A Geometrically-exact rod model incorporating shear and torsion-warping deformation

A theory of rods 2 or, equivalently, a one-dimensional theory of solids is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a certain curve and the time as the only independent variables. The formidable mathematical obstacles presented by three-dimensional continuum theories dictate the need for tractable, accurate, and illuminating one-dimensional models.

The Theory of Rods

ii(t)+Au(t)+p(t) B(u(t)) =O. Here A is a densely defined positive selfadjoint linear operator on a real Hilbert space H with inner product ( , ), B is a locally Lipschitz map from D(A”2) (endowed with the graph norm) into H, and p(t) is a real-valued control. The finite-dimensional stabilization problem H = IW” has been considered in the recent papers of Jurdjevic and Quinn [20] and Slemrod [24]. In the case when (1.1) is bilinear, i.e., B linear, these papers give simple criteria for feedback stabilization. Specifically, it is a consequence of both [20] and [24] that if the only solution of the uncontrolled system

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Nonharmonic fourier series and the stabilization of distributed semi‐linear control systems

The objective of this paper is to present an account of recent developments in the direct formation of theories of rods and shells based on 1- and 2-dimensional continuum models originating in the works of Duhem and E. and F. Cosserat. Following some preliminaries and description of (3-dimensional) shell-like and rod-like bodies, the rest of the paper is arranged in two parts namely Part A (for shells) and Part B (for rods) and can be read independently of each other. In each part, after providing the main ingredients of the direct model and a statement of the conservation laws, a rapid outline is given of the derivation of the basic equations and nonlinear constitutive equations for elastic materials. Each part also includes a discussion of constrained theories and an account of recent developments on the subject.

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R.J. Knops

Papers

Large deformations, superposed small deformations and stability of elastic rods