Advertisement

Annals of Solid and Structural Mechanics

, Volume 6, Issue 1–2, pp 17–28 | Cite as

On shakedown of shape memory alloys structures

  • Michaël PeigneyEmail author
Original Article

Abstract

This paper is concerned with the large-time behaviour of shape-memory alloys structures when they are submitted to a given loading history. Extending the approach introduced by Koiter in plasticity, we state sufficient conditions for the energy dissipation to remain bounded in time, independently on the initial state. Such a behavior is classically referred to as shakedown and is associated with the idea that the evolution becomes elastic in the large-time limit. The study of a particular example shows that the large-time behaviour of shape-memory alloys structures exhibit some complex features which are not found in standard plasticity.

Keywords

Shape-memory alloys Shakedown Direct methods  Non-smooth mechanics 

Notes

Acknowledgments

Part of this work has been motivated by discussions with Dr.-Ing. J.W. Simon during the Euromech Colloquium ’Direct and variational methods for non smooth problems in mechanics’ (Amboise, 24-26 June 2013), organized by Pr. G. De Saxcé et Pr. G. Del Piero.

References

  1. 1.
    Akel S, Nguyen Q (1989) Determination of the cyclic response in cyclic plasticity. In: Owen DRJ et al (eds) Computational plasticity: models, software and applications. Pineridge Press, SwanseaGoogle Scholar
  2. 2.
    Auricchio F, Petrini L (2004) A three-dimensional model describing stress-temperature induced solid phase transformations: solution algorithm and boundary value problems. Int J Num Meth Eng 61:807–836CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brézis H (1972) Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de hilbert. North-Holland, AmsterdamGoogle Scholar
  4. 4.
    Constantinescu A, Van Dang K, Maitournam M (2003) A unified approach for high and low cycle fatigue based on shakedown concepts. Eng Mater Struct 26(6):561–568CrossRefGoogle Scholar
  5. 5.
    Feng X, Sun Q (2007) Shakedown analysis of shape memory alloy structures. Int J Plasticity 23:183–206CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Frémond M (2002) Non-smooth thermomechanics. Springer, New YorkCrossRefzbMATHGoogle Scholar
  7. 7.
    Govindjee S, Miehe C (2001) A multi-variant martensitic phase transformation model: formulation and numerical implementation. Comput Meth Appl Mech Engrg 191:215–238CrossRefzbMATHGoogle Scholar
  8. 8.
    Hackl K, Heinen R (2008) An upper bound to the free energy of \(n-\)variant polycrystalline shape memory alloys. J Mech Phys Solids 56:2832–2843CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Halphen B (1978) Accommodation et adaptation des structures élastoviscoplastiques et plastiques. Association amicale des ingénieurs anciens élèves de l’ENPCGoogle Scholar
  10. 10.
    Halphen B, Nguyen QS (1975) Sur les matériaux standards généralisés. J Mécanique 14:1–37Google Scholar
  11. 11.
    Koiter WT (1960) General problems for elastic solids. Progress in solid mechanicsGoogle Scholar
  12. 12.
    Kružík M, Mielke A, Roubícek T (2005) Modelling of microstructure and its evolution in shape-memory alloy single crystals, in particular in cualni. Meccanica 40:389–418CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Maitournam H, Pommier B, Thomas JJ (2002) Détermination de la réponse asymptotique d’une structure anélastique sous chargement cyclique. C R Mecanique 330:703–708CrossRefzbMATHGoogle Scholar
  14. 14.
    Melan E (1936) Theorie statisch unbestimmter systeme aus ideal-plastischen baustoff. Sitz Berl Ak Wiss 145:195–218zbMATHGoogle Scholar
  15. 15.
    Nguyen QS (2003) On shakedown analysis in hardening plasticity. J Mech Phys Solids 51:101–125CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Peigney M (2006) A time-integration scheme for thermomechanical evolutions of shape-memory alloys. C R Mecanique 334(4):266–271CrossRefzbMATHGoogle Scholar
  17. 17.
    Peigney M (2009) A non-convex lower bound on the effective free energy of polycrystalline shape memory alloys. J Mech Phys Solids 57:970–986CrossRefMathSciNetGoogle Scholar
  18. 18.
    Peigney M (2010) Shakedown theorems and asymptotic behaviour of solids in nonsmooth mechanics. Eur J Mech A 29:785793CrossRefMathSciNetGoogle Scholar
  19. 19.
    Peigney M, Seguin J (2013) An incremental variational approach to coupled thermo-mechanical problems in anelastic solids. application to shape-memory alloys. Int J Sol Struct 50(24):4043–4054CrossRefGoogle Scholar
  20. 20.
    Peigney M, Seguin J, Hervé-Luanco E (2011) Numerical simulation of shape memory alloys structures using interior-point methods. Int J Sol Struct 48(20):2791–2799CrossRefGoogle Scholar
  21. 21.
    Peigney M, Stolz C (2001) Approche par contrôle optimal des structures élastoviscoplastiques sous chargement cyclique. C R Acad Sci Paris Série II 329:643–648zbMATHGoogle Scholar
  22. 22.
    Peigney M, Stolz C (2003) An optimal control approach to the analysis of inelastic structures under cyclic loading. J Mech Phys Solids 51:575–605CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Pham D (2008) On shakedown theory for elastic-plastic materials and extensions. J Mech Phys Solids 56:1905–1915CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Simon JW (2013) Direct evaluation of the limit states of engineering structures exhibiting limited, nonlinear kinematical hardening. Int J Plasticity 42:141–167CrossRefGoogle Scholar
  25. 25.
    Simon JW, Weichert D (2012) Shakedown analysis of engineering structures with limited kinematical hardening. Int J Sol Struct 49(15):2177–2186CrossRefMathSciNetGoogle Scholar
  26. 26.
    Souza A, Mamiya E, Zouain N (1998) Three-dimensional model for solids undergoing stress-induced phase transformations. Eur J Mech A 17:789–806CrossRefzbMATHGoogle Scholar
  27. 27.
    Spiliopoulos KV, Panagiotou KD (2012) A direct method to predict cyclic steady states of elastoplastic structures. Comput Methods Appl Mech Engrg 223:186–198CrossRefMathSciNetGoogle Scholar
  28. 28.
    Wesfreid E (1980) Etude du comportement asymptotique pour un modèle viscoplastique. C R Acad Sci Paris A 290:297–300zbMATHMathSciNetGoogle Scholar
  29. 29.
    Zarka J, Frelat J, Inglebert G (1988) A new approach to inelastic analysis of structures. Martinus Nijhoff Publishers, DordrechtGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Laboratoire Navier (Ecole des Ponts ParisTech, IFSTTAR, CNRS)Université Paris-EstMarne-la-Vallée Cedex 2France

Personalised recommendations