ON 2 CIRCULAR INCLUSIONS IN HARMONIC PROBLEMS

In this paper, we derive the solution for two circular cylindrical elastic inclusions perfectly bonded to an elastic matrix of infinite extent, under anti-plane deformation. The two inclusions have different radii and possess different elastic properties. The matrix is subjected to arbitrary loading. The solution is obtained, via iterations of Mobius transformations, as a rapidly convergent series with an explicit general term involving the complex potential of the corresponding homogeneous problem, i.e., when the inclusions are absent and the matrix material occupies the entire space and is subjected to the same loading. This procedure has been termed "heterogenization." The technique used can be applied to problems governed by Laplace's equation. Finally some remarks are included concerning the relation of our solution to the theory of discontinuous groups and automorphic functions and possible generalizations to multiple inclusions.