JournalofElasticity 40: 107-122, 1995. 107 © 1995 KluwerAcademic Publishers. Printed in the Netherlands. Internal Dissipation, Relaxation Property, and Free Energy in Materials with Fading Memory M. FABRIZIO 1, C. GIORGI 2 and A. MORRO 3 1Mathematics Department, Piazza di Porta S. Donato 5, 4012Z Bologna, Italy 2Department of Electronics and Automation, Universityof Brescia, 25060 Brescia, Italy 3DIBE, University, Via Opera Pia l la, 16145 Genova, Italy Received 12 April 1995 Abstract. This paper examines some features of the standard theory of materials with fading memory and emphasizes that the commonly-accepted notion of dissipation yields unexpectedconsequences. First, application of the Clausius-Duhem inequality to linear viscoelasticityshows that there is a free energy functional such that the so-calledinternal dissipation vanishes in spite of the dissipative character of the model. Second,upon the choice of a suitablefunction norm, the relaxation property is proved not to hold for viscoelasticsolids. Finally,the particular case is consideredwhen the relaxation function is a superposition of exponentials. Different descriptions of state are then possible which prove to be inequivalent as far as the free energy is concerned. 1. Introduction Mathematical theories of materials with fading memory have been developed to a great extent in the last three decades. In this regard, we mention the books [1-3] for expository accounts of the subject. In spite of the considerable progress about mod- elling and mathematical development, the analysis of relatively simpler models, such as linear viscoelasticity, has shown the need for revisiting the theory. Some papers by Fichera on non-existence or non-uniqueness of the solution to problems in linear viscoelasticity [4, 5] show that even the constitutive properties of the body are affected by the choice of the norm for the underlying fading memory space. Also, recent investigations of ours [6] on linear viscoelasticity have shown that some aspects may be better understood by reexamining carefully the notion of fading memory and relaxation property along with the content of the second law. More generally, Coleman and Mizel [7] say that a theory of materials with memory which does not rest upon L p spaces has the advantage of 'eliminating the arbitrary numbers p and deemphasizing the influence function k which are not experimentally determinable and appear in a physical theory as analytical encumbrances rather than as aids to understanding'. In addition, the occurrence of a subjective influence function in the norm and hence in the statement of general properties and principles, such as the second law for approximate cycles, makes the pertinent statements of dubious meaning. Nevertheless, to our mind little has been