Finite plane strain

A general theory of plane strain, valid for large elastic deformations of isotropic materials, is developed using a general system of co-ordinates. No restriction is imposed upon the form of the strain-energy function in the formulation of the basic theory, apart from that arising naturally from the assumption of plane strain. In applications, attention is confined to incompressible materials, and the general method of approach is illustrated by the examination of a number of problems which are capable of exact solution. These include the flexure of a cuboid, and of an initially curved cuboid, and a generalization of the shear problem. A method of successive approximation is then evolved, suitable for application to problems for which exact solutions are not readily obtainable. Attention is again confined to incompressible materials, and the approximation process is terminated when the second-order terms have been obtained. In considering problems in plane strain, complex variable techniques are employed and the stress and displacement functions are expressed in terms of complex potential functions. In dealing with finite elastic deformations, a complex co-ordinate system may be chosen which is related either to points in the deformed body or to points in the undeformed body, and in the present paper both methods are developed. The theory is applied to obtain solutions for an infinite body which contains either a circular hole or a circular rigid inclusion, and which is under a uniform tension at infinity.