Circular-arc inclusion at isotropic bimaterial interface

Abstract

A general solution to the circular-arc inclusion problem in bonded dissimilar media is provided. The proposed analysis is based on the use of Hilbert problem formulation, and a special technique of analytical continuation. The general expressions of the complex potentials are derived explicitly in both the circular inhomogeneity and the surrounding matrix. Several specific solutions are provided in closed form which are verified by comparison with existing ones. Numerical examples of composite materials under uniform remote load or a point force acting on the inclusion surface are examined and detailed results are presented. The oscillatory behavior in tractions along the interface is also discussed. HE characteristics of the stress field near the tip of a line inclusion have been extensively studied. A review on the related subject was given by Mura.1 Based on the equivalent inclusion approach of Eshelby 2 and the complex variable method of Muskhelishvil i,3 Wang et al. 4 have considered the problem of a rigid line inhomogeneity in an isotropic elastic body under remote loading. They found analytical expressions of the corresponding elastic field and examined the characteristics of the resulting stress singularity. The same problem has also been considered by Erdogan and Gupta,5 Atkinson,6 and Ballarini,7 among others. Based on Stroh 8 formalism for anisotropic elasticity, a line inclusion in an anisotropic elastic solid has been treated by Li and Ting.9 Wu 10 studied a line inclusion at the interface of an anisotropic bimaterial and found that the neartip stress field exhibits oscillatory singularities of the type r~2±y* with r being the distance measured from the tips of the line inclusion and y* a bimaterial constant. In Wu's paper,10 the strain intensity factors are introduced to characterize the near-tip fields instead of using the stress singularity coefficients proposed by Wang et al. 4 All of the aforementioned investigators have concentrated on the plane problem of a line inclusion in an infinite medium. The corresponding problem associated with curvilinear inclusions has been rather limited. Recently, Chao and Shen11 gave exact solutions of the elastic and thermoelastic problem with a rigid circular-arc inclusion. They showed that near-tip stress field exhibits a square-root singularity similar to the case of a traction-free crack. In this paper, a circular-arc inclusion at an isotropic bimaterial interface is investigated. The Hilbert problem formulation and a special technique of analaytic continuation are employed to derive the stress and displacement fields in an explicit form. The general solution of the present problem is provided in Sec. II. The external loads considered in this study consist of a remote uniform load and concentrated force acting on the inclusion surface which are discussed separately in Sec. Ill and Sec. IV, respectively. In Sec. V, numerical examples for commonly used fiber reinforced composite such as carbon/alumin um and tungsten/aluminum systems are given to illustrate the use of the present approach. Finally, Sec. VI concludes the article. II. Formulation and Solution of the Problem Consider two homogeneous, isotropic elastic materials. Let one occupy the region S+, interior to the unit circle, r = 1, whereas the other occupies the infinite region S~, exterior to the unit circle, Fig. 1. The elastic properties of the material in S+ can be specified by the constants IJL and K and those of the material S~ by ^ and K2 where p,j are the shear modulus, and K]-, = (3 - v/)/U + v/) for generalized plane stress and KJ = (3 - 4v;) for plane strain with