Understanding Non-Linear Strain-Displacement Relations

The first monograph, titled Understanding the Elastic Stress Field Around an Elliptical Hole in a Thin Plate (in the deformed configuration)- ISBN 978-1-48359-262-6, utilized stress equilibrium conditions to develop expressions for stress components which contain a parameter a_e. This second monograph does not deal with stress equilibrium conditions. It deals with strain-displacement relations as well as stress-strain relations and shows how to evaluate a_e for linear elastic materials.Consequently, the first monograph is useful for courses in stress analysis. It helps teach the basics of elliptical co-ordinate system which is very useful in dealing with "extreme limits of form which an ellipse can assume i.e. a circular hole to a fine straight crack". It focuses on understanding the impact of geometrical non-linearity without having to worry about the concepts of strain and displacement. Complex variables are used to develop the Airy stress functions so as to satisfy the field equations of equilibrium, expressed in the current (deformed) configuration.The second monograph is useful for courses that deal with strain and displacements. Co-ordinate transformations are utilized to develop expressions for displacement and deformation gradients in an elliptical co-ordinte system. Linear and non-linear strain-displacement relations with constitutive (stress-strain) relation for isotropic linear elastic materials are used to develop a clearer evaluation of stresses and strains. It is proved that Singh-Glinka-Dubey model is a special case of the general model developed using the Engineering strain definition. Various other models are also developed using definitions for Green, logarithmic, and Almansi strains.It is intended to publish the third monograph dealing with applications based on concepts developed in the first two monographs.