Elastic snap-through instabilities are governed by geometric symmetries

TL;DR

This paper investigates how geometric symmetries influence elastic snap-through instabilities, revealing universal principles that govern shape transitions in elastic structures through analytical and numerical analysis.

Contribution

It demonstrates that bifurcation types in elastic shape transitions are governed by geometric symmetries, providing universal design rules for predicting and controlling these instabilities.

Findings

Elastic snap-through is governed by geometric symmetries.

Bifurcation types can be predicted from symmetry considerations.

The analysis applies to various elastic systems and transitions.

Abstract

Many elastic structures exhibit rapid shape transitions between two possible equilibrium states: umbrellas become inverted in strong wind and hopper popper toys jump when turned inside-out. This snap-through is a general motif for the storage and rapid release of elastic energy, and it is exploited by many biological and engineered systems from the Venus flytrap to mechanical metamaterials. Shape transitions are known to be related to the type of bifurcation the system undergoes, however, to date, there is no general understanding of the mechanisms that select these bifurcations. Here we analyze numerically and analytically two systems proposed in recent literature in which an elastic strip, initially in a buckled state, is driven through shape transitions by either rotating or translating its boundaries. We show that the two systems are mathematically equivalent, and identify three cases that illustrate the entire range of transitions described by previous authors. Importantly, using reduction order methods, we establish the nature of the underlying bifurcations and explain how these bifurcations can be predicted from geometric symmetries and symmetry-breaking mechanisms, thus providing universal design rules for elastic shape transitions.