The static deformation of a twisted elastic rod constrained to lie on a cylinder
G.H.M. van der Heijden
The Cosserat director theory is used to formulate the problem of a long thin
weightless rod constrained, by suitable distributed forces, to lie on a
cylinder while being held by end tension and twisting moment. Applications of
this problem are found, for instance, in the buckling of drill strings inside
a cylindrical hole. In the case of a rod of isotropic cross-section the
equilibrium equations can be reduced to those of a one-degree-of-freedom
oscillator in terms of the angle the local tangent to the rod makes with the
axis of the cylinder. Depending on the radius of the cylinder and the applied
load, the oscillator has several fixed points each of which corresponds to a
different helical solution of the rod. More complicated shapes are also
possible, and special attention is given to localised configurations
described by homoclinic orbits of the oscillator. Heteroclinic saddle
connections are found to play an important role in the post-buckling
behaviour by defining critical loads at which a straight rod may coil up
into a helix.
keywords: elastic rod, cylindrical constraint, localized solutions,
homoclinic orbits, helical collapse, drill string
Proc. R. Soc. Lond. A 457, 695-715 (2001)