Geometry and equilibria of parallel bundles of elastic rods

Geometrical conditions of existence of curved bundles of hexagonally packed rods are presented.Closure of the bundle results in a group of automorphisms of the cross-sectional lattice. The elastic energy accountsfor bending and torsion of the rods. Equilibria of the bundle correspond to solutions of a variational problemformulated for the axis of the bundle. The Euler-Lagrange equations are obtained in terms of curvature andtorsion and the problem is reduced to the quadrature. The Ilyukhin-type equations describe the shape of thebundle in special cylindrical coordinates. The results are of universal nature and are applicable to various fibrousstructures, including DNA and nanotubes

A formula for the minimal coordination number of a parallel bundle

An exact formula for the minimal coordination numbers of the parallel packed bundle of rods is presented based on an optimal thickening scenario. Hexagonal and square lattices are considered.Comment: 12 pages, 4 figures, to appear in J. Chem. Phy

Equilibria of elastic cable knots and links

We present a theory for equilibria of geometrically exact braids made of two thin, uniform, homogeneous, isotropic, initially-straight, inextensible and unshear- able elastic rods of circular cross-section. We formulate a second-order variational problem for an action functional whose Euler–Lagrange equations, partly in Euler– Poincaré form, yield a compact system of ODEs for which we define boundary-value problems for braids closed into knots or links. The purpose of the chapter is to present a pathway of deformations leading to braids with a knotted axis, thereby offering a way to systematically compute elastic cable knots and links. A representative bifurca- tion diagram and selected numerical solutions illustrate our approach

Forceless folding of thin annular strips

Thin strips or sheets with in-plane curvature have a natural tendency to adopt highly symmetric shapes when forced into closed structures and to spontaneously fold into compact multi-covered configurations under feed-in of more length or change of intrinsic curvature. This disposition is exploited in nature as well as in the design of everyday items such as foldable containers. We formulate boundary-value problems (for an ODE) for symmetric equilibrium solutions of unstretchable circular annular strips and present sequences of numerical solutions that mimic different folding modes. Because of the high-order symmetry, closed solutions cannot have an internal force, i.e., the strips are forceless. We consider both wide and narrow (strictly zero-width) strips. Narrow strips cannot have inflections, but wide strips can be either inflectional or non-inflectional. Inflectional solutions are found to feature stress localisations, with divergent strain energy density, on the edge of the strip at inflections of the surface. ‘Regular’ folding gives these singularities on the inside of the annulus, while ‘inverted’ folding gives them predominantly on the outside of the annulus. No new inflections are created in the folding process as more length is inserted. We end with a discussion of an intriguing apparent connection with a deep result on the topology of curves on surfaces

Biased statistical ensembles for developable ribbons

Letter to the edito

Writhe formulas and antipodal points in plectonemic DNA configurations

The linking and writhing numbers are key quantities when characterizing the structure of a piece of supercoiled DNA. Defined as double integrals over the shape of the double-helix, these numbers are not always straightforward to compute, though a simplified formula exists. We examine the range of applicability of this widely-used simplified formula, and show that it cannot be employed for plectonemic DNA. We show that inapplicability is due to a hypothesis of Fuller theorem that is not met. The hypothesis seems to have been overlooked in many works.Comment: 20 pages, 7 figures, 47 reference