On the stability of Euler’s elastica with natural curvature: Out-of-plane bifurcation with twisting

Abstract

Elastic rods can exhibit distinct buckling behavior in the presence of natural curvature due to the induced prestress. Recent work has shown that prestressed, clamped rods can undergo a secondary bifurcation from in-plane Euler buckling and subsequently exhibit twisting or snapping under axial compression. In this work, a bifurcation analysis for the secondary out-of-plane bifurcation of Euler’s elastica is performed based on Kirchhoff’s three-dimensional theory of inextensional rods. The critical end-shortening, critical load, and associated eigenmode at the onset of bifurcation are determined, and the role of natural curvature in the out-of-plane bifurcation from the Euler solution is systematically examined. It is found that the effect of natural curvature depends on the bending direction of Euler buckling. When the Euler buckling direction is consistent with the rod’s naturally curved direction, the critical end-shortening decreases as the natural curvature increases, and vice versa. The imperfection sensitivity of the out-of-plane bifurcation is further investigated. The results reveal that the primary role of in-plane imperfections is to select the in-plane Euler buckling direction, which in turn has a major effect in the presence of substantial natural curvature. By contrast, the primary role of out-of-plane imperfections is to trigger the out-of-plane bifurcation and induce three-dimensional twisting or snapping.