A thermodynamic frame work for rate type fluid models
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abstract
In this paper, we develop a thermodynamic approach for modeling a class of viscoelastic fluids based on the notion of an 'evolving natural configuration'. The material has a family of elastic (or non-dissipative) responses governed by a stored energy function that is parametrized by the 'natural configurations'. Changes in the current natural configuration result in dissipative behavior that is determined by a rate of dissipation function. Specifically, we assume that the material possesses an infinity of possible natural (or stress-free) configurations. The way in which the current natural configuration changes is determined by a 'maximum rate of dissipation' criterion subject to the constraint that the difference between the stress power and the rate of change of the stored energy is equal to the rate of dissipation. By choosing different forms for the stored energy function and the rate of dissipation function , a whole plethora of energetically consistent rate type models can be developed. We show that the choice of a neo-Hookean type stored energy function and a rate of dissipation function that is quadratic, leads to a Maxwell-like fluid response. By using this procedure with a different choice for the rate of dissipation, we also derive a model that is similar to the Oldroyd-B model. We also discuss several limiting cases, including the limit of small elastic strains, but arbitrarily large total strains, which leads to the classical upper convected Maxwell model as well as the Oldroyd-B model.
