Chapter 1 - Mathematical Preliminaries

Abstract

Elasticity theory is formulated in terms of a variety of variables including scalar, vector, and tensor fields, and this calls for the use of tensor notation along with tensor algebra and calculus. Through the use of particular principles from continuum mechanics, the theory is formulated as a system of partial differential field equations that are to be solved in a region of space coinciding with the body under study. Solution techniques used on these field equations commonly employ Fourier methods, variational techniques, integral transforms, complex variables, potential theory, finite differences, and finite and boundary elements. Therefore, in order to develop proper formulation methods and solution techniques for elasticity problems, it is necessary to have an appropriate mathematical background. The purpose of this initial chapter is to provide this background primarily for the formulation part of our study. The chapter includes material on scalars, vectors, matrices, tensors, index notation, coordinate transformation, matrix principal value problem, calculus of Cartesian tensors, and curvilinear coordinates.