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IssuesArchive of Issues2001-6pp.113-119

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I. V. Andrianov and O. G. Samoilenko, "On the Ishlinskii-Leibenzon method in the theory of elastic stability," Mech. Solids. 36 (6), 113-119 (2001)
Year 2001 Volume 36 Number 6 Pages 113-119
Title On the Ishlinskii-Leibenzon method in the theory of elastic stability
Author(s) I. V. Andrianov (Dnepropetrovsk)
O. G. Samoilenko (Dnepropetrovsk)
Abstract A. Yu. Ishlinskii [1] and L. S. Leibenzon [2] suggested a method for the stability analysis of elastic systems applicable to structures with free edges (or solids with free surfaces). The essence of this method is that the parametric terms are neglected in the stability equations, but are preserved only in the boundary conditions. This method has become widely used [3-10] (see also the list of bibliography in [11-15]) and has been repeatedly discussed in the literature [13-15]; a number of reasonable critical remarks have been stated as regards the accuracy of this method. For example, in [13], it is mentioned that "numerous publications based on the approximate method of [1, 2] need an additional study to define the limits of their applicability."

In the present paper we show that these remarks can be removed by regarding this approach as a zeroth-order approximation of an asymptotic process admitting refinement.
References
1.  A. Yu. Ishlinskii, "Considering the stability of equilibrium of elastic bodies from the point of view of the mathematical theory of elasticity," Ukrainskii Matematicheskii Zhurnal, Vol. 6, No. 2, pp. 140-146, 1954.
2.  L. S. Leibenzon, "On the application of harmonic functions to the stability of spherical and cylindrical shells," in L. S. Leibenzon, Collected Works [in Russian], Vol. 1, pp. 50-86, Izd-vo AN SSSR, Moscow, 1951.
3.  M. T. Alimzhanov, Stability of Equilibrium of Solids and Problems of Mechanics of Rocks [in Russian], Nauka, Alma-Ata, 1982.
4.  L. V. Ershov, "On the statement of the problems for stability of mining shafts," Doklady AN SSSR, Vol. 143, No. 2, pp. 305-307, 1962.
5.  N. F. Voitsekhovskaya, "A stability of cylindrical shells from the point of view of the mathematical theory of elasticity," Doklady AN SSSR, Vol. 266, No. 1, pp. 59-63, 1982.
6.  L. V. Ershov and D. D. Ivlev, "On the buckling of a thick-walled pipe under internal pressure," Izv. AN SSSR. OTN, No. 8, pp. 149-152, 1957.
7.  I. D. Legenya, "On the stability of thick rectangular simply supported plate under compressive loads," Doklady AN SSSR, Vol. 140, No. 4, pp. 776-779, 1961.
8.  Zh. S. Erzhanov and A. K. Egorov, Theory of Folding Process in Rock Mass (Mathematical Description) [in Russian], Nauka, Alma-Ata, 1968.
9.  Zh. S. Erzhanov, A. K. Egorov, I. A. Garagash et al. (Editors) Theory of Folding in the Earth's Crust [in Russian], Nauka, Moscow, 1975.
10.  O. V. Gendelman and L. I. Manevitch, "Local buckling of reinforced cylindrical shells," in VII Sympozjum Statecznosci Konstrukcji, pp. 27-31, Bielsko-Biala, 1994.
11.  A. N. Guz', Stability of Three-dimensional Solids [in Russian], Naukova dumka, Kiev, 1971.
12.  A. N. Guz' and A. N. Skorykhin, "Three-dimensional theory of inelastic stability," Prikladnaya Mekhanika, Vol. 18, No. 7, pp. 3-22, No. 8, pp. 3-27, 1982.
13.  A. N. Guz', "On the three-dimensional theory of stability of deformable solids. Internal instability," Prikladnaya Mekhanika, Vol. 21, No. 11, pp. 3-17, 1985.
14.  A. N. Guz', "On the three-dimensional theory of stability of deformable solids. Surface stability," Prikladnaya Mekhanika, Vol. 22, No. 1, pp. 24-35, 1986.
15.  A. N. Guz', "On the three-dimensional theory of stability of deferrable solids. Stability of structural components," Prikladnaya Mekhanika, Vol. 22, No. 2, pp. 3-17, 1986.
16.  V. L. Berdichevskii, "A variational asymptotic method for constructing the theory of shells," PMM [Applied Mathematics and Mechanics], Vol. 43, No. 4, pp. 664-687, 1979.
17.  V. L. Berdichevskii, Variational Principles in Mechanics of a Continuous Medium [in Russian], Nauka, Moscow, 1983.
18.  N. A. Alfutov, Fundamentals of Stability Analysis of Elastic Systems [in Russian], Mashinostroenie, Moscow, 1978.
19.  I. F. Obraztsov, B. V. Nerubailo, and I. V. Andrianov, Asymptotic Methods in Mechanics of Thin-walled Structures [in Russian], Mashinostroenie, Moscow, 1991.
20.  J. Baker and P. Graves-Morris, Padé Approximations [Russian translation], Mir, Moscow, 1982.
21.  I. L. Birger and Ya. G. Panovko (Editors), Strength, Stability, and Vibrations. Volume 3 [in Russian], Mashinostroenie, Moscow, 1968.
22.  D. V. Vainberg and E. D.Vainberg, Plates, Disks, and Thin-walled Beams (Strength, Stability, and Vibrations) [in Russian], Gosstroiizdat UkrSSR, Kiev, 1959.
Received 02 June 1997
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