Spatial and planar deformations of a thin elastic wire under prescribed end rotations

Abstract

We consider the deformation of a thin elastic wire with circular cross section when the distance of its two ends and its two end rotation angles are prescribed. The two ends are limited to rotate about axes perpendicular to the line passing through the two ends. The load–deflection (end moment-end angle) relations of two examples are calculated numerically and verified experimentally. It is found that both spatial and planar deformations are possible. By employing the same method for all possible rotation angle pairs, the boundary between spatial and planar deformations in the plane of the two end angles can be established. We do not find the phenomenon of multiple solutions corresponding to a specified end angle pair. The only exception is the case when the two end tangents become collinear. Snapping motion occurs when the two end clamps face each other or face back to back. Finally the boundaries between spatial and planar deformations are established for some typical end distances between 0.1 and 0.67. When the end distance is smaller than 0.67, the load–deflection curve repeats itself for every 2π of the end angles. When end distance is greater than 0.67, there is no solution when prescribed end angles are beyond certain limits. As a consequence, the load–deflection curve does not repeat itself when end distance is greater than 0.67.