A general solution to the antiplane problem of two concentric circular-ave inclusions in bonded dissimilar materials is provided. Based upon the Hilbert problem formulation and the technique of analytical continuation the expressions of the complex potentials are derived explicitly in both the unit disk and the surrounding medium. Both the stress singularity coefficients and contact stress are provided in an explicit form and the former are verified by comparison with the existing ones. The effects of material and geometrical parameters upon the contact stress and stress singularity coefficients are examined and shown in graphic form. Failure initiation of an infinite bi-material plate with two are inclusions is also discussed.