The methods based on the properties of the two-state integrals allow one to calculate the amplitude of singular and the other terms of the Williams’ asymptotic expansion. The paper is focused on the use of the Y-integral, whose application is conditioned by the knowledge of the so-called auxiliary solution of the solved problem. On the other hand, the Y-integral can be applied to the analysis of the problems with various geometries, e.g. the analysis of the bi-material notches. The application of the Y-integral can be also extended to the matched asymptotic procedure, which allows one to predict the behavior of the cracked notches or following crack growth near the bi-material interfaces.