New Amplitude Equations for Thin Elastic Rods

Publisher Website
Google Scholar
Abstract
Filaments are fundamental physical structures that can be found in many guises on many different scales. They appear in various problems in biology, chemistry, physics, and engineering (1). One way to model these structures is to assume that they are made of an elastic material obeying the appropriate laws of elasticity. The Kirchhoff model for rods describes the dynamics of (thin) elastic filaments within the approximation of linear elasticity the- ory. The stability of stationary filaments under external (or internal) constraints is one of the oldest fundamental problems in the theory of elasticity, dating back to Eu- ler. The general problem is to understand and describe the postbifurcation behavior of stationary structures. To answer this question, many authors have considered the linear (and nonlinear) analysis of phenomenological equa- tions such as the (one-dimensional) "beam equation" (2). However, these simplified equations lack crucial informa- tion on the three-dimensional structure of the solution af- ter bifurcation. In addition, these analyses have mainly been confined to stationary perturbations. Here, we study the dynamical (i.e., time-dependent) stability of the full three-dimensional Kirchhoff equations for the twisted straight rod. We first develop a new perturbation scheme to study the stability of stationary solutions. These perturbation expansions are performed at the level of a local basis (the director basis) attached to the central axis of the curve. We use these expansions to perform both linear and nonlinear analyses with the latter leading to a new amplitude equation describing how the rod deformation amplitude is coupled to the twist density for the solutions after, but close to, bifurcation. We first consider a simple space curve, x, parametrized by arc length, s, whose position may vary in time, i.e., x › xss, td. We assume that x is at least twice differentiable. In what follows, sd 0 denotes differentiation with respect to s and Ÿ sd differentiation with respect to time. At each point of the curve, one can define a local orthonormal basis di › diss, td, i › 1, 2, 3 ,b y introducing the tangent vector d3 › x 0 ss, td and choosing two unit vectors d1, d2 in the plane normal to d3 such that sd1, d2, d3d forms a right-handed orthonormal basis for each value of ss, td. By construction there exist a twist vector k › k1d1 1k 2 d 2 1k 3 d 3 and a spin vector v › v1d1 1v 2 d 2 1v 3 d 3 which control the space and time evolution of the basis along the curve via the spin and twist equations, d 0 i › k3 d i , Ÿ d i › v3 d i , i ›1, 2, 3 . (1) Knowledge of k › kss, td and v › vss, td is enough to construct the position and motion of the curve in space (up to a rigid translation), since the solution of the spin and twist equations determines d3 › d3ss, td which can be integrated once to give x. If one chooses d1 to be the normal vector to the curve, then d2 is the binormal and the local basis reduces to the well-known Frenet frame (3). The Kirchhoff model of rod dynamics considers rods whose length is much greater than the cross sectional radius. Moreover, here it is assumed that the rod is in- extensible and of circular cross section. These additional assumptions are not essential to the procedure de- scribed below and could be relaxed if required. A one-dimensional theory can be derived in which all the relevant physical quantities are averaged over the cross sections and attached to the central axis. It follows that the total force F › Fss, td and the total moment M › Mss, td can be expressed locally in terms of the

This publication has 11 references indexed in Scilit: