Compressible viscoelasticity: stability of solutions for homogeneous plane-Earth models

Recently, Purcell has discussed the influence of gravity on load-induced perturbations of compressible, viscoelastic plane-Earth models, where the contributions arising from initial stress and internal buoyancy have been distinguished in the equation of motion. According to his results, the consideration of initial stress is mandatory, whereas that of internal buoyancy has only a minor influence on the solution. We re-examine his study using a different approach. In particular, we present analytical solutions for a homogeneous half-space and discuss the associated relaxation spectra. We show that the solution to the problem involves singularities and branch cuts in the complex s-plane in addition to the singularities caused by roots of the determinant function. Furthermore, Rayleigh-Taylor instabilities arising from internal buoyancy cannot be completely balanced by initial stress. Also, the stability margin introduced by Love for an elastic continuum cannot be ignored when applying the solution. Finally, we show that the relaxation spectra can be directly related to the spectral response of a homogeneous, compressible, viscoelastic sphere as studied in recent papers by Vermeersen and Hanyk.