Nonlinear Mechanics of Elastic Rods Constrained to Deform on Rigid Tubular Surfaces

Slender, elastic rod-like structures on or inside constrained rigid surfaces are prevalent in a wide range of engineering (drill strings in borewells, pipelines under the seabed, ocean cables), medical (stents in angioplasty of arteries), biological (DNA toroidal condensates, bacterial flagella), electronic (carbon nanotubes) and robotic (soft robots for in-pipe inspection) applications. The tendency of such rods to deform by bending, twisting, buckling or coiling into complex geometrical shapes when subjected to applied end loads can be investigated using techniques from mathematical modelling and nonlinear dynamics. However, despite some work being done in this field, a comprehensive study analysing the bifurcations and configurations of surface-constrained elastic rods is still lacking. Past research on the subject has primarily used the special Cosserat director theory of rods to exclusively obtain specialised sinusoidal, helical or localised homoclinic solutions. In light of this, the purpose of this thesis is to employ a more comprehensive variational theory of elastic two-strand braids to investigate the post-buckling behaviour of end-loaded rods subject to cylindrical and toroidal surface constraints. Methods comprising the calculus of variations and Lagrangian and Hamiltonian mechanics are utilised to procure more general types of solutions to various nonlinear boundary value problems, using both analytical and numerical approaches. The key results obtained include the determination of critical compressive and torsional buckling loads for straight rods on a cylinder and an open torus, physically realistic and purely helical solutions for helical rods on a cylinder, critical surface lift-off loads and threshold twist values for heavy rods on a cylinder and applied torsion and twist-bend instability solutions for curved rods on a closed torus. The work also comprises the development of two, new frictional contact models to investigate physically meaningful problems for rods lying on frictional rigid surfaces and a self-contained Euler-Lagrange theoretical formulation to acquire all governing equations from a single constraint-inclusive energy functional