Post-buckling behavior of a nonlinearly hyperelastic thin rod with cross- section invariant under the dihedral group \(D_ n\)
type: Item
instance of: scholarly article
title: Post-buckling behavior of a nonlinearly hyperelastic thin rod with cross- section invariant under the dihedral group \(D_ n\)
zbMATH Open document ID: 0568.73048
DOI: 10.1007/BF00250729
author: E. Buzano
author: Giuseppe Geymonat
author: Timothy Poston
published in: Archive for Rational Mechanics and Analysis
publication date: 1985-01-01T00:00:00Z
review text: Making use of the usual director approach to rod theory the authors study the spatial buckled states of naturally straight, uniform, nonlinearly hyperelastic rods, whose cross-sections are regular n-gons, subject to terminal loads. In order to use the results of singularity theory a variational bifurcation problem is formulated which then is reduced by means of the so-called splitting lemma, which according to the authors has certain advantages over the classical Liapunov-Schmidt method, to bifurcation equations which are invariant under the action of \(D_ n\). The case \(D_ 3\) is treated in detail by giving all invariant perturbations. Applied to a rod clamped at both ends and a cross-section with \(D_ 3\) symmetry, very interesting results are obtained if a prismatic rod is compared with a tapered rod. For the tapered rod secondary bifurcation are shown to occur where the two buckling modes involved are due to bending and torsion. The clever approach used by the authors to study the case with a codimension as big as possible in order to obtain a globalization of the bifurcation diagrams seems to be in general important for bifurcation problems.
Mathematics Subject Classification ID: 74G60
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74K10
zbMATH DE Number: 3907897
zbMATH Keywords: invariant under dihedral group
zbMATH Keywords: transverse isotropy condition
zbMATH Keywords: director approach to rod theory
zbMATH Keywords: spatial buckled states
zbMATH Keywords: naturally straight, uniform, nonlinearly hyperelastic rods
zbMATH Keywords: cross-sections are regular n- gons
zbMATH Keywords: terminal loads
zbMATH Keywords: singularity theory
zbMATH Keywords: variational bifurcation problem
zbMATH Keywords: splitting lemma
zbMATH Keywords: invariant perturbations
zbMATH Keywords: rod clamped at both ends
zbMATH Keywords: prismatic rod
zbMATH Keywords: tapered rod
zbMATH Keywords: secondary bifurcation
zbMATH Keywords: buckling modes
zbMATH Keywords: bending
zbMATH Keywords: torsion
zbMATH Keywords: codimension as big as possible
reviewed by: Hans Troger
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3907897