Parametric instabilities of the radial motions of non-linearly viscoelastic shells under pulsating pressures

This paper treats the classical problem of radial motions of cylindrical and spherical shells under pulsating pressures. The novelty in this work is that the shells are taken to be non-linearly viscoelastic (of strain-rate type). It is remarkable that this classical problem, which does not treat the loss of stability to non-radial motions (but which is essential for such treatments), has such a rich dynamics due to the often neglected effects of non-linear material response, to the role of prestress under the action of the mean pressure, and to the different effects of pressure on cylindrical and spherical shells. The study of radial motions near primary resonance (when the frequency of the pulsating pressure is near the natural frequency about an equilibrium state under a constant pressure) gives formulas ensuring that the motions are of hardening or softening type depending on the constitutive functions and whether the constant mean pressure is compressive or inflational. The method of multiple scales gives asymptotic formulas for the principal parametric instability regions (Mathieu tongues) and for the stable and unstable motions at twice the forcing frequency, which closely agree with those obtained by numerical continuation methods. The dependence of frequency on amplitude and the form of instability regions are critically influenced by deviations (even very slight deviations) of material response from that of linearly viscoelastic shells, by the constant mean pressure, and by the type of shell. This paper exhibits the rich diversity of postcritical periodic motions.

► Non-linearly viscoelastic shells under pulsating pressures are studied asymptotically.
► The material non-linearities and the prestress influence critically the parametric instability regions.
► The mentioned effects vary from cylindrical to spherical shells.