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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 special 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, |κ_i|≪t/a^2, and Sadowsky's inextensible ribbon model for |κ_i|≫t/a^2; we point out that Sadowsky's inextensible model may be a poor approximation even for ribbons having a very thin cross-section, a/t∼50≫1. 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.
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https://hal.archives-ouvertes.fr/hal-03132668
Contributor : Basile Audoly <>
Submitted on : Friday, February 5, 2021 - 12:11:15 PM
Last modification on : Wednesday, February 10, 2021 - 3:34:03 AM

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  • HAL Id : hal-03132668, version 1

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Basile Audoly, Sébastien Neukirch. A one-dimensional model for elastic ribbons: a little stretching makes a big difference. 2021. ⟨hal-03132668⟩

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