Localisation, looping and pop-out in rod-like structures

Long slender structures such as textile yarns and ocean cables show localised buckling behaviour under torsional loading. In a 1D elastic theory these localised solutions are described by homoclinic orbits in a dynamical system in which arclength along the structure is the independent variable. Upon continued loading after localisation a snap into self-contact and looping may occur. We discuss the intricate bifurcation behaviour that unfolds after the initial jump and also consider the problem of loop pop-out. We also briefly consider the mechanics of the plied structure that forms and show that the problem is governed by the equations for an elastic rod constrained to lie on a cylinder. This work is relevant for supercoiling DNA molecules and staple fibre yarns. Introduction Long thin structures under torsional loading are well known to undergo localised deformation. If the applied tension is insufficiently high the structure (a cable or wire) may ‘throw a loop’. In many applications loop formation is undesirable. In marine cables, if tension is reapplied after looping this may or may not cause the loop to pop out. As the loop is tightened the cable may undergo plastic deformation to an extent that it is permanently damaged. In textile yarns the formation of loops (here often called snarls) during spinning may lead to yarn breakage and a costly waste of time as well as to non-uniformities in the final fabric. Coyne [3] studied cable looping and pop-out by using the homoclinic solution which describes localised deformation in an infinitely long rod. Here we use an exact formulation for a self-contacting clamped finite-length incompressible circular tube obeying the Kirchhoff equations for a linearly elastic inextensible and unshearable rod. The contact force is obtained numerically as part of the solution of the equilibrium equations and pop-out is inferred when the contact force drops to zero. Multiple points of contact are allowed and we find a bifurcation sequence in which the rod jumps to states of increasing number of self-contacts as a control parameter (end rotation or end shortening) is increased. In the highly twisted case the final configuration is a plied structure known, for instance, from DNA supercoiling [2] and textile yarn twisting [4]. This process of ply formation, although remarkably delicate in its theoretical detail, can easily be observed in a rubber rod. The beginning of ply formation can be seen in Fig. 6. The fully fledged ply has line contact and we show that its mechanics can be idealised as a pair of rods winding around a cylinder. Formulation of a clamped rod Consider a thin elastic rod of length L held by end forces and moments. If we denote the position of the rod’s centreline by the vector function R, then the internal force, F , and moment, M , in an element of the rod are governed by the balance equations [1] F ′ = 0, (1) M ′ + r′ × F = 0, (2) where ( ) = d/ds, s being the arclength parameter measured along the centreline of the rod. We assume the rod to be perfect (i.e., intrinsically straight and untwisted), inextensible and unshearable.