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The Influence of Limited Kinematical Hardening on Shakedown of Materials and Structures

  • Jaan -W. SimonEmail author
Chapter

Abstract

One of the most important tasks in design for construction engineers is the determination of the load bearing capacity of the considered engineering structure. This can be particularly challenging when the applied thermo-mechanical loads vary with time and are high enough to exceed the material’s elastic regime. In these cases, the lower bound shakedown analysis provides a convenient tool. Since accounting for the realistic material behavior is inevitable to achieve reliable results, it is highly relevant to consider limited kinematical hardening. Although there exist different formulations in the literature, in which limited kinematical hardening is incorporated, these usually do not take into account the underlying hardening law in an explicit manner. The most important question in that context is whether such formulations can cover both linear and nonlinear hardening laws. In consequence, the aim of this paper is to investigate the effect of nonlinearity of the hardening law by showing that in certain scenarios the introduction of an explicit hardening law as a subsidiary constraint is unavoidable.

Keywords

Yield Surface Kinematical Hardening Residual Stress Field Associate Flow Rule Shakedown Analysis 
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.

References

  1. 1.
    Abdel-Karim M (2005) Shakedown of complex structures according to various hardening rules. Int J Press Vessel Pip 82(5):427–458CrossRefGoogle Scholar
  2. 2.
    Armstrong P-J, Frederick C-O (1966) A mathematical representation of the multiaxial Bauschinger effect. Technical report, C.E.G.B. Report No. RD/B/N907Google Scholar
  3. 3.
    Bodovillé G, De Saxcé GD (2001) Plasticity with nonlinear kinematic hardening: modelling and shakedown analysis by the bipotential approach. Eur J Mech A/Solids 20:99–112CrossRefzbMATHGoogle Scholar
  4. 4.
    Bouby C, De Saxcé G, Tritsch J-B (2006) A comparison between analytical calculations of the shakedown load by the bipotential approach and step-by-step computations for elastoplastic materials with nonlinear kinematic hardening. Int J Solids Struct 43:2670–2692CrossRefzbMATHGoogle Scholar
  5. 5.
    Bouby C, De Saxcé G, Tritsch J-B (2009) Shakedown analysis: comparison between models with the linear unlimited, linear limited and non-linear kinematic hardening. Mech Res Commun 36:556–562CrossRefzbMATHGoogle Scholar
  6. 6.
    Corigliano A, Maier G, Pycko S (1995) Kinematic criteria of dynamic shakedown extended to nonassociate constitutive laws with saturation nonlinear hardening. Redic Accademia Lincei IX 6:55–64zbMATHMathSciNetGoogle Scholar
  7. 7.
    De Saxcé GD, Tritsch J-B, Hjiaj M (2000) Shakedown of elastic-plastic structures with nonlinear kinematic hardening by the bipotential approach. In: Weichert D, Maier G (eds) Inelastic analysis of structures under variable loads. Kluwer Academic Publishers, DordrechtGoogle Scholar
  8. 8.
    Fuschi P (1999) Structural shakedown for elastic-plastic materials with hardening saturation surface. Int J Solids Struct 36:219–240CrossRefzbMATHGoogle Scholar
  9. 9.
    Groß-Weege J (1997) On the numerical assessment of the safety factor of elastic-plastic structures under variable loading. Int J Mech Sci 39(4):417–433CrossRefGoogle Scholar
  10. 10.
    Groß-Weege J, Weichert D (1992) Elastic-plastic shells under variable mechanical and thermal loads. Int J Mech Sci 34:863–880CrossRefzbMATHGoogle Scholar
  11. 11.
    Halphen B, Nguyen QS (1975) Sur les materiaux standards generalises. J Mech 14:39–63zbMATHGoogle Scholar
  12. 12.
    Heitzer M (1999) Traglast- und Einspielanalyse zur Bewertung der Sicherheit passiver Komponenten, PhD thesis, Forschungszentrum Jülich, RWTH Aachen, GermanyGoogle Scholar
  13. 13.
    Heitzer M, Pop G, Staat M (2000) Basis reduction for the shakedown problem for bounded kinematical hardening material. J Glob Optim 17:185–200CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Heitzer M, Staat M, Reiners H, Schubert F (2003) Shakedown and ratcheting under tension-torsion loadings: analysis and experiments. Nucl Eng Des 225:11–26CrossRefGoogle Scholar
  15. 15.
    Koiter WT (1960) General theorems for elastic-plastic solids. In: Sneddon IN, Hill R (eds) Progress in solid mechanics. North-Holland Publisher, Amsterdam, pp 165–221Google Scholar
  16. 16.
    König JA (1987) Shakedown of elastic-plastic structures. Elsevier, AmsterdamGoogle Scholar
  17. 17.
    Lemaitre J, Chaboche J-L (1990) Mechanics of solid materials. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  18. 18.
    Mandel J (1976) Adaptation d’une structure plastique ecrouissable et approximations. Mech Res Commun 3:483–488CrossRefzbMATHGoogle Scholar
  19. 19.
    Melan E (1938) Der Spannungszustand eines “Mises-Hencky’schen” Kontinuums bei veränderlicher Belastung. Sitzungsber Akad Wiss Wien, Math-Nat Kl, Abt IIa 147:73–87zbMATHGoogle Scholar
  20. 20.
    Melan E (1938) Zur Plastizität des räumlichen Kontinuums. Ing-Arch 9:116–126CrossRefzbMATHGoogle Scholar
  21. 21.
    Nguyen Q-S (2003) On shakedown analysis in hardening plasticity. J Mech Phys Solids 51:101–125CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Pham DC (2007) Shakedown theory for elastic plastic kinematic hardening bodies. Int J Plast 23:1240–1259CrossRefzbMATHGoogle Scholar
  23. 23.
    Pham DC (2008) On shakedown theory for elastic-plastic materials and extensions. J Mech Phys Solids 56:1905–1915CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Pham DC, Weichert D (2001) Shakedown analysis for elastic-plastic bodies with limited kinematical hardening. Proc R Soc Lond A 457:1097–1110CrossRefzbMATHGoogle Scholar
  25. 25.
    Pham PT, Vu DK, Tran TN, Staat M (2010) An upper bound algorithm for shakedown analysis of elastic-plastic bounded linearly kinematic hardening bodies. In: Proceedings  of the ECCM 2010Google Scholar
  26. 26.
    Polizzotto Castrenze (1986) A convergent bounding principle for a class of elastoplastic strain-hardening solids. Int J Plast 2(4):359–370CrossRefzbMATHGoogle Scholar
  27. 27.
    Polizzotto Castrenze (2010) Shakedown analysis for a class of strengthening materials within the framework of gradient plasticity. Int J Plast 26(7):1050–1069CrossRefzbMATHGoogle Scholar
  28. 28.
    Portier L, Challoch S, Marquis D, Geyer P (2000) Ratchetting under tension-torsion loadings: experiments and modelling. Int J Plast 16:303–335CrossRefzbMATHGoogle Scholar
  29. 29.
    Prager W (1959) An introduction to plasticity. Addison-Wesley, LondonzbMATHGoogle Scholar
  30. 30.
    Pycko S, Maier G (1995) Shakedown theorems for some classes of nonassociative hardening elastic-plastic material models. Int J Plast 11(4):367–395CrossRefzbMATHGoogle Scholar
  31. 31.
    Simon J-W (2013) Direct evaluation of the limit states of engineering structures exhibiting limited, nonlinear kinematical hardening. Int J Plast 42:141–167CrossRefGoogle Scholar
  32. 32.
    Simon J-W, Weichert D (2011) Numerical lower bound shakedown analysis of engineering structures. Comput Methods Appl Mech Eng 200:2828–2839CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Simon J-W, Weichert D (2012) Interior-point method for lower bound shakedown analysis of von Mises-type materials. In: de Saxcé G, Oueslati A, Charkaluk E, Tritsch J-B (eds) Limit states of materials and structures—direct methods 2. Springer, Berlin, pp 103–128Google Scholar
  34. 34.
    Simon J-W, Weichert D (2012) Shakedown analysis of engineering structures with limited kinematical hardening. Int J Solids Struct 49(4):2177–2186CrossRefMathSciNetGoogle Scholar
  35. 35.
    Simon J-W, Weichert D (2012) Shakedown analysis with multidimensional loading spaces. Comput Mech 49(4):477–485CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Staat M, Heitzer M (2002) The restricted influence of kinematical hardening on shakedown loads. In: Proceedings of the WCCM VGoogle Scholar
  37. 37.
    Stein E, Zhang G, König JA (1992) Shakedown with nonlinear strain-hardening including structural computation using finite element method. Int J Plast 8(1):1–31CrossRefzbMATHGoogle Scholar
  38. 38.
    Stein E, Zhang G, Huang Y (1993) Modeling and computation of shakedown problems for nonlinear hardening materials. Comput Methods Appl Mech Eng 103(1–2):247–272CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Stein E, Zhang G, Mahnken R, König JA (1990) Micromechanical modelling and computation of shakedown with nonlinear kinematic hardening including examples for 2-D problems. In: Axelard DR, Muschik W (eds) Recent developments of micromechanics. Springer, BerlinGoogle Scholar
  40. 40.
    Weichert D, Groß-Weege J (1988) The numerical assessment of elastic-plastic sheets under variable mechanical and thermal loads using a simplified two-surface yield condition. Int J Mech Sci 30(10):757–767CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Applied MechanicsRWTH Aachen UniversityAachenGermany

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