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The Kachanov Method for a Rigid-Plastic Rolling Problem

  • Todor Angelov Angelov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2542)

Abstract

In this work, the method of successive linearization, proposed by L. M. Kachanov for solving nonlinear variational problems, arizing in the deformation theory of plasticity, is applied to a steady state, hot strip rolling problem. The material behaviour is described by a rigid-plastic, incompressible, strain rate dependent material model and for the roll-workpiece interface a constant friction law is used. The problem is stated in the form of a variational inequality with strongly nonlinear and nondifferentiable terms. The equivalent minimization problem is also given. Under certain restrictions on the material characteristics, existence and uniqueness results are obtained and the convergence of the method is proved.

Keywords

Variational Inequality Deformation Theory Strain Rate Tensor Dilatation Strain Successive Linearization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Washizu, K.: Variational Methods in Elasticity and Plasticity. Pergamon Press (1982)Google Scholar
  2. 2.
    Zienkiewicz, O.C.: Flow formulation for numerical solution of forming processes. In: Pittman, J.F.T., Zienkiewicz, O.C., Wood, R.D., Alexander, J.M. (eds.): Numerical Analysis of Forming Processes. John Wiley & Sons (1984) 1–44.Google Scholar
  3. 3.
    Perzyna, P.: Fundamental problems in viscoplasticity. Adv. Appl. Mech. 9 (1966) 243–377.CrossRefGoogle Scholar
  4. 4.
    Cristescu, N., Suliciu, I.: Viscoplasticity. Martinus Nijhoff Publ., Bucharest (1982)Google Scholar
  5. 5.
    Mosolov, P.P., Myasnikov, V.P.: Mechanics of Rigid-Plastic Media. Nauka, Moscow (1981) (in Russian).Google Scholar
  6. 6.
    Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976)zbMATHGoogle Scholar
  7. 7.
    Awbi, B., Shillor, M., Sofonea, M.: A contact problem for Bingham fluid with friction. Appl. Analysis, 72 (1999) 469–484.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Le Tallec, P.: Numerical solution of viscoplastic flow problems by augmented lagrangians. IMA, J. Num. Analysis, 6 (1986) 185–219.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Angelov, T.A.: A secant-modulus method for a rigid-plastic rolling problem. Int. J. Nonlinear Mech. 30 (1995) 169–178.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, Berlin (1984)zbMATHGoogle Scholar
  11. 11.
    Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadephia PA (1988)Google Scholar
  12. 12.
    Fučik, S., Kufner, A.: Nonlinear Differential Equations. Elsevier, Amsterdam (1980)Google Scholar
  13. 13.
    Mikhlin, S.G.: The Numerical Performance of Variational Methods. Walters-Noordhoff, The Netherlands (1971)Google Scholar
  14. 14.
    Nečas, J., Hlavaček, I.: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Elsevier, Amsterdam (1981)Google Scholar
  15. 15.
    Chen, J., Han, W., Huang, H.: On the Kačanov method for a quasi-Newtonian flow problem. Numer. Funct. Anal. and Optimization, 19 (9 & 10) (1998) 961–970.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Todor Angelov Angelov
    • 1
  1. 1.Bulgarian Academy of SciencesInstitute of MechanicsBulgaria

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