From helix to localized writhing in the torsional post-buckling of elastic rods

In this paper pie study the competition between helical and localizing modes in the torsional buckling of stretched and twisted elastic rods. Within the Love-Kirchhoff formulation, we make comparative studies of the helical deformation of Love (1927) and the homoclinic localizing solution of Coyne (1990). Plots of the loads against their corresponding deflections allo-rv the energetically preferred mode to be identified: it is found that preference switches from the helix to the localized mode early in the postbuckling range. These plots also allow us to predict the jumps that are observed under a variety of dead and rigid loading processes: these dynamic jumps take the rod from the spatially localized form to the familiar writhing state. Preliminary experiments confirm this preference for the localizing mode. They also reveal a second type of helical deformation, at a shorter wavelength, that is not predicted by the above long-rod analyses. A programme of further experimental and theoretical studies is suggested, and in a companion paper we lay the mathematical foundations for a numerical investigation of the complex and spatially chaotic deformations of a wider class of elastic rods.