The minimum free energy for continuous spectrum materials

A general closed expression is given for the isothermal minimum free energy of a linear viscoelastic material with continuous spectrum response. Two quite distinct approaches are adopted, which give the same final result. The first involves expressing a positive quantity, closely related to the loss modulus of the material, defined on the frequency domain, as a product of two factors with specified analyticity properties. The second is the continuous spectrum version of a method used in [S. Breuer and E. T. Onat, Z. Angew. Math. Phys., 15 (1964), pp. 13-21] for materials with relaxation function given by sums of exponentials. It is further shown that minimal energy states are uniquely related to histories and that the work function is the maximum free energy with the property that it is a function of state.