On the Planar Elastica, Stress, and Material Stress

Abstract

We revisit the classical problem of the planar Euler elastica with applied forces and moments, and present a classification of the shapes in terms of tangentially conserved quantities associated with spatial and material symmetries. We compare commonly used director, variational, and dynamical systems representations, and present several illustrative physical examples. We remark that an approach that employs only the shape equation for the tangential angle obscures physical information about the tension in the body.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Notes

  1. 1.

    Material forces and closely related quantities, which in our one-dimensional system have just one component, appear under many names in the literature, including Eshelbian force, quasimomentum, pseudomomentum, (Kelvin) impulse, and configurational force [312]. In the present case, one can derive everything from consideration of conventional force balance and its projection onto the tangents of the body, but the concept of material force is useful as a descriptive term, and also corresponds to an important symmetry of the Lagrangian. We generally prefer the term “spatial” to “conventional”, but try to avoid it in this note because of its alternative meaning as “non-planar” in the context of rods embedded in three dimensions rather than two.

  2. 2.

    Similar multipliers can be found in Burchard and Thomas [14], Singer [15], Tornberg and Shelley [16], who do not employ a variational derivation, and Guven and Vázquez-Montejo [17], who use multiple redundant multipliers. The multipliers in [14] and [16] are misidentified as the tension. For seemingly similar but qualitatively different multipliers, see the following footnote.

  3. 3.

    The tension \({\boldsymbol{n}}\cdot\partial_{s} {\boldsymbol{x}}\) can be associated with the multiplier \(T\) appearing in Nordgren [21] and Shelley and Ueda [22], who do not employ variational derivations, and Audoly [23], whose multiplier is different than our \(\sigma\) despite appearing similarly in a Lagrangian. The difference with Audoly arises due to his splitting of the variation in terms of an angular variation \(\delta\theta\) on the boundary, rather than \(\partial_{s}\delta{\boldsymbol{x}}\) as done here. A similar issue arises in plate and shell theories [24].

References

  1. 1.

    Oldfather, W.A., Ellis, C.A., Brown, D.M.: Leonhard Euler’s elastic curves. Isis 20(1), 72–160 (1933)

    Article  MATH  Google Scholar 

  2. 2.

    Mladenov, I.M., Hadzhilazova, M.: The Many Faces of Elastica. Springer, Cham (2017)

    Google Scholar 

  3. 3.

    Rogula, D.: Forces in material space. Arch. Mech. 29, 705–713 (1977)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Herrmann, A.G.: On conservation laws of continuum mechanics. Int. J. Solids Struct. 17, 1–9 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Benjamin, T.B.: Impulse, flow force and variational principles. IMA J. Appl. Math. 32, 3–68 (1984)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Gurevich, V.L., Thellung, A.: Quasimomentum in the theory of elasticity and its conservation. Phys. Rev. B 42, 7345–7349 (1990)

    ADS  Article  Google Scholar 

  7. 7.

    Nelson, D.F.: Momentum, pseudomomentum, and wave momentum: toward resolving the Minkowski-Abraham controversy. Phys. Rev. A 44, 3985–3996 (1991)

    ADS  Article  Google Scholar 

  8. 8.

    Maugin, G.A.: Material forces: concepts and applications. Appl. Mech. Rev. 48, 213–245 (1991)

    ADS  Article  Google Scholar 

  9. 9.

    Kienzler, R., Herrmann, G.: Mechanics in Material Space. Springer, Berlin (2000)

    Google Scholar 

  10. 10.

    Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, New York (2000)

    Google Scholar 

  11. 11.

    Yavari, A., Marsden, J.E., Ortiz, M.: On spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 042903 (2006)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    O’Reilly, O.M.: Modeling Nonlinear Problems in the Mechanics of Strings and Rods. Springer, New York (2017)

    Google Scholar 

  13. 13.

    Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (2005)

    Google Scholar 

  14. 14.

    Burchard, A., Thomas, L.E.: On the Cauchy problem for a dynamical Euler’s elastica. Commun. Partial Differ. Equ. 28, 271–300 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Singer, D.A.: Lectures on elastic curves and rods. AIP Conf. Proc. 1002, 3–32 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    Tornberg, A.-K., Shelley, M.J.: Simulating the dynamics and interactions of flexible fibers in Stokes flows. J. Comput. Phys. 196, 8–40 (2004)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Guven, J., Vázquez-Montejo, P.: Confinement of semiflexible polymers. Phys. Rev. E 85, 026603 (2012)

    ADS  Article  Google Scholar 

  18. 18.

    Tsuru, H.: Nonlinear dynamics for thin elastic rod. J. Phys. Soc. Jpn. 55, 2177–2182 (1986)

    ADS  Article  Google Scholar 

  19. 19.

    Steigmann, D.J., Faulkner, M.G.: Variational theory for spatial rods. J. Elast. 33, 1–26 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Kehrbaum, S., Maddocks, J.H.: Elastic rods, rigid bodies, quaternions and the last quadrature. Philos. Trans. R. Soc. Lond. A 355, 2117–2136 (1997)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Nordgren, R.P.: On computation of the motion of elastic rods. J. Appl. Mech. 41, 777–780 (1974)

    ADS  Article  MATH  Google Scholar 

  22. 22.

    Shelley, M.J., Ueda, T.: The Stokesian hydrodynamics of flexing, stretching filaments. Physica D 146, 221–245 (2000)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Audoly, B.: Introduction to the elasticity of rods. In: Duprat, C., Stone, H.A. (eds.) Fluid-Structure Interactions in Low-Reynolds-Number Flows, pp. 1–24. The Royal Society of Chemistry, Cambridge (2016)

    Google Scholar 

  24. 24.

    Steigmann, D.J.: Extension of Koiter’s linear shell theory to materials exhibiting arbitrary symmetry. Int. J. Eng. Sci. 51, 216–232 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Noether, E., Tavel, M.A.: Invariant variation problems and Noether’s theorem. Transp. Theory Stat. Phys. 3, 183–207 (1971)

    MATH  Google Scholar 

  26. 26.

    Hill, E.L.: Hamilton’s principle and the conservation theorems of mathematical physics. Rev. Mod. Phys. 23, 253–260 (1951)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    O’Reilly, O.M.: A material momentum balance law for rods. J. Elast. 86, 155–172 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Broer, L.J.F.: On the dynamics of strings. J. Eng. Math. 4, 195–202 (1970)

    Article  Google Scholar 

  29. 29.

    Maddocks, J.H., Dichmann, D.J.: Conservation laws in the dynamics of rods. J. Elast. 34, 83–96 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Ericksen, J.L.: Simpler static problems in nonlinear theories of rods. Int. J. Solids Struct. 6, 371–377 (1970)

    Article  MATH  Google Scholar 

  31. 31.

    Antman, S.S., Jordan, K.B.: Qualitative aspects of the spatial deformation of non-linearly elastic rods. Proc. R. Soc. Edinb. 73A(5), 85–105 (1974/75)

  32. 32.

    Nizette, M., Goriely, A.: Towards a classification of Euler-Kirchhoff filaments. J. Math. Phys. 40(6), 2830–2866 (1999)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Van Der Heijden, G.H.M., Thompson, J.M.T.: Helical and localised buckling in twisted rods: a unified analysis of the symmetric case. Nonlinear Dyn. 21, 71–99 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Goldstein, R.E., Langer, S.A.: Nonlinear dynamics of stiff polymers. Phys. Rev. Lett. 75(6), 1094–1097 (1995)

    ADS  Article  Google Scholar 

  35. 35.

    Capovilla, R., Chryssomalakos, C., Guven, J.: Hamiltonians for curves. J. Phys. A 35, 6571–6587 (2002)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Lu, C.-L., Perkins, N.C.: Nonlinear spatial equilibria and stability of cables under uni-axial torque and thrust. J. Appl. Mech. 61, 879–886 (1994)

    ADS  Article  MATH  Google Scholar 

  37. 37.

    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944)

    Google Scholar 

  38. 38.

    Craig, G.F.: Mathematical technique and physical conception in Euler’s investigation of the elastica. Centaurus 34, 211–246 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Levien, R.: The elastica: a mathematical history. Technical Report No. UCB/EECS-2008-103, University of California, Berkeley (2008)

  40. 40.

    Hubbard, M.: An iterative numerical solution for the elastica with causally mixed inputs. J. Appl. Mech. 47, 200–202 (1980)

    ADS  Article  MATH  Google Scholar 

  41. 41.

    Griner, G.M.: A parametric solution to the elastic pole-vaulting pole problem. J. Appl. Mech. 51, 409–414 (1984)

    ADS  Article  Google Scholar 

  42. 42.

    O’Reilly, O.M.: Some perspectives on Eshelby-like forces in the elastica arm scale. Proc. R. Soc. Lond. A 471, 20140785 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Bigoni, D., Dal Corso, F., Bosi, F., Misseroni, D.: Eshelby-like forces acting on elastic structures: theoretical and experimental proof. Mech. Mater. 80, 368–374 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

We thank E.G. Virga for alerting us to the interesting features of the sleeve example, and J.H. Maddocks and O.M. O’Reilly for helpful discussions. This work was supported by U.S. National Science Foundation grant CMMI-1462501. This work has been available free of peer review on the arXiv since 6/2017.

Author information

Affiliations

Authors

Corresponding author

Correspondence to H. Singh.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Singh, H., Hanna, J.A. On the Planar Elastica, Stress, and Material Stress. J Elast 136, 87–101 (2019). https://doi.org/10.1007/s10659-018-9690-5

Download citation

Keywords

  • Elasticity
  • Rods
  • Symmetry
  • Material force

Mathematics Subject Classification

  • 74K10