In this paper the mode III crack problem for two bonded homogeneous half planes is considered. The interfacial zone is modeled by a nonhomogeneous strip in such a way that the shear modulus is a continuous function throughout the composite medium and has discontinuous derivatives along the boundaries of the interfacial zone. The problem is formulated for cracks perpendicular to the nominal interface and is solved for various crack locations in and around the interfacial region. The asymptotic stress field near the tip of a crack terminating at an interface is examined and it is shown that, unlike the corresponding stress field in piecewise homogeneous materials, in this case the stresses have the standard square root singularity and their variation is identical to that of a crack in a homogeneous medium. With application to the subcritical crack growth process in mind, the results given include mostly the stress intensity factors for some typical crack geometries and various material combinations.