Integrable dynamics and geometric conservation laws of hyperelastic strips

Gözde Özkan Tükel; Gözde Özkan Tükel

doi:10.3934/math.20241186

AIMS Mathematics

2024, Volume 9, Issue 9: 24372-24384. doi: 10.3934/math.20241186

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Research article

Integrable dynamics and geometric conservation laws of hyperelastic strips

Gözde Özkan Tükel ^,

Isparta University of Applied Sciences, Faculty of Technology, Department of Basic Sciences, Türkiye

Received: 14 June 2024 Revised: 02 August 2024 Accepted: 13 August 2024 Published: 19 August 2024
MSC : 37K25, 53A04, 53A05

We consider the energy-minimizing configuration of the Sadowsky-type functional for narrow rectifying strips. We show that the functional is proportional to the $ p $-Willmore functional using classical analysis techniques and the geometry of developable surfaces. We introduce hyperelastic strips (or p-elastic strips) as rectifying strips whose base curves are the critical points of the Sadowsky-type functional and find the Euler-Lagrange equations for hyperelastic strips using a variational approach. We show a naturally expected relationship between the planar stationary points of the Sadowsky-type functional and the hyperelastic curves. We derive two conservation vector fields, the internal force and torque, using Euclidean motions and obtain the first and second conservation laws for hyperelastic strips.
- conservation laws,
- hyperelastic strips,
- p-elastic strips,
- Sadowsky-type functional,
- variational calculus
Citation: Gözde Özkan Tükel. Integrable dynamics and geometric conservation laws of hyperelastic strips[J]. AIMS Mathematics, 2024, 9(9): 24372-24384. doi: 10.3934/math.20241186

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Abstract

We consider the energy-minimizing configuration of the Sadowsky-type functional for narrow rectifying strips. We show that the functional is proportional to the $ p $-Willmore functional using classical analysis techniques and the geometry of developable surfaces. We introduce hyperelastic strips (or p-elastic strips) as rectifying strips whose base curves are the critical points of the Sadowsky-type functional and find the Euler-Lagrange equations for hyperelastic strips using a variational approach. We show a naturally expected relationship between the planar stationary points of the Sadowsky-type functional and the hyperelastic curves. We derive two conservation vector fields, the internal force and torque, using Euclidean motions and obtain the first and second conservation laws for hyperelastic strips.

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