Partial Constraint Singularities in Elastic Rods

Abstract

We present a unified classical treatment of partially constrained elastic rods. Partial constraints often entail singularities in both shapes and reactions. Our approach encompasses both sleeve and adhesion problems, and provides simple and unambiguous derivations of counterintuitive results in the literature. Relationships between reaction forces and moments, geometry, and adhesion energies follow from the balance of energy during quasistatic motion. We also relate our approach to the balance of material momentum and the concept of a driving traction. The theory is generalizable and can be applied to a wide array of contact, adhesion, gripping, and locomotion problems.

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Notes

  1. 1.

    This argument is presented for a planar problem, but can be fully generalized.

  2. 2.

    O’Reilly [25, 34] uses terms which can be easily translated into ours as \(\mathsf{C} = -c\) and \(\mathsf{B} = -Y\).

  3. 3.

    Equation (34) of [34] provides a relation between singular sources such that, given our prescription for \(\tilde{E}\), the correct prescription for \(Y\) can be obtained. We thank O.M. O’Reilly for suggesting this approach.

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Acknowledgements

J.A. Hanna and H. Singh were supported by U.S. National Science Foundation grant CMMI-1462501. E.G. Virga acknowledges the kind hospitality of the Oxford Centre for Nonlinear PDE, where part of this work was done while he was visiting the Mathematical Institute at the University of Oxford.

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Correspondence to H. Singh.

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Hanna, J.A., Singh, H. & Virga, E.G. Partial Constraint Singularities in Elastic Rods. J Elast 133, 105–118 (2018). https://doi.org/10.1007/s10659-018-9673-6

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Keywords

  • Rods
  • Constraints
  • Adhesion
  • Jump conditions

Mathematics Subject Classification

  • 74A15
  • 74G70
  • 74K10
  • 74M15