## Spatially complex localisation after one-twist-per-wave equilibria in twisted circular rods with initial curvature

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

In experiments on long rubber rods subject to end tension and moment, a
one-twist-per-wave deformation is often observed on the fundamental path
prior to the onset of localised buckling. An analysis is undertaken here to
account for this observed behaviour. First we derive general equilibrium
equations using the Cosserat theory, incorporating the effects of
non-symmetric cross section, shear deformation, gravity, and a uniform
initial curvature of the unstressed rod. Each of these effects in turn can
be expressed as a perturbation of the classical completely integrable
Kirchhoff-Love differential equations which are equivalent to those
describing a spinning symmetric top. Non-symmetric cross-section was dealt
with in earlier papers. Here, after demonstrating that shear deformation
alone makes little qualitative difference, the case of initial curvature is
examined in some detail.
It is shown that the straight configuration of the rod is replaced by
a one-twist-per wave equilibrium whose amplitude varies with pre-buckling
load. Superimposed on this equilibrium is a localised buckling mode, which
can be described as a homoclinic orbit to the new fundamental path. The
dependence is measured of the pre-buckled state and critical buckling load
on the amount of initial curvature. Numerical techniques are used to explore
the multiplicity of localised buckling modes, given that non-zero initial
curvature breaks the complete integrability of the differential equations,
and also one of a pair of reversibilities.
Finally, the physical implications of the results are assessed and are
shown to match qualitatively what is observed in an experiment.

*Phil. Trans. R. Soc. Lond. A* **355**, 2151-2174 (1997).