The mechanics and morphology evolutions in stretched ribbons under torsion: A 3D phase diagram

Slender structures ubiquitously undergo complex shape transformations in both natural and artificial systems. Soft ribbons can exhibit rich morphologies when subjected to loading and geometric constraints. The competition between the macroscopic, microscopic, and transverse buckling modes results in multiple morphological transitions, revealing distinct evolution paths among helicoid, crease, cylinder, and scrolled yarn configurations. However, the roles of tension, twist, and geometry in determining the buckling modes of a stretched ribbon under torsion remain unclear. In this study, we comprehensively investigate the torsional instabilities of elastic ribbons through experiments, theory, and finite element simulations. A one-dimensional model is employed to describe the nonlinear torsional responses in the helicoid stage. We present a modeling approach to describe the crease configuration and derive a semi-analytical solution based on the finite element analysis. We find that the competition between the macroscopic and microscopic buckling modes is governed by the thickness-to-width ratio, which determines the relative magnitudes of the membrane and bending strain energy in the helicoid. Additionally, as the applied tension increases, the buckling mode shifts from the longitudinal to the transverse direction. A three-dimensional phase diagram, as a function of twist, tension, and thickness-to-width ratio, is constructed to identify the morphology transitions of elastic ribbons. This study has implications for understanding the morphological complexity and achieving precise control of slender structures.