s. neukirch, g.h.m. van der heijden and j.m.t. thompson- writhing instabilities of twisted rods:...

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Journal of the Mechanics and Physics of Solids 50 (2002) 1175–1191 www.elsevier.com/locate/jmps Writhing instabilities of twisted rods: from inÿnite to ÿnite length S. Neukirch ∗ , G.H.M. van der Heijden, J.M.T. Thompson Centre for Nonlinear Dynamics and its Applications, University College London, Gower Street, London WC1E 6BT, UK Received 31 May 2001; received in revised form 10 October 2001; accepted 26 October 2001 Abstract We use three dierent approaches to describe the static spatial conÿgurations of a twisted rod as well as its stability during rigid loading experiments. The ÿrst approach considers the rod as inÿnite in length and predicts an instability causing a jump to self-contact at a certain point of the expe riment. Semi-ÿ nit e cor rect ions, take n int o account as a seco nd appr oach , reve al some possible experiments in which the conÿguration of a very long rod will be stable through out. Finally, in a third approach, we consider a rod of real ÿnite length and we show that another type of ins tabi lity may occu r, lead ing to pos sible hys ter esi s beha vior . As we go from inÿnit e to ÿnit e lengt h, we compare the di er ent informat ion given by the three approaches on the possible equilibrium conÿgurations of the rod and their stability. These ÿnite size eects studied here in a 1D elasticity problem could help us guess what are the stability features of other more complicated (2D elastic shells for example) problems for which only the inÿnite length approach is understood. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Stability and bifurcations; Buckling; Finite deections; Elastic material; Beams and columns 1. Introd uction We study equili br ium solutions of long and thin elasti c rods (or ÿl aments). We use the Cosser at rod theory [1] to de sc ri be the st at e of the rod by its ce nt er li ne together with a ÿeld of directors. At each point along the center line curve, a set of 3 orthogonal vectors, the directors, provides a way to describe local bending, twisting, stretc hing and she ari ng of the ela sti c mat eri al. Consti tut ive rel ati ons express the way these elastic deformations are related to the stress applied to the rod. ∗ Corresponding author. Tel.: +41-21-693-29-06; fax: +41-21-693-42-50. E-mail address: [email protected] (S. Neukirch). 0022-5096/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S0022-5 096(01) 00130-2

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Page 1: S. Neukirch, G.H.M. van der Heijden and J.M.T. Thompson- Writhing instabilities of twisted rods: from infinite to finite length

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