Abstract

This paper presents numerical simulations and comparisons between different approaches concerning elastic thin rods. Elastic rods are ideal for modeling the stretching, bending, and twisting deformations of such long and thin elastic materials. The static solution of Kirchhoff's equations [2] is produced using ODE45 solver where Kirchhoff and reference system equations are combined instantaneously. Solutions using formulations are based on Euler's elastica theory [1] which determines the deformed centerline of the rod by solving a boundary-value problem, on the Discreet Elastic Rod method using Bishop frame (DER) [5,6] which is based on discrete differential geometry, it starts with a discrete energy formulation and obtains the forces and equations of motion by taking the derivative of energies. Instead of discretizing smooth equations, DER solves discrete equations and obeys geometrical exactness. Using DER we measure torsion as the difference of angles between the material and the Bishop frame of the rod so that no additional degree of freedom is needed to represent the torsional behavior. We found excellent agreement between our Kirchhoff-based solution and numerical results obtained by the other methods. In our numerical results, we include simulation of the rod under the action of the terminal moment and illustrations of the gravity effects.

Highlights

  • Elastic rods have many interesting applications, at large length scale they are used to study the mechanics of macro structural elements such as marine cables

  • These theories have been used to model biological threads, hair beams [15], DNA molecules [16] rubber bands and microfibers [17]. They are ideal for modeling the stretching, bending, and twisting deformations of such long, thin elastic materials

  • The rod theory is considered as an example of a Cosserat rod theory

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Summary

A Simulation of an Elastic Filament Using Kirchhoff Model

Received September 8, 2021; Revised November 25, 2021; Accepted December 13, 2021. Cite This Paper in the following Citation Styles (a): [1] Saimir Tola, Alfred Daci, Gentian Zavalani , "A Simulation of an Elastic Filament Using Kirchhoff Model," Mathematics and Statistics, Vol 10, No 1, pp. (b): Saimir Tola, Alfred Daci, Gentian Zavalani (2022). A Simulation of an Elastic Filament Using Kirchhoff Model. Mathematics and Statistics, 10(1), 25 - 34.

Introduction
Kirchhoff Rod Model
Geometric Description
Geometry of Deformation
Mechanics of the Rod
Implementation and Numerical Results
Conclusion
Full Text
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