## The spatial complexity of localised buckling in rods with non-circular cross-section

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

We study the post-buckling behaviour of long, thin, elastic rods subject
to end moment and tension. This problem in statics has the well-known
Kirchhoff dynamic analogy in rigid body mechanics consisting of a reversible
three-degrees-of-freedom Hamiltonian system. For rods with non-circular
cross-section this dynamical system is in general non-integrable and in
dimensionless form depends on two parameters: a unified load parameter and
a geometric parameter measuring the anisotropy of the cross-section.
Previous work has given strong evidence of the existence of a countable
infinity of localised buckling modes which in the dynamic analogy correspond
to N-pulse homoclinic orbits to the trivial solution representing the
straight rod. This paper presents a systematic numerical study of a large
sample of these buckling modes. The solutions are found by applying a
recently developed shooting method which exploits the reversibility of the
system. Subsequent continuation of the homoclinic orbits as parameters are
varied then yields load-deflection diagrams for rods with varying load and
anisotropy. From these results some structure in the multitude of buckling
modes can be found, allowing us to present a coherent picture of localised
buckling in twisted rods.
keywords: anisotropic rods, torsional buckling, spatial chaos, homoclinic
orbits, bifurcation

*SIAM J. Appl. Math.* **59**, 198-221 (1999)