On two circular inclusions in harmonic problems

Abstract

Abstract. 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.

A Method of Images for the Evaluation of Electrostatic Fields in Systems of Closely Spaced Conducting Cylinders

“…Rather than include higher-order multipole corrections as in Rayleigh's method, the method of images [10,18,22,24] proceeds by representing the field in terms of a series of dipole fields. For ease of notation, we equate the plane R 2 with the complex plane C. LEMMA 2.1.…”

Analysis of In-Plane Problems with Singular Disturbances for an Isotropic Elastic Medium with Two Circular Holes or Rigid Inclusions

“…Note that the boundary condition on L 1 is not satisfied by the previous steps on L 2 . For this reason, we may setf 3 (z) andĝ 3 (z) to satisfy the continuity on L 1 .…”

“…The purpose of the present study is to apply the reflection principle of Moriguchi (1) , who investigated a single hole in in-plane problems, and we use the techniques of Honein (2) and Hirashima (4)∼ (6) to consider anti-plane problems. Using these techniques, we expand this single hole problem to a problem involving two circular holes or rigid inclusions.…”

Analysis of In-Plane Problems for an Isotropic Elastic Medium with Many Circular Elastic Inclusions

“…The purpose of the present study is to apply the reflection principle of Moriguchi (1) , Dunders (2) and Sendeckyj (3) who investigated a single hole or inclusion in the in-plane problems, and the techniques of Honein (4) and Hirashima (5)∼ (7) to consider anti-plane multi-inclusion problems. We obtained general solutions (8) (9) for up to two circular inclusions.…”