Post-buckling behavior of a nonlinearly hyperelastic thin rod with cross- section invariant under the dihedral group \(D_ n\)
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.
naturally straight, uniform, nonlinearly hyperelastic rods
scientific article; zbMATH DE number 3907897
scientific article