International Journal of Non-Linear Mechanics 40 (2005) 833 – 843 www.elsevier.com/locate/nlm Twist buckling and the foldable cylinder: an exercise in origami Giles W. Hunt a , Ichiro Ario b, ∗ a Centre of Nonlinear Mechanics, University of Bath, Bath BA2 7AY, UK b Department of Civil Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-hiroshima 739-8527, Japan Received 28 May 2004; accepted 20 August 2004 Abstract Basic mechanisms for the buckling of a thin cylindrical shell under torsional loading are reviewed from a post-buckling perspective. Deflections are considered so far into the large-deflection range that the shell is allowed to fold to a flat two- dimensional form, in a mechanism reminiscent of a deployable structure. Critical and initial post-buckling effects are explored through concepts of energy minimization and hidden symmetries. For comparisons with the final large-deflection folded shape, a truss element program is employed. It is shown that, as buckling develops, the mode shape must change to accommodate both the symmetry-breaking aspects of the predominately inwards deflection, and the rotation of peak and valley lines of the buckle pattern necessary to accommodate the geometry of the final folded shape.  2004 Elsevier Ltd. All rights reserved. Keywords: Deployable structures; Hidden symmetry; Shell theory; Post-buckling; Yoshimura pattern 1. Introduction As with buckling under axial compression, the buck- ling of the thin circular cylindrical shell under torsion is recognized as one of the fundamental problems of elastic stability [1–3]. But the two contrasting load- ing situations show quite different characteristics of load/deflection response as deformation continues into the large-deflection range. While the former has ini- tially a highly unstable response, followed by resta- bilization [4] as it settles into a localized form of the ∗ Corresponding author. E-mail address: mario@hiroshima-u.ac.jp (I. Ario). 0020-7462/$ - see front matter  2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2004.08.011 well-known Yoshimura or diamond pattern [5,6] the latter forms similar but oblique shapes, which are ca- pable of folding entirely flat in the axial direction with- out significant in-plane stretching. Fig. 1 shows the de- velopment of these folds in a paper specimen twisted between two inner plastic mandrels [7]. Following an initial buckling stage that involves both bending and membrane (in-plane) stretching, a pattern more like a mechanism from origami develops, after which the system can be folded completely flat as shown. Such folding is denied to the axially loaded Yoshimura pat- tern, where significant stretching occurs at the resta- bilization stage [8]. Folding without stretching is, of course, of considerable interest to the field of deploy- able structures [9–11].