Finite strain in elastic problems

Abstract

The mathematical theory of Elasticity, as at present developed, is based on the assumption that the displacements that have to be considered in elastic solid bodies are so small that the squares and products of the first differential coefficients of m, v, w with respect to x, y, z can be neglected in comparison with their first powers. That is why we cannot use it to get a satisfactory solution of many a problem in elasticity in which the displacement is finite and the strain produced is not small enough to justify the above assumption. For example, we can bend any rectangular plate in the form of a cylinder by couples applied to the edges only. As far as I know there exists no exact solution for this simple problem. In Section III I have attempted to give its solution based on the theory of finite strain. The first step towards the solution of the type of problems we have in mind is to get the components of strain corresponding with any displacement. Like the body-stress equations, these should be referred to the actual position of a point P of the material in the strained condition, and not to the position of a point considered before strain. The importance of this point, overlooked by various authors, cannot be exaggerated. Apparently Filon and Coker were the first to notice it and stress its importance. To a first approximation it is immaterial which method of reference is adopted, for the values of the strain components in the two cases differ only in the second order terms.