Helical Birods: An Elastic Model of Helically Wound Double-Stranded Rods

Abstract

We consider a geometrically accurate model for a helically wound rope constructed from two intertwined elastic rods. The line of contact has an arbitrary smooth shape which is obtained under the action of an arbitrary set of applied forces and moments. We discuss the general form the theory should take along with an insight into the necessary geometric or constitutive laws which must be detailed in order for the system to be complete. This includes a number of contact laws for the interaction of the two rods, in order to fit various relevant physical scenarios. This discussion also extends to the boundary and how this composite system can be acted upon by a single moment and force pair. A second strand of inquiry concerns the linear response of an initially helical rope to an arbitrary set of forces and moments. In particular we show that if the rope has the dimensions assumed of a rod in the Kirchhoff rod theory then it can be accurately treated as an isotropic inextensible elastic rod. An important consideration in this demonstration is the possible effect of varying the geometric boundary constraints; it is shown the effect of this choice becomes negligible in this limit in which the rope has dimensions similar to those of a Kirchhoff rod. Finally we derive the bending and twisting coefficients of this effective rod.