Harmonic waves in heat conducting bars: a comparison of constrained and unconstrained thermoelasticity

The thermodynamics of heat conducting bars of elastic material, which may d eform subject to a deformation-temperature constraint, is outlined. The lin earised equations governing infinitesimal thermoelastic deformations in bar s are derived, and the conditions necessary for their extension to semi-inf inite bars are emphasized. Then the problem of a semi-infinite bar subject to temperature and/or displacement variations at the free end which are tim e periodic is solved by Laplace transforms for the cases of constrained and unconstrained materials. The solutions contain harmonic waves upon which a re imposed transients. In the case of the constrained material both of the harmonic waves classified as 'unstable' and one of the waves classified as 'stable', according to a definition originating elsewhere (loc. cit.), are excluded by the boundary conditions, so that only a single wave of diffusio n is present. This is in contrast to the unconstrained case which has a wav e headed by a wavefront, in addition to a wave of diffusion.