Bifurcations of buckled, clamped anisotropic rods and thin bands under lateral end translations

Abstract

Motivated by observations of snap-through phenomena in buckled elastic strips subject to clamping and lateral end translations, we experimentally explore the multi-stability and bifurcations of thin bands of various widths and compare these results with numerical continuation of a perfectly anisotropic Kirchhoff rod. Our choice of boundary conditions is not easily satisfied by the anisotropic structures, forcing a cooperation between bending and twisting deformations. We find that, despite clear physical differences between rods and strips, a naive Kirchhoff model works surprisingly well as an organizing framework for the experimental observations. In the context of this model, we observe that anisotropy creates new states and alters the connectivity between existing states. Our results are a preliminary look at relatively unstudied boundary conditions for rods and strips that may arise in a variety of engineering applications, and may guide the avoidance of jump phenomena in such settings. We also briefly comment on the limitations of current strip models. * jhyutian@vt.edu † hannaj@vt.edu arXiv:1708.05968v2 [cond-mat.soft] 31 Jan 2018 the backbone of the behavior of wider bands. We reveal connections between various states, including higher-order unstable elastica modes and stable twisted states created by the rod's anisotropy.While the Kirchhoff equations show themselves to be a surprisingly useful tool in the analysis of strip behavior, we wish to emphasize that there is no reason to assume that such a model, which assumes that cross sections remain perpendicular to the centerline, would be appropriate for strips. On the other hand, the common assumption that transverse bending of strips is governed by the constraint of developability can lead to difficulties of its own, particularly for narrow strips, issues that we will briefly touch upon in an appendix. Until such issues are resolved, it is advantageous to employ an easily implemented rod model from which the strips inherit most, or even all, of their bifurcations. However, use of such a model should not be taken to imply that a narrow strip is equivalent to a rod.Boundary conditions like those we explore here are potentially of interest in helping to avoid violent snap-throughs of connectors, hinges, and umbilicals in flexible and deployable systems. Geometries similar to ours appear as slipping folds [3] in deployable space membranes, buckled elements in flexible electronics and robotics, and decorative streamers in childrens' toys [4]. Multi-stable structures find use in compliant mechanisms [5] at all length scales. The behavior of strips under our loading conditions is likely related to the phenomenon of lateral-torsional buckling, known to structural engineers [6].There is much prior work on the configurations of naturally straight rods. Work on the general behavior and classification of solutions includes that of Antman [7,8], Maddocks [9], Nizette and Goriely [10], and Cognet and co-workers [11]. Neukirch and Henderson made a detailed investigation...

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