Anti-plane shear stress problem of two bonded dissimilar half spaces with an elliptical hole or rigid inclusion on the interface

The problem of anti-plane shear stress of two bonded dissimilar half spaces with an elliptical hole or a rigid inclusion at the interface and having interfacial cracks is presented. Uniform anti-plane shear stresses and the stress free or zero displacement boundary conditions on the elliptical hole are considered. The two cases are reduced to Riemann-Hilbert problems and closed form solutions are obtained by use of the complex stress function and the conformal mapping approaches. Stress distributions, as well as stress intensity factor, are shown. When the elliptical hole collapses, the known solutions of the interfacial crack and thin rigid fiber can be obtained. If the coordinates in the Plemelj function are changed, a debonding length can be determined.