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IssuesArchive of Issues2004-4pp.68-76

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M. Ya. Brovman, "Solution of equations of a plastic flow in curvilinear coordinates," Mech. Solids. 39 (4), 68-76 (2004)
Year 2004 Volume 39 Number 4 Pages 68-76
Title Solution of equations of a plastic flow in curvilinear coordinates
Author(s) M. Ya. Brovman (Tver)
Abstract The plane deformation of a perfect rigid-plastic medium is considered in curvilinear orthogonal coordinates. Methods for solving the equations of the plastic flow are considered in [1-3]. As a rule, such problems are solved in the traditional systems of coordinates-Cartesian, cylindrical or spherical. However, for metal forming, tools with more complex curved surfaces are frequently utilized and, for this reason, the development of methods for solving such problems is desired. In [4, 5], the curvilinear coordinates in which the flow lines coincide with the coordinate lines are utilized. However, it is difficult to obtain exact solutions for each of these coordinate systems. In the present paper, we consider the plane deformation in orthogonal coordinates with new functions being introduced for the coefficients of the first quadratic form and their derivatives. In some cases, this makes it possible to obtain solutions for an arbitrary orthogonal system of coordinates.
References
1.  V. V. Sokolovskii, Theory of Plasticity [in Russian], Vysshaya Shkola, Moscow, 1969.
2.  D. D. Ivlev, Theory of Perfect Plasticity [in Russian], Nauka, Moscow, 1966.
3.  I. A. Kiiko, "Theory of plastic flow (as applied to metal forming processes), in Issues of Strength and Plasticity [in Russian], Izd-vo MGU, Moscow, 1984.
4.  M. Ya. Brovman, "The flow lines in the case of plane plastic deformation," Izv. AN SSSR. MTT [Mechanics of Solids], No. 2, pp. 185-187, 1989.
5.  M. Ya. Brovman, Applications of the Theory of Plasticity in Rolling [in Russian], Metallurgiya, Moscow, 1991.
6.  P. K. Rashevskii, Riemannian Geometry and Tensor Analysis [in Russian], Nauka, Moscow, 1964.
7.  V. V. Vasil'ev, "Stress state in solids and some geometric effects," Izv. AN SSSR. MTT [Mechanics of Solids], No. 5, pp. 30-34, 1989.
8.  V. A. Gatsev, A Course in Elasticity and Fundamentals of Plasticity [in Russian], Izd-vo LGU, Leningrad, 1973.
9.  M. Ya. Brovman, "A method for the analysis of heat transfer using isothermal coordinates," Inzhener. Fizich. Zh., Vol. 68, No. 4, pp. 651-659, 1995.
Received 14 March 2002
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