© 1954 by Oxford University Press
ELLIPTIC ELASTIC INCLUSION IN AN INFINITE ELASTIC PLATE
( Westfield College, University of London )
It is shown that constant stresses at infinity in an elastic plate containing an elliptic inclusion of different material induce constant stresses in the elliptic inclusion, a result similar to the known fact that a uniform two-dimensional electric field will induce in an elliptic dielectric cylinder placed in it a uniform field in the dielectric. It is interesting that similar results should hold in the electric and elastic problems, since the first is essentially a harmonic problem, whilst the second involves biharmonic functions.
The plate is assumed to be in a state of generalized plane stress, and the method consists in finding the complex potentials 
(z) and 
(z) inside and outside the inclusion which give
- appropriate stresses at infinity or in the inclusion.
- continuity of displacement and mean stresses nn and ns across the interface of arc s and normal n. There will, in general, be an interesting discontinuity in the peripheral stress 88 round the interface.
Since these solutions were obtained 
my attention has been drawn to some earlier results in this field obtained by Donnell (1) by the Airy stress function technique in terms of elliptic curvilinear coordinates. A comparison will reveal considerable advantage in using the complex potential. Moreover Donnell's solutions are all for the case in which the rigidities of the materials are the same—an unnecessary restriction. Further, Donnell's treatment does not bring out the simple constant nature of the stress fields in the elliptic inclusion. This seems to the present writer to be one of the most interesting features of the solutions presented here.