## 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)