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IssuesArchive of Issues2007-1pp.57-63

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A. G. Bagdoev, A. N. Martirosyan, G. A. Martirosyan, and S. M. Pogosyan, "Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space," Mech. Solids. 42 (1), 57-63 (2007)
Year 2007 Volume 42 Number 1 Pages 57-63
Title Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space
Author(s) A. G. Bagdoev (Institute of Mechanics, National Academy of Sciences of the Republic of Armenia, pr-t Marshla Baghramiana 24B, Yerevan, 375019, Republic of Armenia, mechins@sci.am)
A. N. Martirosyan (Goris State University, Avangardi 4, Goris, Syunik Marz, 3204, Republic of Armenia, seuagoris@mail.ru)
G. A. Martirosyan (Goris State University, Avangardi 4, Goris, Syunik Marz, 3204, Republic of Armenia)
S. M. Pogosyan (Goris State University, Avangardi 4, Goris, Syunik Marz, 3204, Republic of Armenia, sam41po@rambler.ru)
Abstract We consider the problem on the motion of an isotropic elastic body occupying the half-space z≥0 on whose boundary, along the half-plane x≥0, the horizontal components of displacement are given, while the remaining part of the boundary is stress-free. We seek the solution by the method of integral Laplace transforms with respect to time t and Fourier transforms with respect to the coordinates x, y; the problem is reduced to a system of Wiener-Hopf equations, which can be solved by the methods of singular-integral equations and circulants. We invert the integral transforms and reduce the solution to the Smirnov-Sobolev form. We calculate the tangential stress intensity coefficients near the boundary z=0, x=0, |y|<∞ of the half-plane. The circulant method for solving the Wiener-Hopf system was proposed in [1]. A static problem similar to that considered in the present paper was solved earlier. The Hilbert problem was reduced to a system of Fredholm integral equations in [2]. In the present paper, we solve the above problem by reducing the solution to quadratures and a quasiregular system of Fredholm integral equations. We give a numerical solution of the Fredholm equations and calculate the integrals for the tangential stress intensity coefficients.
References
1.  V. S. Sarkisyan and I. M. Karakhanyan, "Diffraction of Shear Elastic Harmonic Waves on Half-Infinite Inclusions," in Problems of Mechanics of Thin Bodies. Dedicated to the 80th Anniversary of S. A. Ambartsumyan, Academician of Armenian NAS (Erevan, 2002), pp. 266-280 [in Russian]
2.  N. P. Vekua, Systems of Singular Integral Equations and Some Boundary Value Problems (Nauka, Moscow, 1970) [in Russian].
3.  B. Noble, Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations (Izd-vo Inostr. Lit., Moscow, 1962) [in Russian].
4.  A. N. Martirosyan, "Solution of a Mixed Dynamic Boundary Value Problem for an Elastic Half-Space," Inform. Tekhnologii i Upravlenie, No. 3 (2003).
Received 05 October 2004
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