Lock-on to tape-like behaviour in the torsional buckling of anisotropic rods

Through nonlinear normal form analysis of the equilibrium equations developed in a neighbourhood of the straight rod solution we show the existence of a 'phase transition' from mildly anisotropic to full 'tape-like' buckling in non-symmetric rods subject to terminal loads, provided the material has a sufficiently low Poisson's ratio. For a solid elliptical rod with a Poisson's ratio of 0.2 the critical point is found to occur at a cross-sectional aspect ratio of just over 3. For larger aspect ratios the rod locks on to a one-twist-per-wave buckling mode with no internal twist. In the mildly anisotropic regime there turn out to be two physically distinct pairs of symmetric so-called primary localised buckling modes, differing in the phase of the internal twist. These four modes (homoclinic orbits) remain of the full circle of localised buckling modes existing in the symmetric case after breaking of the circular symmetry of the rod's cross-section. in the strongly anisotropic regime only the energetically favourable pair of 'flat' buckling modes survives.