Helices

Abstract

Helices are among the simplest shapes that are observed in the filamentary and molecular structures of nature. The local mechanical properties of such structures are often modeled by a uniform elastic potential energy dependent on bending and twist, which is what we term a rod model. Our first result is to complete the semi-inverse classification, initiated by Kirchhoff, of all infinite, helical equilibria of inextensible, unshearable uniform rods with elastic energies that are a general quadratic function of the flexures and twist. Specifically, we demonstrate that all uniform helical equilibria can be found by means of an explicit planar construction in terms of the intersections of certain circles and hyperbolas. Second, we demonstrate that the same helical centerlines persist as equilibria in the presence of realistic distributed forces modeling nonlocal interactions as those that arise, for example, for charged linear molecules and for filaments of finite thickness exhibiting self-contact. Third, in the absence of any external loading, we demonstrate how to construct explicitly two helical equilibria, precisely one of each handedness, that are the only local energy minimizers subject to a nonconvex constraint of self-avoidance.

Fig. 1.

The hyperboloid in curvature space (u₁, u₂, u₃) on which all helical solutions for a quadratic energy W are located (case shown: K₁ = 1, K₂ = 3/2, K₂₃ = K₁₃ = 1/2, û₁ = 1, û₂ = û₃ = 0). We show various helical solutions for different points on the hyperboloid. (Inset) A planar horizontal section of the hyperboloid (case shown: u₃ = 3/2).

Fig. 2.

With body forces created by pairwise central nonlocal interactions, the contributions to the total body force on the cross section at s by points located at s± δ is along the principal normal ν(s).

Fig. 3.

The geometry of helical tubes with circular cross sections. (A) In the curvature–torsion plane, helices are in contact at the boundary between the clear and opaque region. (B) The helix H₀ corresponds to a helical tube with self-penetration while H₁ and H₂ are in self-contact. The configuration H₀ is the stress-free configuration for the elastic energy W₁ = u₁² + (u₂ − 9/10)² + 3/4(u₃ − 2/10)² when self-avoidance is not taken into account. Helical configurations H₁ and H₂ are two minimum energy configurations of opposite-handedness respecting self-avoidance. For reasons of geometric clarity, the point H₀ has been chosen comparatively close to the boundary κ = 1 where for some materials the validity of a rod model might already be considered questionable, but the same geometric phenomena persist for choices of H₀ arbitrarily close to the origin.

Fig. 4.

The contact surface in u-space is obtained by rotating the curve in Fig. 3A around the vertical axis. Given an unstressed configuration in the forbidden region (due to self-penetration) represented by a point H₀, there exist two helices with self-contact and no external loading obtained by finding the tangency points of the contact surface with a level set of the energy W₁ = C in the upper (right-handed) and lower (left-handed), half space. The two tangency points correspond to the solutions H_1,2 of Fig. 3.