Dynamic buckling of rings and annuli

When an elastic ring is subject to an external pressure, it is known to buckle as a 'figure of 8'; similarly an annulus subject to a strong internal tension is known to form two folds. Here we explore how these examples of mode-2 buckling change under dynamic loading. In particular, we show that the presence of inertia means that higher modes are observed at the onset of instability, and present results for the observed mode number as a function of the driving pressure. Our results are in accord with experiments in which an elastic ring is rapidly pulled inwards by surface tension. Surprisingly, we find that while inertia is required to observe higher modes, the actual selected mode does not depend on inertia. We explain this, and reconcile our observation of higher modes with earlier experiments that exhibited mode 2, by understanding the growth rate of instability.