# CAIMS Prize Award

Department of Mathematics and Statistics, Simon Fraser University

Supervisor: G. A. C. Graham

Thesis title: Some Elastic Multi-Crack and Multi-Punch Problems

Abstract:

In this thesis, two types of problems involving a homogeneous isotropic linear elastic medium are studied. The first problem is that an elastic body, which could be an infinite solid or a sem-infinite solid or a layer, containing two or more parallel penny-shaped cracks whose upper and lower surfaces are loaded by equal and oposite arbitrary tractions. The second problem is concened with two more more cicular punches, whose faces are of arbitary shape, indenting the surface of an elastic layer which rests on a rigid foundation. Both normal indentation and tangential indentation are examined.

The method used in this thesis is based on a formulation, which is similar to that given by Muki[24], for general three-dimensional asymmetric elasticity problems and the superposition principle of linear elasticity. For example, the solution to the problem of an elastic layer containing two penny-shaped cracks under arbitrary loadings can be considered as superposition of the solutions of two layer problem, each containing one penny-shaped crack. Furthermore, the solution to the problem of layer weakened by one penny-shaped crck can be decomposed into the sume of solutions fo two basic problems, namely problem with an infinite solid containining one crack and problem of a layer without any crack. With the aid of the solutions to the basic problems and the addition formula for Bessel functions, we can superpose solutions in different cylindrical co-ordinates and then reduce the original problem to a system of Fredholm integral equations of the second kind, which can be solved by iteration for some special cases.

All the solutions given in this thesis can be extended to the case where the elastic solid is transversely isotropic. More important, the method used in this thesis applies to any other problem if the problem can be decompsed into several basic problems and analytic solutions to the basic problems in terms of some potential functions are vailable.

[24] R. Muki, Asymmetric problems of the theory of elasticity for a semi-infinite solid and a thick plate, Progress in Solid Mechanics Vol.1 (editors: I. N. Sneddon and R. Hill), pp401-439, North-Holland Publishing Company, Armsterdam (19961).