General Theories of Rods and Shells

Abstract

In Chap. N we defined a theory of rods to be the characterization of the motion of slender solid bodies by a finite number of equations in which there is but one independent spatial variable, which we denote by s. There are several kinds of rod theories, reflecting different ways to construct them. Perhaps the most elegant are the intrinsic(ally one-dimensional) theories (Cosserat Theories), the simplest example of which is that presented in Chap. N. In intrinsic theories, the configuration of a rod is defined as a geometric entity, equations of motion are laid down, and constitutive equations relating mechanical variables to geometrical variables are prescribed. But, as we saw in Chap. VIII, there are parts of the theory that are best developed under the inspiration of the three-dimensional theory. In the special Cosserat theory of Chap. VIII, the classical equations of motion, namely, the balances of linear and angular momentum, suffice to produce a complete theory. They are inadequate for more refined intrinsic theories. In our treatment of refined intrinsic theories in Sec. 7, we discuss the construction of the requisite additional equations of motion.