Helicoids, wrinkles, and loops in twisted ribbons.

We investigate the instabilities of a flat elastic ribbon subject to twist under tension and develop an integrated phase diagram of the observed shapes and transitions. We find that the primary buckling mode switches from being localized longitudinally along the length of the ribbon to transverse above a triple point characterized by a crossover tension that scales with ribbon elasticity and aspect ratio. Far from threshold, the longitudinally buckled ribbon evolves continuously into a self-creased helicoid with focusing of the curvature along the triangular edges. Further twist causes an anomalous transition to loops compared with rods due to the self-rigidity induced by the creases. When the ribbon is twisted under high tension, transverse wrinkles are observed due to the development of compressive stresses with higher harmonics for greater width-to-length ratios. Our results can be used to develop functional structures using a wide range of elastic materials and length scales.