Parametric resonances of nonlinearly viscoelastic rings subject to a pulsating pressure

Refined theories are employed for accurate predictions of the onset of the parametric instabilities of nonlinearly viscoelastic rings subject to a hydrostatic pressure. The breathing motions are first investigated and closed-form conditions ensuring motions of the hardening or softening type are derived showing that this mode nonlinearity depends on the constitutive law and the static part of the hydrostatic pressure. The parametric instability regions and the ensuing 2-period breathing motions are obtained employing an asymptotic approach. Thereafter, the parametric instabilities of the flexural motions are investigated via a direct asymptotic approach and it is shown that these motions can also be softening or hardening depending on the low-order parts of the constitutive law. The parametric instabilities cause high-amplitude coupled-mode flexural motions involving simultaneously the directly excited mode and its companion mode.