Free Energies and Minimal States for Scalar Linear Viscoelasticity

The concept of a minimal state was introduced in recent decades, based on earlier work by Noll. The property that a given quantity is a functional of the minimal state is of central interest in the present work. Using a standard representation of a free energy associated with a linear memory constitutive relation, a new condition, involving linear functionals, is derived which, if satisfied, ensures that the free energy is a functional of the minimal state. Using this result and recent work on constructing free energy functionals, it is shown that if the kernel of the rate of dissipation functional is given by sums of products, the associated free energy functional is a functional of the minimal state.