Kirchhof弦理论及应用.pdf

Kirchhoff's Theoryof Rods ELLIS HAROLD DILL Communicatedby C. TRUESDELL The purpose of this paper is to examine the classical theory of finite displace- ments 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 three- dimensional 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-LOVEwithin 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-LOVEhypotheses. One even sees statements in the current literature about the mutual contradictions within the KIRCHHOFF-LOVEhypotheses. In