Marzio Lembo

On the dynamics of rods in the theory of Kirchhoff and Clebsch

We discuss here the dynamical equations of a theory of elastic rods that is due to Kmcm-ioFF [1, 2] and CL~Bsc~ [3, 4]. This properly invariant theory is applicable to motions in which the strains relative to an undistorted configuration remain small, although rotations may be large. It is constructed to be a first-order theory, i.e., a theory that is complete to within an error of order two in an appropriate dimensionless measure of thickness, curvature, twist, and extension. In a first-order theory of thin rods, one can treat the rod as inextensible, and we do so here at the outset. Thus, at each time t, the arc-length parameter s for the axial curve ~(t) is employed as a material coordinate, i.e., a parameter whose value at a material point is constant in time, and not only the resultant of the shearing forces on a cross section, but also the tension in the rod, are reactive quantities not given by constitutive equations. Consider for a moment a rod that is naturally prismatic and dynamically symmetric, i.e., a rod that in an undistorted stress-free configuration is a cylinder whose directrix, although not necessarily a circle, bounds a figure with equal principal moments of inertia. A motion of the rod is said to be planar and twist-flee, i.e,, a motion of pure flexure, if it is such that ~(t) lies at all times in a fixed plane ~ which contains a principal axis of inertia of each cross section. For such a motion we employ a fixed Cartesian coordinate system on ~, and, for consistency with a discussion of more general motions to be given later in this paper, we call the abscissa z and the ordinate x. We may write FZ(s, t), FX(s, t) for the z- and x-components of the resultant force F at time t on the cross section with arc-length coordinate s. The motion of the rod may be described by giving the (z, x)-coordinates of the points on ~(t) as functions of s and t. With O(s, t) the counterclockwise angle from the z-axis to the tangent of ~(t) at s, we have

A class of exact dynamical solutions in the elastic rod model of DNA with implications for the theory of fluctuations in the torsional motion of plasmids

New explicit solutions are obtained for the nonlinear equations of Kirchhoff’s theory of the dynamics of inextensible elastic rods without neglect of rotatory inertia. These exact solutions describe a class of motions possible in closed circular rings possessing a uniform distribution of intrinsic curvature ku and intrinsic torsion. When ku≠0, the motions in this class are such that the axial curve of the ring remains stationary while the cross sections rotate about their centers in such a way that the angle ψ of rotation is independent of axial location and is governed by the nonlinear pendulum equation. When ku=0, such uniform rotation of cross sections can occur at an arbitrary steady rate. The methods of classical equilibrium statistical mechanics yield the following conclusions for canonical ensembles of rings for which the motion is this type of pure homogeneous torsion. When 1/ku=11.85 nm (i.e., when the intrinsic curvature ku is among the highest observed in naturally occurring, approximately unif...

On the stability of elastic annular rods

Abstract The stability of equilibrium of non-linearly elastic rods, whose deformations obey the classical Kirchhoff’s equations, is considered. A variational formulation of the equilibrium problem is given, and the equilibrium equations for infinitesimal deformations superimposed to a finite transformation of a rod are deduced. The stability of annular rings, in which the twisting strain is non-null, is investigated by study of the second variation of the energy functional.

On the free shapes of elastic rods

The problem of free shape consists in finding the form that an elastic body must have in a natural state in order that it shall assume a given form in an equilibrium configuration under the action of assigned loads. The problem, that is of interest in itself, arises in some practical applications and can constitute a preliminary step in the study of some mechanical properties of classes of equilibrium configurations that are not natural states. This paper examines the problem of free shape for inextensible elastic rods which in equilibrium are subject only to the action of forces and couples applied to the ends, and whose deformations can be described by the theory of finite displacements of thin rods due to Kirchhoff. After the general equations governing the problem have been deduced, they are employed to give a classification of the free shapes of rods that in equilibrium are circular rings.

In-plane Strain and Stress Fields in Theories of Shearable Laminated Plates Subject to Transverse Loads

The predictions of a new theory of orthotropic laminated plates are compared with those of two other theories equally based on a Reissner-Mindlin Ansatz for the displacement field, either layer by layer [5] or for the whole plate [4]. A well-known merit of such an Ansatz is to allow and account for transverse shearings. What we are after here is to determine how well in-plane strain and stress fields are described. For definiteness, we consider circular plates that are axi-symmetrically loaded, whose layers are made of transversely isotropic materials and are symmetrically located with respect to the midplane of the plate. The new theory allows for an explicit analytic solution of this problem, as the simpler of the two theories considered for comparison does, but shows an accuracy closer to the other more complex theory, whose governing equations we solve numerically; as a benchmark, we use a numerical solution of the corresponding three-dimensional equilibrium problem; the results of our comparison are summarized graphically in the final section. While the new theory allows for an explicit analytic solution of this simple problem, the governing equations of the other two theories are solved numerically; as a benchmark, we use a numerical solution of the corresponding three-dimensional equilibrium problem; the results of our comparison are summarized graphically in the final section.

On the Linearization of Stress in Constrained Elasticity

The linearization of stress in an elastic body subject to internal constraints is discussed. It is shown that, as in the nonlinear case, the stress resulting from the linearization process is composed of a reactive part that does no work in each admissible motion, and an active part that lies in the complement of the reaction space.

On the dynamics of multilayered plates

Abstract The dynamics of multilayered plates is considered from a new point of view that regards the well-known fundamental hypotheses of the classical plate theory as constitutive assumptions reflecting the presence of internal constraints in the material of which the body is formed. The motion of laminated plates, composed of an arbitrary number of layers made of general monoclinic materials, is examined. The equations of motion and the boundary conditions are obtained from the stationarity of a two-dimensional Hamiltonian functional. Expressions of the main formulae in terms of displacements are given.