Bifurcation reordering programs snap-through symmetry in folded elastic ribbons

Abstract

Snap-through in slender elastic structures is often viewed as a sudden transition between stable configurations, yet the pathway taken during this transition can differ fundamentally. A structure may snap while preserving symmetry, or first lose symmetry and pass through an asymmetric state before reaching its final configuration. What selects between these pathways remains less well understood, especially when the geometry and loading are themselves symmetric. Here, we show that snap-through symmetry can be programmed by reordering competing bifurcations in folded elastic ribbons. We study an elastic ribbon with two localized folds placed symmetrically about the midpoint and show that varying the fold position changes the relative order of two instabilities: a symmetry-breaking pitchfork bifurcation and a saddle-node bifurcation on the symmetry-preserving branch. When the pitchfork bifurcation occurs first, the ribbon loses symmetry before snapping and follows an asymmetric pathway. Conversely, when the saddle-node bifurcation occurs first, the ribbon loses stability while remaining on the symmetric branch, resulting in a symmetry-preserving transition. Combining experiments, discrete differential geometry simulations and numerical continuation, we map this exchange in bifurcation ordering and construct a phase diagram that predicts the switch between asymmetric and symmetric snap-through regimes. A reduced-order double-mass von Mises truss model captures the same mechanism as a generic competition between symmetry-breaking and symmetry-preserving instabilities. These results establish bifurcation reordering as a geometric mechanism for programming snap-through pathways in slender elastic structures, offering a design principle for multistable systems, morphing structures and instability-based mechanical devices.