Hierarchical Adaptive FEM at Finite Elastoplastic Deformations

  • Reiner Kreißig
  • Anke Bucher
  • Uwe-Jens Görke
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 52)


The simulation of non-linear problems of continuum mechanics was a crucial point within the framework of the subproject “Efficient parallel algorithms for the simulation of the deformation behaviour of components of inelastic materials”. Nonlinearity appears with the occurence of finite deformations as well as with special material behaviour as e.g. elastoplasticity.


Plastic Zone Yield Condition Coarse Mesh Gauss Point Free Energy Density 
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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  1. 1.
    1. T. Apel, R. Mücke and J.R. Whiteman. An adaptive finite element technique with a-priori mesh grading. Report 9, BICOM Institute of Computational Mathematics, 1993.Google Scholar
  2. 2.
    2. N. Aravas. Finite strain anisotropic plasticity: Constitutive equations and computational issues. Advances in finite deformation problems in materials processing and structures. ASME 125 : 25–32, 1991.Google Scholar
  3. 3.
    3. R. Asaro. Crystal plasticity. J Appl Mech 50 : 921–934, 1983.zbMATHCrossRefGoogle Scholar
  4. 4.
    4. I. Babuška, W.C. Rheinboldt. A-posteriori error estimates for the finite element method. Int J Numerical Meth Eng 12 : 1597–1615, 1978.CrossRefzbMATHGoogle Scholar
  5. 5.
    5. I. Babuška, A. Miller. A-posteriori error estimates and adaptive techniques for the finite element method. Technical report, Institute for Physical Science and Technology, University of Maryland, 1981.Google Scholar
  6. 6.
    6. D. Besdo. Zur anisotropen Verfestigung anfangs isotroper starrplastischer Medien. ZAMM 51 : T97–T98, 1971.CrossRefGoogle Scholar
  7. 7.
    7. A. Bucher. Deformationsgesetze für große elastisch-plastische Verzerrungen unter Berücksichtigung einer Substruktur. PhD thesis, TU Chemnitz, 2001.Google Scholar
  8. 8.
    8. A. Bucher, U.-J. Görke and R. Kreißig. A material model for finite elasto-plastic deformations considering a substructure. Int. J. Plasticity 20 : 619–642, 2004.zbMATHCrossRefGoogle Scholar
  9. 9.
    9. A. Bucher, A. Meyer, U.-J. Görke, R. Kreißig. A contribution to error estimation and mapping algorithms for a hierarchical adaptive FE-strategy in finite elastoplasticity. Comput. Mech. 36 : 182–195, 2005.zbMATHCrossRefGoogle Scholar
  10. 10.
    10. C. Carstensen, J. Alberty. Averaging techniques for reliable a posteriori FEerror control in elastoplasticity with hardening. Comput. Method. Appl. Mech. Engrg. 192 : 1435–1450, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    11. H. Cramer, M. Rudolph, G. Steinl, W. Wunderlich. A hierarchical adaptive finite element strategy for elastic-plastic problems. Comput. Struct. 73 : 61–72, 1999.zbMATHCrossRefGoogle Scholar
  12. 12.
    12. Y.F. Dafalias. A missing link in the macroscopic constitutive formulation of large plastic deformations. In: Plasticity today, (Sawczuk, A., Bianchi, G. eds.), from the International Symposium on Recent Trends and Results in Plasticity, Elsevier, Udine, p.135, 1983.Google Scholar
  13. 13.
    13. Y.F. Dafalias. On the microscopic origin of the plastic spin. Acta Mech. 82 : 31– 48, 1990.CrossRefMathSciNetGoogle Scholar
  14. 14.
    14. J.P. Gago, D. Kelly, O.C. Zienkiewicz and I. Babuška. A-posteriori error analysis and adaptive processes in the finite element method. Part I: Error analysis. Part II: Adaptive processes. Int. J. Num. Meth. Engng., 19/83 : 1593–1656, 1983.Google Scholar
  15. 15.
    15. P. Haupt. On the concept of an intermediate configuration and its application to a representation of viscoelastic-plastic material behavior. Int. J. Plasticity, 1 : 303–316.Google Scholar
  16. 16.
    16. R. Hill. A theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. of London A 193, 281–297, 1948.zbMATHCrossRefGoogle Scholar
  17. 17.
    17. C. Johnson, P. Hansbo. Adaptive finite element methods in computational mechanics. Comput. Methods Appl. Mech. Engrg. 101 : 143–181, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    18. J. Kratochvil. Finite strain theory of cristalline elastic-plastic materials. Acta Mech. 16 : 127–142, 1973.CrossRefzbMATHGoogle Scholar
  19. 19.
    19. G. Kunert. Error estimation for anisotropic tetrahedral and triangular finite element meshes. Preprint SFB393/97–16, TU Chemnitz, 1997.Google Scholar
  20. 20.
    20. G. Kunert. An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math., 86(3) : 471–490, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    21. G. Kunert. A local problem error estimator for anisotropic tetrahedral finite element meshes. SIAM J. Numer. Anal. 39 : 668–689, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    22. G. Kunert. A posteriori l2 error estimation on anisotropic tetrahedral finite element meshes. IMA J. Numer. Anal. 21 : 503–523, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    23. G. Kunert and R. Verfürth. Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes. Numer. Math., 86(2):283–303, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    24. J. Mandel. Plasticité classique et viscoplasticité. CISM Courses and Lectures No.97, Springer-Verlag, Berlin, 1971.Google Scholar
  25. 25.
    25. C. Miehe, J. Schotte, J. Schröder. Computational micro-macro transitions and overall moduli in the analysis of polycristals at large strains. Comp. Mat. Sci. 16 : 372–382, 1999.CrossRefGoogle Scholar
  26. 26.
    26. J. Ning, E.C. Aifantis. On anisotropic finite deformation plasticity, Part I. A two-back stress model. Acta Mech. 106 : 55–72, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    27. J. Ning, E.C. Aifantis. Anisotropic yield and plastic flow of polycristalline solids. Int. J. Plasticity 12 : 1221–1240, 1996.zbMATHCrossRefGoogle Scholar
  28. 28.
    28. E.T. Onat. Representation of inelastic behaviour in the presence of anisotropy and of finite deformations. In: Recent advances in creep and fracture of engineering materials and structures, Pineridge Press, Swansea, 1982.Google Scholar
  29. 29.
    29. S.S. Rao. Engineering Optimization. John Wiley and Sons, New York, 1996.Google Scholar
  30. 30.
    30. C. Sansour. On anisotropy at the actual configuration and the adequate formulation of a free energy function. IUTAM Symposium on Anisotropy, Inhomogenity and Nonlinearity in Solid Mechanics, 43–50, 1995.Google Scholar
  31. 31.
    31. J.C. Simo. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation. Comput. Methods Appl. Mech. Engrg. 66 : 199–219, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    32. E. Steck. Zur Berücksichtigung von Vorgängen im Mikrobereich metallischer Werkstoffe bei der Entwicklung von Stoffmodellen. ZAMM 75 : 331–341, 1995.zbMATHGoogle Scholar
  33. 33.
    33. E. Stein (Ed.). Error-controlled Adaptive Finite Elements in Solid Mechanics. Wiley, Chichester, 2003.Google Scholar
  34. 34.
    34. V. Ulbricht, H. Röhle. Berechnung von Rotationsschalen bei nichtlinearem Deformationsverhalten, PhD thesis, TU Dresden, 1975.Google Scholar
  35. 35.
    35. H. Ziegler, D. Mac Vean. On the notion of an elastic solid. In: B. Broberg, J. Hult and F. Niordson, eds., Recent Progress in Appl. Mech. The Folke Odquist Volume, Eds. B. Broberg, J. Hult, F. Niordson, Almquist & Wiksell, Stockholm, 561–572, 1965.Google Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Reiner Kreißig
    • 1
  • Anke Bucher
    • 1
  • Uwe-Jens Görke
    • 1
  1. 1.Institut für MechanikTechnische Universität ChemnitzChemnitzGermany

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