Twisted rods, helices and buckling solutions in three dimensions

The study of slender elastic structures is an archetypical problem in continuum mechanics, dynamical systems and bifurcation theory, with a rich history dating back to Euler's seminal work in the 18th century. These filamentary elastic structures have widespread applications in engineering and biology, examples of which include cables, textile industry, DNA experiments, collagen modelling etc. One is typically interested in the equilibrium configurations of these rod-like structures, their stability and dynamic evolution and all three questions have been extensively addressed in the literature. However, it is generally recognized that there are still several open non-trivial questions related to three-dimensional analysis of rod equilibria, inclusion of topological and positional constraints and different kinds of boundary conditions

Buckling between soft walls: sequential stabilization through contact

Motivated by applications of soft-contact problems such as guidewires used in medical and engineering applications, we consider a compressed rod deforming between two parallel elastic walls. Free elastica buckling modes other than the first are known to be unstable. We find the soft constraining walls to have the effect of sequentially stabilizing higher modes in multiple contacts by a series of bifurcations, in each of which the degree of instability (the index) is decreased by one. Further symmetry-breaking bifurcations in the stabilization process generate solutions with different contact patterns that allow for a classification in terms of binary symbol sequences. In the hard-contact limit, all these bifurcations collapse into highly degenerate ‘contact bifurcations’. For any given wall separation at most a finite number of modes can be stabilized and eventually, under large enough compression, the rod jumps into the inverted straight state. We chart the sequence of events, under increasing compression, leading from the initial straight state in compression to the final straight state in tension, in effect the process of pushing a rod through a cavity. Our results also give new insight into universal features of symmetry-breaking in higher mode elastic deformations. We present this study also as a showcase for a practical approach to stability analysis based on numerical bifurcation theory and without the intimidating mathematical technicalities often accompanying stability analysis in the literature. The method delivers the stability index and can be straightforwardly applied to other elastic stability problems

Stability of Discontinuous Elastic Rods with Applications to Nanotube Junctions

Buckling and post-buckling stability of elastic rods with discontinuities in bending stiffness and curvature, as well as rods lying on an interactive surface is investigated using the theory of conjugate points. Second order matching conditions at points of discontinuity are formulated, which allow the classic Jacobi condition to be extended to incorporate calculus of variations problems with discontinuous integrands. Static equilibrium equations for intrinsically straight, inextensible and unshearable discontinuous rods are formulated. Conjugate points are found by numerically solving the Jacobi equation as an initial value problem. For the case of a rod interacting with a surface, an external force potential is added to the energy functional, causing the Jacobi operator to be an integrodifferential operator. Morse index theory is used to find expressions for critical buckling values of load pa- rameters with respect to parameters measuring jumps in bending stiffness, or parameters measuring the strength of the rod-surface interaction. Bifurcation diagrams of buckled rod solutions are presented, with the Morse stability index calculated for each solution branch. These are found to be consistent with the theory of stability exchange at folds for distinguished diagrams. The presence of a jump in bending stiffness is shown, in some cases, to cause an extra stable solution branch. Numerical continuation of folds in two parameters is used to find the parameter space for which these stable branches exist. The rod equilibrium equations are solved numerically using parameter continua- tion for a boundary value problem. Clamped boundary conditions are considered, as well as pinned boundary conditions, which require a more robust adaptation of the classic Jacobi condition. The theory is applied to the modelling of carbon nanotube intramolecular junctions, in which the bonding of two or more carbon nanotubes causes a jump in the diameter, chirality or cross-section shape of the resulting tube, as well as a possible kink (jump in the centreline curvature) in the tube. The effects of van der Waals forces between a nanotube undergoing compression, and a substrate are modelled