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

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 localized 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 localized 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 localized 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.