Thermodynamic Restrictions in Linear Viscoelasticity

The thermodynamic consistency of linear viscoelastic models is investigated. First, the classical Boltzmann law of stress-strain is considered. The kernel (Boltzmann function) is shown to be consistent only if the half-range sine transform is negative definite. The existence of free-energy functionals is shown to place further restrictions. Next, the Boltzmann function is examined in the unbounded power law form. The consistency is found to hold if the stress functional involves the strain history, not the strain-rate history. The stress is next taken to be given by a fractional order derivative of the strain. In addition to the constitutive equations involving strain-rate histories, finding a free-energy functional, consistent with the second law, seems to be an open problem.