Stability and bifurcation of compressed elastic cylindrical tubes A. Dorfmann a , D.M. Haughton b, * a Department of Civil and Environmental Engineering, Tufts University, Medford, MA 02155, USA b Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK Received 1 March 2006; received in revised form 22 June 2006; accepted 22 June 2006 Available online 12 October 2006 Abstract The axial compression of cylindrical tubes is considered from the point of view of both bifurcation and stability. The paper considers mainly compressible hyperelastic materials and looks at all modes of bifurcation. Some general results are obtained and other aspects are illustrated with numerical results based on a compressible neo-Hookean material. In particular it is shown that in certain circumstances higher order modes can be the dominant mode. It is also shown that there are two distinct forms of the barrelling mode for shorter cylinders. This is shown by calculating the eigenfunction associated with the two distinct bifurcation points. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Stability; Bifurcation; Nonlinear elasticity; 73C; 73G; 73H 1. Introduction The bifurcation behaviour of elastic cylinders under axial compression has been studied by many authors and the response is well known. Roughly speaking, long slender cylinders will undergo Euler buckling at some critical loading while short cylinders will undergo a barrelling mode for sufficiently large loading. In most cases it is solid cylinders, often composed of incompressible neo-Hookean material, that have been investigated. See for example Wilkes [1], (who also considers cylindrical tubes), Simpson and Spector [2,3], Davies [4,5], and references therein. Recently Healey and Montes-Pizarro [6] have looked at some compressible materials using a novel approach to the problem. The corresponding problem for cylindrical tubes has received less attention. Wilkes [1] considered the neo-Hookean material while Pan and Beatty [7] extended the theory to look at a selection of incompressible materials. The present work has been motivated by a description in the review article by Beatty [8] of some experimen- tal work of Willis [9]. Here, very short thick-walled tubes are compressed in the axial direction. The typical dimensions of the tubes considered in [9] are an outer diameter of 3 in., an inner diameter of 2 in. and a height of 3 in. An axial load is applied to reduce the initial height to a final value of (down to) 1.5 in., a compression 0020-7225/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2006.06.014 * Corresponding author. Tel.: +44 141 330 4748; fax: +44 41 330 4111. E-mail addresses: Luis.Dorfmann@tufts.edu (A. Dorfmann), dmh@maths.gla.ac.uk (D.M. Haughton). International Journal of Engineering Science 44 (2006) 1353–1365 www.elsevier.com/locate/ijengsci