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# Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

Hoffman, Kathleen A. and Manning, Robert S. and Paffenroth, Randy C. (2002) Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut. SIAM Journal on Applied Dynamical Systems, 1 (1). pp. 115-145. ISSN 1536-0040. http://resolver.caltech.edu/CaltechAUTHORS:HOFsiamjads02

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## Abstract

We consider the problem of minimizing the energy of an inextensible elastic strut with length 1 subject to an imposed twist angle and force. In a standard calculus of variations approach, one first locates equilibria by solving the Euler--Lagrange ODE with boundary conditions at arclength values 0 and 1. Then one classifies each equilibrium by counting conjugate points, with local minima corresponding to equilibria with no conjugate points. These conjugate points are arclength values $\sigma \le 1$ at which a second ODE (the Jacobi equation) has a solution vanishing at $0$ and $\sigma$. Finding conjugate points normally involves the numerical solution of a set of initial value problems for the Jacobi equation. For problems involving a parameter $\lambda$, such as the force or twist angle in the elastic strut, this computation must be repeated for every value of $\lambda$ of interest. Here we present an alternative approach that takes advantage of the presence of a parameter $\lambda$. Rather than search for conjugate points $\sigma \le 1$ at a fixed value of $\lambda$, we search for a set of special parameter values $\lambda_m$ (with corresponding Jacobi solution $\bfzeta^m$) for which $\sigma=1$ is a conjugate point. We show that, under appropriate assumptions, the index of an equilibrium at any $\lambda$ equals the number of these $\bfzeta^m$ for which $\langle \bfzeta^m, \Op \bfzeta^m \rangle < 0$, where $\Op$ is the Jacobi differential operator at $\lambda$. This computation is particularly simple when $\lambda$ appears linearly in $\Op$. We apply this approach to the elastic strut, in which the force appears linearly in $\Op$, and, as a result, we locate the conjugate points for any twisted unbuckled rod configuration without resorting to numerical solution of differential equations. In addition, we numerically compute two-dimensional sheets of buckled equilibria (as the two parameters of force and twist are varied) via a coordinated family of one-dimensional parameter continuation computations. Conjugate points for these buckled equilibria are determined by numerical solution of the Jacobi ODE.

Item Type:Article
Related URLs:
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http://dx.doi.org/10.1137/S1111111101396622DOIUNSPECIFIED
Additional Information:© 2002 Society for Industrial and Applied Mathematics. Received by the editors October 17, 2001; accepted for publication (in revised form) by P. Holmes May 6, 2002; published electronically July 23, 2002. We thank John Maddocks for helpful discussions on some of these questions. Physical configurations and bifurcation surfaces were produced in POV-Ray. The research of this author [R.S.M.] was supported by NSF grant DMS-9973258. The research of this author [R.C.P.] was supported by NSF grant KDI/NCC Molecular information and computer modeling in electrophysiology, SBR-9873173.
Funders:
Funding AgencyGrant Number
National Science FoundationDMS-9973258
National Science FoundationSBR-9873173
Subject Keywords:elastic rods, anisotropy, stability index, conjugate points, buckling, parameter continuation, isoperimetric constraints