On the free shapes of elastic rods

The problem of free shape consists in finding the form that an elastic body must have in a natural state in order that it shall assume a given form in an equilibrium configuration under the action of assigned loads. The problem, that is of interest in itself, arises in some practical applications and can constitute a preliminary step in the study of some mechanical properties of classes of equilibrium configurations that are not natural states. This paper examines the problem of free shape for inextensible elastic rods which in equilibrium are subject only to the action of forces and couples applied to the ends, and whose deformations can be described by the theory of finite displacements of thin rods due to Kirchhoff. After the general equations governing the problem have been deduced, they are employed to give a classification of the free shapes of rods that in equilibrium are circular rings.