Archive of Applied Mechanics

, Volume 88, Issue 1–2, pp 65–81 | Cite as

Critical stresses estimation by crystal viscoplasticity modeling of rate-dependent anisotropy of Al-rich TiAl alloys at high temperature

  • Helal Chowdhury
  • Konstantin Naumenko
  • Holm Altenbach
  • Manja Krüger
Special

Abstract

Determining critical stresses for different slip systems is one of the most important parts in crystal plasticity modeling of anisotropy. However, the task of finding individual critical resolved shear stress (CRSS) for every single slip system, if not impossible, is formidable and a delicate one especially if the microstructure is very complex. Slip family-based, mechanism-based and morphology-based (e.g., phase interface) slip systems classification and hence determining CRSS consistent with experimental measurements are often used in crystal plasticity. In this work, a novel approach to determining CRSS at high homologous temperature has been proposed by crystal plasticity modeling of rate-dependent anisotropy. Two-internal-variable-based phenomenological crystal viscoplasticity model is adopted for simulating isothermal, two-phase, single-crystal-like Al-rich lamellar Ti–61.8at.%Al binary alloy at high-temperature compression state (\(1050\,^\circ \hbox {C}\)) by employing finite strain and finite rotation framework. To the best of authors’ knowledge, this is the first micromechanical modeling attempt with long-period superstructures. Conventional approaches related to CRSS estimation are also compared with the proposed one. Our material parameters are based on calibrating three different sets of compressive stain rate-controlled plasticity data taken from the loading of two different lamellar directions. It is revealed that the proposed approach works fine for rate-dependent anisotropy modeling, while other conventional approaches highly under- or overestimate available anisotropic experimental behavior of this alloy.

Keywords

Crystal viscoplasticity CRSS Al-rich TiAl alloy Rate-dependent anisotropy 

Notes

Acknowledgements

This work has been supported by DFG within the Ph.D. school GRK1554. We would like to gratefully acknowledge Prof. Albrecht Bertram (Magdeburg) for his suggestion regarding modeling approach and would like to thank our GRK fellow Mr. Philipp Thiem for his valuable comments from the material science perspectives. We also would like to express our special thanks of gratitude to Dr. Mokarram Hossain from Swansea University and to Mr. Mamun Al-Siraj from TU Darmstadt for their valuable comments after line-by-line proofreading.

References

  1. 1.
    Agnew, S.R., Duygulu, O.: Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B. Int. J. Plast. 21(6), 1161–1193 (2005)CrossRefMATHGoogle Scholar
  2. 2.
    Nakano, T., Hayashi, K., Umakoshi, Y., Chiu, Y.L., Veyssiere, P.: Effects of Al concentration and resulting long-period superstructures on the plastic properties at room temperature of Al-rich TiAl single crystals. Philos. Mag. 85(22), 2527–2548 (2005)CrossRefGoogle Scholar
  3. 3.
    Oesterle, W., Bettge, D., Fedelich, B., Klingelhoeffer, H.: Modelling the orientation and direction dependence of the critical resolved shear stress of nickel-base superalloy single crystals. Acta Mater. 48(3), 689–700 (2000)CrossRefGoogle Scholar
  4. 4.
    Zupan, M., Hemker, K.J.: Yielding behavior of aluminum-rich single crystalline \(\gamma \)-TiAl. Acta Mater. 51(20), 6277–6290 (2003)CrossRefGoogle Scholar
  5. 5.
    Werwer, M., Cornec, A.: The role of superdislocations for modeling plastic deformation of lamellar TiAl. Int. J. Plast. 22(9), 1683–1698 (2006)CrossRefMATHGoogle Scholar
  6. 6.
    Sanchez-Martin, R., Perez-Prado, M.T., Segurado, J., Bohlen, J., Gutierrez-Urrutia, I., Llorca, J., Molina-Aldareguia, J.M.: Measuring the critical resolved shear stresses in Mg alloys by instrumented nanoindentation. Acta Mater. 71, 283–292 (2014)CrossRefGoogle Scholar
  7. 7.
    Li, H., Mason, D.E., Bieler, T.R., Boehlert, C.J., Crimp, M.A.: Methodology for estimating the critical resolved shear stress ratios of \(\alpha \)-phase Ti using EBSD-based trace analysis. Acta Mater. 61(20), 7555–7567 (2013)CrossRefGoogle Scholar
  8. 8.
    Herrera-Solaz, V., Hidalgo-Manrique, P., Perez-Prado, M.T., Letzig, D., Llorca, J., Segurado, J.: Effect of rare earth additions on the critical resolved shear stresses of magnesium alloys. Mater. Lett. 128, 199–203 (2014)CrossRefGoogle Scholar
  9. 9.
    Hutchinson, W.B., Barnett, M.R.: Effective values of critical resolved shear stress for slip in polycrystalline magnesium and other HCP metals. Scr. Mater. 63(7), 737–740 (2010)CrossRefGoogle Scholar
  10. 10.
    Hamelin, C.J., Diak, B.J., Pilkey, A.K.: Multiscale modelling of the induced plastic anisotropy in BCC metals. Int. J. Plast. 27(8), 1185–1202 (2011)CrossRefMATHGoogle Scholar
  11. 11.
    Grujicic, M., Batchu, S.: A crystal plasticity materials constitutive model for polysynthetically-twinned \(\gamma \text{-TiAl } + \alpha _2 \text{-Ti }_{3} \text{ Al }\) single crystals. J. Mater. Sci. 36, 2851–2863 (2001)CrossRefGoogle Scholar
  12. 12.
    Nitz, A., Nembach, E.: Critical resolved shear stress anomalies of the \(\text{ L1 }_{2}\)-long-range ordered \(\gamma \)-phase of the superalloy NIMONIC 105. Mater. Sci. Eng. A 263(1), 15–22 (1999)CrossRefGoogle Scholar
  13. 13.
    Kumar, M.A., Beyerlein, I.J., Tome, C.N.: Effect of local stress fields on twin characteristics in HCP metals. Acta Mater. 116, 143–154 (2016)CrossRefGoogle Scholar
  14. 14.
    Lebensohn, R., Uhlenhut, H., Hartig, C., Mecking, H.: Plastic flow of \(\gamma \)-TiAl-based polysynthetically twinned crystals: micromechanical modeling and experimental validation. Acta Mater. 46(13), 4701–4709 (1998)CrossRefGoogle Scholar
  15. 15.
    Naumenko, K., Altenbach, H.: Modeling High Temperature Materials Behavior for Structural Analysis: Part I: Continuum Mechanics Foundations and Constitutive Models, vol. 28. Springer, Cham (2016)Google Scholar
  16. 16.
    Huh, J., Huh, H., Lee, C.S.: Effect of strain rate on plastic anisotropy of advanced high strength steel sheets. Int. J. Plast. 44, 23–46 (2013)CrossRefGoogle Scholar
  17. 17.
    Meredith, C.S., Khan, A.S.: Texture evolution and anisotropy in the thermo-mechanical response of UFG Ti processed via equal channel angular pressing. Int. J. Plast. 3031, 202–217 (2012)CrossRefGoogle Scholar
  18. 18.
    Nixon, M.E., Cazacu, O., Lebensohn, R.A.: Anisotropic response of high-purity \(\alpha \)-titanium: Experimental characterization and constitutive modeling. Int. J. Plast. 26(4), 516–532 (2015)CrossRefMATHGoogle Scholar
  19. 19.
    Naumenko, K., Gariboldi, E.: A phase mixture model for anisotropic creep of forged Al–Cu–Mg–Si alloy. Mater. Sci. Eng. A 618, 368–376 (2014)CrossRefGoogle Scholar
  20. 20.
    Gariboldi, E., Naumenko, K., Ozhoga-Maslovskaja, O., Zappa, E.: Analysis of anisotropic damage in forged al-cu-mg-si alloy based on creep tests, micrographs of fractured specimen and digital image correlations. Mater. Sci. Eng. A 652, 175–185 (2016)CrossRefGoogle Scholar
  21. 21.
    Truszkowski, W.: The Plastic Anisotropy in Single Crystals and Polycrystalline Metals. Springer, Dordrecht (2001)CrossRefMATHGoogle Scholar
  22. 22.
    Fujiwara, T., Nakamuraa, A., Hosomia, M., Nishitania, S.R., Shiraia, Y., Yamaguchia, M.: Deformation of polysynthetically twinned crystals of TiAl with a nearly stoichiometric composition. Philos. Mag. A 61(4), 591–606 (1990)CrossRefGoogle Scholar
  23. 23.
    Brockman, R.A.: Analysis of elastic–plastic deformation in TiAl polycrystals. Int. J. Plast. 19(10), 1749–1772 (2003)CrossRefMATHGoogle Scholar
  24. 24.
    Zambaldi, C., Raabe, D.: Plastic anisotropy of \(\gamma \)-tial revealed by axisymmetric indentation. Acta Mater. 58(9), 3516–3530 (2010)CrossRefGoogle Scholar
  25. 25.
    Wegmann, G., Suda, T., Maruyama, K.: Deformation characteristics of polysynthetically twinned (PST) crystals during creep at 1150K. Intermetallics 8(2), 165–177 (2000)CrossRefGoogle Scholar
  26. 26.
    Appel, F., Clemens, H., Fischer, F.D.: Modeling concepts for intermetallic titanium aluminides. Prog. Mater. Sci. 81, 55–124 (2016)CrossRefGoogle Scholar
  27. 27.
    Chen, G., Peng, Y., Zheng, G., Qi, Z., Wang, M., Yu, H., Dong, C., Liu, C.T.: Polysynthetic twinned TiAl single crystals for high-temperature applications. Nat. Mater. 15, 876–881 (2016)CrossRefGoogle Scholar
  28. 28.
    Palm, M., Engberding, N., Stein, F., Kelm, K., Irsen, S.: Phases and evolution of microstructures in Ti–60at.%Al. Acta Mater. 60, 3559–3569 (2012)CrossRefGoogle Scholar
  29. 29.
    Li, J., Weng, G.J.: Time-dependent creep of a dual-phase viscoplastic material with lamellar structure. Int. J. Plast. 14(8), 755–770 (1998)CrossRefMATHGoogle Scholar
  30. 30.
    Sturm, D.: Herstellung und eigenschaften al-reicher tial legierungen. Ph.D. thesis, OvGU Magdeburg (2010)Google Scholar
  31. 31.
    Lei, C., Xu, Q., Sun, Y.: Phase orientation relationships in the \(\text{ TiAl-TiAl }_{2}\) region. Mater. Sci. Eng. A 313, 227–236 (2001)CrossRefGoogle Scholar
  32. 32.
    Stein, F., Zhang, L.C., Sauthoff, G., Palm, M.: TEM and DTA study on the stability of \(\text{ Al }_{5} \text{ Ti }_{3-}\) and \(\text{ h-Al }_{2}\)Ti-superstructures in aluminium-rich TiAl alloys. Acta Mater. 49(15), 2919–2932 (2001)CrossRefGoogle Scholar
  33. 33.
    Sturm, D., Heilmaier, M., Saage, H., Aguilar, J., Schmitz, G.J., Drevermann, A., Palm, M., Stein, F., Engberding, N., Kelm, K., Irsen, S.: Creep strength of a binary \(\text{ Al }_{62} \text{ Ti }_{38}\) alloy. Int. J. Mater. Res. 101(5), 676–679 (2010)CrossRefGoogle Scholar
  34. 34.
    Hayashi, K., Nakano, T., Umakoshi, Y.: Plastic deformation behaviour and deformation substructure in Al-rich TiAl single crystals deformed at high temperatures. Sci. Technol. Adv. Mater. 2, 433–441 (2001)CrossRefGoogle Scholar
  35. 35.
    Nakano, T., Hayashi, K., Umakoshi, Y.: Formation and stability of transitional long-period superstructures in Al-rich TiAl single crystals. Philos. Mag. A 82, 763–777 (2002)CrossRefGoogle Scholar
  36. 36.
    Zhang, L.C., Palm, M., Stein, F., Sauthoff, G.: Formation lamellar microstructures Al-rich TiAl alloys between 900 to \(1100\,^\circ \text{ C }\). Intermetallics 9, 229–238 (2001)CrossRefGoogle Scholar
  37. 37.
    Nakano, T., Hata, S., Hayashi, K., Umakoshi, Y.: Some Long-Period Superstructures and the Related Motion of Dislocations in Al-Rich TiAl Single Crystals, vol. 2, pp. 797–804. Wiley, New York (2012)Google Scholar
  38. 38.
    Umakoshi, Y., Nakano, T., Ashida, K.: High-temperature deformation in Ti–62.5at.%Al single crystals containing small \(\text{ Al }_{2}\)Ti-type precipitates. Mater. Sci. Forum 304–306, 163–168 (1999)CrossRefGoogle Scholar
  39. 39.
    Nakano, T., Hayashi, K., Ashida, K., Umakoshi, Y.: Effect of \(\text{ Al }_{2}\text{ Ti }\) phase on plastic behavior in Ti-62.5 At%Al single crystals. In: Symposium: High-Temperature Ordered Intermetallic Alloys VIII. MRS Proceedings, vol. 552 (1998)Google Scholar
  40. 40.
    Hata, S., Nakano, T., Kuwano, N., Itakura, M., Matsumura, S., Umakoshi, Y.: Microscopic properties of long-period ordering in Al-rich TiAl alloys. Metal. Mater. Trans. A 39(7), 1610–1617 (2008)CrossRefGoogle Scholar
  41. 41.
    Appel, F., Paul, J., Oehring, M.: Gamma Titanium Aluminide Alloys: Science and Technology. Wiley, Weinheim (2011)CrossRefGoogle Scholar
  42. 42.
    Asaro, R.J., Needleman, A.: Overview no. 42 texture development and strain hardening in rate dependent polycrystals. Acta Metal. 33(6), 923–953 (1985)CrossRefGoogle Scholar
  43. 43.
    Mathur, K.K., Dawson, P.R.: On modeling the development of crystallographic texture in bulk forming processes. Int. J. Plast. 5(1), 67–94 (1989)CrossRefGoogle Scholar
  44. 44.
    Zhang, M., Zhang, J., McDowell, D.L.: Microstructure-based crystal plasticity modeling of cyclic deformation of Ti–6Al–4V. Int. J. Plast. 23(8), 1328–1348 (2007)CrossRefMATHGoogle Scholar
  45. 45.
    Brown, S.B., Kim, K.H., Anand, L.: An internal variable constitutive model for hot working of metals. Int. J. Plast. 5(2), 95–130 (1989)CrossRefMATHGoogle Scholar
  46. 46.
    Conti, S., Hackl, K.: Analysis and Computation of Microstructure in Finite Plasticity. Springer, Cham (2015)CrossRefMATHGoogle Scholar
  47. 47.
    Asaro, R.J., Rice, J.R.: Strain localization in ductile single crystals. J. Mech. Phys. Solids 25(5), 309–338 (1977)CrossRefMATHGoogle Scholar
  48. 48.
    Asaro, R.J.: Micromechanics of Crystals and Polycrystals. Advances in Applied Mechanics, vol. 23, pp. 1–115. Elsevier, Amsterdam (1983)Google Scholar
  49. 49.
    Meric, L., Poubanne, P., Cailletaud, G.: Single crystal modeling for structural calculations: part 1—model presentation. ASME J. Eng. Mater. Technol. 113(1), 162–170 (1991)CrossRefGoogle Scholar
  50. 50.
    le Graverend, J.B., Cormier, J., Gallerneau, F., Villechaise, P., Kruch, S., Mendez, J.: A microstructure-sensitive constitutive modeling of the inelastic behavior of single crystal nickel-based superalloys at very high temperature. Int. J. Plast. 59, 55–83 (2014)CrossRefGoogle Scholar
  51. 51.
    Tanaka, K.: Single-crystal elastic constants of gamma-TiAl. Philos. Mag. Lett. 73(2), 71–78 (1996)CrossRefGoogle Scholar
  52. 52.
    He, Y., Schwarz, R., Darling, T., Hundley, M., Whang, S., Wang, Z.: Elastic constants and thermal expansion of single crystal \(\gamma \)-TiAl from 300 to 750K. Mater. Sci. Eng. A 239–240, 157–163 (1997)CrossRefGoogle Scholar
  53. 53.
    Yoo, M.H., Fu, C.L.: Physical constants, deformation twinning, and microcracking of titanium aluminides. Metal. Mater. Trans. A 29, 49–63 (1998)CrossRefGoogle Scholar
  54. 54.
    Tang, P., Tang, B., Su, X.: First-principles studies of typical long-period superstructures \(\text{ Al }_{5}\text{ Ti }_{3}\), \(\text{ h-Al }_{2}\text{ Ti }\) and \(\text{ r-Al }_{2} \text{ Ti }\) in Al-rich TiAl alloys. Comput. Mater. Sci. 50(4), 1467–1476 (2011)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Inui, H., Chikugo, K., Nomura, K., Yamaguchi, M.: Lattice defects and their influence on the deformation behavior of single crystals of TiAl. Mater. Sci. Eng. A 329–331, 377–387 (2002)CrossRefGoogle Scholar
  56. 56.
    Nakano, T., Matsumoto, K., Seno, T., Oma, K., Umakoshi, Y.: Effect of chemical ordering on the deformation mode of Al-rich Ti–Al single crystals. Philos. Mag. A 74(1), 251–268 (1996)CrossRefGoogle Scholar
  57. 57.
    Inui, H., Matsumuro, M., Wu, D.H., Yamaguchi, M.: Temperature dependence of yield stress, deformation mode and deformation structure in single crystals of TiAl (Ti–56 at.% Al). Philos. Mag. A 75(2), 395–423 (1997)CrossRefGoogle Scholar
  58. 58.
    Feng, Q., Whang, S.H.: Deformation of Ti-56at.%Al single crystals oriented for single slip by \(\frac{1}{2}<110]\) ordinary dislocations. Acta Mater. 48(17), 4307–4321 (2000)CrossRefGoogle Scholar
  59. 59.
    Kassner, M.E., Perez-Prado, M.T.: Five-power-law creep in single phase metals and alloys. Prog. Mater. Sci. 45(1), 1–102 (2000)CrossRefGoogle Scholar
  60. 60.
    Khadyko, M., Dumoulin, S., Cailletaud, G., Hopperstad, O.S.: Latent hardening and plastic anisotropy evolution in AA6060 aluminium alloy. Int. J. Plast. 76, 51–74 (2016)CrossRefGoogle Scholar
  61. 61.
    Groh, S., Marin, E.B., Horstemeyer, M.F., Zbib, H.M.: Multiscale modeling of the plasticity in an aluminum single crystal. Int. J. Plast. 25(8), 1456–1473 (2009)CrossRefMATHGoogle Scholar
  62. 62.
    Lim, H., Carroll, J.D., Battaile, C.C., Boyce, B.L., Weinberger, C.R.: Quantitative comparison between experimental measurements and CP-FEM predictions of plastic deformation in a tantalum oligocrystal. Int. J. Mech. Sci. 92, 98–108 (2015)CrossRefGoogle Scholar
  63. 63.
    Nakada, Y., Keh, A.S.: Latent hardening in iron single crystals. Acta Metal. 14(8), 961–973 (1966)CrossRefGoogle Scholar
  64. 64.
    Horstemeyer, M.F., McDowell, D.L., McGinty, R.D.: Design of experiments for constitutive model selection: application to polycrystal elastoviscoplasticity. Model. Simul. Mater. Sci. Eng. 7(2), 253–273 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Helal Chowdhury
    • 1
  • Konstantin Naumenko
    • 2
  • Holm Altenbach
    • 2
  • Manja Krüger
    • 3
  1. 1.Graduate School for ‘Micro-Macro Interactions in Structured Media and Particle Systems’Otto-von-Guericke UniversityMagdeburgGermany
  2. 2.Institute of MechanicsOtto-von-Guericke UniversityMagdeburgGermany
  3. 3.Institute of Materials and Joining TechnologyOtto-von-Guericke UniversityMagdeburgGermany

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