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 centreline-based parametrization of the tape-spring midsurface with the assumption that the strain varies slowly along the length of the tape spring. The one-dimensional model is effectively a higher-order 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. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.