Circularly Cylindrical and Plane Layered Media in Antiplane Elastostatics

Abstract

In this paper we consider, within the framework of the linear theory of elasticity, the problem of circularly cylindrical and plane layered media under antiplane deformations. The layers are, in the first instance, coaxial cylinders of annular crosssections with arbitrary radii and different shear moduli. The number of layers is arbitrary and the system is subjected to arbitrary loading (singularities). The solution is derived by applying the heterogenization technique recently developed by the authors. Our formulation reduces the problem to solving linear functional equations and leads naturally to a group structure on the set t of real numbers such that −1 < t < 1. This allows us to write down the solution explicitly in terms of the solution of a corresponding homogeneous problem subjected to the same loading. In the course of these developments, it is discovered that certain types of inclusions do not disturb a uniform longitudinal shear. That these inclusions, which may be termed “stealth,” are important in design and hole reinforcements is pointed out. By considering a limiting case of the aforementioned governing equations, the solution of plane layered media can be obtained. Alternatively, our formulation leads, in the case of plane layered media, to linear functional equations of the finite difference type which can be solved by several standard techniques.