Algebraic and Numerical Exploration of Free Energies for Materials with Memory

We study the forms of a range of free energy functionals for materials with memory for two types of strain history, namely sinusoidal and exponential behaviours. The work deals with discrete spectrum materials, which are those with relaxation functions given by sums of decaying exponentials. Various standard free energy functionals are explored, including the minimum free energy and related quantities. It is shown that quite different formulae are obtained, depending on the manner in which these functionals are evaluated, particularly for those related to the minimum free energy. Such differences are resolved. Various numerical plots of free energies are presented and discussed, along with the rates of dissipation associated with them. This is the first comprehensive exploration of quantitative comparisons between the various free energy functionals. The results both illustrate certain properties that have been deduced theoretically and expose an interesting proximity between apparently unconnected functionals.