© 1952 by Oxford University Press
UNIQUENESS THEOREMS OF TWO-DIMENSIONAL ELASTICITY THEORY
( Birkbeck College London )
This paper is concerned with elastic material occupying a multiply-connected region and in a state of plane strain or generalized plane stress. For a finite material it is shown that a solution giving specified stresses over all boundaries is effectively unique, i.e. the stresses are uniquely determined throughout the material and the displacements are unique except for rigid body displacements, and that a solution giving specified displacements over all boundaries is unique. The theorems are stated so that they retain their significance if the boundary loading is discontinuous, and the proofs make no appeal to special forms of the complex potential functions. The hitherto neglected problem of material for which the exterior contour is partly at infinity and partly in the finite part of the plane is considered, and a solution giving required stresses over all boundaries in the finite part of the plane and specified stresses at infinity of orders higher than o (r–1), where r is the distance from the origin, is proved to be effectively unique. The simple requirement of vanishing stresses at infinity is shown to be, in general, insufficient to define a unique elastic problem, whilst on the other hand stresses at infinity which are o (r–1) may not be arbitrarily specified. A solution giving specified displacements over all finite boundaries and prescribed finite and infinite displacements at infinity is shown to be unique, but an arbitrary assignment of infinitesimal displacements at infinity is not permissible. The special cases of material extending to infinity in all directions and of material occupying a half-plane are considered in detail.