Integration of kinks and creases enables tunable folding in meta-ribbons

Foldable structures find diverse applications. Folding of thin structures into compact shapes involves the interplay of nonlinear mechanics and topology. In this study, we employ discrete models, theoretical analysis, and tabletop experiments to systematically investigate the geometrically nonlinear folding process of ring-shape elastic ribbons through in-plane kinks and out-of-plane creases. We find that kinks initiate continuous folding through supercritical bifurcation, while creases trigger abrupt snapping via subcritical bifurcation. Master curves that summarize energy landscapes for ribbons with varying numbers of kinks and creases are obtained. By integrating kinks and creases, a "meta-ribbon" can be created, which shows the tunable folding behavior, transitioning from continuous to snapping, or vice versa, by strategically engineering the in-plane and out-of-plane angles guided by the constructed energy map. As a product of folding, we demonstrate the snapping-induced vibration accomplished with dynamic folding, as well as the multistability of meta-ribbons with saddle-like configurations and their transformation.