On the dynamics of rods in the theory of Kirchhoff and Clebsch

Abstract

We discuss here the dynamical equations of a theory of elastic rods that is due to Kmcm-ioFF [1, 2] and CL~Bsc~ [3, 4]. This properly invariant theory is applicable to motions in which the strains relative to an undistorted configuration remain small, although rotations may be large. It is constructed to be a first-order theory, i.e., a theory that is complete to within an error of order two in an appropriate dimensionless measure of thickness, curvature, twist, and extension. In a first-order theory of thin rods, one can treat the rod as inextensible, and we do so here at the outset. Thus, at each time t, the arc-length parameter s for the axial curve ~(t) is employed as a material coordinate, i.e., a parameter whose value at a material point is constant in time, and not only the resultant of the shearing forces on a cross section, but also the tension in the rod, are reactive quantities not given by constitutive equations. Consider for a moment a rod that is naturally prismatic and dynamically symmetric, i.e., a rod that in an undistorted stress-free configuration is a cylinder whose directrix, although not necessarily a circle, bounds a figure with equal principal moments of inertia. A motion of the rod is said to be planar and twist-flee, i.e,, a motion of pure flexure, if it is such that ~(t) lies at all times in a fixed plane ~ which contains a principal axis of inertia of each cross section. For such a motion we employ a fixed Cartesian coordinate system on ~, and, for consistency with a discussion of more general motions to be given later in this paper, we call the abscissa z and the ordinate x. We may write FZ(s, t), FX(s, t) for the z- and x-components of the resultant force F at time t on the cross section with arc-length coordinate s. The motion of the rod may be described by giving the (z, x)-coordinates of the points on ~(t) as functions of s and t. With O(s, t) the counterclockwise angle from the z-axis to the tangent of ~(t) at s, we have