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.
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.
In this paper we are concerned with the problem of two circular piezoelectric fibers of different radii and distinct material properties, perfectly bonded to a host intelligent material, of infinite extent. The matrix material may be piezoelectric or nonpiezoelectric but, together with the fibers' materials, it possesses the symmetry of a hexagonal crystal in the 6 mm class. The system is subjected to electromechanical loadings (singularities) which produce out-of-plane displacement and in-plane electric fields, but are otherwise arbitrary. Within the framework of the procedure of heterogenization, recently developed by the authors, the solution is sought as a transformation applied to the solution of the corresponding homogeneous problem (i.e., the problem of the host material occupying the full space and subjected to the same sources). The solution is formulated in a manner which leads to some exact results. Universal formulae are derived for the electromechanical field at the point of contact of two piezoelectric fibers. Some quantities which are invariant under the transformation, i.e., quantities which take the same values in the heterogeneous as in the corresponding homogeneous problems, are also discovered. The ramifications of this discovery are investigated. Moreover, the asymptotic behavior of the electromechanical field at the closest points of two plated circular holes or rigid conductors, approaching each other in an intelligent matrix material is also studied and given by universal formulae, i.e., formulae which are independent of the electromechanical sources. The interaction of the fibers with host-material microdefects, such as dislocations, electric line charges and microvoids, is scrutinized. The possibility of manipulating the electrical potential to reduce the high stress level is also discussed.
The heterogenization technique, recently developed by the authors, is applied to the problem, in antiplane elastostatics, of two circular inclusions of arbitrary radii and of different shear moduli, and perfectly bonded to a matrix, of infinite extent, subjected to arbitrary loading. The solution is formulated in a manner which leads to some exact results. Universal formulae are derived for the stress field at the point of contact between two elastic inclusions. It is also discovered that the difference in the displacement field, at the limit points of the Apollonius family of circles to which the boundaries of the inclusions belong, is the same for the heterogeneous problem as for the corresponding homogeneous one. This discovery leads to a universal formula for the average stress between two circular holes or rigid inclusions. Moreover, the asymptotic behavior of the stress field at the closest points of two circular holes or rigid inclusions approaching each other is also studied and given by universal formulae, i.e., formulae which are independent of the loading being considered.
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