Static Kirchhoff Rods under the Action of External Forces: Integration via Runge-Kutta Method

This paper shows how to apply a simple Runge-Kutta algorithm to get solutions of Kirchhoff equations for static filaments subjected to arbitrary external and static forces. This is done by suitably integrating at once Kirchhoff and filament reference system equations under appropriate initial boundary conditions. To show the application of the method, we display several numerical solutions for filaments including cases showing the effect of gravity. 1. Introduction Filamentary bodies such as rods, ribbons, and wires are present in many scales from microscopic to astrophysical bodies. Marine cables [1], hair beams [2], rubber bands, microfibers [3], nanowires [4], and DNA molecules [5] share common dynamics, which is testified by the onset of rich elastic phenomena such as buckling, coiling, and looping [6]. The writhing instability for example [7, 8] is a looping process in which the spatial configuration of an elastic filament changes remarkably as a response to an applied external torque, the change itself as a minimization process of the filament elastic energy. The dynamics of filamentary bodies has been the subject of extensive research in continuum mechanics with the aim of establishing either exact or approximate solutions for thin filaments. The theory itself started with Kirchhoff [9, 10] in 1892 and Clebsch and Love [11] who considered small deformation of thin rods within the elastic range. The theory was developed under special assumptions that later became the approximations adopted by the so-called simple bending and torsion theory [12]. Although the deformations of infinitesimal portions of a rod in the theory may be small, the overall deformation is quite large and allows the coiling and looping phenomena above mentioned. Exact solutions of Kirchhoff equations for static rods are well known. Shi and Hearst found the most general analytical solution to the static Kirchhoff rod, which they called helix on a linear helix [13]. Nizzete and Goriely [14] used the equivalence between the static Kirchhoff equations for rods with circular cross-sections and the Euler equations for spinning tops to provide a classification of all different equilibrium solutions for a filament. Chouaieb et al. [15] completed the study of existence and stability of helical structures within the Kirchhoff rod model. Goriely and Tabor [7, 16] developed a dynamical method to study the stability of equilibrium solutions of the Kirchhoff rod model, based on a time-dependent perturbation scheme, thus providing a framework for classification of new solutions. This method