Euler Buckling in a Potential Field

Summary.

We consider elastic buckling of an inextensible rod with free ends, confined to the plane, and in the presence of distributed body forces derived from a potential. We formulate the geometrically nonlinear (Euler) problem with nonzero preferred curvature, and show that it may be written as a three-degree-of-freedom Hamiltonian system. We focus on the special case of an initially straight rod subject to body forces derived from a quadratic potential uniform in one direction; in this case the system reduces to two degrees of freedom. We find two classes of trivial (straight) solutions and study the primary non-trivial branches bifurcating from one of these classes, as a load parameter, or the rod's length, increases. We show that the primary branches may be followed to large loads (lengths) and that segments derived from primary solutions may be concatenated to create secondary solutions, including closed loops, implying the existence of disconnected branches. At large loads all finite energy solutions approach homoclinic and heteroclinic orbits to the other class of straight states, and we prove the existence of an infinite set of such `spatially chaotic' solutions, corresponding to arbitrary concatenations of `simple' homoclinic and heteroclinic orbits. We illustrate our results with numerically computed equilibria and global bifurcation diagrams.