(~) 1996 Kluwer Academic Publishers. Printed in the Netherlands. On the Analytic Expression of the Free Energy in Linear Viscoelasticity

Abstract. Two definitions of free energy for a linear viscoelastic material, due to Graffi and to Coleman and Owen, are considered, and the compatibility of these definitions with some xpressions of the free energy proposed in the literature is examined. For the expressions of Staverman d Schwarzl and of Breuer and Onat, the two definitions are proved to be equivalent, and the set of all relaxation functions for which the two expressions are indeed free energies i determined. Two more expressions, proposed by Volterra nd Graffi and by Morro and Vianello, are taken into consideration. For them, only the classes of relaxation functions for which they are free energies according tothe first definition, iscompletely characterized. All results are established under regularity assumptions weaker than those usually made in the literature. Key words: linear viscoelasticity, constitutive equations AMS Mathematics Subject Classification (1991): 73F05. 1. Introduct ion In his theory of simple materials with memory, Coleman [3] assumes that the functions of state depend on a list of variables which, in the isothermal case, reduce to the relative history and to the current value of the deformation gradient