Stress Intensity Factor for an Interphase Griffith Crack Interacting with Two Imperfect Interfaces

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 work, a semi-analytical method is presented for the problem of a pre-existing radial Griffith crack embedded within the interphase layer surrounding a circular inclusion. Novel to this work is that the bonding at the inclusion-interphase interface and the interphase-matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. Employing 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. The advantage of the series method over other methods, such as the dislocation density method, is that in the former case the resulting expressions are linear and can be solved readily whereas in the latter case the method leads to cumbersome integral equations which are often numerically difficult to solve. Stress intensity factor (SIF) calculations are performed at the crack tips for different material property combinations, imperfect interface conditions and crack locations under mode I loading. The results not only provide for a quantitative description of the interaction between a radial interphase crack and a three-phase inclusion with imperfect interfaces but the results clearly demonstrate the significance of how two imperfect boundaries can influence crack behavior.