On Spatial Behavior in Linear Viscoelasticity

Within the framework of linear viscoelasticity this paper deals with the study of spatial behavior of solutions describing harmonic vibrations in a right cylinder of finite extent. Some exponential decay estimates of Saint–Venant type, in terms of the distance from the excited end of the cylinder are obtained from a first-order differential inequality concerning an appropriate measure associated with the amplitude of the steady-state vibration. The dissipative mechanism guarantees the validity of the result for every value of the frequency of vibration and for the class of viscoelastic materials compatible with thermodynamics whose relaxation tensor is supposed to be symmetric and sufficiently regular. The case of a semi-infinite cylinder is also studied, and some alternatives of Phragmén–Lindelöf type are established. Introduction. The present paper is concerned with the study of the spatial behavior of solutions in a right cylinder made of an anisotropic and homogeneous viscoelastic solid. We consider a finite cylinder subject to boundary data varying harmonically in time on one end, while the other end and lateral surface are clamped. The history of the displacement up to time t = 0 is supposed to be known and the body force is assumed to be absent. Initial boundary value problems of this type have been treated by Flavin and Knops [1] in the framework of the linearly damped wave equation and the linearly elastic damped cylinder. They proved that in both cases the existence of damping gives rise ultimately to a steady-state oscillation, whose amplitude decays exponentially from the excited end provided the exciting frequency is less than a certain critical value. The latter case has Received May 16, 2008. 2000 Mathematics Subject Classification. Primary 74D05, 74G50; Secondary 74H45, 74E10.