Mechanics of Solids (about journal) Mechanics of Solids
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in January 1966
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IssuesArchive of Issues2008-1pp.117-123

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V. G. Zadorozhnyi, A. V. Kovalev, and A. N. Sporykhin, "Analyticity of the solution of a plane elastoplastic problem," Mech. Solids. 43 (1), 117-123 (2008)
Year 2008 Volume 43 Number 1 Pages 117-123
DOI 10.3103/S0025654408010111
Title Analyticity of the solution of a plane elastoplastic problem
Author(s) V. G. Zadorozhnyi (Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia,
A. V. Kovalev (Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia,
A. N. Sporykhin (Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia,
Abstract Elastoplastic problems of the theory of ideal plasticity were studied numerically by the method of small parameter in the monographs [1-3], and strengthening elastoplastic problems were studied in the monograph [4], where a rather complete bibliography of studies in this direction is given. The conditions for the continuous dependence of the the stress-strain state characteristics on the boundary conditions and the material inhomogeneity are defined using the implicit function theorem in [5, 6]. In the present paper, in the framework of the perturbation method, we determine the stress state in a cylindrical tube with nearly circular boundaries under the action of pressure. Using the implicit function theorem, we examine the existence and uniqueness of the solution of the problem. We show that its solution can be obtained in the case of any analytic shape of the boundary and any number of material physical-mechanical characteristics depending on finitely many small parameters. The results can be generalized to more complicated models of media. The results obtained in this paper are compared with already known results.
1.  D. D. Ivlev and L. V. Ershov, Perturbation Method in Theory of Elastoplastic Body (Nauka, Moscow, 1978) [in Russian].
2.  A. Yu. Ishlinskii and D. D. Ivlev, Mathematical Theory of Plasticity (Fizmatlit, Moscow, 2001) [in Russian].
3.  D. D. Ivlev, Mechanics of Plastic Media, Vol. 2 (Fizmatlit, Moscow, 2002) [in Russian].
4.  A. N. Sporykhin, A. V. Kovalev, and Yu. D. Shcheglova, Inhomogeneous Problems of Elastoviscoplasticity with Unknown Boundaries (Izd-vo Voronezh Univ., Voronezh, 2004) [in Russian].
5.  N. V. Minaeva, Perturbation Method in Mechanics of Deformable Bodies (Nauchnaya Kniga, Moscow, 2002) [in Russian].
6.  N. B. Kostyrin, N. V. Minaeva, and Yu. M. Myasnyankin, "Stress-Strain State of Nearly Homogeneous Elastoplastic Bodies," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 90-95 (2004) [Mech. Solids (Engl. Transl.) 39 (5), 71-75 (2004)].
7.  A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1976; Dover Publications, New York, 1999).
8.  K. B. Bitsenko and R. Grammel, Technical Dynamics, Vol. 1 (Gostekhizdat, Leningrad, 1950) [in Russian].
9.  L. E. Elsgolts, Differential Equations and the Calculus of Variations (Nauka, Moscow, 1969; University Press of the Pacific, Miami, 2003).
10.  A. N. Sporykhin, Perturbation Method in Stability Problems for Complex Media (Izd-vo Voronezh Univ., Voronezh, 1997) [in Russian].
11.  G. I. Bykovtsev and D. D. Ivlev, Theory of Plasticity (Dalnauka, Vladivostok, 1998) [in Russian].
Received 25 February 2005
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