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IssuesArchive of Issues2006-1pp.35-45

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D. V. Georgievskii, "The Prandtl problem for a plastic layer weakly inhomogeneous with respect to the yield strength," Mech. Solids. 41 (1), 35-45 (2006)
Year 2006 Volume 41 Number 1 Pages 35-45
Title The Prandtl problem for a plastic layer weakly inhomogeneous with respect to the yield strength
Author(s) D. V. Georgievskii (Moscow)
Abstract The analysis of the sensitivity of the deformation parameters to perturbations of material functions plays an important role (in both academic and applied aspects) in continuum mechanics, especially in problems with mixed boundary conditions [1]. The statement of the problem for an inhomogeneous continuous medium in the case of Eulerian description of the motion involves, in particular, the determination of the law of motion of Lagrangian particles along their trajectories and the inversion of this law. If at the initial time instant the difference of the material functions in an inhomogeneous material from some familiar distributions (e.g., constant distributions) is small, one can use the asymptotic method of [2]. This method is tested on the problem of compression of a thin perfectly plastic layer weakly inhomogeneous with respect to the yield strength by rigid plates. As the basic process we choose the quasistatic deformation of a homogeneous medium. This process corresponds to Prandtl's solution. A linear initial-boundary value problem is formulated for the perturbations. For some particular cases of the initial inhomogeneity this problem can be solved analytically.
References
1.  V. M. Alexandrov and E. V. Kovalenko, Problems of Cotinuum Mechanics with Mixed Boundary Conditions [in Russian], Nauka, Moscow, 1986.
2.  D. V. Georgievskii, "Modeling of weak inhomogeneity in the case of the Eulerian description of a continuous medium," Doklady RAN, Vol. 405, No. 4, pp. 497-483, 2005.
3.  A. A. Il'yushin, Plasticity. Part 1. Elastoplastic Deformations [in Russian], Logos, Moscow, 2004.
4.  V. V. Sokolovskii, Theory of Plasticity [in Russian], Vysshaya Shkola, Moscow, 1969.
5.  M. I. Erkhov, Theory of Perfectly Plastic Bodies and Structures [in Russian], Nauka, Moscow, 1978.
6.  M. A. Zadoyan, Spatial Problems of Plasticity [in Russian], Nauka, Moscow, 1992.
7.  A. Yu. Ishlinskii and D. D. Ivlev, Mathematical Theory of Plasticity [in Russian], Fizmatlit, Moscow, 2001.
8.  V. G. Zubchaninov, Mathematical Theory of Plasticity [in Russian], Izd-vo TvGTU, Tver, 2002.
9.  D. A. Georgievskaya and D. V. Georgievskii, "Saint-Venant plastic flows weakly inhomogeneous with respect to the yield strength," Izv. RAN. MTT [Mechanics of Solids], No. 6, pp. 11-25, 2005.
10.  A. A. Il'yushin, "Deformation of a viscoplastic body," Uchenye Zapiski MGU. Mekhanika, Issue 39, pp. 3-81, 1940.
11.  A. G. Petrov, "An asymptotic method for constructing the Poincaré mapping to describe the transition to the dynamic chaos in Hamiltonian systems," Doklady RAN, Vol. 382, No. 1, pp. 15-19, 2002.
12.  I. A. Kiiko, "A generalization of L. Prandtl's problem of the settlement of a strip from a compressible material," Vestnik MGU [Bulletin of the Moscow State University], Ser. 1. Matematika. Mekhanika, No. 4, pp. 47-52, 2002.
13.  I. A. Kiiko and V. A. Kadymov, "A generalization of L. Prandtl's problem of the compression of a strip," Vestnik MGU [Bulletin of the Moscow State University], Ser. 1. Matematika. Mekhanika, No. 4, pp. 50-56, 2003.
14.  V. L. Kolmogorov, Mechanics of Metal Plastic Working [in Russian], Izd-vo UrGTU-UPI, Ekaterinburg, 2001.
15.  B. E. Pobedrya, Mechanics of Composite Materials [in Russian], Izd-vo MGU, Moscow, 1984.
16.  B. E. Pobedrya and I. L. Guzei, "Mathematical modeling of deformation of composites with the thermal diffusion being taken into account," Matematicheskoe Modelirovanie Sistem i Protsessov, No. 6, pp. 82-91, 1998.
17.  V. Kadymov and R. Wille, "Plastic flow in piecewise-homogeneous layers," ZAMM, B. 75, No. 1, S. 293-294, 1995.
Received 05 June 2005
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