Further Aspects of the Elastic Field for Two Circular Inclusions in Antiplane Elastostatics

Abstract

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.

Explicit Solutions of Plane Elastostatics Problems in Heterogeneous Solids

“…Until now we have completed the derivation of the solution, in plane elastostatics, for an infinite domain containing two circular inclusions, which is obtained from the solution of the corresponding homogeneous problem merely by substitution into a simple algebraic expression. This was termed "heterogenization" by Honein et al [11]. Keeping in mind that the relation is universal in the sense of being independent of the loading considered.…”

“…3). The transformation functions in (11) and (23) After immediate application of the expressions given in (25) ~ (35), the solution of the heterogeneous problem can be readily obtained. In order to demonstrate the accuracy of the present derived solution, the problem of two nearby holes under remote uniform tension is considered first.…”

On multiple circular inclusions in plane magnetoelasticity

“…The elasticity problems of a finite number of arbitrarily located inclusions within an infinitely extended medium were treated by Moschovidis and Mura (1975), Tandon and Weng (1986), Rodin and Hwang (1991), Gong and Meguid (1993). Several works have devoted to the elastic fields in antiplane elastostatic problems of two circular inclusions, such as Gorce and Wilson (1967), Budiansky and Carrier (1984), Steif (1989), 0020 Honein et al (1992). All the studies mentioned above have focused on the elastic fields induced by pure mechanical loading.…”

Heterogeneous Problems in Plane Thermoelasticity

“…The thermal loadings considered in this paper include a remote uniform heat flux and a point heat source (or sink). We propose to express the solution of this problem in terms of the solution of the corresponding much simpler problem of an infinite homogeneous body subjected to the same loading, a methodology which was termed "heterogenization" [1]. Heterogenization refers roughly to the methodology whereby the solution of a heterogeneous problem is sought in terms of a transformation performed on the solution of the corresponding homogeneous problem.…”