Some spatial decay estimates in continuum dynamics

Spatial evolution of harmonic vibrations in linear elasticity

“…In this respect, for vibrations in the low frequency range, our expected results describe exponential spatial estimates similar with those previously established by Flavin et al [1987;1990]. Moreover, for harmonic vibrations with appropriate high frequencies, the present results predict some algebraic spatial estimates, confirming the foregoing observations made by Boley in related context.…”

“…It is outlined in [Horgan and Knowles 1983] that one would not expect to find unqualified decay estimates of the kind concerning Saint-Venant's principle in problems involving elastic wave propagation, even if the end loads are self-equilibrated at each instant. In this connection, Flavin and Knops [1987] have carried out an analysis of spatial decay for certain damped acoustic and elastodynamic problems in the low frequency range which substantiates the early work of Boley. These results are extended to linear anisotropic Keywords: spatial behavior, harmonic vibrations, linear elasticity, strongly elliptic elasticity tensor.…”

“…We note that the spatial behavior of the transient solution U r can be described by the methods developed in [Chirit ,ȃ and Ciarletta 1999;Tibullo and Vaccaro 2008]. The exponential spatial decay of the amplitude v r of the forced oscillation has been established in [Flavin and Knops 1987;Flavin et al 1990;Knops 1991] for isotropic and anisotropic elastic materials with a positive definite elasticity tensor, provided the exciting frequency is less than a certain critical value. Considering an appropriate region filled with an isotropic elastic material an algebraical spatial decay of the amplitude of vibration has been established in [Chirit ,ȃ and Quintanilla 1996] under the assumption that the elasticity tensor is positive definite and without any restriction upon the frequency of vibration.…”

Evolution of solutions for dipolar bodies in Thermoelasticity without energy dissipation

“…We must emphasize that the spatial behavior of the harmonic in time vibrations has been studied by Chirita [3] in the classical linear thermoelasticity, by using a technique developed by Flavin and Knops [6] in the low frequency range. The author establishes some exponential estimates for spatial evolution of the amplitude of vibration, provided the positive definiteness of the constitutive coefficients is assumed.…”

On spatial behavior in linear viscoelasticity

“…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.…”