Spatially complex localization after one-twist-per-wave equilibria in twisted circular rods with initial curvature
- 15 November 1997
- journal article
- research article
- Published by The Royal Society in Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Vol. 355 (1732), 2151-2174
- https://doi.org/10.1098/rsta.1997.0115
Abstract
In experiments on long rubber rods subject to end tension and moment, a one–twist–per–wave deformation is often observed on the fundamental path prior to the onset of localized buckling. An analysis is undertaken here to account for this observed behaviour. First we derive general equilibrium equations using the Cosserat theory, incorporating the effects of non–symmetric cross–section, shear deformation, gravity and a uniform initial curvature of the unstressed rod. Each of these effects in turn can be expressed as a perturbation of the classical completely integrable Kirchhoff–Love differential equations which are equivalent to those describing a spinning symmetric top. Non–symmetric cross–section was dealt with in earlier papers. Here, after demonstrating that shear deformation alone makes little qualitative difference, the case of initial curvature is examined in some detail. It is shown that the straight configuration of the rod is replaced by a one–twist–per–wave equilibrium whose amplitude varies with pre–buckling load. Superimposed on this equilibrium is a localized buckling mode, which can be described as a homoclinic orbit to the new fundamental path. The dependence is measured of the pre–buckled state and critical buckling load on the amount of initial curvature. Numerical techniques are used to explore the multiplicity of localized buckling modes, given that non–zero initial curvature breaks the complete integrability of the differential equations, and also one of a pair of reversibilities. Finally, the physical implications of the results are assessed and are shown to match qualitatively what is observed in an experiment.This publication has 12 references indexed in Scilit:
- Lock-on to tape-like behaviour in the torsional buckling of anisotropic rodsPhysica D: Nonlinear Phenomena, 1998
- From helix to localized writhing in the torsional post-buckling of elastic rodsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1996
- A multiplicity of localized buckling modes for twisted rod equationsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1996
- Perturbed Homoclinic Solutions in Reversible 1:1 Resonance Vector FieldsJournal of Differential Equations, 1993
- Hunting for homoclinic orbits in reversible systems: A shooting techniqueAdvances in Computational Mathematics, 1993
- NUMERICAL ANALYSIS AND CONTROL OF BIFURCATION PROBLEMS (II): BIFURCATION IN INFINITE DIMENSIONSInternational Journal of Bifurcation and Chaos, 1991
- NUMERICAL ANALYSIS AND CONTROL OF BIFURCATION PROBLEMS (I): BIFURCATION IN FINITE DIMENSIONSInternational Journal of Bifurcation and Chaos, 1991
- Spatially complex equilibria of buckled rodsArchive for Rational Mechanics and Analysis, 1988
- 5.—Qualitative Aspects of the Spatial Deformation of Non-linearly Elastic Rods.Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1975
- Ueber das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes.Journal für die reine und angewandte Mathematik (Crelles Journal), 1859