ABSTRACT
In this chapter, the authors consider mathematical models of cracks and inclusions in elastic media. Each problem analyzed here involves a small positive parameter characterizing the geometry of the domain. The objective is to construct the asymptotic representation of the stress and displacement fields and, in some cases, to evaluate the critical load which provides the brittle fracture of the inhomogeneous elastic medium. It allows one to take into account the longitudinal external load. The general case of a uni-axial load, applied at infinity, is considered. For the case of a longitudinal compression they shall need two terms of the asymptotic expansion of the displacement field in a neighbourhood of the tips of a crack, and it enables us to explain the effect of a rock fracture where the region of failure is characterized by a triangular shape.
