Straight-to-helicoid transition of twisted cords

Abstract When a stretched rubber cord is twisted, it eventually develops elaborate instability patterns. For a sufficiently slender cord, the first step in this sequence is the emergence of helical shapes, which later destabilize into knots as the torque grows. Capturing this onset analytically is notoriously difficult owing to geometric and constitutive nonlinearities. Here, we present an energy-based route that focuses on the first transition: the formation of a helicoid from the straight, twisted state. By adopting reduced kinematics for a slender, volume-preserving, hyperelastic cord, we place straight and helicoidal configurations on the same footing under combined axial stretch and torsion. This reduction turns an otherwise differential problem into a purely algebraic one, allowing us to pinpoint the critical torsional load at which the transition occurs. A key advantage of this formulation is that it yields a fully analytical description of the post-buckled helicoidal state, valid even for arbitrarily large applied torque and axial pre-stretch. Beyond the instability threshold, the helicoidal amplitude grows smoothly with torque, while the pitch decreases; features consistent with our experiments and with previous observations.