Mohr circles for large and small strains in two-dimensional deformations

Abstract

The mathematical apparatus is provided for the analysis of large deformations in two dimensions. Deformations are considered as co-ordinate transformations. For homogeneous deformations, a method of finding the directions and magnitudes of the principal strains is presented. The formulae used may be expressed either in terms of the elements of a transformation matrix or in those of displacement gradients. It is shown that, when the displacements are small, the results are reduced to the familiar formulae for small strains in linear elasticity. The Mohr circles for Green's and Cauchy's deformation tensors are also discussed, and their relations with the Mohr circle for the matrix of pure deformation shown. Examples of Mohr circles are provided for some commonly known types of deformation.