Flexural vibrations of nonlinearly elastic circular rings

This paper treats free nonlinear flexural vibrations of circular elastic rings in the context of a geometrically exact formulation accounting for the effects of nonlinear material behavior. A direct asymptotic approach based on the method of multiple scales is used to investigate such vibrations. It is shown that the flexural motions are softening for linearly elastic rings, in agreement with previous results in the literature, while there are nonlinearly elastic rings for which the motions are hardening. There are thresholds in the nonlinear constitutive laws separating softening from hardening behaviors.