Kirchhoff's theory of rods

The purpose of this paper is to examine the classical theory of finite displacements of thin rods as developed by KIRCHHOFF [1859, 1876] and CLEBSCH [1862], and presented by LOVE [1892, 1906]. In their work, a rod is a threedimensional body with two dimensions which are very small compared to the third. Their objective was to find an approximate solution of the three-dimensional equations of nonlinear elasticity which is satisfactory for thin rods, within a certain class of boundary conditions, and which is applicable to motions such that the strains relative to the initial configuration are very small, although rotations may be large. Their work suffered from the lack of the well developed nonlinear theory of elasticity which is available today. We will re-examine in this paper the general ideas of KIRCNHOFF-CLEBSCH-LOVE within the framework of modern continuum mechanics 1. The reader who intends to follow through the derivations should begin with appendix A. Contemporary treatments of the bending and twisting of rods are frequently based on some set of hypotheses about the motion and the state of stress such as the following ones. (i) Cross-sections remain plane, undistorted, and normal to the axis of the rod. (ii) The transverse stress is zero. (iii) The bending moments and the twisting moment are proportional to the components of curvature and twist of the axis. Such assumptions are called the KmCHHOFF-LOVE hypotheses. One even sees statements in the current literature about the mutual contradictions within the KIRCHHOFF-LOVE hypotheses. In fact, neither author made such assumptions. KIRCHHOFF viewed the rod as an assembly of short segments. Each segment was regarded as loaded by the contact forces from the adjacent segments. The displacement within each segment was assumed to be small. Continuity between segments was expressed with the help of a redundant system of four space coordi-