Spatially complex localisation in twisted elastic rods constrained to a cylinder

We consider the problem of a long thin weightless rod constrained to lie on a cylinder while being held by end tension and twisting moment. Applications of this problem are found, for instance, in the buckling of drill strings inside a cylindrical hole. In a previous paper the general geometrically exact formulation was given and the case of a rod of isotropic cross-section analysed in detail. It was shown that in that case the static equilibrium equations are completely integrable and can be reduced to those of a one-degree-of-freedom oscillator whose non-trivial fixed points correspond to helical solutions of the rod. A critical load was found where the rod coils up into a helix. Here the anisotropic case is studied. It is shown that the equations are no longer integrable and give rise to spatial chaos with infinitely many multi-loop localised solutions. Helices become slightly modulated. We study the bifurcations of the simplest single-loop solution and a representative multi-loop as the aspect ratio of the rod’s cross-section is varied. It is shown how the anisotropy unfolds the ‘coiling bifurcation’. The resulting post-buckling behaviour is of the softening–hardening–softening type typically seen in the cellular buckling of long structures, and can be interpreted in terms of a so-called Maxwell effective failure load