  Slender elastic objects such as a column tend to buckle under loads. While
static buckling is well understood as a bifurcation problem, the evolution of
shapes during dynamic buckling is much harder to study. Elastic rings under
normal pressure have emerged as a theoretical and experimental paradigm for the
study of dynamic buckling with controlled loads. Experimentally, an elastic
ring is placed within a soap film. When the film outside the ring is removed,
surface tension pulls the ring inward, mimicking an external pressurization.
Here we present a theoretical analysis of this process by performing a
post-bifurcation analysis of an elastic ring under pressure. This analysis
allows us to understand how inertia, material properties, and loading affect
the observed shape. In particular, we combine direct numerical solutions with a
post-bifurcation asymptotic analysis to show that inertia drives the system
towards higher modes that cannot be selected in static buckling. Our
theoretical results explain experimental observations that cannot be captured
by a standard linear stability analysis.
