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The three-dimensional dynamic problem of coplanar circular cracks in an elastic half-space with a clamped boundary condition is considered. The crack faces are subjected to harmonic loads. The problem is reduced to a system of two-dimensional boundary integral equations of the type of the Helmholtz potential for unknown discontinuities in the displacements of the opposite faces of the cracks. The stress intensity factors at the crack contours are obtained and discussed.Introduction. It is well known that the strength of real bodies depends heavily on the presence of structural defects such as cracks and inclusions, which are often stress concentrators. In studies of the stress-strain state of bodies with defects, particular attention is paid to the inertia effects caused by dynamic loads. Thus, in the case of infinite and semi-infinite bodies with cracks under harmonic and impact loading, the stress concentrations arising near the defects can far exceed the static values [1-4]. The above-mentioned effects are also influenced by the presence of the outer surface of the body [5][6][7]. One of the effective methods for solving dynamic problems of threedimensional theory of elasticity is the method of boundary integral equations (BIE) [8][9][10][11][12][13]. In the present paper, the BIE method is used to study the inertia effects near the contours of circular cracks in a half-space subjected to harmonic loads with a clamped boundary condition.Formulation of the Problem. We consider an isotropic elastic half-space whose boundary surface S 0 is clamped. The half-space contains K plane circular cracks of radius a, which occupy regions S k (k = 1, K) and are at equal depths d = |O 0 O 1 | in a plane perpendicular to the boundary S 0 . The opposite faces of the cracks S ± k are loaded by self-equilibrated harmonic tearing forces

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We propose a method of computing certain double integrals that are encountered in dynamic problems of the theory of elasticity for semi-infinite bodies with cracks. Application of this method makes it possible to reduce a system of boundary integral equations for the interaction of a crack with the boundary of the half-space to a form that contains only integrals over the regions occupied by the cracks.In reducing dynamic problems of the theory of elasticity for a half-space with cracks to a system of boundary integral equations it is necessary to overcome significant mathematical difficulties connected with the fact that these equations contain integrals over the unbounded region So that coincides with the boundary of the half-space. The occurrence of such an integral results from satisfying the boundary conditions on the boundary of the half-space. This greatly complicates the application of numerical methods for solving such equations. For that reason lowering the dimension of the system of integral equations by eliminating the integral equations that contain an infinite limit of integration is important.Consider an unbounded body weakened by N + l planar cracks, arbitrarily situated, whose opposite surfaces S~, n = 0, N, are subject to the action of self-balancing forces, varying harmonically in time with frequency w.Choose local coordinate systems Onxln, x2m x3n, n = O, N, in such a way that the domain Sn occupied by the nth crack lies in the xl,~x2n-coordinate plane, and the values x3n = :t:0 correspond to the opposite surfaces S~ of the nth crack. Then the location of the cracks in the body is completely determined by giving the direction cosines of the coordinate axes and the distances between the cracks. We denote by ljkn, mjk,~, and njkm k,n = 0, N, the direction cosines of the Xjn-aXiS in the Okxlkx2kxak-coordinate system. We denote by dkn the distance between the origins of the kth and nth coordinate systems, and by ejkn the direction cosines of the vector dkn in the kth coordinate system. Then an arbitrary point x* with coordinates (xl~, x2n, x3~) in the nth coordinate system has the following coordinates (xlkn, x2kn, x3kn) in the kth coordinate system: 3 3 3 j=l j=l j=1(1)To reduce the system of boundary integral equations of a dynamic problem for a half-space with cracks to a form not containing integrals over an unbounded domain So, it is necessary to compute the following double integrals:where x~n and 4 are the points with coordinates (XlOn, X2On, ::g3On) and (

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