Skip to main content
SpringerLink
Log in

Material Frames and Measures of Twists

  • Chapter
  • First Online:
A Primer on the Kinematics of Discrete Elastic Rods

Part of the book series: SpringerBriefs in Applied Sciences and Technology ((BRIEFSTHERMAL))

Abstract

The material frames associated with an edge of a discretized curve is discussed in this chapter. For the discrete curve, a pair of material vectors play the role of directors in Kirchhoff’s rod theory. The motion of these vectors is calibrated using either a Bishop frame or a reference frame that is continually being updated. The pair of parallel transport operators defined in the previous chapter combined with a rotation about the tangent vector to an edge are used to define the motions the material vectors. Of particular interest is the difference in an angle of twist between two adjacent edges. This angle will be identified with the torsion of the rod-like body that the discrete elastic rod is modeling. In this chapter the bending strains will also be defined. The developments in this chapter and, in particular, the twist of the reference frame are illustrated using the example of a rod uncoiling under its own weight.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In some works, the material vectors are identified as the discrete directors: \(\mathbf {m}^k_1 = \mathbf {d}^k_1\) and \(\mathbf {m}^k_2 = \mathbf {d}^k_2\).

  2. 2.

    In the code, \(\varDelta m^{k+1}_{\mathrm{ref}}\) is known by the variable name signang.

  3. 3.

    The easiest method to compute this representation is to use the relative angular velocity vector proposed in Casey and Lam [7]. This relative angular velocity vector was discussed earlier in Sect. 2.3.

References

  1. Antman, S.S.: Nonlinear Problems of Elasticity, Applied Mathematical Sciences, vol. 107, second edn. Springer-Verlag, New York (2005). URL http://dx.doi.org/10.1007/0-387-27649-1

  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. Journal of Computational Physics 253, 18–49 (2013). URL http://dx.doi.org/10.1016/j.jcp.2013.06.034

  3. Bergou, M., Audoly, B., Vouga, E., Wardetzky, M., Grinspun, E.: Discrete viscous threads. ACM Transactions on Graphics (SIGGRAPH) 29(4), 116:1–116:10 (2010). URL http://dx.doi.org/10.1145/1778765.1778853

  4. Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., Grinspun, E.: Discrete elastic rods. ACM Transactions on Graphics (SIGGRAPH) 27(3), 63:1–63:12 (2008). URL http://dx.doi.org/10.1145/1360612.1360662

  5. Casey, J., Lam, V.C.: On the relative angular velocity tensor. ASME Journal of Mechanisms, Transmissions, and Automation in Design 108, 399–400 (1986). URL http://dx.doi.org/10.1115/1.3258746

  6. Evangelista, D., Hotton, S., Dumais, J.: The mechanics of explosive dispersal and self-burial in the seeds of the filaree, Erodium cicutarium (Geraniaceae). Journal of Experimental Biology 214(4), 521–529 (2011). URL http://dx.doi.org/10.1242/jeb.050567

  7. Kaldor, J.M., James, D.L., Marschner, S.: Efficient yarn-based cloth with adaptive contact linearization. In: ACM SIGGRAPH 2010 Papers, SIGGRAPH ’10, pp. 105:1–105:10. ACM, New York, NY, USA (2010). URL http://doi.acm.org/10.1145/1833349.1778842

  8. Kirchhoff, G.: Über des gleichgewicht und die Bewegung eines unendlich dünnen elastichen Stabes. Crelles Journal für die reine und angewandte Mathematik 56, 285–313 (1859). URL http://dx.doi.org/10.1515/crll.1859.56.285

  9. O’Reilly, O.M.: Modeling Nonlinear Problems in the Mechanics of Strings and Rods: The Role of the Balance Laws. Interaction of Mechanics and Mathematics. Springer International Publishing, New York (2017). URL http://dx.doi.org/10.1007/978-3-319-50598-5

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, CA, USA

    M. Khalid Jawed

  2. Department of Mechanical Engineering, University of California, Berkeley, Berkeley, CA, USA

    Alyssa Novelia & Oliver M. O’Reilly

Authors
  1. M. Khalid Jawed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alyssa Novelia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oliver M. O’Reilly
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 The Author(s), under exclusive licence to Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Jawed, M.K., Novelia, A., O’Reilly, O.M. (2018). Material Frames and Measures of Twists. In: A Primer on the Kinematics of Discrete Elastic Rods. SpringerBriefs in Applied Sciences and Technology(). Springer, Cham. https://doi.org/10.1007/978-3-319-76965-3_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-76965-3_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-76964-6

  • Online ISBN: 978-3-319-76965-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation