The nonlinear mechanics of slender structures undergoing large deformations

The deformation of slender structures (i.e., structures that are much longer in one direction than in the other two directions) may be described by elastic rod theory if one is interested in phenomena on length scales much larger than the lateral dimensions of the structure. A simplified version of this theory, beam theory, suffices if only small deformations need to be considered. Many traditional structural engineering applications are adequately modelled by a beam: bridges and buildings are simply not designed to suffer large deformations. But in other applications of structural mechanics deformations may be large, meaning that nonlinear geometric effects have to be taken into account. In these cases full (nonlinear) rod theory is the theory of choice. Besides engineering applications this approach is also increasingly being applied to molecular biology, for instance in the study of DNA packing and supercoiling. Aided by the improved experimental techniques for manipulating structures at the microand nanoscale, there is great activity in the application of mechanical models to biological filaments. Mathematically, a rod is modelled as a curve in space, i.e., a one-dimensional object, with effective mechanical properties such as bending and torsional stiffnesses [1]. Although the problems considered are often statics problems, mathematically they have a strong relation with problems in dynamics: arclength along the rod plays a role similar to that of time in a dynamical system (ordinary differential equation). In its simplest form this analogy (so called Kirchhoff’s Dynamic Analogy [19]) is seen between the deformations of an elastic strut (or Euler elastica) and the motions of a swinging pendulum. A similar, albeit less well known, analogy exists between a twisted rod and a spinning top, and precession of the top then corresponds to a helical deformation of the rod. Thus the extensive field of rigid-body dynamics can be brought to bear on the deformation of slender structures. Important tools in this research are therefore modern analytical as well as numerical techniques from nonlinear dynamics. This includes techniques for dealing with chaotic dynamics, which in rods takes the form of ‘spatial chaos’ [12]. For a sufficiently long intrinsically straight rod the energetically favourable deformation is a localised one in which the structure tends to the trivial, flat, state towards the ends of the rod. This state is described by a homoclinic solution of the dynamical system, i.e., a solution that connects a saddle point (describing the straight rod) to itself [13, 10]. Such homoclinic solutions are characteristic of many chaotic systems and have been the focus of intensive research in recent years (e.g., [3, 4]). Engineering problems where rod theory is applied include the looping of ocean cables [7] and the buckling of oilwell dring strings [26]. Drill strings nowadays can be over 5 km long; they are confined to narrow (possibly curved) boreholes, and therefore present the problem of structural deformation and buckling in the presence of a constraining surface. Transitions in the buckling shape of the drill string, for instance from a snaky to a helical pattern, are known to occur but still poorly understood. For a twisted rod in permanent contact with a cylinder critical loads have been found at which the rod collapses into a helical shape [10]. The same model for a rod on a cylinder describes the supercoiling of a DNA molecule into a so-called plectoneme, a plied structure in which two strands wind around each other. The problem of this ply is one of a rod winding on the outside (rather than the inside) of a cylinder, namely the cylinder, of radius equal to the radius of the rod, on which the strand centrelines wind [6, 24]. There is also a relation with chromatin fibres in which DNA coils around protein (histone) cores in the first stage of the DNA compaction process [21].