Two-dimensional theory of elasticity for finite deformations

Abstract

A general theory of plane stress, 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, which follows similar lines to the treatment by Adkins, Green & Shield (1953) of finite plane strain. The reduction of the equations to two-dimensional form subsequent to the assumption of plane stress enables the theory to be presented in complex variable notation. A method of successive approximation is evolved, similar to that developed for problems in plane strain, which may be applied when exact solutions are not readily obtainable. The stress and displacement functions are expressed in terms of complex potential functions, and in the present paper the approximation process is terminated when the second-order terms have been obtained. The theory is formulated initially in terms of a complex co-ordinate system related to points in the deformed body, and the corresponding results for complex co-ordinates in the undeformed body are then obtained by a simple change of independent variable. Approximation methods are also applied to compressible materials in plane strain, and it is shown that the second-order terms for plane stress and plane strain can be expressed in similar forms. This leads to a general formulation of the second-order theory for two-dimensional problems, the results for plane stress or plane strain being derived by introducing the appropriate constants into the expressions thus obtained.