Skip to main content
SpringerLink
Log in

On shakedown of shape memory alloys structures

  • Original Article
  • Published:
Annals of Solid and Structural Mechanics

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  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, Swansea

    Google Scholar 

  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 61:807–836

    Article  MATH  MathSciNet  Google Scholar 

  3. Brézis H (1972) Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de hilbert. North-Holland, Amsterdam

    Google Scholar 

  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):561–568

    Article  Google Scholar 

  5. Feng X, Sun Q (2007) Shakedown analysis of shape memory alloy structures. Int J Plasticity 23:183–206

    Article  MATH  MathSciNet  Google Scholar 

  6. Frémond M (2002) Non-smooth thermomechanics. Springer, New York

    Book  MATH  Google Scholar 

  7. Govindjee S, Miehe C (2001) A multi-variant martensitic phase transformation model: formulation and numerical implementation. Comput Meth Appl Mech Engrg 191:215–238

    Article  MATH  Google Scholar 

  8. Hackl K, Heinen R (2008) An upper bound to the free energy of \(n-\)variant polycrystalline shape memory alloys. J Mech Phys Solids 56:2832–2843

    Article  MATH  MathSciNet  Google Scholar 

  9. Halphen B (1978) Accommodation et adaptation des structures élastoviscoplastiques et plastiques. Association amicale des ingénieurs anciens élèves de l’ENPC

  10. Halphen B, Nguyen QS (1975) Sur les matériaux standards généralisés. J Mécanique 14:1–37

    Google Scholar 

  11. Koiter WT (1960) General problems for elastic solids. Progress in solid mechanics

  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 40:389–418

    Article  MATH  MathSciNet  Google Scholar 

  13. Maitournam H, Pommier B, Thomas JJ (2002) Détermination de la réponse asymptotique d’une structure anélastique sous chargement cyclique. C R Mecanique 330:703–708

    Article  MATH  Google Scholar 

  14. Melan E (1936) Theorie statisch unbestimmter systeme aus ideal-plastischen baustoff. Sitz Berl Ak Wiss 145:195–218

    MATH  Google Scholar 

  15. Nguyen QS (2003) On shakedown analysis in hardening plasticity. J Mech Phys Solids 51:101–125

    Article  MATH  MathSciNet  Google Scholar 

  16. Peigney M (2006) A time-integration scheme for thermomechanical evolutions of shape-memory alloys. C R Mecanique 334(4):266–271

    Article  MATH  Google Scholar 

  17. Peigney M (2009) A non-convex lower bound on the effective free energy of polycrystalline shape memory alloys. J Mech Phys Solids 57:970–986

    Article  MathSciNet  Google Scholar 

  18. Peigney M (2010) Shakedown theorems and asymptotic behaviour of solids in nonsmooth mechanics. Eur J Mech A 29:785793

    Article  MathSciNet  Google Scholar 

  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):4043–4054

    Article  Google Scholar 

  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):2791–2799

    Article  Google Scholar 

  21. Peigney M, Stolz C (2001) Approche par contrôle optimal des structures élastoviscoplastiques sous chargement cyclique. C R Acad Sci Paris Série II 329:643–648

    MATH  Google Scholar 

  22. Peigney M, Stolz C (2003) An optimal control approach to the analysis of inelastic structures under cyclic loading. J Mech Phys Solids 51:575–605

    Article  MATH  MathSciNet  Google Scholar 

  23. Pham D (2008) On shakedown theory for elastic-plastic materials and extensions. J Mech Phys Solids 56:1905–1915

    Article  MATH  MathSciNet  Google Scholar 

  24. Simon JW (2013) Direct evaluation of the limit states of engineering structures exhibiting limited, nonlinear kinematical hardening. Int J Plasticity 42:141–167

    Article  Google Scholar 

  25. Simon JW, Weichert D (2012) Shakedown analysis of engineering structures with limited kinematical hardening. Int J Sol Struct 49(15):2177–2186

    Article  MathSciNet  Google Scholar 

  26. Souza A, Mamiya E, Zouain N (1998) Three-dimensional model for solids undergoing stress-induced phase transformations. Eur J Mech A 17:789–806

    Article  MATH  Google Scholar 

  27. Spiliopoulos KV, Panagiotou KD (2012) A direct method to predict cyclic steady states of elastoplastic structures. Comput Methods Appl Mech Engrg 223:186–198

    Article  MathSciNet  Google Scholar 

  28. Wesfreid E (1980) Etude du comportement asymptotique pour un modèle viscoplastique. C R Acad Sci Paris A 290:297–300

    MATH  MathSciNet  Google Scholar 

  29. Zarka J, Frelat J, Inglebert G (1988) A new approach to inelastic analysis of structures. Martinus Nijhoff Publishers, Dordrecht

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Laboratoire Navier (Ecole des Ponts ParisTech, IFSTTAR, CNRS), Université Paris-Est, 77455, Marne-la-Vallée Cedex 2, France

    Michaël Peigney

Authors
  1. Michaël Peigney
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michaël Peigney.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peigney, M. On shakedown of shape memory alloys structures. Ann. Solid Struct. Mech. 6, 17–28 (2014). https://doi.org/10.1007/s12356-014-0035-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12356-014-0035-1

Keywords

Search

Navigation