Path-independent integral analyses for microcrack damage in dissimilar anisotropic materials

This paper deals with microcrack damage below an interface in a dissimilar anisotropic material. The interaction problem among arbitrarily oriented and located subinterface microcracks is studied in detail by solving the associated singular integral equations derived from the dislocation technique and superimposing technique. The path-independent integral analyses are performed by adopting theJk-integral vector and theM-integral, which are evaluated, respectively, along specially introduced contours. It is found that the total contributions of the subinterface microcracks to the first component of theJk-vector vanish, provided that the chosen contour encloses all the microcracks. This leads to a conservation law of theJ1-integral in the microcrack damage problem, from which a consistency check or a necessary condition for this kind of problem is presented. Similar conclusions are also given for the second component of the vector under the more strict assumption that the closed contour chosen to calculate theJ2-integral is infinite large and encloses not only all the microcracks, but also the whole interface. Therefore, theM-integral analysis could be performed, which is proved to be divided into two parts. One of them, called the net part, is the simple summation of the distinct contribution induced from the stress intensity factors at all microcrack tips, while the other part, called the additional part, is induced from the global coordinates of each microcrack center and theJk-vectorevaluated along a specially introduced contour surrounding only one single microcrack completely. Numerical results are given for a particular dissimilar anistropic material whose upper half part shows anisotropy (the fiber direction is parallel to the interface) and the lower half part shows isotropic (the fiber direction is perpendicular to the global coordinate plane). It is concluded that the numerical results do actually meet the consistency check. Moreover, it is found that the values of theM-integral for the present dissimilar material is always smaller than those for the corresponding isotropic material, no matter how the subinterface matrix microcracks are located and oriented. It is concluded that theM-integral plays an important role in the present microcrack damage problem and that any aray of matrix microcracks below the interface shows less unstable nature than the same array in the corresponding isotropic material.