Saint-Venant's principle in linear viscoelasticity
We consider an anisotropic and inhomogeneous viseoelastic 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 dynamic linear viscoelasticity is considered, we prove that, for each \(t\in[0,T]\), there exists a bounded domain \(D_{ct}\) so that the whole activity vanishes in the outside of \(D_{ct}\); while into the region \(D_{ct} \setminus D_T\) 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.
spatial decay estimate of exponential type
scientific article; zbMATH DE number 1162234