular, a sub-class of implicit constitutive relations, namely relations
wherein the left Cauchy-Green tensor is expressed as a function of stress is of
interest. We show how to use this new perspective it the development of
mathematical models for viscoelastic fluids, and we provide a discussion of the
thermodynamic underpinnings of such models. We focus on the use of Gibbs free
energy instead of the Helmholtz free energy, and using the standard
Giesekus/Oldroyd-B models, we show how the alternative approach works in the
case of well-known models. The proposed approach is straightforward to
generalise to more complex setting wherein the classical approach might be
impractical of even inapplicable.
