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IssuesArchive of Issues2002-3pp.77-88

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N. D. Vaisfel'd and G. Ya. Popov, "Nonstationary dynamic problems of elastic stress concentration near a spherical imperfection," Mech. Solids. 37 (3), 77-88 (2002)
Year 2002 Volume 37 Number 3 Pages 77-88
Title Nonstationary dynamic problems of elastic stress concentration near a spherical imperfection
Author(s) N. D. Vaisfel'd (Odessa)
G. Ya. Popov (Odessa)
Abstract By using the method of discontinuous solutions we solve nonstationary dynamic problems of the elastic stress concentration near a spherical imperfection which is either a spherical crack or a thin rigid spherical inclusion. The approach suggested allows one to reduce these problems to a system of one-dimensional integro-differential or integral equations in the space of Laplace transforms. The subsequent inversion of these transforms and using the convolution theorem reduce the equations to two-dimensional ones which can be solved by using time-discretization or the method of orthogonal polynomials. As a result, the problem is reduced to the solution of the sequence of infinite systems of linear algebraic equations. The detailed development of this approach is given for the problems of diffraction of nonstationary elastic torsional waves by the spherical inclusion adhered to an elastic medium (which is fixed or only adhered to the elastic medium). In the latter case we apply also the asymptotic method to solve the problem. The numerical analysis of the applicability area of this method is performed. Time histories of important mechanical characteristics are calculated.
References
1.  G. Ya. Popov, Concentration of Elastic Stresses Near Punches, Cuts, Thin Inclusions and Reinforcements [in Russian], Nauka, Moscow, 1982.
2.  A. N. Guz', V. D. Kubenko, and M. A. Cherevko, Diffraction of Elastic Waves [in Russian], Naukova Dumka, Kiev, 1978.
3.  G. Ya. Popov, "On a new representation of the solution of the Lamé equations in the spherical coordinate system," Doklady AN, Vol. 356, No. 1, pp. 47-49, 1997.
4.  D. V. Grilitskii and A. P. Poddubnyak, "Scattering of nonstationary torsional wave on an immovable sphere in an elastic medium," Izv. AN SSSR. MTT [Mechanics of Solids], No. 5, pp. 86-92, 1980.
5.  A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marichev, Integrals and Series. Special Functions [in Russian], Nauka, Moscow, 1983.
6.  M. Abramovitz and I. A. Stegun (Editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [Russian translation], Nauka, Moscow, 1979.
7.  H. Bateman and A. Erdélyi, Tables of Integral Transformations, Vol. 1 [Russian translation], Nauka, Moscow, 1984.
8.  L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow, 1984.
9.  G. A. Morar', Method of Discontinuous Solutions in the Mechanics of Solids [in Russian], Shtiintsa, Kishinev, 1990.
Received 27 March 2000
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