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)