Kirchhoff’s Rod Theory

Abstract

The theory of an elastic rod whose centerline is inextensible and whose cross sections remain plane and normal to the centerline is discussed. This theory, which is known as Kirchhoff rod theory, is presented in the modern context of a Cosserat rod theory. The governing equations for this widely used theory result in a set of equations to determine a rotation tensor P and a position vector r. This theory has a celebrated history in part because of Kirchhoff’s discovery that the equations governing static deformations of the rod are analogous to those for the rotational motion of a rigid body. A range of applications of the theory is also presented in this chapter. These examples include a terminally loaded rod which is bent and twisted and an initially curved rod which is straightened.