On spatial behavior in linear viscoelasticity

Abstract

Abstract. 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

On thermo-electro-viscoelastic relaxation functions in a Green–Naghdi type theory

“…where Lin denotes the linear space of second-order tensors on IR 3 . We assume that "The memory of a simple material fades in time" [1], and we describe how the material has a fading memory by an obliviator or influence function [10].…”

“…In literature there have been many articles on this topic, with different settings, and here we mention only a few. Gales ¸and Chirit ¸ǎ in [3] study the spatial behavior of solutions in a right cylinder made of an anisotropic and homogeneous viscoelastic solid in a class of linear viscoelastic materials compatible with thermodynamics in the sense of Fabrizio and Morro [4], [5]. Huang [6] proposes a thermo-viscoelastic constitutive theory at finite deformation in a framework of irreversible thermodynamics and introducing the internal variables in the constitutive relations.…”

Spatial Behavior of the Steady State Vibrations in a Dual-Phase-Lag Rigid Conductor

“…We have to outline that the corresponding studies for the classical elastic and thermoelastic theories have been carried out in [7][8][9][10][11][12][13]. It is established there that, for appropriate values of the frequency of the harmonic excitation and by using some differential inequalities, the spatial decay of effects with distance from the excited end of the right cylinder is described by some exponential decay estimates of Saint Venant type.…”