From Phenomenology to a Nonlinear Model of Dynamic Snap-Through of an Elastica

Abstract: Rotating the clamped ends of a buckled elastica induces a snap-through instability. Predicting the limit point and determining the equilibria at the start and end of the snap are routine computations in the quasi-static setting. The instability itself, however, is dynamic, and quite violently so. We propose an energy-preserving nonlinear single degree of freedom model for this dynamic phenomenon in the case of a symmetrically deforming elastica. The model hinges on a surprising observation relating elastica profiles during the free dynamic snap with a specific sequence of geometrically-constrained elastic energy minimizing configurations. We corroborate this phenomenological observation over a significant range of arch depths through experiments and finite element simulations. The resulting model does not rely on modal expansions, explicit slowness assumptions, or linearization of the arch's kinematics. Instead, the model is effective because its solutions approximate the action integral well. The model provides distinctive computational benefits and new insights into the snap-through phenomenon. Our study is motivated by an application harnessing snap-through instabilities in submerged ribbons for underwater propulsion. We briefly describe its novel working principle and discuss its relationship to the problem studied.