The interaction of two circular multi-layered coated fibers in a composite under arbitrary loading/singularities in anti-plane elasticity

Abstract

In this paper we present a solution, within the framework of the linear theory of elasticity, to the problem of an infinite matrix in which two parallel infinitely long circularly cylindrical inclusions are embedded. The inclusions have different radii and are formed of materials of different shear moduli. The interfaces between the matrix and the inclusions are formed of two different multi-layered interphases, The layers are coaxial cylinders of annular cross-sections with arbitrary radii and different shear moduli. The number of layers may also be arbitrary. Continuity of traction and displacement across all interfaces is assumed. The matrix is subjected to loading/singularities, which induce anti-plane deformation, but are otherwise arbitrary. The exact, analytical solution to this heterogeneous problem, when only one multi-layered fiber is present, was derived some years ago by the authors. This general solution, which is valid under arbitrary loading, to a single multi-layered coated fiber is used in this paper as a building block for a methodology of successive approximation which yields through a recursive scheme the solution of two multi-layered coated fibers. The convergence is rapid and is assured even in the extreme case when the fibers are replaced with holes touching each other. Furthermore, the accuracy can be set a priori and the series will be automatically truncated once the desired accuracy is achieved. Examples are provided, including the interaction of a screw dislocation located in the matrix with two multi-layered coated fibers. This last example can serve as a tool to model a crack as a pile-up of screw dislocations in such composite material.