A multiplicity of localized buckling modes for twisted rod equations

The Kirchhoff-Love equations governing the spatial equilibria of long thin elastic rods subject to end tension and moment are reviewed and used to examine the existence of localized buckling solutions. The effects of shear and axial extension are not considered, but the model. does additionally allow for nonlinear constitutive laws. Under the assumption of infinite length, the dynamical phase space analogy allows one to use techniques from dynamical systems theory to characterize many possible equilibrium paths. Localizing solutions correspond to homoclinic orbits of the dynamical system. Under non-dimensionalization the twisted rod equations are shown to depend on a single load parameter, and the bifurcation behaviour of localizing solutions of this problem is investigated using analytical and numerical techniques. First, in the case of a rod with equal principal bending stiffnesses, where the equilibrium equations are completely integrable, a known one-parameter family of localizing solutions is computed for a variety of subcritical loads. Load-deflection diagrams are computed for this family and certain materially nonlinear constitutive laws are shown to make little difference to the qualitative picture. The breaking of the geometrical circular symmetry destroys complete integrability and, in particular, breaks the non-transverse intersection of the stable and unstable manifolds of the trivial steady state. The resulting transverse intersection, which is already known to lead to spatial chaos, is explicitly demonstrated to imply a multitude of localized buckling modes. A sample of primary and multi-modal solutions are computed numerically, aided by the reversibility of the differential equations. Finally, parallels are drawn with the conceptually simpler problem of a strut resting on a (nonlinear) elastic foundation, for which much more information is known about the global behaviour of localized buckling modes.