Modelling of the mechanical response of Zr–Nb and Ti–Nb alloys in a wide temperature range

  • Vladimir A. Skripnyak
  • Vladimir V. SkripnyakEmail author
  • Evgeniya G. Skripnyak
  • Nataliya V. Skripnyak


This article presents the results of modeling the mechanical behavior of Zr–Nb and Ti–Nb alloys in a range of strain rates from 0.001 to 1000 1/s and temperature range 297–1273 K. A modification of constitutive equations describing the mechanical response of fine-grained and coarse-grained Zr–1Nb and Ti–13Nb–13Zr alloys in a wide temperature range is proposed. It was shown that the phase transition between the hexagonal closed packed and body-centered cubic crystal structure at elevated temperatures leads to a sharp change in strain rate sensitivity of the yield strength of Zr–Nb and Ti–Nb alloys. The proposed modifications of constitutive equations make it possible to describe the strain hardening and the strain rate sensitivity of the plastic flow stress over a wide temperature range in the coarse-crystalline and ultrafine-grained Zr–Nb and Ti–Nb alloys. The results can be used for engineering analysis of structural elements of technical systems and design of manufacturing technologies for biomedical products.


Yield stress Strain rate sensitivity Ultra-fine grained structure Elevated temperature Zirconium–niobium alloys Titanium–niobium alloys 



This work was supported by the Russian Science Foundation (RSF), Grant No. 18-71-00117. The authors are grateful for the support of this research. Authors thank V. A. Serbenta and S.D. Rudakov for the help in work.


  1. Abed, F., Voyiadijs, G.Z.: A consistent modified Zerilli–Armstrong flow stress model for BCC and FCC metals for elevated temperatures. Acta Mech. 175, 1–18 (2005)CrossRefzbMATHGoogle Scholar
  2. Behera, A.N., Chaudhuri, A., Kapoor, R., Chakravartty, J.K., Suwas, S.: High temperature deformation behavior of Nb–1 wt% Zr alloy. Mater. Des. 92, 750–759 (2016)CrossRefGoogle Scholar
  3. Blokhin, D.A., Chernov, V.M., Blokhin, A.I., Demin, N.A., Sipachev, I.V.: Nuclear and physics properties of zirconium alloys E-110 and E-635 under long time neutron irradiation in the VVER-1000 reactor. Adv. Mater. 5, 23–29 (2011)Google Scholar
  4. Bobbili, R., Madhu, V.: Constitutive modeling and fracture behavior of a biomedical Ti–13Nb–13Zr alloy. Mater. Sci. Eng., A 700, 82–91 (2017)CrossRefGoogle Scholar
  5. Bonisch, M., Calin, M., Waitz, T., Panigrahi, A., Zehetbauer, M., Gebert, A., Skrotzki, W., Eckert, J.: Thermal stability and phase transformations of martensitic Ti–Nb alloys. Sci. Technol. Adv. Mater. 14, 055004 (2013)CrossRefGoogle Scholar
  6. Cao, W.Q., Yu, S.H., Chun, Y.B., Yoo, Y.C., Lee, C.M., Shin, D.H., Hwang, S.K.: Strain path effects on the microstructure evolution and mechanical properties of Zr702. Mater. Sci. Eng. A395, 77–86 (2005)CrossRefGoogle Scholar
  7. Chui, P.: Near β-type Zr–Nb–Ti biomedical alloys with high strength and low modulus. Vacuum 143, 54–58 (2017)CrossRefGoogle Scholar
  8. Clouet, E., Cottura, M.: Solubility in Zr–Nb alloys from first-principles. Acta Mater. 144, 21–30 (2018)CrossRefGoogle Scholar
  9. Dafang, W., Fei, S., Chengxiang, L., Ronghai, M., Chinan, C., Yuewu, W., Liang, H.: Experimental study on mechanical behaviors of Al-alloys under transient aerodynamic heating. Int. J. Mech. Mater. Des. 6, 331–340 (2010)CrossRefGoogle Scholar
  10. Dar, U.A., Zhang, W.H., Xu, Y.J.: Numerical implementation of strain rate dependent thermo viscoelastic constitutive relation to simulate the mechanical behavior of PMMA. Int. J. Mech. Mater. Des. 10, 93–107 (2014)CrossRefGoogle Scholar
  11. Duan, Z., Yang, Y., Satoh, Y., Murakami, K., Kano, S., Zhao, Z., Shen, J., Abe, H.: Current status of materials development of nuclear fuel cladding tubes for light water reactors. Nucl. Eng. Des. 316, 131–150 (2017)CrossRefGoogle Scholar
  12. Fong, R.W.L.: Anisotropic deformation of Zr–2.5Nb pressure tube material at high temperatures. J. Nucl. Mater. 440, 467–476 (2013)CrossRefGoogle Scholar
  13. Gao, C.Y., Zhang, L.C., Yan, H.X.: A new constitutive model for HCP metals. Mater. Sci. Eng. A528, 4445–4452 (2011)CrossRefGoogle Scholar
  14. Guo, D., Zhang, Z., Zhang, G., Li, M., Shi, Y., Ma, T., Zhang, X.: An extraordinary enhancement of strain hardening in fine-grained zirconium. Mater. Sci. Eng. A591, 167–172 (2014)CrossRefGoogle Scholar
  15. Hahn, E.N., Meyers, M.A.: Grain-size dependent mechanical behavior of nanocrystalline metals. Mater. Sci. Eng. A646, 101–134 (2015)CrossRefGoogle Scholar
  16. Hatt, B.A., Rivlin, V.G.: Phase transformations in superconducting Ti–Nb alloys. J. Phys. D Appl. Phys. 1(9), 1145–1149 (1968)CrossRefGoogle Scholar
  17. Huh, H., Ahn, K., Lim, J.H., Kim, H.W., Park, L.J.: Evaluation of dynamic hardening models for BCC, FCC, and HCP metals at a wide range of strain rates. J. Mater. Process. Technol. 214, 1326–1340 (2014)CrossRefGoogle Scholar
  18. Hynowska, A., Pellicer, E., Fornell, J., González, S., van Steenberge, N., Suriñach, S., Gebert, A., Calin, V., Eckert, J., Baró, M.D., Sort, J.: Nanostructured β-phase Ti–31.0Fe–9.0Sn and sub-μm structured Ti–39.3Nb–13.3Zr–10.7Ta alloys for biomedical applications. Microstructure benefits on the mechanical and corrosion performances. Mater. Sci. Eng. C32, 2418–2425 (2012)CrossRefGoogle Scholar
  19. Johnson, G.R., Cook, W.H.: Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng. Fract. Mech. 21, 31–48 (1985)CrossRefGoogle Scholar
  20. Kazakov, D.N., Kozelkov, O.E., Mayorova, A.S.: Dynamic behavior of zirconium alloy E110 under submicrosecond shock-wave loading. EPJ Web Conf. 94, 1–5 (2015)CrossRefGoogle Scholar
  21. Li, J., Weng, G.J.: A micromechanical approach to the stress–strain relations, strain-rate sensitivity and activation volume of nanocrystalline materials. Int. J. Mech. Mater. Des. 9, 141–152 (2013)CrossRefGoogle Scholar
  22. Moffat, D.L., Kattner, U.R.: Stable and metastable Ti–Nb phase diagrams. Metall. Mater. Trans. A 19(10), 2389–2397 (1988)CrossRefGoogle Scholar
  23. Motta, A.T., Yilmazbayhan, A., Gomes da Silva, M., Comstock, R.J., Busby, J., Gartner, E.: Zirconium alloys for supercritical water reactor applications: challenges and possibilities. J. Nucl. Mater. 371, 61–75 (2007)CrossRefGoogle Scholar
  24. Nikonov, AYu., Zharmukhambetova, A.M., Skripnyak, N.V., Ponomareva, A.V., Abrikosov, I.A., Barannikova, S.A., Dmitriev, A.I.: Calculation of mechanical properties of BCC Ti–Nb alloys. AIP Conf. Proc. 1683, 020165 (2015)CrossRefGoogle Scholar
  25. Rodchenkov, B.S., Semenov, A.N.: High temperature mechanical behavior of Zr–2.5% Nb alloy. Nucl. Eng. Des. 235, 2009–2018 (2005)CrossRefGoogle Scholar
  26. Sarkar, A., Chandanshive, S.A., Thota, M.K., Kapoor, R.: High temperature deformation behaviour of Zr-1Nb alloy. J. Alloys Compd. 703, 56–66 (2017)CrossRefGoogle Scholar
  27. Skripnyak, V.A., Skripnyak, E.G.: Mechanical behavior of nanostructured and ultrafine-grained metal alloy under intensive dynamic loading. In: Vakhrushev, A. (ed.) Nanotechnology and Nanomaterials, Chapter 2. IntechOpen, London (2017)Google Scholar
  28. Skripnyak, N.V., Skripnyak, V.A., Skripnyak, V.V.: Fracture of thin metal sheets with distribution of grain sizes in the layers. In: Papadrakakis, M., Papadopoulos, V., Stefanou, G., Plevris, V. (eds.) ECCOMAS Congress 2016 VII European Congress on Computational Methods in Applied Sciences and Engineering, Crete Island, Greece, 5–10 June 2016, vol. 1, pp. 355–365 (2016)Google Scholar
  29. Skripnyak, V.A., Skripnyak, N.V., Skripnyak, E.G., Skripnyak, V.V.: Influence of grain size distribution on the mechanical behaviour of light alloys in wide range of strain rates. AIP Conf. Proc. 1793, 110001 (2017)CrossRefGoogle Scholar
  30. Tengen, T.B.: The response of the statistics of the cumulative features on grains in nanomaterials to different grain growth phenomena. Int. J. Mech. Mater. Des. 8, 101–112 (2012)CrossRefGoogle Scholar
  31. Toyama, T., Matsukawa, Y., Saito, K., Satoh, Y., Abe, H., Shinohara, Y., Nagai, Y.: Microstructural analysis of impurity segregation around β-Nb precipitates in Zr–Nb alloy using positron annihilation spectroscopy and atom probe tomography. Scr. Mater. 108, 156–159 (2015)CrossRefGoogle Scholar
  32. van Liempt, P., Bos, C., Sietsma, J.: A physically based yield criterion II. Incorporation of Hall Petch effect and resistance due to thermally activated dislocation glide. Mater. Sci. Eng. A 652, 7–13 (2016)CrossRefGoogle Scholar
  33. Xiao, D., Li, Y., Hu, S.: High strain rate deformation behavior of zirconium at elevated temperatures. J. Mater. Sci. Technol. 26, 878–882 (2010)CrossRefGoogle Scholar
  34. Yang, Y., Wu, S.Q., Li, G.P., Li, Y.L., Lu, Y.F., Yang, K., Ge, P.: Evolution of deformation mechanisms of Ti–22.4 Nb–0.73Ta–2Zr–1.34O alloy during straining. Acta Mater. 58, 2778–2787 (2010)CrossRefGoogle Scholar
  35. Zain-ul-abdein, M., Nelias, D.: Effect of coherent and incoherent precipitates upon the stress and strain fields of 6xxx aluminium alloys: a numerical analysis. Int. J. Mech. Mater. Des. 12, 255–271 (2016)CrossRefGoogle Scholar
  36. Zerilli, F.J., Armstrong, R.W.: The effect of dislocation drag on the stress–strain behaviour of FCC metals. Acta Metall. Mater. 40, 1803–1808 (1992)CrossRefGoogle Scholar
  37. Zhang, W., Cai, Y.: Continuum Damage Mechanics and Numerical Applications. Springer, Heidelberg (2010)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Vladimir A. Skripnyak
    • 1
  • Vladimir V. Skripnyak
    • 1
    Email author
  • Evgeniya G. Skripnyak
    • 1
  • Nataliya V. Skripnyak
    • 1
    • 2
  1. 1.National Research Tomsk State UniversityTomskRussia
  2. 2.Linköping UniversityLinköpingSweden

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