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.

The material force acting on a screw dislocation in the presence of a multi-layered circular inclusion

“…The material (or Peach-Koehler) force on the screw dislocation will be evaluated using Budiansky and RiceÕs formula (1973) for the J-integral in anti-plane elastostatics. In the case of a screw dislocation residing in the matrix, and apart from a missing factor due to a misprint, our result for the Peach-Koehler force agrees with the one obtained by Honein et al (1994).…”

“…The equations derived in this section agree with the ones obtained by Honein et al (1994). However, their final expression for the Peach-Koehler force contains some misprints.…”

“…it can then be shown (see Honein et al, 1994) that, if the solution to the homogeneous problem is expressed by a Taylor series expansion as / ¼ P 1 k¼1 b k z k , where b k , k=1, . .…”

“…1016/j.ijsolstr.2005.05.054 holomorphic function, Cauchy integral and Laurent series expansion techniques to arrive at explicit series solutions for the two cases when the screw dislocation is located either in the inclusion or in the matrix. It turns out that the latter case is a particular instance of a much more general treatment carried out some years earlier by Honein et al (1994). These authors have treated the case where an arbitrary number of layers are embedded between the inclusion and the matrix, which may be subjected to arbitrary singularities and/or loading.…”

“…Following the same line of reasoning as presented by Honein et al (1994), we express the solution to this heterogeneous problem in terms of the solution of the corresponding homogeneous problem, i.e., when the layers and the matrix are absent and the inclusion, still subjected to the same singularities, occupies the whole space. We show that this can be achieved by exploiting a connection between the solution to the heterogeneous problem and a group structure on the set I = (À1, 1), of real numbers x, such that À1 < x < 1.…”