E. L. Starostin

Triangular buckling patterns of twisted inextensible strips

When twisting a strip of paper or acetate under high longitudinal tension, one observes, at some critical load, a buckling of the strip into a regular triangular pattern. Very similar triangular facets have recently been found in solutions to a new set of geometrically exact equations describing the equilibrium shape of thin inextensible elastic strips. Here, we formulate a modified boundary-value problem for these equations and construct post-buckling solutions in good agreement with the observed pattern in twisted strips. We also study the force–extension and moment–twist behaviour of these strips by varying the mode number n of triangular facets and find critical loads with jumps to higher modes.

Tension-induced multistability in inextensible helical ribbons.

We study the nonmonotonic force-extension behavior of helical ribbons using a new model for inextensible elastic strips. Unlike previous rod models, our model predicts hysteresis behavior for low-pitch ribbons of arbitrary material properties. Associated with it is a first-order transition between two different helical states as observed in experiments with cholesterol ribbons. Numerical solutions show nonuniform uncoiling with hysteresis also occurring under controlled tension. They furthermore reveal a new uncoiling scenario in which a ribbon of very low pitch shears under tension and successively releases a sequence of almost planar loops. Our results may be relevant for nanoscale devices such as force probes.

Force and moment balance equations for geometric variational problems on curves.

We consider geometric variational problems for a functional defined on a curve in a three-dimensional space. The functional is assumed to be written in a form invariant under the group of Euclidean motions. We present the Euler-Lagrange equations as equilibrium equations for the internal force and moment. Examples are discussed to illustrate our approach. This form of the equations particularly serves to promote the study of biofilaments and nanofilaments.

Chiral effects in dual-DNA braiding

The biologically important problem of DNA braiding was studied in the past by means of dual-DNA magnetic tweezer experiments. In such experiments, two DNA molecules are braided about each other using an externally imposed force and torque. Here we develop a theoretical model of molecular braiding that includes interactions between molecules, thermal fluctuations, and the elastic response of molecules, all in a consistent manner. This is useful to study the chiral effects of helix-dependent electrostatic interactions on the braids equilibrium geometrical and mechanical properties. When helix-dependent forces are weak, our model yields a reasonably accurate reproduction of previously measured extension–rotation curves, where only very slight chirality has been observed. On the other hand, when helix-specific electrostatic forces are strong, the model predicts several new features of the extension–rotation curves. These are: (a) a distinct asymmetry between left-handed and right-handed DNA braiding; (b) the emergence, under a critical pulling force, of coexistence regions of tightly and loosely wound DNA; (c) spontaneous formation of left-handed DNA braids at zero external torque (zero bead rotations). Strong chiral forces are expected for braiding experiments conducted in solutions in which there are counter-ions that bind specifically in the DNA grooves.

Cascade unlooping of a low-pitch helical spring under tension

Abstract We study the force vs. extension behaviour of a helical spring made of a thin torsionally stiff anisotropic elastic rod. Our focus is on springs of very low helical pitch. For certain parameters of the problem such a spring is found not to unwind when pulled but rather to form hockles that pop out one by one and lead to a highly non-monotonic force–extension curve. Between abrupt loop pop-outs this curve is well described by the planar elastica whose relevant solutions are classified. Our results may be relevant for tightly coiled nanosprings in future micro- and nano(electro)mechanical devices.

Characterisation of cylindrical curves

We employ moving frames along pairs of curves at constant separation to derive various conditions for a curve to belong to the surface of a circular cylinder.

Tightening elastic (n, 2)-torus knots

We present a theory for equilibria of elastic torus knots made of a single thin, uniform, homogeneous, isotropic, inextensible, unshearable rod of circular cross-section. The theory is formulated as a special case of an elastic theory of geometrically exact braids consisting of two rods winding around each other while remaining at constant distance. We introduce braid strains in terms of which we formulate a second-order variational problem for an action functional that is the sum of the rod elastic energies and constraint terms related to the inextensibility of the rods. The Euler-Lagrange equations for this problem, partly in Euler-Poincare form, yield a compact system of ODEs suitable for numerical solution. By solving an appropriate boundary- value problem for these equations we study knot equilibria as the dimensionless ropelength parameter is varied. We are particularly interested in the approach of the purely geometrical ideal (tightest) limit. For the trefoil knot the tightest shape we could get has a ropelength of 32.85560666, which is remarkably close to the best current estimate. For the pentafoil we find a symmetry-breaking bifurcation.