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The problem of determining the effect of the media interface in a piecewise-homogeneous body on the stress concentration in the vicinity of cracks located in one of the half-spaces is reduced to a system of two-dimensional singular equations of Newtonian potential type. We study the effect of the relations of the elastic constants of the materials of a body composed of two half-spaces on the stress intensity factors in the vicinity of a crack. We study different variants of the crack location relative to the interface.The relations between the elastic constants of the materials of a piecewise-homogeneous body have a strong effect on the stress concentration in the vicinity of cracks located in the body. These problems are mathematically quite complicated and cumbersome, as shown by the limited amount published on them. A piecewise-homogeneons body composed of two half-spaces is the simplest object for which it is possible to study the effect of the relations between the elastic constants of the materials on the stress concentration in the vicinity of cracks.Suppose an infinite piecewise-homogeneous body composed of two half-spaces contains in its lower halfspace a system of N planar cracks, arbitrarily situated. The surfaces of the cracks are subject to given selfbalancing external forces N/., j = 1, 3, n = 1, N. Here NI. and N2. are the shears, and N~. are the normal external forces prescribed on the n th crack. The upper half-space is characterized by its Poisson coefficient v t and its shear modulus G~. The lower half-space, weakened by the system of N cracks is characterized by the respective elastic constants v 2 and G 2 .To solve the problem of determining the stress concentrations in the vicinity of the cracks we choose a basic Cartesian coordinate system OXIoX2oX30 in such a way that the xtoox2o-plane coincides with the interface between the media. We also prescribe local Cartesian coordinate systems Oxl.x2.x3. with origin O. in the region S., n = 1, N, occupied by the nth crack so that the x~.O.x2.-plane coincides with the region of the crack S.. When this is done, the opposite faces of the cracks S. ~ correspond to the values x3. = _+0. The locations of the cracks in the lower half-space are defined by giving the distances between the origins of the coordinate systems, the direction cosines ej0 . , ej,,0, j = 1,---3, n = 1, N, of the vector d.0 joining the points O. and O (Fig. 1), and the direction cosines of the axes of the local coordinate systems O.xj. in the Ox~oX2oX3o coordinate system, which are given by Table 1.The external loads producing a strain on the piecewise-homogeneous body under consideration cause a mutual displacement of the opposite faces of the cracks. These displacements are characterized by functions ctj., j=l, 3, n=l,N. In the process of reducing the problem of determining the stress-strain state of this body with cracks to boundary integral equations it is necessary to have a solution that makes it possible to satisfy the boundary

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We reduce a three-dimensional problem of the theory of elasticity about the interaction of cracks in a body to the solution of a system of boundary integral equations for functions that characterize mutual displacements of the opposite surfaces of the cracks in the process of deformation of the body. For the case of parallel cracks, the regular kernels of these equations taking into account the interaction of cracks across the interface of materials are presented in the explicit form. The deduced boundary inte= gral equations are solved by using a numerical-analytic approach. For two disk-shaped parallel cracks in different materials loaded by normal forces, we obtain the dependences of the stress intensity factors on the ratio of elastic constants of the components of the body for various distances from the crack to the interface of materials.The problem of interaction of plane cracks in homogeneous bodies is fairly well studied by the method of reduction to boundary integral equations [1,2]. The same problems for three-dimensional piecewise-homogeneous bodies are studied insufficiently due to the absence of the corresponding fundamental solutions. In the present work, we consider the interaction of plane parallel cracks across the interface of materials. For this purpose, we use an integral representation of the solution for a continuous piecewise-homogeneous body formed by two half spaces constructed in the form of a combination of harmonic potentials [3]. The problem is reduced to singular integral equations. The expressions for regular kemels taking into account the interaction of cracks across the interface of materials are obtained in the explicit form. The integral equations of the problem are solved numerically.1. Consider an infinite body formed by two soldered half spaces and weakened by a system of K + N arbitrarily located mutually disjoint plane cracks. The crack surfaces are loaded by self-balanced forces tangential ( N~/m ) and "2mJ~r(i) ~ and normal (N~) to the surfaces of the mth crack (i = 1, 2, m = 1, M). We denote by K the number of cracks in the upper half space characterized by the shear modulus G 1 and Poisson's ratio v 1 and by N the number of cracks in the lower half space with elastic constants v 2 and G2. Also let M = K + N. The domains with cracks are denoted by Sin, where m = k in the upper half space and m = n in the lower half space.On the interface of materials, we choose a principal Cartesian coordinate system Oxlox20x30 and K + N local Cartesian coordinate systems OkXlkX2kX3k (k = 1, K) and OnYlnY2nY3n (n = 1, N) (Fig. 1). The opposite surfaces S~ of the cracks correspond to the values Z3m = + 0. Here, Zjm ----Xjk (J = 1~ ) if the subscript m coincides with the subscript k and Zjm = Yjn (J = 1, 3) whenever m = n. The locations of the cracks are specified by the distances between the origins of coordinates, the direction cosines of the vectors connecting the origins of the corresponding coordinate systems, and the direction cosines of the axes O k Xjk and O n Yjn in the system Ox...