Saint venant's principle in linear viscoelasticity

Abstract

In this paper we consider an anisotropic and inhomogeneous viscoelastic body that is subjected on the time interval [0, T] to body forces and initial and boundary data having a bounded support. Then a complete description is given upon what happen in the outside of the support region D T of the data. More precisely, when the genuine dynamic linear viscoelasticity is considered, we prove that, for each tφ E [0, T], there exists a bounded domain D ct so that the whole activity vanishes in the outside of D ct while into the region D ct D T an appropriate measure of the dynamic viscoelastic process in question decays spatially with the distance r from the bounded support D T the decay rate being controlled by the factor (1 − r/ ct). As a direct consequence of this analysis, we establish a uniqueness theorem for linear viscoelastodynamics, that is valid for infinite domains and is free of any kind of a priori assumptions on the orders of growth of the velocity and stress fields at infinity. When the quasi-static linear viscoelasticity is considered for a bounded body, we associate with the quasi-static viscoelastic process an appropriate energetic volume measure and we establish a spatial decay estimate of exponential type similar to that presented by Toupin for linear elastostatics. The analysis of this paper is based on the recent study on the free energies and the maximal free energy in linear viscoelasticity presented by Fabrizio et al. in Ref. [1] ( Archives for Rational Mechanical and Analysis, 1994, 125, 341).