The static deformation of a twisted elastic rod constrained to lie on a cylinder

The Cosserat director theory is used to formulate the problem of a long thi n weightless rod constrained, by suitable distributed forces, to lie on a c ylinder while being held by end tension and twisting moment. Applications o f this problem are found, for instance, in the buckling of drill strings in side a cylindrical hole. In the case of a rod of isotropic cross-section th e equilibrium equations can be reduced to those of a one-degree-of-freedom oscillator in terms of the angle that the local tangent to the rod makes wi th the axis of the cylinder. Depending on the radius of the cylinder and th e applied load, the oscillator has several fixed points, each of which corr esponds to a different helical solution of the rod. More complicated shapes are also possible, and special attention is given to localized configurati ons described by homoclinic orbits of the oscillator. Heteroclinic saddle c onnections are found to play an important role in the post-buckling behavio ur by defining critical loads at which a straight rod may coil up into a he lix.