On rate-type viscoelasticity and the second law of thermodynamics

We deduce an energy identity which must be satisfied by the smooth solutions of the system of equations governing the dynamics of body with quasilinear rate-type constitutive equation. We give conditions when a unique energy function exists for rate-type viscoelasticity. In the semilinear case we give the conditions when a unique, positive and convex energy function exists and we obtain estimates in energy for the smooth solutions of initial-boundary value problems. A viscoelastic approach to nonlinear elasticity is discussed. Finally, an example shows that the second law of thermodynamics does not imply stability.