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 naturallystraight, linearly elastic, inextensible, circular rod suffering bendingand torsion but no shear. Our primary focus is on the post-bucklingbehaviour of such rods when subjected to end moment and tension.Although this is a classic problem with an extensive literature, datingback to Kirchhoff, the usual approach tends to neglect the physicalinterpretation of solutions (i.e., rod configurations) to the modelsproposed. Here, we explicitly compute geometrical properties of buckledrods. In a unified approach, making use of Kirchhoff's dynamic analogy,both the classical helical and the more recently investigated localisedbuckling are considered. Special attention is given to a consistenttreatment of concepts of link, twist and writhe.