A convenient formulation of Sadowsky model for elastic ribbons

Basile Audoly (1, 2) , Sébastien Neukirch (3)

1 LMS - Laboratoire de mécanique des solides
2 IP Paris - Institut Polytechnique de Paris
3 DALEMBERT - Institut Jean Le Rond d'Alembert

Elastic ribbons are elastic structures whose length-to-width and width-to-thickness aspectratios are both large. Sadowsky proposed a one-dimensional model for ribbons featuring a nonlinear constitutive relation for bending and twisting: it brings in both rich behaviors and numerical difficulties. By discarding non-physical solutions to this constitutive relation, we show that it can be inverted; this simplifies the system of differential equations governing the equilibrium of ribbons. Based on the inverted form, we propose a natural regularization of the constitutive law that eases the treatment of singularities often encountered in ribbons. We illustrate the approach with the classical problem of the equilibrium of a Möbius ribbon, and compare our findings to the predictions of the Wunderlich model. Overall, our approach provides a simple method for simulating the statics and the dynamics of elastic ribbons.

Mots clés: boundary value problems; elastic plates; twisted rods

Domaines: Mécanique des solides [physics.class-ph]; Mécanique des structures [physics.class-ph]; Mécanique [physics]

Basile Audoly, Sébastien Neukirch. A convenient formulation of Sadowsky model for elastic ribbons. Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, 2021, 477 (2256), pp.20210548. ⟨10.1098/rspa.2021.0561⟩. ⟨hal-03278697v2⟩