Skip to main content
Book cover

International Congress and Exhibition "Sustainable Civil Infrastructures: Innovative Infrastructure Geotechnology"

GeoMEast 2017: Numerical Analysis of Nonlinear Coupled Problems pp 98–109Cite as

A Posteriori Error Estimation for the Non-associated Plasticity Drucker-Prager Model with Hardening

  • 988 Accesses

Part of the Sustainable Civil Infrastructures book series (SUCI)

Abstract

The numerical solution of non-associated elastoplasticity is still a key aspect of research and development in computational plasticity. Approximate solution procedures are based, in the context of a displacement method, on a weak form of the equilibrium and reply upon two main ingredients: the numerical integration of the rate constitutive relations over a generic time step (local stage) and the iterative algorithm exploited to solve the nonlinear equilibrium equations (global stage). The fully discrete problem is the obtained by performing a spatial discretization of the field equations and a time-integration of the evolution rule. The interest is here given to the discretization errors, which are caused by the numerical discretization of the continuous mathematical model in order to define an adaptive strategy.

The aim of this paper is to extend the concept of error in the constitutive equations to non-associated plasticity Drucker-Prager model to handle non-associative rate-independent plasticity problems solved by employing the incremental displacement conforming finite element method.

Numerical examples by PLSAER2D (a Matlab program) for both the associated and the non-associated cases for Drucker-Prager model with hardening are also presented.

Keywords

  • Non-associated plasticity
  • Drucker-Prager model
  • Hardening
  • Error estimation

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-61905-7_9
  • Chapter length: 12 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   189.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-61905-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   249.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.

References

  1. Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. John Wiley & Son and B G Teubner, Chichester (2000)

    CrossRef  Google Scholar 

  2. Halphen, B., Nguyen, Q.S.: Sur les matériaux standards généralisés. J. Méc. 14, 39–63 (1975)

    Google Scholar 

  3. Babuska, I., Rheinboldt, W.C.: A posteriori error estimates for the finite element method. Int. J. Numer. Methods Eng. 12, 1597–1615 (1978)

    CrossRef  Google Scholar 

  4. Drucker, D.C.: On the postulate of stability of material in the mechanics of continua. J. Méc. 3(2), 235–249 (1964)

    Google Scholar 

  5. Lam, D.D.: A posteriori error estimation for non-associated plasticity problems, Ph. D. thesis, INSA de Rennes (2009)

    Google Scholar 

  6. Lam, D.D.: A bi-potential update algorithm for the non-associated plasticity model with hardening. In: Proceedings of CIGOS-2015: Innovations in Construction, Paris, France (2015)

    Google Scholar 

  7. de Saxcé, G.: Une généralisation de l’inégalité de Fenchel et ses applications aux lois constitutives. Comptes Rendus de l’Académie des Sciences de Paris (1992)

    Google Scholar 

  8. Hjiaj, M.: Sur la classe des matériaux standard implicites: Concept, Aspects discrétisés et Estimation de l’erreur a posteriori. Ph. D. thesis, Polytechnic Faculty of Mons. (1999)

    Google Scholar 

  9. Hjiaj, M., Lam, D.D., de Saxcé, G.: A family of bi-potentials describing the non-associated flow rule of pressure-dependant plastic models. Acta Mech. 220, 237–246 (2011)

    CrossRef  Google Scholar 

  10. Ladevèze, P., Maunder, E.A.W.: A general procedure for recovering equilibrating element tractions. Comput. Methods Appl. Mech. Eng. 137, 111–151 (1996)

    CrossRef  Google Scholar 

  11. Ladevèze, P., Moës, N., Douchin, B.: Constitutive relation error estimators for (visco) plastic finite element analysis with softening. Comput. Methods Appl. Mech. Eng. 176, 247–264 (1999)

    CrossRef  Google Scholar 

  12. Ladevèze, P., Moës, N.: A new a posteriori error estimation for nonlinear time dependent finite element analysis. Comput. Methods Appl. Mech. Eng. 157, 45–68 (1997)

    CrossRef  Google Scholar 

  13. Ladevèze, P., Pelle, J.P.: Mastering Calculations in Linear and Nonlinear Mechanics. Springer, New York (2004)

    Google Scholar 

  14. Ladevèze, P., Passieux, J.-C., Néron, D.: The LATIN multiscale computational method and the proper generalized decomposition. Comput. Methods Appl. Mech. Eng. 199, 1287–1296 (2009)

    CrossRef  Google Scholar 

  15. Maunder, E.A.W., Moitinho de Almeida, J.P.: The stability of stars of triangular equilibrium plate elements. Int. J. Numer. Methods Eng. 77, 922–968 (2009)

    CrossRef  Google Scholar 

  16. Moreau, J.: Evolution problem associated with a moving convex set in a hilbert space. J. Diff. Equ. 26, 347–374 (1977)

    CrossRef  Google Scholar 

  17. Orlando, A., Peri´c, D.: Analysis of transfer procedures in elastoplasticity based on the error in the constitutive equations: Theory and numerical illustration. Int. J. Numer. Methods Eng. 60, 1595–1631 (2004)

    CrossRef  Google Scholar 

  18. Simo, J.C., Hughes, T.J.: Computational Inelasticity. Springer, New York (2000)

    Google Scholar 

  19. Zienkiewicz, O.C., Liu, Y.C., Huang, G.C.: Error estimation and adaptivity in flow formulation for forming problems. Int. J. Numer. Methods Eng. 25, 23–42 (1988)

    CrossRef  Google Scholar 

  20. MATLAB Release: The MathWorks, Inc., Natick, Massachusetts, United States (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Transport and Communications, Hanoi, Vietnam

    Dao Duy Lam

Authors
  1. Dao Duy Lam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dao Duy Lam .

Editor information

Editors and Affiliations

  1. Soil-Structure Interaction Group in Egypt (SSIGE), Cairo, Egypt

    Hany Shehata

  2. Cairo University, Cairo, Egypt

    Youssef Rashed

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Cite this paper

Lam, D.D. (2018). A Posteriori Error Estimation for the Non-associated Plasticity Drucker-Prager Model with Hardening. In: Shehata, H., Rashed, Y. (eds) Numerical Analysis of Nonlinear Coupled Problems. GeoMEast 2017. Sustainable Civil Infrastructures. Springer, Cham. https://doi.org/10.1007/978-3-319-61905-7_9

Download citation