The solution for the elastic three-phase circular inclusion problem plays a fundamental role in many practical and theoretical applications. In particular, it offers the fundamental solution for the generalized self-consistent method in the mechanics of composites materials. In this thesis, a general method is presented for evaluating the interaction between a pre-existing radial matrix crack and a three-phase circular inclusion. The bonding at the inclusion-interphase interface is considered to be imperfect with the assumption that the interface imperfections are constant. On the remaining boundary, that being the interphase-matrix interface, the bonding is considered to be perfect. Using complex variable techniques, we derive series representations for the corresponding stress functions inside the inclusion, in the interphase layer and in the surrounding matrix. The governing boundary value problem is then formulated in such a way that these stress distributions simultaneously satisfy the traction free condition along the crack face, the imperfect interface condition and the prescribed asymptotic loading conditions. Stress intensity factor (SIF) calculations are performed at the crack tips for different material property combinations and crack positions. The results illustrate convincingly the role of an interphase layer as well as the effects of an imperfect interface on crack behavior. Moreover, the conclusions reached in this dissertation provide a quantitative description of the interaction problem between a three-phase circular inclusion with interface imperfections and a radial matrix crack.