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IssuesArchive of Issues2004-1pp.73-80

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V. N. Kukudzhanov, "Decomposition method for elastoplastic equations," Mech. Solids. 39 (1), 73-80 (2004)
Year 2004 Volume 39 Number 1 Pages 73-80
Title Decomposition method for elastoplastic equations
Author(s) V. N. Kukudzhanov (Moscow)
Abstract A new method for the integration of equations governing the behavior of elastoplastic media is proposed. This method is based on the decomposition of the constitutive relations for relaxation media with respect to physical processes. We show that for the classical elastoplastic medium independent of the time scale, the decomposition leads to an algebraic power-law equation for the correction coefficient of the elastic solution. In the case of perfect plasticity, this coefficient coincides with that obtained in [1]. For more general models, this coefficient can also be expressed in closed form.

For elastoviscoplastic media, the decomposition leads to a differential equation for the correction coefficients that in this case are functions of time. The solution of this differential equation is obtained in an analytical form. This allows one to investigate the convergence of the numerical method and to determine asymptotic properties of this method.

The proposed method has computational advantages as compared with the standard iteration methods. At each integration step, this method involves the solution of an elasticity problem and the solution of one equation for the correction coefficients at the corrector stage. In contrast to this, the traditional methods require solving a system of n constitutive equations (ngeq6) at each point of the body.
References
1.  M. L. Wilkins, "Numerical analysis of elastoplastic flows," in B. Alder, S. Fernbach, and M. Rotenberg (Editors), Fundamental Methods in Hydrodynamics [Russian translation], pp. 212-263, Mir, Moscow, 1967.
2.  E. Hinton and D. R. J. Owen, Finite Elements in Plasticity, Pineridge Press, Swaensea, 1981.
3.  V. M. Fomin, A. I. Gulidov, G. A. Sapozhnikov, et al., High-speed Interaction of Bodies [in Russian], Izd-vo SO RAN, Novosibirsk, 1999.
4.  V. V. Sokolovskii, Theory of Plasticity [in Russian], Vysshaya Shkola, Moscow, 1969.
5.  N. G. Burago and V. N. Kukudzhanov, Solution of Elastoplastic Problems by the Finite Element Method. A Software Package "Astra." Preprint No. 326 [in Russian], Institute for Problems in Mechanics of the USSR Academy of Sciences, Moscow, 1988.
6.  V. M. Sadovskii, Discontinuous Solutions in Dynamics of Elastoplastic Media [in Russian], Nauka, Moscow, 1997.
7.  G. Duvant and J.-L. Lions, Inequalities in Mechanics and Physics [Russian translation], Nauka, Moscow, 1980.
8.  V. N. Kukudzanov, Difference Methods for Solving Problems in Mechanics of Solids [in Russian], MFTI, Moscow, 1992.
9.  G. I. Marchuk, Decomposition Methods [in Russian], Nauka, Moscow, 1988.
10.  V. M. Kovenya and N. N. Yanenko, The Decomposition Method in Gas Dynamics [in Russian], Nauka, SO AN SSSR, Novosibirsk.
11.  O. M. Belotserkovskii and Yu. M. Davidov, The Method of Large Particles in Gas Dynamics [in Russian], Nauka, Moscow, 1982.
12.  V. N. Kukudzhanov and Yu. I. Kudryashev, Solution of Mixed Problems of Unsteady Interaction of Gaseous Media with Solids. Preprint No. 472 [in Russian], Institute for Problems in Mechanics of the USSR Academy of Sciences, Moscow, 1990.
13.  V. Yu. Kibardin and V. N. Kukudzhanov, "Modeling of continuum fracture in an elastoviscoplastic material," Izv. AN. MTT [Mechanics of Solids], No. 1, pp. 109-123, 2001.
14.  V. N. Kukudzhanov, "Numerical simulation of dynamical processes of deformation and fracture in elastoplastic media," Uspekhi Mekhaniki [Advances in Mechanics], Vol. 8, No. 4, pp. 21-65, 1985.
15.  O. Zienkiewicz, The Finite Element Method in Engineering Sciences [Russian translation], Mir, Moscow, 1875.
16.  C. A. Brebbia, J. C. F. Telles, and L. C. Wrobel, Boundary Element Techniques [Russian translation], Mir, Moscow, 1987.
17.  V. N. Kukudzhanov, "Wave propagation in elastic-viscoplastic materials with a general stress-strain diagram," Izv. AN. MTT [Mechanics of Solids], No. 5, pp. 96-111, 2001.
Received 16 October 2003
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