## Spatially complex localisation in twisted elastic rods constrained to a cylinder

G.H.M. van der Heijden, A.R. Champneys & J.M.T. Thompson

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.
keywords: elastic rod, anisotropy, cylindrical constraint, localised
solutions, multi-pulse homoclinic orbits, softening-hardening-softening
response, Maxwell failure mode, helical collapse, drill string

*Int. J. Solids Struct.* **39**, 1863-1883 (2002)