On the Planar Elastica, Stress, and Material Stress

  • H. Singh
  • J. A. Hanna


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.


Elasticity Rods Symmetry Material force 

Mathematics Subject Classification




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.


  1. 1.
    Oldfather, W.A., Ellis, C.A., Brown, D.M.: Leonhard Euler’s elastic curves. Isis 20(1), 72–160 (1933) CrossRefGoogle Scholar
  2. 2.
    Mladenov, I.M., Hadzhilazova, M.: The Many Faces of Elastica. Springer, Cham (2017) CrossRefGoogle Scholar
  3. 3.
    Rogula, D.: Forces in material space. Arch. Mech. 29, 705–713 (1977) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Herrmann, A.G.: On conservation laws of continuum mechanics. Int. J. Solids Struct. 17, 1–9 (1981) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Benjamin, T.B.: Impulse, flow force and variational principles. IMA J. Appl. Math. 32, 3–68 (1984) ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Gurevich, V.L., Thellung, A.: Quasimomentum in the theory of elasticity and its conservation. Phys. Rev. B 42, 7345–7349 (1990) ADSCrossRefGoogle Scholar
  7. 7.
    Nelson, D.F.: Momentum, pseudomomentum, and wave momentum: toward resolving the Minkowski-Abraham controversy. Phys. Rev. A 44, 3985–3996 (1991) ADSCrossRefGoogle Scholar
  8. 8.
    Maugin, G.A.: Material forces: concepts and applications. Appl. Mech. Rev. 48, 213–245 (1991) ADSCrossRefGoogle Scholar
  9. 9.
    Kienzler, R., Herrmann, G.: Mechanics in Material Space. Springer, Berlin (2000) CrossRefGoogle Scholar
  10. 10.
    Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, New York (2000) zbMATHGoogle 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) ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    O’Reilly, O.M.: Modeling Nonlinear Problems in the Mechanics of Strings and Rods. Springer, New York (2017) CrossRefGoogle Scholar
  13. 13.
    Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (2005) zbMATHGoogle 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) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Singer, D.A.: Lectures on elastic curves and rods. AIP Conf. Proc. 1002, 3–32 (2008) ADSMathSciNetCrossRefGoogle 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) ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Guven, J., Vázquez-Montejo, P.: Confinement of semiflexible polymers. Phys. Rev. E 85, 026603 (2012) ADSCrossRefGoogle Scholar
  18. 18.
    Tsuru, H.: Nonlinear dynamics for thin elastic rod. J. Phys. Soc. Jpn. 55, 2177–2182 (1986) ADSCrossRefGoogle Scholar
  19. 19.
    Steigmann, D.J., Faulkner, M.G.: Variational theory for spatial rods. J. Elast. 33, 1–26 (1993) MathSciNetCrossRefGoogle 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) ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Nordgren, R.P.: On computation of the motion of elastic rods. J. Appl. Mech. 41, 777–780 (1974) ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Shelley, M.J., Ueda, T.: The Stokesian hydrodynamics of flexing, stretching filaments. Physica D 146, 221–245 (2000) ADSMathSciNetCrossRefGoogle 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) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Noether, E., Tavel, M.A.: Invariant variation problems and Noether’s theorem. Transp. Theory Stat. Phys. 3, 183–207 (1971) zbMATHGoogle Scholar
  26. 26.
    Hill, E.L.: Hamilton’s principle and the conservation theorems of mathematical physics. Rev. Mod. Phys. 23, 253–260 (1951) ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    O’Reilly, O.M.: A material momentum balance law for rods. J. Elast. 86, 155–172 (2007) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Broer, L.J.F.: On the dynamics of strings. J. Eng. Math. 4, 195–202 (1970) CrossRefGoogle Scholar
  29. 29.
    Maddocks, J.H., Dichmann, D.J.: Conservation laws in the dynamics of rods. J. Elast. 34, 83–96 (1994) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ericksen, J.L.: Simpler static problems in nonlinear theories of rods. Int. J. Solids Struct. 6, 371–377 (1970) CrossRefGoogle 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) MathSciNetCrossRefGoogle Scholar
  32. 32.
    Nizette, M., Goriely, A.: Towards a classification of Euler-Kirchhoff filaments. J. Math. Phys. 40(6), 2830–2866 (1999) ADSMathSciNetCrossRefGoogle 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) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Goldstein, R.E., Langer, S.A.: Nonlinear dynamics of stiff polymers. Phys. Rev. Lett. 75(6), 1094–1097 (1995) ADSCrossRefGoogle Scholar
  35. 35.
    Capovilla, R., Chryssomalakos, C., Guven, J.: Hamiltonians for curves. J. Phys. A 35, 6571–6587 (2002) ADSMathSciNetCrossRefGoogle 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) ADSCrossRefGoogle Scholar
  37. 37.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944) zbMATHGoogle Scholar
  38. 38.
    Craig, G.F.: Mathematical technique and physical conception in Euler’s investigation of the elastica. Centaurus 34, 211–246 (1991) MathSciNetCrossRefGoogle Scholar
  39. 39.
    Levien, R.: The elastica: a mathematical history. Technical Report No. UCB/EECS-2008-103, University of California, Berkeley (2008) Google Scholar
  40. 40.
    Hubbard, M.: An iterative numerical solution for the elastica with causally mixed inputs. J. Appl. Mech. 47, 200–202 (1980) ADSCrossRefGoogle Scholar
  41. 41.
    Griner, G.M.: A parametric solution to the elastic pole-vaulting pole problem. J. Appl. Mech. 51, 409–414 (1984) ADSCrossRefGoogle 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) ADSMathSciNetCrossRefGoogle 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) CrossRefGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Biomedical Engineering and MechanicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Department of Biomedical Engineering and Mechanics, Department of Physics, Center for Soft Matter and Biological PhysicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA

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