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Iteration Algorithms for Solving Boundary-Value Problems of the Theory of Small Elastic-Plastic Strains on the Basis of the Mixed Finite Element Method

Abstract

A mixed projection-mesh scheme has been formulated for the solution of nonlinear boundary-value problems of the theory of small elastic-plastic strains in displacements, strains, and stresses. Iteration algorithms of the solution of nonlinear equations of the mixed method have been considered. Numerical results of the solution of two model problems are presented. The data obtained on the basis of the classical and mixed finite element methods are compared.

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REFERENCES

  1. 1.

    A. A. Il’yushin, “On the theory of small elastic-plastic strains,” Prikl. Math. Mekh., 10, No.3, 347–356 (1946).

  2. 2.

    A. A. Il’yushin, Plasticity [in Russian], Gostekhizdat, Moscow (1948).

  3. 3.

    M. M. Vainberg, Variational Method and Method of Monotone Operators [in Russian], Nauka, Moscow (1972).

  4. 4.

    Kh. Gaevskii, K. Greger, and K. Zakharias, Nonlinear Operator Equations and Operator Differential Equations [in Russian], Mir, Moscow (1978).

  5. 5.

    K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press, New York (1975).

  6. 6.

    A. Yu. Chirkov, “Mixed projection-mesh scheme of the finite-element method for the solution of the boundary-value problems of the theory of small elastic-plastic strains,” Strength Mater., 36, No.6, 591–611 (2004).

  7. 7.

    J. Ortega and W. C. Rheinbboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, London (1970).

  8. 8.

    A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1981).

  9. 9.

    I. A. Birger, “Some general methods for solving the problems of the plasticity theory,” Prikl. Mat. Mekh., 15, No.6, 765–770 (1951).

  10. 10.

    E. G. D’yakonov, Minimization of the Computational Work [in Russian], Nauka, Moscow (1989).

  11. 11.

    M. Hestens and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” Nat. Bur. Std. J. Res., 49, 409–436 (1952).

  12. 12.

    A. Yu. Chirkov, “Mixed approximation scheme of the finite-element method for the solution of two-dimensional problems of the elasticity theory,” Strength Mater., 35, No.6, 608–632 (2003).

  13. 13.

    G. S. Pisarenko and N. S. Mozharovskii, Equations and Boundary-Value Problems of the Theories of Plasticity and Creep [in Russian], Naukova Dumka, Kiev (1981).

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Translated from Problemy Prochnosti, No. 3, pp. 111 – 127, May – June, 2005.

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Chirkov, A.Y. Iteration Algorithms for Solving Boundary-Value Problems of the Theory of Small Elastic-Plastic Strains on the Basis of the Mixed Finite Element Method. Strength Mater 37, 310–322 (2005). https://doi.org/10.1007/s11223-005-0044-8

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Keywords

  • plasticity theory
  • finite element method
  • mixed scheme
  • approximation
  • stability
  • convergence
  • accuracy
  • iterative methods