Alternative Spatial Growth and Decay for Constrained Motion in an Elastic Cylinder

Abstract

Alternative spatial growth and decay behavior is established for the motion of a (semi-infinite) cylinder composed of a nonhomogeneous anisotropic linear material and subject to zero body force and zero lateral boundary conditions. The motion, induced by displacement prescribed on the base, is constrained to have the displacement and velocity at a given time proportional to the respective initial values. Neither initial data nor the asymptotic behavior at large axial distance are specified. A differential inequality is derived for certain time integrals of the cross-sectional energy flux and integration leads to estimates that either exponentially grow or decay. Decay estimates are also obtained for the cross-sectional measures of strain energy, strain, displacement and its gradient, while explicit bounds in terms of the data are constructed for the amplitude in each decay estimate.