Twist-Induced bifurcation and path manipulation in compressed ribbons

In this paper, we investigate the nonlinear behaviour of ribbons subjected to coaxial compression and twisting through theoretical, numerical, and experimental approaches. Using anisotropic Kirchhoff rod theory and continuation techniques, we construct global bifurcation diagrams and identify stability transition points via the conjugate-point test. Here, we show that the twist induces supercritical pitchfork bifurcations in ribbons, giving rise to a rich landscape of multi-stability with up to four coexisting stable states. With increasing twist, we observe stability transitions between the fundamental and second Euler buckling modes. Moreover, gravity triggers a global bifurcation reconstruction characterized by the emergence of saddle-node bifurcations. These gravity-induced transformations allow for multiple, controllable snap-through pathways between stable states. We extend a mixed-curvature-based numerical optimization method to predict snap-through destinations and propose a general path-planning framework to navigate between stable configurations. Experiments on ribbons with varied aspect ratios corroborate the theoretical predictions and demonstrate the viability of programmable transitions in multi-stable systems. Our findings provide new insights into bifurcation and snap-through behaviour in slender structures, with potential applications in mechanical metamaterials, flexible electronics, and soft robotics.