Interior Cracking of a Circular Inclusion with Imperfect Interface under Thermal Loading

Abstract

A semi-analytic solution is developed to determine the effect of an imperfect interface on the stress field inside a circular elastic inclusion containing a preexisting interior radial crack, subjected to thermal loading. The inclusion is surrounded by an infinite matrix of different elastic material with the inclusion- matrix interface assumed to be homogeneously imperfect. This is characterized by continuity of tractions and discontinuity of displacements across the interface. Using complex variable methods, we derive series representations of the corresponding stress functions both inside the circular inclusion and in the surrounding matrix. The governing boundary value problem is then formulated in such a way that these stress functions simultaneously satisfy the traction-free condition along the crack face, the imperfect interface conditions, and the prescribed asymptotic conditions. The thermal load is modeled by a prescribed volume eigenstrain which characterizes the stress-free thermal strain of the inclusion. The method is illustrated for a number of crack-inclusion geometries and shear moduli ratios (of inclusion to matrix). We present explicit values of the stress intensity factor at the crack tips. These results can be used to ascertain the direction of initial crack propagation in, for example, fiber-reinforced composites or in components subjected to thermal loading in microelectronic packaging. To demonstrate the accuracy of the results obtained in this paper, we draw comparisons with corresponding cases documented in the literature where analytical results are available.