Asymptotic derivation of a higher-order one-dimensional model for tape springs

We derive a one-dimensional model for tape springs. The derivation starts from nonlinear thin-shell theory and uses a dimension reduction technique that combines a centerline-based parameterization of the tape-spring mid-surface with the assumption that the strain varies slowly along the length of the tape spring. The one-dimensional model is effectively a higherorder rod model: at leading order, the strain energy depends on the extensional, bending and twisting strains and is consistent with classical results from the literature; the two following orders are novel and capture the dependence of the strain energy on the strain gradients. The cross-sectional displacements are solved as part of the dimension reduction process, making the one-dimensional model asymptotically exact. We expect that the model will accurately and efficiently capture the deformations and instabilities in tape springs, including those involving highly localized deformations