Variational Modeling and Finite-Element Simulation of Functional Fatigue in Polycrystalline Shape Memory Alloys

  • Johanna WaimannEmail author
  • Klaus Hackl
  • Philipp Junker


Based on our previous works, we present the finite-element implementation of an energy-based material model that displays the effect of functional fatigue of shape memory alloys during cyclic loading. The functional degradation is included in our model by taking account of irreversible martensitic volume fractions. Three internal variables are used: reversible and irreversible volume fractions for the crystallographic phases and Euler angles for parametrization of the martensite strain orientation. The evolution of the volume fractions is modeled in a rate-independent manner, whereas a viscous approach is employed for the Euler angles, which account for the materials’ polycrystalline structure. For the case of a cyclically loaded wire, we calibrate our model using experimental data. The calibration serves as input for the simulation of two more complex boundary value problems to demonstrate the functionality of our material model for localized phase transformations.


Variational modeling Shape memory alloys Functional fatigue Irreversible phase transformation Finite-element method 



  1. 1.
    Hartl, D.J., Lagoudas, D.C.: Aerospace applications of shape memory alloys. Proc. Inst. Mech. Eng. G J. Aerosp. Eng. 221(4), 535–552 (2007)CrossRefGoogle Scholar
  2. 2.
    Van Humbeeck, J.: Non-medical applications of shape memory alloys. Materi. Sci. Eng. A 273, 134–148 (1999)CrossRefGoogle Scholar
  3. 3.
    Jani, J.M., Leary, M., Subic, A., Gibson, M.A.: A review of shape memory alloy research, applications and opportunities. Mater. Des. 56, 1078–1113 (2014)CrossRefGoogle Scholar
  4. 4.
    Otsuka, K., Wayman, C.M.: Shape Memory Materials. Cambridge University Press, Cambridge (1999)Google Scholar
  5. 5.
    Otsuka, K., Ren, X.: Physical metallurgy of Ti–Ni-based shape memory alloys. Prog. Mater. Sci. 50(5), 511–678 (2005)CrossRefGoogle Scholar
  6. 6.
    Waimann, J., Junker, P., Hackl, K.: Modeling the cyclic behavior of shape memory alloys. Shape Mem. Superelasticity 3, 124–138 (2017)CrossRefGoogle Scholar
  7. 7.
    Ibarra, A., San Juan, J., Bocanegra, E.H., Nó, M.L.: Evolution of microstructure and thermomechanical properties during superelastic compression cycling in Cu–Al–Ni single crystals. Acta Materialia 55(14), 4789–4798 (2007)CrossRefGoogle Scholar
  8. 8.
    Gall, K., Maier, H.J.: Cyclic deformation mechanisms in precipitated NiTi shape memory alloys. Acta Materialia 50(18), 4643–4657 (2002)CrossRefGoogle Scholar
  9. 9.
    Delville, R., Malard, B., Pilch, J., Sittner, P., Schryvers, D.: Transmission electron microscopy investigation of dislocation slip during superelastic cycling of Ni–Ti wires. Int. J. Plasticity 27(2), 282–297 (2011)CrossRefGoogle Scholar
  10. 10.
    Krooß, P., Niendorf, T., Kadletz, P.M., Somsen, C., Gutmann, M.J., Chumlyakov, Y.I., Schmahl, W.W., Eggeler, G., Maier, H.J.: Functional fatigue and tension-compression asymmetry in [001]-oriented Co49Ni21Ga30 high-temperature shape memory alloy single crystals. Shape Mem. Superelasticity 1(1), 6–17 (2015)CrossRefGoogle Scholar
  11. 11.
    Simon, T., Kröger, A., Somsen, C., Dlouhy, A., Eggeler, G.: On the multiplication of dislocations during martensitic transformations in NiTi shape memory alloys. Acta Materialia 58(5), 1850–1860 (2010)CrossRefGoogle Scholar
  12. 12.
    Wagner, M.F.X.: Ein Beitrag zur strukturellen und funktionalen Ermüdung von Drähten und Federn aus NiTi-Formgedächtnislegierungen. Europ. Univ.-Verlag (2005)Google Scholar
  13. 13.
    Morgan, N.B., Friend, C.M.: A review of shape memory stability in NiTi alloys. Le Journal de Physique IV 11(8), 325–332 (2001)Google Scholar
  14. 14.
    Eggeler, G., Hornbogen, E., Yawny, A., Heckmann, A., Wagner, M.F.X.: Structural and functional fatigue of NiTi shape memory alloys. Mater. Sci. Eng. A 378(1), 24–33 (2004)CrossRefGoogle Scholar
  15. 15.
    Burow, J.: Herstellung, eigenschaften und mikrostruktur von ultrafeinkörnigen niti-formgedächtnislegierungen. Ph.D. thesis, Ruhr-Universität Bochum (2010)Google Scholar
  16. 16.
    Wagner, M.F.X., Nayan, N., Ramamurty, U.: Healing of fatigue damage in NiTi shape memory alloys. J. Phys. D Appl. Phys. 41(18), 185,408 (2008)CrossRefGoogle Scholar
  17. 17.
    Brinson, C.: One-dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J. Intell. Mater. Syst. Struct. 4(2), 229–242 (1993)CrossRefGoogle Scholar
  18. 18.
    Bouvet, C., Calloch, S., Lexcellent, C.: A phenomenological model for pseudoelasticity of shape memory alloys under multiaxial proportional and nonproportional loadings. Eur. J. Mech. A Solids 23(1), 37–61 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Govindjee, S., Miehe, C.: A multi-variant martensitic phase transformation model: formulation and numerical implementation. Comput. Methods Appl. Mech. Eng. 191(3), 215–238 (2001)CrossRefzbMATHGoogle Scholar
  20. 20.
    Mielke, A., Theil, F., Levitas, V.I.: A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162(2), 137–177 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Govindjee, S., Hall, G.J.: A computational model for shape memory alloys. Int. J. Solids Struct. 37(5), 735–760 (2000)CrossRefzbMATHGoogle Scholar
  22. 22.
    Govindjee, S., Kasper, E.P.: Computational aspects of one-dimensional shape memory alloy modeling with phase diagrams. Comput. Methods Appl. Mech. Eng. 171(3), 309–326 (1999)CrossRefzbMATHGoogle Scholar
  23. 23.
    Stupkiewicz, S., Petryk, H.: Modelling of laminated microstructures in stress-induced martensitic transformations. J. Mech. Phys. Solids 50(11), 2303–2331 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Stein, E., Sagar, G.: Theory and finite element computation of cyclic martensitic phase transformation at finite strain. Int. J. Numer. Methods Eng. 74(1), 1–31 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Saint-Sulpice, L., Arbab Chirani, S., Calloch, S.: A 3D super-elastic model for shape memory alloys taking into account progressive strain under cyclic loadings. Mech. Mater. 41(1), 12–26 (2009)CrossRefGoogle Scholar
  26. 26.
    Abeyaratne, R., Kim, S.J.: Cyclic effects in shape-memory alloys: a one-dimensional continuum model. Int. J. Solids Struct. 34(25), 3273–3289 (1997)CrossRefzbMATHGoogle Scholar
  27. 27.
    Hartl, D.J., Chatzigeorgiou, G., Lagoudas, D.C.: Three-dimensional modeling and numerical analysis of rate-dependent irrecoverable deformation in shape memory alloys. Int. J. Plasticity 26(10), 1485–1507 (2010)CrossRefzbMATHGoogle Scholar
  28. 28.
    Auricchio, F., Reali, A., Stefanelli, U.: A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity. Int. J. Plasticity 23(2), 207–226 (2007)CrossRefzbMATHGoogle Scholar
  29. 29.
    Bo, Z., Lagoudas, D.C.: Thermomechanical modeling of polycrystalline SMAs under cyclic loading, part III: evolution of plastic strains and two-way shape memory effect. Int. J. Eng. Sci. 37(9), 1175–1203 (1999)CrossRefzbMATHGoogle Scholar
  30. 30.
    Lagoudas, D.C., Entchev, P.B.: Modeling of transformation-induced plasticity and its effect on the behavior of porous shape memory alloys. part I: constitutive model for fully dense SMAs. Mech. Mater. 36(9), 865–892 (2004)CrossRefGoogle Scholar
  31. 31.
    Bartel, T., Osman, M., Menzel, A.: A phenomenological model for the simulation of functional fatigue in shape memory alloy wires. Meccanica 1–16 (2016)Google Scholar
  32. 32.
    Tanaka, K., Nishimura, F., Hayashi, T., Tobushi, H., Lexcellent, C.: Phenomenological analysis on subloops and cyclic behavior in shape memory alloys under mechanical and/or thermal loads. Mech. Mater. 19(4), 281–292 (1995)CrossRefGoogle Scholar
  33. 33.
    Lexcellent, C., Bourbon, G.: Thermodynamical model of cyclic behaviour of Ti–Ni and Cu–Zn–Al shape memory alloys under isothermal undulated tensile tests. Mech. Mater. 24(1), 59–73 (1996)CrossRefGoogle Scholar
  34. 34.
    Carstensen, C., Hackl, K., Mielke, A.: Non-convex potentials and microstructures in finite-strain plasticity. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 458(2018), 299–317 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Hackl, K., Fischer, F.D.: On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials. Proc. R. Soc. A Math. Phys. Eng. Sci. 464(2089), 117–132 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Junker, P., Makowski, J., Hackl, K.: The principle of the minimum of the dissipation potential for non-isothermal processes. Continuum Mech. Thermodyn. 26(3), 259–268 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Junker, P.: A novel approach to representative orientation distribution functions for modeling and simulation of polycrystalline shape memory alloys. Int. J. Numer. Methods Eng. 98(11), 799–818 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Hackl, K., Heinen, R.: A micromechanical model for pretextured polycrystalline shape-memory alloys including elastic anisotropy. Continuum Mech. Thermodyn. 19(8), 499–510 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Govindjee, S., Hackl, K., Heinen, R.: An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations. Continuum Mech. Thermodyn. 18(7–8), 443–453 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Waimann, J., Junker, P., Hackl, K.: A coupled dissipation functional for modeling the functional fatigue in polycrystalline shape memory alloys. Eur. J. Mech. A Solids 55, 110–121 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Junker, P.: An accurate, fast and stable material model for shape memory alloys. Smart Mater. Struct. 23(11), 115,010 (2014)CrossRefGoogle Scholar
  42. 42.
    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method: Solid Mechanics, vol. 2. Butterworth-heinemann (2000)Google Scholar
  43. 43.
    Wriggers, P.: Nonlinear Finite Element Methods, vol. 4. Springer, Berlin (2008)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Applied MechanicsRWTH Aachen UniversityAachenGermany
  2. 2.Institute of Mechanics of MaterialsRuhr-Universität BochumBochumGermany

Personalised recommendations