Abstract
This paper examines the behavior of a class of nonlinearly elastic rods under tension. These rods can suffer longitudinal extension, transverse contraction, shear, and flexure. The deformation is governed by an eighth order, quasilinear system of ordinary differential equations. A convergent perturbation process shows that there is a very rich collection of solutions that bifurcate from the trivial affine configuration. Global results are obtained for hyperelastic rods. These solutions describe instabilities of the shear and necking type and can exhibit hysteresis.
Résumé
On considère le comportement d'une classe de poutres nonlinéairement élastiques en état de tension. Ces poutres peuvent souffrir une extension longitudinale, une contraction transversale, un cisaillement et une fiexion. La déformation est gouvernée par un système quasi-linéaire d'équations différentielles ordinaires du huitième ordre. Un procès convergeant de perturbations montre qu'il y a une collection très riche de solutions qui bifurquent de la configuration banale affine. Dans le cas hypérélastique on obtient des résultats globaux. On montre que ces solutions décrivent des instabilités du type de cisaillement et d'étranglement et peuvent exhiber une hystérésis.
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The research of the first author was supported by National Science Foundation Grant MPS 73-08587 A02. Some of the results reported here were obtained by the second author in his master's thesis.
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Antman, S.S., Carbone, E.R. Shear and necking instabilities in nonlinear elasticity. J Elasticity 7, 125–151 (1977). https://doi.org/10.1007/BF00041087
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DOI: https://doi.org/10.1007/BF00041087