Stability of the non-radial eigenmodes of a uniformly charged elastic globe

The eigenmotions of an electrically charged isotropic elastic material are modeled by self-consistent system of elastodynamic equations (comprising the continuity equation for the density, the Euler equation for the velocity field of elastic displacements coupled with equation for the stress tensor governing evolution of elastic distortions) and Poisson’s equation of electrostatics. To elucidate the question of how the elastodynamic features of bulk elastic substance can be traced in a spherical sample of finite size, the low-frequency electromagnetic response of the sample is studied in the homogeneous model of a uniformly charged elastic globe. The response is described in terms of long wavelength, essentially non-radial, spheroidal (S-mode) and torsional (T-mode) electro-elastic vibrations. The restoring force is presumed to be dominated by the surface cohesive stresses and the volume disruptive stresses of electric origin acting destructively. It is shown that the continuum model in question leads to explicit form for the frequency of the non-radial eigenmodes. The emphasis is placed on the onset of vibrational instability originating from conflicting tendency of the above surface and volume elastic stresses. The elastodynamic criteria of instability for both spheroidal and torsional vibrations are established in analytic form.