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# br Conclusion br Introduction The

Conclusion

Introduction
The interaction between cracks and interfaces of materials is an important problem in the mechanics of heterogeneous structures. The cracks growing in the process of such interaction can either be deflected by the interface with secondary debonding cracks forming or break when penetrating into another medium [1,2].
Starting with the classical article by Zak and Williams [3], numerous publications have been dedicated to the interaction of cracks with a bimaterial interface as a plane problem [1,2,4–8]. These papers considered plane cracks of finite or semi-infinite length, either oriented perpendicularly or inclined to a straight interface. Refs. [1,2,4–6] actually used a model of a perfect interface, in which the material interface is regarded as a layer of zero thickness, i.e., a line along which the elastic moduli of materials have discontinuities, while the displacements and stresses tangential and normal to this PLX 4032 line preserve continuity. Ref. [7] considered an imperfect interface, and in Ref. [8] the interface was simulated by a layer of a functionally gradient material of finite thickness. Of particular note, Ref. [9] obtained an exact solution for a semi-infinite mode I crack approaching a perfect interface, and determined the material parameter controlling the crack\'s stable growth.
Anti-plane cracks have received far less attention in studies in this direction. Apparently, the first study on this subject was Ref. [10], which considered a crack either terminating at a perfect interface or intersecting it. It was established that for a crack with a tip located at the interface between two dissimilar materials, the singularity of the stress field had a power-law behavior. However, its exponent was different from the standard value of 0.5 and was determined by the first root of the characteristic equation. An analysis of an inclined crack meeting a perfect linear interface was carried out in Ref. [11]. Atkinson [12] obtained the asymptotic formula for the stresses in the crack tip at small distances to the straight interface. The breaking of a semi-infinite mode III crack was investigated by Kuliyev [13]. An antiplane crack of finite length, located in the functionally graded coating near the functionally graded substrate, was considered in Ref. [14]. Erdogan [15] established that the singularity exponent had a classical value of 0.5 for a crack terminating perpendicular to the interface between two functionally gradient materials in perfect contact with the direction of the material gradient coinciding with the direction of the crack. In other words, in this case the continuity of elastic moduli was shown to generate a standard square-root singularity, as in the case of a homogeneous material.

Problem statement and conversion to the Wiener–Hopf equation
Let us consider a semi-infinite mode III crack the located in the Ω2∪Ω3 matrix, whose tip lies at a distance ε from the vertex of a wedge-like inclusion Ω1 (Fig. 1). A self-balanced load g(r) is applied to the edges of the crack. The materials of the inclusion and the matrix are assumed to be homogeneous and isotropic with shear moduli μ1 and μ2, respectively. The contact at the material interface is assumed to be perfect.
The geometry of the elastic composite under consideration can be conveniently described by two parameters: the vertex angle of the inclusion, α, (0 < α < 2π) and the angle β at which the crack approaches the inclusion, that is, the angle between the direction of the initial crack and the symmetry axis of the inclusion. It is evident that
Varying the angle β with a fixed α value causes the inclusion to rotate around its vertex. Thus, the angle β characterizes the mutual orientation of the crack and the inclusion. For example, the problem is symmetric for β=0. An interfacial crack corresponds to the values
and the values correspond to the cases when the crack approaches the inclusion along the interphase boundary.