Gaussian curvature-induced localization in tape springs: A one-dimensional nonlinear model

A one-dimensional model for tape springs is proposed to capture their essential mechanical behavior, including membrane and bending effects. A key feature of the model is the incorporation of Gaussian curvature, for which a one-dimensional estimate is provided. The bending energy is derived from shell theory using a suitable kinematic Ansatz. Equilibrium configurations with uniform curvature are analyzed under prescribed end rotations, and critical rotations are determined through analytical estimates of the stability conditions. An inextensible version of the model is also considered, allowing for the prediction of the post-localization bending moment and the explanation of the linear energy growth observed at large rotations. Numerical results are presented and compared with both experimental data and existing numerical predictions from the literature, confirming the model’s capability to accurately capture curvature localization phenomena.