The large deflection theory of circular cross-section elastic rods is used to consider the writhing of long straight rods subjected to tension and torque, such as undersea cables, and to closed loops with inserted twist, such as DNA supercoils.The writhed shape of the long straight rod under tension and torque is easily generated by twisting a piece of string with the fingers and consists of three separate parts: a balanced-ply region, a free end loop, and two tail regions. The solution for the rod shape in each of the regions is found. The results are then joined together to ensure continuity of the position and tangent vectors of the strand centreline through the introduction of point forces and moments at the points where the strands enter and exit the balanced ply. The results of the model are consistent with simple experiments on long braided rope.The writhed shape of the closed loop with twist inserted between the ends prior to closure is modelled as a balanced ply joined to two end loops. The analysis combines the mechanics solution with the conservation of topological link to provide a simple formula which quantitatively predicts the approximate shape and helix angle of the supercoil. The results are in good agreement with simple experiments on rope and with available data on DNA supercoils.

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A dynamical theory of twist insertion in ring-spun yarns that brings together the previous work of the authors and their colleagues on the stability of the ring-spinning balloon, and the recent work on yarn twisting dynamics by Miao & Chen (1993), in a single comprehensive theory is derived.It is shown that, to first order in small terms, the equations governing the yarn path and the equations governing the movement of twist along the yarn path are independent of each other. The general solution of the time-dependent twist flow equations is derived. These results are then used to determine formulae for the twist variation in the ring-spinning balloon and in a false twisting system. Some restrictions on the applicability of the theory are also noted.

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