Sébastien Neukirch
Institut Jean le Rond d'Alembert
Centre National de la Recherche Scientifique
Sorbonne Université, Campus Pierre et Marie Curie
Paris, Francetel: +33 1 44 27 72 61 (secr. 37 90)
e-mail: sebastien.neukirch (-atat-) sorbonne-universite.fr
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Sébastien Neukirch and Florence Bertails-Descoubes
A Noetherian approach to invariants for the statics and dynamics of elastic rods
Int. Journal of Non-Linear Mechanics, in the press (2025) pp---
Abstract : The static-dynamic analogy discovered by Gustav Kirchhoff in 1859 shows that the statics of an elastic beam is equivalent to the dynamics of a spinning top. This analogy, where time and angular velocity are, for example, equivalent to arclength and curvatures, allows the use of Emmy Noether's 1918 theorem on continuous symmetries to unravel a quantity that is invariant along elastic rods at equilibrium. A spinning top having a Lagrangian independent of time will have its mechanical energy constant in time. In the same manner, an elastic rod with uniform elastic properties will have the sum of its curvature energy and its tension force uniform along the structure. This arclength invariance property is known in simple cases, but the present approach generalises it to more complex scenarios where extensibility, shear, conservative loads (\eg gravity), and contact are involved. Moreover, still using Noether's theorem and bringing to light the continuous symmetries of the Lagrangian of the variational approach, we recover all known invariants for the statics and dynamics of rods and ribbons, including configurational invariants. Furthermore, we show how the arclength invariant may be used to obtain pivotal information on some landmark elastic rod problems, including confinement buckling, sliding sleeves, or plectonemes. Finally, we extend the approach to vibrations. Overall, this paper is an attempt to explain, unify and extend all previous results on rod invariants thanks to the beautiful Noetherian formalism, and to show its practical use on a few relevant applications.
download the pre-print version, here PDF or here: hal-05149317