Journal of Elasticity

, Volume 136, Issue 1, pp 87–101 | Cite as

On the Planar Elastica, Stress, and Material Stress

  • H. SinghEmail author
  • 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.


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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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