A one-dimensional model for elastic ribbons: A little stretching makes a big difference

Abstract

Starting from the theory of elastic plates, we derive a non-linear one-dimensional model for elastic ribbons with thickness t, width a and length ℓ, assuming t≪a≪ℓ. It takes the form of a rod model with a specific non-linear constitutive law accounting for both the stretching and the bending of the ribbon mid-surface. The model is asymptotically correct and can handle finite rotations. Two popular theories can be recovered as limiting cases, namely Kirchhoff’s rod model for small bending and twisting strains, |κi|≪t∕a2, and Sadowsky’s inextensible ribbon model for |κi|≫t∕a2; we point out that Sadowsky’s inextensible model may be a poor approximation even for ribbons having a very thin cross-section (say, with t∕a as small as 0.02). By way of illustration, the one-dimensional model is applied (i) to the lateral-torsional instability of a ribbon, showing good agreement with both experiments and finite-element shell simulations, and (ii) to the stability of a twisted ribbon subjected to a tensile force. The non-convexity of the one-dimensional model is discussed; it is addressed by a convexification argument.