Helical and localised buckling in twisted rods: A unified analysis of the symmetric case

We review the geometric rod theory for the case of a naturally straight, linearly elastic, inextensible, circular rod suffering bending and torsion but no shear. Our primary focus is on the post-buckling behaviour of such rods when subjected to end moment and tension. Although this is a classic problem with an extensive literature, dating back to Kirchhoff, the usual approach tends to neglect the physical interpretation of solutions (i.e., rod configurations) to the models proposed. Here, we explicitly compute geometrical properties of buckled rods. In a unified approach, making use of Kirchhoff's dynamic analogy, both the classical helical and the more recently investigated localised buckling are considered. Special attention is given to a consistent treatment of concepts of link, twist and writhe.