Spatially complex localisation in twisted elastic rods constrained to lie in the plane

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

Equilibrium configurations are considered of a long rod constrained to lie in a plane, subject to end conditions constituting a wrench. Using the Cosserat theory, a formulation of the problem is proposed using a reduced angular description of the director basis. On the assumption of an isotropic cross-section, it is found that flexure and torsion decouple so that the rod buckles like a planar elastica. For rods held under gravity, a condition is derived for the applied end loads required for lift-off of the localised mode under tension. For anisotropic rods, flexure and torsion are coupled and additional more complex equilibrium shapes are possible including multi-loop localised modes. Using specially adapted numerical shooting techniques such solutions, which are mathematically represented by homoclinic orbits to a periodic solution, are computed, and conditions for lift-off of the single-loop solutions are calculated as a function of the applied loads and an anisotropy parameter.

J. Mech. Phys. Solids 47, 59-79 (1999)