Interaction between a radial matrix crack and a three-phase circular inclusion with imperfect interface in plane elasticity

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 paper, 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 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 conditions and the prescribed asymptotic loading conditions. Stress intensity factor (SIF) calculations are performed at the crack tips for different material property combinations, imperfect interface conditions 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. For instance, when the interphase layer is softer than the inclusion and matrix, the results show that the radial matrix crack will propagate from the nearby crack tip regardless of the level of the imperfect (spring-layer) interface parameter. In comparison, when the interphase layer is stiffer than the inclusion and matrix, the interphase layer will shield the crack from effects of the imperfect (spring-layer) interface. Hence, these results provide a quantitative description of the interaction problem between a three-phase inclusion with interface imperfections and a radial matrix crack.