Skip to main content
SpringerLink
Log in

On a planar theory of a discrete nonlinearly elastic rod

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A discrete model for elastic rods undergoing planar motions based on the theory of a directed (or Cosserat) rod is presented. Edge vectors and a director are used to capture cross section deformations including stretch, stretch gradients, shear, shear gradients, and the Poisson effect. In addition, deformations such as longitudinal stretch and bending are also incorporated. The model is validated with the help of known analytical solutions to benchmark problems using Green and Naghdi’s rod theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. Some authors [9, 34] use the descriptor “Cosserat” to classify a theory where a rigid orthonormal triad is attached at each cross section. The authors whose work most of this article is based of [15, 16] use “Cosserat” more generally to refer to a directed rod theory, where the directors are deformable and not necessarily orthonormal.

  2. For an introduction to the continuous directed rod theories and Kirchhoff’s rod theory, see [1, 25], and [32].

  3. That is, the bending moment was given by the constitutive relation \(EI \frac{\partial \theta }{\partial \xi }\) where \(\xi \) is the arc length parameter of the extensible rod in a reference configuration and \(\theta \) is the angle subtended by the tangent vector to the rod’s centerline with a fixed direction. The curvature of the rod in the present configuration is \(\kappa = \frac{\partial \theta }{\partial s} = \mu ^{-1} \frac{\partial \theta }{\partial \xi }\) where \(\mu = \frac{\partial s}{\partial \xi }\) is the stretch of the rod.

  4. These constants were established in a series of works by Green and Naghdi and are summarized in [24, 25].

References

  1. Antman, S.S.: Nonlinear Problems of Elasticity, Applied Mathematical Sciences, vol. 107, 2nd edn. Springer, New York (2005). https://doi.org/10.1007/0-387-27649-1

    Book  Google Scholar 

  2. Audoly, B., Clauvelin, N., Brun, P.T., Bergou, M., Grinspun, B., Wardetzky, M.: A discrete geometric approach for simulating the dynamics of thin viscous threads. J. Comput. Phys. 253, 18–49 (2013). https://doi.org/10.1016/j.jcp.2013.06.034

    Article  MathSciNet  MATH  Google Scholar 

  3. Bergou, M., Audoly, B., Vouga, E., Wardetzky, M., Grinspun, E.: Discrete viscous threads. ACM Trans. Gr. (SIGGRAPH) 29(4), 116:1–116:10 (2010). https://doi.org/10.1145/1778765.1778853

    Article  Google Scholar 

  4. Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., Grinspun, E.: Discrete elastic rods. ACM Trans. Gr. (SIGGRAPH) 27(3), 63:1–63:12 (2008). https://doi.org/10.1145/1360612.1360662

    Article  Google Scholar 

  5. Bishop, R.L.: There is more than one way to frame a curve. Am. Math. Mon. 82(3), 246–251 (1975). https://doi.org/10.2307/2319846

    Article  MathSciNet  MATH  Google Scholar 

  6. Bobenko, A.I.: Geometry II: Discrete Differential Geometry (2015). http://page.math.tu-berlin.de/~bobenko/Lehre/Skripte/DDG_Lectures.pdf

  7. Brand, M., Rubin, M.: A constrained theory of a cosserat point for the numerical solution of dynamic problems of non-linear elastic rods with rigid cross-sections. Int. J. Non-Linear Mech. 42(2), 216–232 (2007). https://doi.org/10.1016/j.ijnonlinmec.2006.10.002

    Article  Google Scholar 

  8. Bukowicki, M., Ekiel-Jeżewska, M.L.: Different bending models predict different dynamics of sedimenting elastic trumbbells. Soft Matter 14(28), 5786–5799 (2018). https://doi.org/10.1039/C8SM00604K

    Article  Google Scholar 

  9. Cao, D., Tucker, R.W.: Nonlinear dynamics of elastic rods using the Cosserat theory: modelling and simulation. Int. J. Solids Struct. 45(2), 460–477 (2008). https://doi.org/10.1016/j.ijsolstr.2007.08.016

    Article  MATH  Google Scholar 

  10. Cowper, G.R.: The shear coefficient in Timoshenko’s beam theory. ASME J. Appl. Mech. 33(2), 335–340 (1966). https://doi.org/10.1115/1.3625046

    Article  MATH  Google Scholar 

  11. Gerstmayr, J., Matikainen, M.K., Mikkola, A.M.: A geometrically exact beam element based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 20(4), 359 (2008). https://doi.org/10.1007/s11044-008-9125-3

    Article  MathSciNet  MATH  Google Scholar 

  12. Goldberg, N.N., Huang, X., Majidi, C., Novelia, A., O’Reilly, O.M., Paley, D.A., Scott, W.L.: On planar discrete elastic rod models for the locomotion of soft robots. Soft Robot. 6(5), 595–610 (2019). https://doi.org/10.1089/soro.2018.0104

    Article  Google Scholar 

  13. Green, A.E., Laws, N., Naghdi, P.M.: A linear theory of straight elastic rods. Arch. Ration. Mech. Anal. 25(4), 285–298 (1967). https://doi.org/10.1007/BF00250931

    Article  MathSciNet  MATH  Google Scholar 

  14. Green, A.E., Naghdi, P.M.: On thermal effects in the theory of rods. Int. J. Solids Struct. 15(11), 829–853 (1979). https://doi.org/10.1016/0020-7683(79)90053-2

    Article  MathSciNet  MATH  Google Scholar 

  15. Green, A.E., Naghdi, P.M., Wenner, M.L.: On the theory of rods. I Derivations from three-dimensional equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 337(1611), 451–483 (1974). https://doi.org/10.1098/rspa.1974.0061

    Article  MathSciNet  MATH  Google Scholar 

  16. Green, A.E., Naghdi, P.M., Wenner, M.L.: On the theory of rods. II Developments by direct approach. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 337(1611), 485–507 (1974). https://doi.org/10.1098/rspa.1974.0062

    Article  MathSciNet  MATH  Google Scholar 

  17. Green, A.E., Zerna, W.T.: Theoretical Elasticity, 2nd edn. Clarendon Press, Oxford (1968)

    MATH  Google Scholar 

  18. Hanson, A.J., Ma, H.: Parallel transport approach to curve framing. Indiana University, Techreports-TR425, vol. 11, pp. 3–7 (1995). ftp://ftp.cs.indiana.edu/pub/hanson/iucs-tr425.ps

  19. Jawed, M.K., Novelia, A., O’Reilly, O.M.: A Primer on the Kinematics of Discrete Elastic Rods. Springer Briefs in Applied Sciences and Technology. Springer, New York (2018). https://doi.org/10.1007/978-3-319-76965-3

    Book  Google Scholar 

  20. Mofid, M., Akin, J.E.: Discrete element response of beams with traveling mass. Adv. Eng. Softw. 25(2–3), 321–331 (1996). https://doi.org/10.1016/0965-9978(95)00099-2

    Article  Google Scholar 

  21. Naghdi, P.M., Rubin, M.B.: Constrained theories of rods. J. Elast. 14, 343–361 (1984). https://doi.org/10.1007/BF00125605

    Article  MATH  Google Scholar 

  22. Naghdi, P.M., Rubin, M.B.: On the significance of normal cross-sectional extension in beam theory with application to contact problems. Int. J. Solids Struct. 25(3), 249–265 (1989). https://doi.org/10.1016/0020-7683(89)90047-4

    Article  Google Scholar 

  23. Neukirch, S., Frelat, J., Goriely, A., Maurini, C.: Vibrations of post-buckled rods: the singular inextensible limit. J. Sound Vib. 331(3), 704–720 (2012). https://doi.org/10.1016/j.jsv.2011.09.021

    Article  Google Scholar 

  24. O’Reilly, O.M.: On constitutive relations for elastic rods. Int. J. Solids Struct. 35(11), 1009–1024 (1998). https://doi.org/10.1016/S0020-7683(97)00100-5

    Article  MathSciNet  MATH  Google Scholar 

  25. O’Reilly, O.M.: Modeling Nonlinear Problems in the Mechanics of Strings and Rods. Springer, New York (2017). https://doi.org/10.1007/978-3-319-50598-5

    Book  MATH  Google Scholar 

  26. Orzechowski, G., Shabana, A.A.: Analysis of warping deformation modes using higher order ANCF beam element. J. Sound Vib. 363, 428–445 (2016). https://doi.org/10.1016/j.jsv.2015.10.013

    Article  Google Scholar 

  27. Rubin, M.: On the numerical solution of one-dimensional continuum problems using the theory of a Cosserat point. J. Appl. Mech. 52(2), 373–378 (1985). https://doi.org/10.1115/1.3169056

    Article  MATH  Google Scholar 

  28. Rubin, M.: On the theory of a Cosserat point and its application to the numerical solution of continuum problems. J. Appl. Mech. 52(2), 368–372 (1985). https://doi.org/10.1115/1.3169055

    Article  MATH  Google Scholar 

  29. Rubin, M.: Numerical solution of two-and three-dimensional thermomechanical problems using the theory of a Cosserat point. In: Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids, pp. 308–334. Springer (1995). https://doi.org/10.1007/978-3-0348-9229-2_17

    Chapter  Google Scholar 

  30. Rubin, M.: Numerical solution procedures for nonlinear elastic rods using the theory of a Cosserat point. Int. J. Solids Struct. 38(24–25), 4395–4437 (2001). https://doi.org/10.1016/S0020-7683(00)00271-7

    Article  MATH  Google Scholar 

  31. Rubin, M.B.: Restrictions on nonlinear constitutive equations for elastic rods. J. Elast. 44(1), 9–36 (1996). https://doi.org/10.1007/BF00042190

    Article  MathSciNet  MATH  Google Scholar 

  32. Rubin, M.B.: Cosserat Theories: Shells, Rods, and Points. Kluwer Academic Press, Dordrecht (2000). https://doi.org/10.1007/978-94-015-9379-3

    Book  MATH  Google Scholar 

  33. Shabana, A.A., Patel, M.: Coupling between shear and bending in the analysis of beam problems: planar case. J. Sound Vib. 419, 510–525 (2018). https://doi.org/10.1016/j.jsv.2017.12.006

    Article  Google Scholar 

  34. Simo, J.C., Vu-Quoc, L.: On the dynamics in space of rods undergoing large motions. A geometrically exact approach. Comput. Methods Appl. Mech. Eng. 66(2), 125–161 (1988). https://doi.org/10.1016/0045-7825(88)90073-4

    Article  MathSciNet  MATH  Google Scholar 

  35. Sokolnikoff, I.S.: Mathematical Theory of Elasticity. McGraw-Hill, New York (1956)

    MATH  Google Scholar 

  36. Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)

    MATH  Google Scholar 

  37. Yavari, A., Nouri, M., Mofid, M.: Discrete element analysis of dynamic response of Timoshenko beams under moving mass. Adv. Eng. Softw. 33(3), 143–153 (2002). https://doi.org/10.1016/S0965-9978(02)00003-0

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The work of Evan Hemingway was supported by a Berkeley Fellowship from the University of California at Berkeley and a U.S. National Science Foundation Graduate Research Fellowship. The work of Oliver O’Reilly was supported by grant number W911NF-16-1-0242 from the U.S. Army Research Organization administered by Dr. Samuel C. Stanton.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA, 94720-1740, USA

    Evan G. Hemingway & Oliver M. O’Reilly

Authors
  1. Evan G. Hemingway
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oliver M. O’Reilly
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oliver M. O’Reilly.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 12 KB)

Supplementary material 2 (mov 211 KB)

Supplementary material 3 (mov 404 KB)

Supplementary material 4 (mov 139 KB)

Supplementary material 5 (mov 222 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hemingway, E.G., O’Reilly, O.M. On a planar theory of a discrete nonlinearly elastic rod. Acta Mech 231, 1217–1240 (2020). https://doi.org/10.1007/s00707-019-02581-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-019-02581-x

Search

Navigation