From static buckling to nonlinear dynamics of circular rings

The dynamic buckling of circular rings is a pervasive instability problem with a major impact in various fields, such as structural, nuclear and offshore engineering, robotics, electromechanics, and biomechanics. This phenomenon may be simply seen as the complex motion that occurs deviating from the original circular shape under, for instance, any kind of time-dependent forcing load. Despite the fact that this topic has progressively gained importance since the mid-20th century, it seems that the same points have not been made completely clear. In fact, even some subtleties in the derivation of classical static buckling load may still give rise to misinterpretations and lead to misleading results. A fortiori, research concerning the nonlinear dynamics of rings still suffers the inherent difficulties associated with different possible analytical formulations of post-buckling dynamics. Advancement in this respect would be relevant, both from a theoretical and a practical point of view, since the applications are endless, with countless possibilities, especially in the biomedical and biotechnological fields: buckling-driven transformations of thin-film materials for applications in electronic microsystems, self-excited oscillations in collapsible tubes and pliable fluid-carrying shells, vocal-fold oscillations during phonation and snoring, pulse wave propagation in arteries, closure and reopening of pulmonary airways, stability of cardiac and venous valves during vascular surgery, stability of annuloplasty devices, flow-induced deformation and ultimate rupture of a cerebral aneurysm, and much more. The present article, in the framework of a critical review of the classic formulation of elastic ring buckling, proposes a straightforward approach for the nonlinear dynamics of an elastic ring that leads to a Mathieu–Duffing equation. In such a manner, some possible evolutions of the system under pulsing loads are analyzed and discussed, showing the inherent complexity of its dynamic behavior.