On the equations governing nonlinear symmetric Kirchhoff’s elastic rods

Abstract

A systematic study of the equations governing the nonlinear symmetric Kirchhoff elastic rods is proposed. Symmetric rods are characterized by the conservation of the contact torque along the tangent of the centre line of the rod. This additional conservation law enables the formal reduction to quadratures of the governing equations and a systematic study of general (or Lancret’s) helical solutions. This improves previous analyses where only solutions relative to the case when the centre line is a circular helix have been investigated. Moroever, we consider the general helical solution for inhomogeneous symmetric rods and, conclusively, we study a special solution where the contact torque along the tangent of the centre line is zero. This solution is valid also for asymmetric Kirchhoff rods.