ure only the compressible material with the relaxation-time ratio s ¼ 1 is plotted alongside the incompressible result. The
long-time dissipated energy over a single cycle is shown in Fig. 4 for a range of frequencies x. It can be clearly observed that
the energy dissipated by the compressible material is greater than that for the neo-Hookean solid. The energy dissipation
appears constant at moderate and higher oscillation frequencies, except for the case with s ¼ 1, which only becomes constant
at very high frequencies. The frequency independence of the dissipated energy at higher frequencies appears to be consistent
with other theory, e.g. Fig. 10 of Lion, 1997 where, for time-harmonic oscillations superposed on an initial
deformation, plots of the temperature against frequency are plotted. One set of curves in that figure does not level off for
the frequency range considered but we expect that at higher frequencies, beyond the range plotted, constant values will
be achieved.
6. Conclusions