Skip to main content
SpringerLink
Log in

A model for high temperature creep of single crystal superalloys based on nonlocal damage and viscoplastic material behavior

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

A model for high temperature creep of single crystal superalloys is developed, which includes constitutive laws for nonlocal damage and viscoplasticity. It is based on a variational formulation, employing potentials for free energy, and dissipation originating from plasticity and damage. Evolution equations for plastic strain and damage variables are derived from the well-established minimum principle for the dissipation potential. The model is capable of describing the different stages of creep in a unified way. Plastic deformation in superalloys incorporates the evolution of dislocation densities of the different phases present. It results in a time dependence of the creep rate in primary and secondary creep. Tertiary creep is taken into account by introducing local and nonlocal damage. Herein, the nonlocal one is included in order to model strain localization as well as to remove mesh dependence of finite element calculations. Numerical results and comparisons with experimental data of the single crystal superalloy LEK94 are shown.

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

Similar content being viewed by others

References

  1. Durand-Charre M.: The Microstructure of Superalloys. Gordon and Breach, New York (1997)

    Google Scholar 

  2. Mughrabi H., Tetzlaff U.: Microstructure and high-temperature strength of monocrystalline nickel-base superalloys. Adv. Eng. Mater. 2, 319–326 (2000)

    Article  Google Scholar 

  3. Prasad S.C., Rao I.J., Rajagopal K.R.: A continuum model for the creep of single crystal nickel-base superalloys. Acta Mater. 53, 669–679 (2005)

    Article  Google Scholar 

  4. Pollock T.M., Argon A.S.: Creep resistance of CMSX-3 nickel base superalloy single crystals. Acta Metall. Mater. 40, 1–30 (1992)

    Article  Google Scholar 

  5. Svoboda J., Lukáš P.: Modelling of kinetics of directional coarsening in Ni-Superalloys. Acta Mater. 44, 2557–2565 (1996)

    Article  Google Scholar 

  6. Svoboda J., Lukáš P.: Modelling of recovery controlled creep in nickel-base superalloy single crystals. Acta Mater. 45, 125–135 (1997)

    Article  Google Scholar 

  7. Svoboda J., Lukáš P.: Model of creep in 〈001〉-oriented superalloy single crystals. Acta Mater. 46, 3421–3431 (1998)

    Article  Google Scholar 

  8. Svoboda J., Lukáš P.: Creep deformation modelling of superalloy single crystals. Acta Mater. 48, 2519–2528 (2000)

    Article  Google Scholar 

  9. Becker A.A., Hyde T.H., Sun W., Andersson P.: Benchmarks for finite element analysis of creep continuum damage mechanics. Comput. Mater. Sci. 25, 34–41 (2002)

    Article  Google Scholar 

  10. Dunne F.P.E., Otheman A.M., Hall F.R., Hayhurst D.R.: Representation of uniaxial creep curves using continuum damage mechanics. Int. J. Mech. Sci. 32, 945–957 (1990)

    Article  Google Scholar 

  11. Kowalewski Z.L., Hayhurst D.R., Dyson B.F.: Mechanisms-based creep constitutive equations for an aluminium alloy. J. Strain Anal. 29, 309–316 (1994)

    Article  Google Scholar 

  12. Naumenko K., Altenbach H.: Modeling of Creep for Structural Analysis. Springer, Berlin (2007)

    Book  Google Scholar 

  13. Kachanov L.M., Hayhurst D.R., Dyson B.F.: Time of the fracture process under creep conditions. Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk 8, 26–31 (1958)

    Google Scholar 

  14. Hayhurst R.R.: Stress redistribution and rupture due to creep in a uniformly stretched thin plate containing a circular hole. J. Appl. Mech. Tech. Phys. 20, 381–390 (1972)

    Article  Google Scholar 

  15. Zyczkowski M.: Creep damage evolution equations expressed in terms of dissipated power. Int. J. Mech. Sci. 42, 755–769 (2000)

    Article  MATH  Google Scholar 

  16. Onsager L.: Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426 (1931)

    Article  ADS  Google Scholar 

  17. Onsager L.: Theories and problems of liquid diffusion. Ann. N.Y. Acad. Sci. 46, 241–265 (1945)

    Article  ADS  Google Scholar 

  18. Ziegler H.: Some extremum principles in irreversible thermodynamics with applications to continuum mechanics. In: Sneddon, I.N., Hill, R. (eds) Progress in Solid Mechanics, pp. 92–193. North-Holland, Amsterdam (1963)

    Google Scholar 

  19. Krajcinovic D.: Constitutive equations for damage materials. J. Appl. Mech. 50, 355–380 (1983)

    Article  ADS  MATH  Google Scholar 

  20. Ziegler H.: Continuum damage mechanics: Part 1-general concepts. J. Appl. Mech. 55, 59–64 (1988)

    Article  Google Scholar 

  21. Murakami S., Kamiya K.: Constitutive and damage evolution equations of elastic-brittle materials based on irreversible thermodynamics. Int. J. Mech. Sci. 39, 473–486 (1997)

    Article  MATH  Google Scholar 

  22. Hesebeck O.: Irreversibility, stability and maximum dissipation in elastoplastic damage. Int. J. Damage Mech. 9, 329–351 (2000)

    Article  Google Scholar 

  23. Ibrahimbegovic A., Jehel P., Davenne L.: Coupled damage-plasticity constitutive model and direct stress interpolation. Comput. Mech. 42, 1–11 (2008)

    Article  MATH  Google Scholar 

  24. Svoboda J., Fischer F.D., Fratzl P., Kroupa A.: Diffusion in multi-component systems with no or dense sources and sinks for vacancies. Acta Mater. 50, 1369–1381 (2002)

    Article  Google Scholar 

  25. Svoboda J., Fischer F.D., Fratzl P.: Diffusion and creep in multi-component alloys with non-ideal sources and sinks for vacancies. Acta Mater. 54, 3043–3053 (2006)

    Article  Google Scholar 

  26. Martin J.B., Ponter A.R.S.: A note on a work inequality in linear viscoelasticity. Q. Appl. Math. 24, 161–165 (1966)

    Google Scholar 

  27. Maier G.: Some theorems for plastic strain rates and plastic strains. J. Mecanique 8, 5–19 (1969)

    MATH  Google Scholar 

  28. Halphen B., Nguyen Q.S.: Sur les matériaux standards généralisés. J. Mecanique 14, 39–63 (1975)

    MATH  MathSciNet  Google Scholar 

  29. Maugin G.A.: The Thermomechanics of Plasticity and Fracture. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  30. Hackl K., Fischer F.D.: On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials. Proc. R. Soc. Lond. A 464, 117–132 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  31. Hackl K., Fischer F.D., Svoboda J.: A study on the principle of maximum dissipation for coupled and non-coupled non-isothermal processes in materials. Proc. R. Soc. Lond. A 467, 1186–1196 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  32. Hackl K., Fischer F.D., Svoboda J.: A study on the principle of maximum dissipation for coupled and non-coupled non-isothermal processes in materials. Proc. R. Soc. Lond. A 467, 2422–2426 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Pollock T.M., Argon A.S.: Directional coarsening in nickel-base single crystals with high volume fractions of coherent precipitates. Acta Metall. Mater. 42, 1859–1874 (1994)

    Article  Google Scholar 

  34. Nathal M.V., Ebert L.J.: The influence of cobalt, tantalum, and tungsten on the microstructure of single crystal nickel-base superalloys. Metall. Mater. Trans. A 16, 1849–1862 (1985)

    Article  ADS  Google Scholar 

  35. Nabarro F.R.N.: Rafting in superalloys. Metall. Mater. Trans. A 27, 513–530 (1996)

    Article  Google Scholar 

  36. Coujou A., Benyoucef M., Legros M., Clément N.: Role of the γ/γ′ interface on the mechanical properties of single crystals nickel base superalloys. Solid State Pheno. 59(60), 185–200 (1998)

    Article  Google Scholar 

  37. Mälzer G., Hayes R.W., MacK T., Eggeler G.: Miniature specimen assessment of creep of the single-crystal superalloy LEK94 in the 1000° C temperature range. Metall. Mater. Trans. A 38, 314–327 (2007)

    Article  Google Scholar 

  38. Hackl K., Heinen R.: A micromechanical model for pretextured polycrystalline shape-memory alloys including elastic anisotropy. Continuum Mech. Therm. 19, 499–510 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Dimitrijevic B.J., Hackl K.: A method for gradient enhancement of continuum damage models. Tech. Mechanik 28, 43–52 (2008)

    Google Scholar 

  40. Dimitrijevic B.J., Hackl K.: A regularization framework for damage-plasticity models via gradient enhancement of the free energy. Int. J. Numer. Method Biomed. Eng. 27, 1199–1210 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  41. Estrin Y., Mecking H.: Unified phenomenological description of work hardening and creep based on one-parameter models. Acta Mater. 32, 57–70 (1984)

    Article  Google Scholar 

  42. Ghosh R.N., Curtis R.V., McLean M.: Creep deformation of single crystal superalloys—modelling the crystallographic anisotropy. Acta Metall. Mater. 38, 1977–1992 (1990)

    Article  Google Scholar 

  43. Méric L., Poubanne P., Cailletaud G.: Single crystal modeling for structural calculations: Part 1—model presentation. J. Eng. Mater. Tech. 113, 162–170 (1991)

    Article  Google Scholar 

  44. Méric L., Cailletaud G.: Single crystal modeling for structural calculations: Part 2—finite element implementation. J. Eng. Mater. Tech. 113, 171–182 (1991)

    Article  Google Scholar 

  45. Zienkiewicz O.C., Taylor R.L.: The Finite Element Method. Volume 1: The Basis. Butterworth-Heinmann, London (2000)

    Google Scholar 

  46. Taylor, R.L.: A Finite Element Analysis Program, Programmer Manual. University of California, Berkeley. http://www.ce.berkeley.edu/projects/feap/ (2008)

  47. Vladimirov I.N., Reese S., Eggeler G.: Constitutive modelling of the anisotropic creep behaviour of nickel-base single crystal superalloys. Int. J. Mech. Sci 51, 305–313 (2009)

    Article  MATH  Google Scholar 

  48. Srinivasan R., Eggeler G., Mills M.J.: γ′-cutting as rate-controlling recovery process during high-temperature and low-stress creep of superalloy single crystals. Acta Mater. 48, 4867–4878 (2000)

    Article  Google Scholar 

  49. Schreuer, J., Mazzolai, G., Agudo, J., Eggeler, G., Hackl, K.: Correlation between microstructure and elastic properties of the single crystal superalloy LEK94 at temperatures up to 1400K. Poster, European Symposium on Superalloys and Their Application 25–28 May 2010, Wildbad Kreuth, Germany

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl für Mechanik-Materialtheorie, Ruhr-Universität Bochum, Universitätsstr. 150, 44801, Bochum, Germany

    B. T. Trinh & K. Hackl

Authors
  1. B. T. Trinh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Hackl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Hackl.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Trinh, B.T., Hackl, K. A model for high temperature creep of single crystal superalloys based on nonlocal damage and viscoplastic material behavior. Continuum Mech. Thermodyn. 26, 551–562 (2014). https://doi.org/10.1007/s00161-013-0317-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-013-0317-6

Keywords

Mathematics Subject Classification (2010)

Search

Navigation