The Use of Various Transforms in the Solution of Boundary Value Problems in Linear Elasticity

This thesis uses combinations of Fourier, Hankel and Weber-Orr transforms to solve boundary value problems in linear elasticity for compressible and incompressible isotropic elastic materials occupying two different geometries. The basic equation of elasticity are developed using the theory of successive approximations as originally described by Green and Adkins [16]. The first problem studies the deformation of an elastic annulus formed by punching a circular hole axially through the centre of a circular disk of uniform thickness. A rigid shaft is passed through this hole and bonded to the annulus. Deformation is induced by applying static and dynamic forces to the shaft. General solutions to this problem are obtained using Weber-Orr transforms in the radial variable and Fourier transforms in the axial variable. The exact problem is thereby reduced to a pair of integral equations which are then solved by representing the two unknown functions using a suitable spectral expansion. The coefficients of these expansions are obtained by solving a system of linear equations. In fact, the solution of the static problem has been approximated by Adkins & Gent [11]. Very good agreement is obtained with their approximate model for a thick annulus but the agreement deteriorates as the annulus becomes thinner principally due to the increasing presence of boundary layer effect. The second problem investigates the deformation of a right circular column under axial load. The plane ends of the column are covered by rigid plates that are bonded to it while its curved surface is stress-free. Moghe and Neff's [27] construct two mathematically different but physically equivalent series solutions to this problem. The first solution is based on the roots of the Bessel function J0(X) while the second uses the roots of the Bessel function J1(x). Unfortunately both "equivalent" solutions predict different shapes for the radial displacement of the curved free surface. The discrepancy is most striking where the curved surface joins the rigid plates. This problem is converted into a pair of integral equations using Fourier transforms in the axial variable and Hankel transforms in the radial variable. Our analysis predicts a curved surface that most resembles that derived by Moghe and Neff using the zeros of J0(X)