Abstract
In this paper the rigorous formulation of the motion of elastic rods which take an arbitrary shape in space has been obtained from the three-dimensional theory of elasticity. All the expressions are referred to a certain natural state. The derived general theory can deal with the large displacements and large rotaions and further the deformation of the cross section of rods, and it is applicable to the rod for the geometrical nonlinear and physical linear. The general theory is the approximation of a rigorous derivation of the curved and twisted elastic rods in reference (25). From this point of view considering the main displacement, it is assumed that the displacement consists of the mean displacement of section and the warping. The warping is expressed by the power expansion. A fully consistent set of foundamental rod equations has been derived systematically through the modified Hellinger-Reissner's variational principle. Using the assumptions of thinness, the simplified equation is considered for practical uses. Also, the linear theory of rod is represented the two kinds of expression corresponding to the nonlinear theories and is discussed the correspondence of classical theories. As the resultants, the assumption of the rigid displacement of the cross section in classical theories can be introduced independent of the thinness of rod and of the curvature and torsion of axial curve. Also it is obtained that the classical theory of rod rest on the basis of the Bernoulli-Euler hypothesis can be not obtained the exact expression unless the assumption of thinness is used in addition to the Bernoulli-Euler hypothesis. In this paper, the straight line and plane curve of axial curves are contained.
