ht. 1. Engng Sci. Vol. 16. pp. 681-705 @ Pergamon Press Ltd.. 1978. Printed in Great Britain zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA LINEARIZED THERMOVISCOELASTICITY WITH HIGH TEMPERATURE VARIATIONS AND RELATED PERIODIC PROBLEMS R. BOUC Laboratoire de Mtcanique et d’Acoustique, C.N.R.S., Marseille, France G. GEYMONAT Politecnico di Torino. Italy (Communicated by B. NAYROLES) Abstract-In the first part of this paper we develop a linearization of the equations of the thermovisco- elastic field in the case of great temperature variations. The possibility of uncoupling the heat equation from the motion equation is discussed in Section 3. After recalling some results on duality and virtual work principle we then study the motion equation with temperature as data, i.e. a given function of time and space variables. More precisely we study existence, uniqueness, regularity and asymptotic stability of a T-periodic (stress) solution of the motion equation in the dynamical case (Section 7) and in the quasi-static case (Section S), when the temperature field is T-periodic in time and with a constitutive equation of Maxwell type where the stiffness and viscosity matrix are temperature dependent and thus are T-periodic functions of time. In the proof of the theorems we use frequently an inequality of monotony which means that the material is dissipative on a period. This inequality hold if the stiffness is a slowly varying function of time (the temperature has a little effect on the stiffness), on the other hand, fortunately, there is no condition on the viscosity. INTRODUCTION FOI_I,OW~NG the fundamental work of Volterra[ 1,2], hereditary phenomena in mechanics have been deeply studied. A great part of the work that has been done treats the case called, by Volterra himself, “the case of closed cycle” (see Vogel[3]) which corresponds to the case where the relaxation function in viscoelasticity is of the type G(t - T). However, in 1907, Hatt[4] has discovered the phenomenon of creep in concrete which presents stress-independent deformations including shrinkage, in addition to thermal dilatation; the material properties of concrete change indeed as a result of internal chemical reactions and the deformation problem is coupled with complicate moisture diffusion through the material, as well as heat conduction. For these reasons, in a first approximation, concrete may be regarded as an aging viscoelastic material whose creep law can be written in a rate-type form, i.e. as a system of first-order differential equations, involving hidden strains, with time-dependent coefficients. For a review of the basic facts on this subject see Barant[5]. More recently it appears that also for other materials, especially polymers in a temperature depending situation, the relaxation function is not of type G(t - T) but following a fundamental remark of Morland and Lee[61, the relaxation function can be written as G(t- 5’) where 6 = t(O) is the reduced time (see also Pipkin[7]). From another point of view the extension of phenomenological laws based on spring and dashpot models to the temperature depending case has been proposed by many experimen- talists (see e.g. [8] specially for metals). As far as we know the study of periodic problems in viscoelasticit’j seems quite new; the main purpose of this paper is to study the periodic dynamical problem in the case of a Maxwell body (which seems the most difficult from a mathematical point of view) when the phenomenological law is temperature depending. In order to give a coherent description of the problem we recall in Section I some consequences of the Clausius-Duhem inequality on the constitutive equation of the continuum mechanics of materials with hidden coordinates (indeed hidden strains) mainly due to C&man- Gut-tin[9]. UES Vol. 16, No. IO-A 681