A centerline model for bending-dominated configurations of elastic strips

Abstract We propose a one-dimensional centerline model to describe bending-dominated configurations of thin elastic strips undergoing large displacements and rotations. The model is derived by dimensional reduction from the Kirchhoff plate theory by incorporating physically-inspired kinematic assumptions on midsurface deformations. The model effectively characterizes the strip as a rod with enriched kinematics. Three crucial considerations underlie the model. First, the ansatz on midsurface kinematics accounts for small but nontrivial curvatures along the width of the strip. Second, the bending energy in the model is derived as an approximation of that in the plate theory over the restricted class of midsurface deformations. Third, the model purposefully avoids imposing strict inextensibility constraints on the midsurface. Instead, the restricted kinematics adopted warrants a judicious choice of constraints deduced from membrane energy contributions in the plate theory. These constraints have direct geometric interpretations, are weakly imposed through a meaningful set of stiffness parameters, and recover bend-twist coupling observed in elastic strip deformations. The closed-form expression for the energy density, the choice of degrees of freedom, and the absence of strict constraints render the model amenable to straightforward finite element discretization. We examine the model's efficacy in describing large deflections and rotations using Mobius strips and annular ribbons, and find excellent agreement with experimental measurements and finite element plate simulations. Predictable discrepancies emerge with increasing strip width and under loading scenarios accompanied by localization of deformation.