Extreme contractility and torsional compliance of soft ribbons under high twist.

We investigate experimentally and model the mechanical response of a soft Hookean ribbon submitted to large twist η and longitudinal tension T, under clamped boundary conditions. We derive a formula for the torque M using the Föppl-von Kàrmàn equations up to third order in twist, incorporating a twist-tension coupling. In the stable helicoid regime, quantitative agreement with experimental data is obtained. When twisted above a critical twist η_{L}(T), ribbons develop wrinkles and folds which modify qualitatively the mechanical behavior. We show a surprisingly large longitudinal contraction upon twist, reminiscent of a Poynting effect, and a much lower torsional stiffness. Far from threshold, we identify two regimes depending on the applied T. In a high-T regime, we find that the torque scales as ηT and the contraction as η^{2}, in agreement with a far from threshold analysis where compression and bending stresses are neglected. In a low-T regime, the contraction still scales as η^{2} but the torque appears T independent and linear with η. We argue that the large curvature of the folds now contributes significantly to the torque. This regime is discussed in the context of asymptotic isometry for very thin plates submitted to vanishing tension but large change of shape, as in crumpling.