Skip to main content

Singular Shear-Force States in Elementary Plate Theory

Abstract

We show that the most classical Kirchhoff-Love theory of thin plates is compatible with the occurrence of a specific singular shear-force state in the interior of the body. It is well-known from Kirchhoff that, on the edge boundary of the plate, the specific shear-forces and the curve-gradient of the specific twisting-moments, measured per unit length, are statically inter-related. We observe and prove that a similar static equivalence holds for the edge boundary of any sub-body, and this allows many interpretations of the contact interactions that may take place between the parts of the plate. In particular, a specific shear-force acting on a smooth part of the edge boundary of a sub-body may depend upon its curvature, tending to a concentrated force at a sharp corner. The possibility of developing concentrated contact interactions is a general characteristic of non-simple continua, of which the theory of thin plates is but one representative example.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    See [2], Sects. 4 and 5, for an exact derivation of the Kirchhoff-Love equations and boundary conditions from three-dimensional linear elasticity with internal constraints. In the absence of confirmation, it seems likely that the no-shear internal constraint of [2] may be coupled with a special form of the Principle of Virtual Power, similar to what is presented in [3], to show that the concentrated forces at corners are associated with the constraint reactions.

References

  1. 1.

    Thomson, W., Tait, P.G.: Treatise on Natural Philosophy, 8th edn. Cambridge University Press, Cambridge (1923)

    Google Scholar 

  2. 2.

    Podio-Guidugli, P.: An exact derivation of thin plate equation. J. Elast. 22, 121–133 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. 3.

    Podio-Guidugli, P., Vianello, M.: Hypertractions and hyperstresses convey the same mechanical information. Contin. Mech. Thermodyn. 22, 163–176 (2010)

    ADS  Article  MathSciNet  MATH  Google Scholar 

  4. 4.

    Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells, 2nd edn. McGraw-Hill, New York (1959)

    Google Scholar 

  5. 5.

    Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech. 18, 31–38 (1951)

    MATH  Google Scholar 

Download references

Acknowledgements

We thank Paolo Podio-Guidugli for his helpful comments on our first draft. GRC gratefully acknowledges the support of the European Community under grant RFCS-CT-2012-00026 “S+G”.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Roger Fosdick.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fosdick, R., Royer-Carfagni, G. Singular Shear-Force States in Elementary Plate Theory. J Elast 118, 89–99 (2015). https://doi.org/10.1007/s10659-014-9480-7

Download citation

Keywords

  • Kirchhoff-Love
  • Plate theory
  • Shear-force singularity

Mathematics Subject Classification (2000)

  • 74A30
  • 74K20