On the Planar Elastica, Stress, and Material Stress


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.

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  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].


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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.

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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

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  • Elasticity
  • Rods
  • Symmetry
  • Material force

Mathematics Subject Classification

  • 74K10