Stability of helical elastic rods with twist-bend coupling

A classic result in the theory of elastic rods due to Born is that an equilibrium configuration of a planar Euler elastica is stable if its centerline does not contain inflection points, regardless of its length. This property does not generally extend to three-dimensional rods since, for example, an inextensible, unshearable, isotropic, and uniform Kirchhoff rod with a circular centerline becomes unstable if it is sufficiently twisted, a phenomenon known as Michell's instability. In this talk, we apply methods from optimal control theory and symplectic geometry to show that the inclusion of twist-bend coupling in the Kirchhoff rod's elastic energy allows for helical configurations that are stable for arbitrary length. Two cases of twist-bend coupling are considered—symmetric (with the twist equally coupled to both bending curvatures) and asymmetric (with the twist coupled to only one bending curvature, as is the case for rod-like models of DNA)—and we show that both coupling types produce infinitely long stable helices.