On Linear Viscoelastic Fluids: Free Energy and Asymptotic Behavior

In this paper we consider isotropic, homogeneous, incompressible viscoelastic fluids, for which the constitutive equation for the symmetric stress tensor is a local functional of the relative history of the strain. Free energies of materials with memory have been investigated by many authors over the years. It was recognized that free energies are not in general uniquely defined for materials with memory. In fact, the free energies associated to a material with memory form a bounded, convex set with a maximum and a minimum element. By virtue of general theorems, the minimum free energy has been shown to be equal to the maximum recoverable work. Explicit formulae for the maximum and various intermediate free energies have been given for linear models. In this paper we recall the definition of minimal state for an incompressible fluid and examine certain expressions for the free energy. Furthermore, it is well known in linear viscoelasticity, that the dissipation effects due to memory lead to results of stability and decay of the energy under the hypothesis that the constitutive equations satisfy the thermodynamic restrictions. This property allows to prove existence, uniqueness and asymptotic stability for the evolutive problem. To obtain the exponential decay of the energy further hypotheses must be required. These hypotheses ask the positive definiteness of suitable linear combinations of the memory kernel with its derivatives that imply the convexity and the exponential decay of the kernel. In this paper we give a sufficient condition on the relaxation function to get the exponential decay of the energy, furthermore, we shall prove that the exponential decay of the kernel is a necessary condition for the exponential decay of the solution. The method used here is based on the study of the Laplace transform of the solution. We end this paper with an application to a model of viscoelastic material, which kernel is given by a sum of exponential functions. This example presents clearly the opportuneness to represent the material system in terms of minimal states instead of using the classical representation trough the history of the strain tensor, because this approach reduces the complexity of the system.