Twisting instabilities of inextensible helical rods

We develop a variational framework for curves whose energies depend on their bend and twist degrees of freedom, which we describe in terms of their material curvatures. We derive the equilibrium equations representing the balance of forces and torques on the curve. The conservation laws of the force and torque on the curve, stemming from the Euclidean invariance of the energy, allow us to obtain first integrals of the equilibrium equations. We apply this framework to the study of isotropic and anisotropic Kirchhoff elastic rods, whose energies are quadratic in the material curvatures. We analyze perturbatively the deformations of helices resulting from their twisting. We examine three kinds of twisting instabilities on unstretchable helices, characterized by their wavenumbers, depending on whether their boundaries are fixed, displaced along the radial direction or orthogonally to it. We also analyze perturbatively the effect of the bending anisotropy on the deformed states, which introduces a coupling between deformation modes with different wavenumbers.