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Singular Shear-Force States in Elementary Plate Theory


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.

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  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.


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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”.

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Correspondence to Roger Fosdick.

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Fosdick, R., Royer-Carfagni, G. Singular Shear-Force States in Elementary Plate Theory. J Elast 118, 89–99 (2015).

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

Mathematics Subject Classification (2000)

  • 74A30
  • 74K20