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**Decaying states of plane strain in a semi-infinite strip and boundary conditions for plate theory.**
*(English)*
Zbl 0536.73047

In this paper a new method is described for obtaining the appropriate boundary conditions for the interior solution of plate problems. An eigenfunction expansion is given for the conventional class of plane strain states in a semi-infinite elastic strip in the absence of distributed or concentrated loads in the strip interior. This result is then used to relate the non-decaying components of the plane strain states to the edge data. The necessary and sufficient conditions for a decaying state of plane strain follow from these relations. To illustrate how these conditions for a decaying state may be used to obtain boundary conditions for the interior solution, a particular problem involving displacement edge-data is solved. This problem deals with the plane strain deformation of an infinite rectangular block whose sides suffer equal and opposite uniform displacements transverse to the upper and lower faces of the block. The interior solution is calculated and the resultant transverse force required at each edge to produce the prescribed edge displacement is determined.

The authors described the general interior expansion for an infinite strip plate in plane strain deformation, with its upper and lower faces free of tractions and derived the boundary conditions satisfied by the terms of this interior expansion for four different types of edge-data. The boundary conditions obtained for the semi-infinite plate case are rigorously correct and the result for the stress data case rigorously justifies the application of St. Venant’s principle. Applications of the displacement boundary conditions obtained are illustrated by the shearing of an infinitely long rectangular block and a clamped infinite plate strip under uniform face pressure.

The authors described the general interior expansion for an infinite strip plate in plane strain deformation, with its upper and lower faces free of tractions and derived the boundary conditions satisfied by the terms of this interior expansion for four different types of edge-data. The boundary conditions obtained for the semi-infinite plate case are rigorously correct and the result for the stress data case rigorously justifies the application of St. Venant’s principle. Applications of the displacement boundary conditions obtained are illustrated by the shearing of an infinitely long rectangular block and a clamped infinite plate strip under uniform face pressure.

Reviewer: W.A.Bassali

### Keywords:

Betti-Rayleigh reciprocity theorem; boundary conditions; interior solution; eigenfunction expansion; plane strain states; semi-infinite elastic strip; absence of distributed or concentrated loads; strip interior; non-decaying components; edge data; general interior expansion; infinite strip plate; plane strain deformation; upper and lower faces free of tractions; St. Venant’s principle
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\textit{R. D. Gregory} and \textit{F. Y. M. Wan}, J. Elasticity 14, 27--64 (1984; Zbl 0536.73047)

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### References:

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