In this paper we study the competition between helical and localizing modes in the torsional buckling of stretched and twisted elastic rods. Within the Love-Kirchhoff formulation, we make comparative studies of the helical deformation of Love (1927) and the homoclinic localizing solution of Coyne (1990). Plots of the loads against their corresponding deflections allow the energetically preferred mode to be identified: it is found that preference switches from the helix to the localized mode early in the post-buckling range. These plots also allow us to predict the jumps that are observed under a variety of dead and rigid loading processes: these dynamic jumps take the rod from the spatially localized form to the familiar writhing state. Preliminary experiments confirm this preference for the localizing mode. They also reveal a second type of helical deformation, at a shorter wavelength, that is not predicted by the above long-rod analyses. A programme of further experimental and theoretical studies is suggested, and in a companion paper we lay the mathematical foundations for a numerical investigation of the complex and spatially chaotic deformations of a wider class of elastic rods.

From Helix to Localized Writhing in the Torsional Post-Buckling of Elastic Rods

An introduction to the analysis of symmetric structures

The use of group theory in simplifying the study of problems involving symmetry is a well-established approach in various branches of physics and chemistry, and major applications in these areas date back more than 70 years. Within the engineering disciplines, the search for more systematic and more efficient strategies for exploiting symmetry in the computational problems of solid and structural mechanics has led to the development of group-theoretic methods over the past 40 years. This paper reviews the advances made in the application of group theory in areas such as bifurcation analysis, vibration analysis and finite element analysis, and summarizes the various implementation procedures currently available. Illustrative examples of typical solution procedures are drawn from recent work of the author. It is shown how the group-theoretic approach, through the characteristic vector-space decomposition, enables considerable simplifications and reductions in computational effort to be achieved. In many cases, group-theoretic considerations also allow valuable insights on the behaviour or properties of a system to be gained, before any actual calculations are carried out. Copyright © 2009 John Wiley & Sons, Ltd.

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Group-theoretic exploitations of symmetry in computational solid and structural mechanics

A group-theoretic method to trace bifurcation behavior of truss structures with regular-polygonal symmetry is proposed. By means of a suitable ‘local’ coordinate transformation, the present method decomposes the tangent-stiffness matrix into a block-diagonal form with possible occurrences of twice-repeated matrices. An efficient way to compute the block-diagonal form is described in view of the sparseness of the local transformation matrix. This achieves computational efficiency and moreover leads to systematic categorization of critical points. The present method is applied to a series of regular-polygonal truss domes to demonstrate its usefulness.

Bifurcation analysis of symmetric structures using block-diagonalization

http://www.plan.civil.tohoku.ac.jp/~akamatsu/Publications/PDF/2012-JEDC-36-p0754.pdf

Ikeda Kiyohiro

Papers

Bifurcation hierarchy of symmetric structures

Critical initial imperfection of structures

Multiple equilibria for unlinked and weakly-linked cellular structural forms

Random initial imperfections of structures