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Viscoelastic fluids are non-Newtonian fluids that exhibit both "viscous" and"elastic" characteristics in virtue of mechanisms to store energy and produceentropy. Usually the energy storage properties of such fluids are modelledusing the same concepts as in the classical theory of nonlinear solids.Recently new models for elastic solids have been successfully developed byappealing to implicit constitutive relations, and these new models offer adifferent perspective to the old topic of elastic response of materials. Inparticular, a sub-class of implicit constitutive relations, namely relationswherein the left Cauchy-Green tensor is expressed as a function of stress is ofinterest. We show how to use this new perspective it the development ofmathematical models for viscoelastic fluids, and we provide a discussion of thethermodynamic underpinnings of such models. We focus on the use of Gibbs freeenergy instead of the Helmholtz free energy, and using the standardGiesekus/Oldroyd-B models, we show how the alternative approach works in thecase of well-known models. The proposed approach is straightforward togeneralise to more complex setting wherein the classical approach might beimpractical of even inapplicable.

Implicit type constitutive relations for elastic solids and their use in the development of mathematical models for viscoelastic fluids