An overview will be given of research of the "Rutgers Group" on the dynamics of rods and the theory of DNA supercoiling. Areas of research to be mentioned in the talk are: the attainment of exact and explicit expressions for traveling waves; analogues of particle-antiparticle pair production and annihilation in the theory of solitary waves; exact representations of equilibrium configurations showing isolated points and intervals of self-contact; bifurcation diagrams for knotted and unknotted cyclized DNA and linear DNA segments subject to twist and tension; the nature of the isolas occurring in such diagrams; and the development of methods of determining the elastic stability of supercoiled configurations.

The variational structure of the elastic rod model will be exploited to predict which equilibrium configurations are stable, that is, correspond to local minima. Three techniques to determine the stability of the equilibrium configurations will be discussed and illustrated using two examples: the twisted elastic loop and an untwisted elastic loop with inherent curvature. The stability properties of equilibria depend upon the material parameters of the rod.

Exploiting the natural frame of space curves, we formulate an intrinsic dynamics of a twisted elastic filament in a viscous fluid. Coupled nonlinear equations describing the temporal evolution of the filament's complex curvature and twist density capture the dynamic interplay of twist and writhe. These equations an used to illustrate a remarkable nonlinear phenomenon: geometric untwisting of open laments, whereby twisting strains relax through a transient writhing instability without axial rotation. Experimentally observed writhing motions of fibers of the bacterium B. subtilis [N. Mendelson et al., J. Bacteriol. 177 , 7060 (1995)] may be examples of this untwisting process.

Motivated by the production process of non-wowen fabrics, we investigate the elastic buckling of rods between parallel, rigid walls. By introducing the concepts of "sheets" and using ideas from the theory of hybride dynamical systems, we draw extremely rich global bifurcation diagrams exhibiting regular and generalized bifurcation points. We derive the diagrams partially on analytical, partially on numerical basis. For the latter purpose we use a global "scanning" algorithm combined with traditional path continuation.

In this paper the large deflection theory of thin elastic rods of uniform circular cross-section is used to find the relationships between the ply-helix angle, the strand convergence angle, and the applied tensions and torques required to hold the plied strands in equilibrium. The solution of this problem is of importance to the textile yarn manufacturing industry where singles yarns, made from long staple fibers, such as wool, are plied together in order to bind the surface fibers more effectively into the plied yarn structure. This produces warp yarns that are more abrasive resistant. A formula for the relationship between the pre-twist in two initially straight strands, and the helix angle of the plied structure obtained when they are allowed to twist together into a balanced ply structure is also derived. A balanced two-ply structure is one that will maintain its configuration without the application of external tension or torque. Balanced ply structures are also important in textile manufacturing processes.

In this talk, I will consider models for long thin intrinsically curved filaments. These models are used in a variety of biological systems ranging from bacterial flagella to climbing plants. As tension is decreased or intrinsic curvature is increased, these filaments bifurcate to solutions connecting asymptotically helices with opposite handedness. The statics of these heteroclinic structures is studied within the framework of the Kirchhoff theory with linear and nonlinear constitutive relkations.A center manifold reduction and a normal form transformation for a triple zero eigenvalue reduce the dynamics to a third order reversible dynamical system. The analysis of this reduced system reveals that the heteroclinic connection representing the physical solution results from the collapse of pairs of symmetric homoclinic orbits. The dynamics of moving fronts and the possibilty of multi-heteroclinic orbits will be discussed.

In experiments on long rubber rods subject to end tension and moment, a periodic helical deformation is often observed on the fundamental path prior to the onset of localised buckling and the rod often sags under gravity. An analysis is undertaken here to account for this observed behaviour. First we derive general equilibrium equations, using the Cosserat theory, of an infintely long rod held under tension and torision, incorporating the effects of non-symmetric cross section, shear deformation, gravity, and a uniform intrinsic curvature of the unstressed rod. The perfect problem, without any of these effects, is complete integrable and corresponds (in the Kirchhoff dynamic analogy) to the motion of a symmetric top. Non-symmetric cross-section is already known to lead to spatial complexity, whereas nonlinear constitutive laws alone lead to another integrable case. This talk deals with the other two cases. First with intrinsic curvature, it is shown that the straight configuration of the rod is replaced by a one-twist-per wave equilibrium whose amplitude varies with pre-buckling load. Superimposed on this equilibrium is a localised buckling mode, which can be described as a homoclinic orbit to the new fundamental path. Numerical techniques are used to explore the multiplicity of localised buckling modes, given that non-zero initial curvature breaks the complete integrability of the differential equations, and also one of a pair of reversibilities. For the case of gravity, a finite piece of rod must be taken, which we assume to be simply-supported and hanging in a shallow catenary. A mixture of asymptotic and numerical methods are used to describe the effect of applying tangential twist Tw. A small parameter ?? measures the relative sizes of bending and gravitational forces. For small values of Tw, the rod shape consists of an outer catenary with an inner boundary layer. Large twist ?? causes buckling into a modulated helix-like spiral with period of ?? superimposed onto the catenary. As Tw is further increased, the helix tightens to take up the slack. Eventually at very high twist, the deformation may localise and the rod jump into a self-intersecting writhed shape.

[Research supported by the Royal Society]
The Cosserat director theory is used to formulate the problem of a long rod constrained to lie on a cylinder while being held by end tension and twisting moment. Applications of this problem are found in the buckling of drill strings inside a cylindrical hole, and also in optical fibres. In the case of a rod of isotropic cross-section the equilibrium equations can be reduced to those of a one-degree-of-freedom equivalent oscillator whose fixed points correspond to helical solutions of the rod. More complicated shapes are also possible, and special attention is given to localised configurations described by homoclinic orbits of the oscillator. Heteroclinic saddle connections are found to play an important role by defining critical loads at which a straight rod may coil up into a helix. For anisotropic rods the equations are no longer integrable and we find spatial chaos with an infinity of (multi-hump) localised solutions, a sample of which is computed using a shooting method. The behaviour of these solutions is investigated as the aspect ratio of the rod's cross-section is varied. The coiling bifurcation is unfolded and the resulting bifurcation behaviour is interpreted in terms of the so-called Maxwell load of a structure.

Motivated by applications to continuum models of tertiary structures of DNA, we use averaging theory to determine effective isotropic bending laws for non-isotropic elastic rods with high intrinsic twist.

Motivated by the application of modelling DNA by an elastic rod, we use the technique of two-scale homogenization on a two-point boundary value problem (BVP) involving buckling of a rod with a high intrinsic twist. The basic question is to understand the errors generated in replacing a rod with a high, but finite, intrinsic twist, by an effective rod in the infinite intrinsic twist limit. For our buckling problem the effective problem is isotropic. As a consequence there is a continuous family of buckled planar solutions. We show that the buckled configurations of the rod with high intrinsic twist are close to certain special planar buckled configurations of the isotropic rod that are selected from among all the solutions to the effective problem. The selected planes are independent of the values of the (bending and twisting) stiffnesses and of the load.

Several different physical systems, for example super-coiled DNA molecules, have been successfully modeled by a curve or line embedded in three dimensions with an associated energy that is to be minimized subject to the constraint that the curve not pass through itself. For closed curves the knot type may therefore be specified a priori, and minima of the energy often appear to involve regions of self-contact, that is, regions in which points that are distant along the curve are close in space. While this phenomenon of self-contact is familiar through every day experience with string, rope and wire, the idea is surprisingly difficult to define in a way that is simultaneously physically reasonable, mathematically precise and analytically tractable. Here we use the notion of global curvature of a space curve in a new formulation of the self-contact constraint, and exploit our formulation to derive existence results for of a variety of elastic energies defined on curves.

Nonsmooth calculus is used to derive the Euler equations for the variational formulation of the self-contact constraint described in the previous talk (Self contact of nonlinearly elastic rods I. Existence). These necessary conditions provide information concerning regularity of the minimizers and the contact forces.