Implicit constitutive theory and stress-rate type viscoelastic models

In the constitutive equations for the classical viscoelastic models, stress tensor is expressed in terms of the strain tensor and its time derivative. However, the classical elastic and viscoelastic models are incapable of explaining the phenomena when the stress is big and the strain is small which is in fact observed in engineering applications. Rajagopal (2003, 2007) proposed a new class of models that are able to explain this kind of behaviour for elastic materials and he called them strain-limiting models. In the case of elastic materials, the main idea, as opposed to the classical approach, is that the strain tensor corresponding to small strains is expressed as a function of the stress tensor in the constitutive equation.

Recently, Erbay ve Şengül (2020) introduced a stress-rate type constitutive relation and showed that the one-dimensional viscoelastic model derived from this relation is thermodynamically consistent. In this project, we would like to consider the Cauchy problem corresponding to the one-dimensional stress-rate type model introduced by Erbay ve Şengül (2020) from the view point of implicit constitutive theories.