Helical configurations of elastic rods in the presence of a long-range interaction potential

Recently, the integrability of the stationary Kirchhoff equations describing an elastic rod folded in the shape of a circular helix was proven. In this paper we explicitly work out the solutions to the stationary Kirchhoff equations in the presence of a long-range potential which describes the average constant force due to a Morse-type interaction acting among the points of the rod. The average constant force results to be parallel to the normal vector to the central line of the folded rod; this condition remarkably permits to preserve the integrability (indeed the solvability) of the corresponding Kirchhoff equations if the elastic rod features constant or periodic stiffnesses and vanishing intrinsic twist. Furthermore, we discuss the elastic energy density with respect to the radius and pitch of the helix, showing the existence of stationary points, namely stable and unstable configurations, for plausible choices of the featured parameters corresponding to a real bio-polymer.