On the realization of a second buckling mode in a periodically-constrained heavy elastica

Abstract

Structural instabilities are often considered a failure state and therefore engineers are trained to avoid them. Recently, flexible buckled structures have gone from merely describing a phenomenon (e.g., twisting of vines, DNA mechanics), to being exploited for novel functionalities (e.g., stretchable electronics, energy harvesting). Herein we present results from a study of the static deformation of a periodically-constrained plate proposed by Muriel et al. (2016) as a hydrokinetic energy harvester. Experimentally, a polycarbonate sheet is placed under longitudinal compression with external forcing provided by equally spaced tension lines anchored in a frame, inducing a second mode wave-like deformation. No additional constraints are placed on the material. To predict this deformation, we proposed a model based on the two-dimensional heavy elastica, where the dependence of the distributed weight to the curvature prevents a closed form solution. Therefore, we solve numerically the nonlinear system of equations of an elemental strip, with the assumptions of uniform cross-section, elasticity, and density, and consider only cylindrical bending and the plate to be thin enough such that the Euler assumption holds. Tension lines are treated with a constitutive equation consistent with Hook’s law, and contact points between lines and the plate are considered as passive (i.e., no friction, no moment, and no elastic singularity). The system is discretized using a second-order centered difference scheme, and solved using trust-region and Levenberg–Marquardt algorithms. The numerical solutions predict the deformation well and confirm the experimental finding of elastic bifurcation (solutions of second mode deformations), buckling instability, and points towards the possibility of deformations resembling higher order modes not seen in the experiments. The sensitivity of the constitutive relation to small strain variations poses questions about convergence of such higher order modes. This structural model is the first step towards a fully coupled fluid–structure model of a deformation that shows promise on reducing the external forcing needed to start structural oscillations.