Nonlinear dynamics of hydrogen concentration in high-strength and high-entropy alloys

  • A. K. Belyaev
  • V. A. Polyanskiy
  • A. V. PorubovEmail author
Original Article


A new nonlinear governing equation is obtained for the dynamics of hydrogen concentration. Numerical solution of the equation allows us to describe evolution of localized disturbance of the hydrogen concentration and the significant influence of nonlinearity on the shape of localized waves of the hydrogen concentration.


Hydrogen Bi-continuum Nonlinear Dynamics Alloy 


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The work was performed in Peter the Great St. Petersburg Polytechnic University (SPbPU) and is supported solely by the Russian Science Foundation (Grant No.18-19-00413).


  1. 1.
    Alvaro, A., Olden, V., Akselsen, O.M.: 3D cohesive modelling of hydrogen embrittlement in the heat affected zone of an X70 pipeline steel. Int. J. Hydrog. Energy 38(18), 7539–7549 (2013)CrossRefGoogle Scholar
  2. 2.
    Alvaro, A., Olden, V., Akselsen, O.M.: 3D cohesive modelling of hydrogen embrittlement in the heat affected zone of an X70 pipeline steel. Part II. Int. J. Hydrog. Energy 39(7), 3528–3541 (2014)CrossRefGoogle Scholar
  3. 3.
    Andrianov, I., Awrejcewicz, J., Danishevs’kyy, V., Ivankov, A.: Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions. Wiley, Chichester (2014)CrossRefGoogle Scholar
  4. 4.
    Belyaev, A.K., Polyanskiy, A.M., Polyanskiy, V.A., Yakovlev, Y.A.: Parametric instability in cyclic loading as the cause of fracture of hydrogenous materials. Mech. Solids 47(5), 533–537 (2012)ADSCrossRefGoogle Scholar
  5. 5.
    Birnbaum, H.K., Sofronis, P.: Hydrogen-enhanced localized plasticity: a mechanism for hydrogen-related fracture. Mater. Sci. Eng. A 176(1–2), 191–202 (1994)CrossRefGoogle Scholar
  6. 6.
    Brouwer, R.C., Jong, E.C.J.N., Mul, L.M., Handel, G.: Modelling Hydrogen Induced Crack Growth: Validation by Comparison with Experiment. NACE International, Houston, TX (1995)Google Scholar
  7. 7.
    Chen, Y.Y., Duval, T., Hung, U.D., Yeh, J.W., Shih, H.C.: Microstructure and electrochemical properties of high entropy alloys—a comparison with type-304 stainless steel. Corros. Sci. 47(9), 2257–2279 (2005)CrossRefGoogle Scholar
  8. 8.
    Delafosse, D., Magnin, T.: Interfaces in stress corrosion cracking: a case study in duplex stainless steels. In: Solid State Phenomena, vol. 59, pp. 221–250. (Trans Tech Publ) (1998)CrossRefGoogle Scholar
  9. 9.
    Gorsky, W.S.: Theorie der elastischen nachwirkung in ungeordneten mischkristallen von CuAu. Physikalische Zeitschrift der Sowjetunion 8, 443–456 (1935)Google Scholar
  10. 10.
    Grossbeck, M.L., Birnbaum, H.K.: Low temperature hydrogen embrittlement of niobium II—microscopic observations. Acta Metall. 25(2), 135–147 (1977)CrossRefGoogle Scholar
  11. 11.
    Haferkamp, H., Meier, O., Harley, K.: Laser beam welding of new high strength steels for auto body construction. In: Sheet Metal 2007, Key Engineering Materials, vol. 344, pp. 723–730. (Trans Tech Publications) (2007)CrossRefGoogle Scholar
  12. 12.
    Hirth, J.P.: Effects of hydrogen on the properties of iron and steel. Metall. Trans. A 11(6), 861–890 (1980)CrossRefGoogle Scholar
  13. 13.
    Hsu, C.Y., Yeh, J.W., Chen, S.K., Shun, T.T.: Wear resistance and high-temperature compression strength of Fcc CuCoNiCrAl\(_{0.5}\)Fe alloy with boron addition. Metall. Mater. Trans. A 35(5), 1465–1469 (2004)CrossRefGoogle Scholar
  14. 14.
    Huang, P.K., Yeh, J.W., Shun, T.T., Chen, S.K.: Multi-principal-element alloys with improved oxidation and wear resistance for thermal spray coating. Adv. Eng. Mater. 6(1–2), 74–78 (2004)CrossRefGoogle Scholar
  15. 15.
    Ignatenko, A.V., Pokhodnya, I.K., Paltsevich, A.P., Sinyuk, V.S.: Dislocation model of hydrogen-enhanced localizing of plasticity in metals with BCC latttice. Paton Weld. J. 3, 15–19 (2012)Google Scholar
  16. 16.
    Indeitsev, D.A., Semenov, B.N.: About a model of structural-phase transformations under hydrogen influence. Acta Mech. 195(1), 295–304 (2008)CrossRefGoogle Scholar
  17. 17.
    Koyama, M., Springer, H., Merzlikin, S.V., Tsuzaki, K., Akiyama, E., Raabe, D.: Hydrogen embrittlement associated with strain localization in a precipitation-hardened Fe–Mn–Al–C light weight austenitic steel. Int. J. Hydrog. Energy 39(9), 4634–4646 (2014)CrossRefGoogle Scholar
  18. 18.
    Leeuwen, H.P.V.: The kinetics of hydrogen embrittlement: a quantitative diffusion model. Eng. Fract. Mech. 6(1), 141–161 (1974)CrossRefGoogle Scholar
  19. 19.
    Lynch, S.: Hydrogen embrittlement phenomena and mechanisms. Corros. Rev. 30(3–4), 105–123 (2012)Google Scholar
  20. 20.
    McVeigh, C., Vernerey, F., Liu, W.K., Moran, B., Olson, G.: An interactive micro-void shear localization mechanism in high strength steels. J. Mech. Phys. Solids 55(2), 225–244 (2007)ADSCrossRefGoogle Scholar
  21. 21.
    Nagumo, M.: Function of hydrogen in embrittlement of high-strength steels. ISIJ Int. 41(6), 590–598 (2001)CrossRefGoogle Scholar
  22. 22.
    Pan, Y., Guan, W., Wen, M., Zhang, J., Wang, C., Tan, Z.: Hydrogen embrittlement of Pt\(_3\)Zr compound from first-principles. J. Alloys Compd. 585, 549–554 (2014)CrossRefGoogle Scholar
  23. 23.
    Polyanskiy, A.M., Popov-Diumin, D.B., Polyanskiy, V.A.: Determination of hydrogen binding energy in various materials by means of absolute measurements of its concentration in solid probe. In: Veziroglu, T.N., Zaginaichenko, S.Y., Schur, D.V., Baranowski, B., Shpak, A.P., Skorokhod, V.V., Kale, A. (eds.) Hydrogen Materials Science and Chemistry of Carbon Nanomaterials, pp. 681–692. Springer, Dordrecht (2007)CrossRefGoogle Scholar
  24. 24.
    Robertson, I.M., Sofronis, P., Nagao, A., Martin, M.L., Wang, S., Gross, D.W., Nygren, K.E.: Hydrogen embrittlement understood. Metall. Mater. Trans. B 46(3), 1085–1103 (2015)CrossRefGoogle Scholar
  25. 25.
    Sofronis, P., Liang, Y., Aravas, N.: Hydrogen induced shear localization of the plastic flow in metals and alloys. Eur. J. Mech. A Solids 20(6), 857–872 (2001)CrossRefGoogle Scholar
  26. 26.
    Taha, A., Sofronis, P.: A micromechanics approach to the study of hydrogen transport and embrittlement. Eng. Fract. Mech. 68(6), 803–837 (2001)CrossRefGoogle Scholar
  27. 27.
    Tasan, C.C., Deng, Y., Pradeep, K.G., Yao, M.J., Springer, H., Raabe, D.: Composition dependence of phase stability, deformation mechanisms, and mechanical properties of the CoCrFeMnNi high-entropy alloy system. JOM 66(10), 1993–2001 (2014)CrossRefGoogle Scholar
  28. 28.
    Tong, C.J., Chen, M.R., Yeh, J.W., Lin, S.J., Chen, S.K., Shun, T.T., Chang, S.Y.: Mechanical performance of the Al\(_x\)CoCrCuFeNi high-entropy alloy system with multiprincipal elements. Metall. Mater. Trans. A 36(5), 1263–1271 (2005)CrossRefGoogle Scholar
  29. 29.
    Traidia, A., Alfano, M., Lubineau, G., Duval, S., Sherik, A.: An effective finite element model for the prediction of hydrogen induced cracking in steel pipelines. Int. J. Hydrog. Energy 37(21), 16214–16230 (2012)CrossRefGoogle Scholar
  30. 30.
    Varias, A.G., Massih, A.R.: Simulation of hydrogen embrittlement in zirconium alloys under stress and temperature gradients. J. Nucl. Mater. 279(2–3), 273–285 (2000)ADSCrossRefGoogle Scholar
  31. 31.
    Yeh, J.W., Chen, S.K., Lin, S.J., Gan, J.Y., Chin, T.S., Shun, T.T., Tsau, C.H., Chang, S.Y.: Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes. Adv. Eng. Mater. 6(5), 299–303 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • A. K. Belyaev
    • 1
    • 2
  • V. A. Polyanskiy
    • 1
    • 2
  • A. V. Porubov
    • 1
    • 2
    Email author
  1. 1.Peter the Great St. Petersburg Polytechnic University (SPbPU)Saint PetersburgRussia
  2. 2.Institute for Problems in Mechanical EngineeringSaint PetersburgRussia

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