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A model of damaged media used for describing the process of non-stationary creep and long-term strength of polycrystalline structural alloys


The main laws of the processes of creep and long-term strength of polycrystalline structural alloys are considered. From the viewpoint of continuum damaged media (CDM), a mathematical model is developed that describes the processes of viscoplastic deformation and damage accumulation under creep. The problem of determining material parameters and scalar functions of the developed constitutive relations based on the results of specially set basic experiments is discussed. An experimental–theoretical methodology for determining material parameters of the derived constitutive relations of CDM is developed based on analyzing the viscoplastic deformation and failure processes of laboratory specimens in the conditions of soft loading (stress controlled). Experimental results of short-term creep of the VZh-159 heat-resistant alloy are presented. The obtained numerical results are compared with the test data using the numerical modeling method of experimental processes. Qualitative and quantitative agreement between numerical results and experimental data is shown. It is concluded that the developed constitutive relations are reliable, and that the proposed methodology accurately determines the material parameters of the model under degradation of initial strength properties of structural materials according to the long-term strength mechanism.

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  1. Mitenkov, F.M., Kaydalov, V.F., Korotkikh, Yu.G.: at all. Mashinostroenie. Methods for substantiating the resource of nuclear power plants. (2007) (in Russian)

  2. Volkov, I.A., Korotkikh, Yu.G.: Equations of state for viscoelastic-plastic media with damage. (2008) (in Russian)

  3. Lokoshchenko, A.M: Creep and long-term strength of metals. (2016) (in Russian)

  4. Lemaitre, J.: Damage modeling for prediction of plastic or creep fatigue failure in structures. Trans. 5th Int. Conf. SMRiT, North Holland, L5/1b (1979)

  5. Murakami, S., Imaizumi, T.: Mechanical description of creep damage and its experimental verification. J. Mech. Theor. Appl. 1, 743–761 (1982)

    MATH  Google Scholar 

  6. Manson, S., Ansign, A.: A quarter-century of progress in the development of correlation and extrapolation methods for creep rupture data. J. Eng. Mater. Technol. 101(4), 317–325 (1979)

    Article  Google Scholar 

  7. May, Le.: Developments in parametric methods for handling creep and creep-rupture data. J. Eng. Mater. Technol. 101(4), 326–330 (1979)

    Article  Google Scholar 

  8. Larson, P.R., Miller, J.A.: A time-temperature relationship for rupture and creep stress. J. Trans. ASME 74, 539–605 (1952)

    Google Scholar 

  9. Nikitenko, A.F.: Experimental substantiation of the hypothesis of the existence of a creep surface under conditions of complex loading: Message 1, 2. Probl. Prochn. 8, 3–11 (1984). ([in Russian])

    Google Scholar 

  10. Woodford, D.A.: Creep damage and the remaining life concept. ASME. J. Eng. Mater. Technol. 101(4), 311–316 (1979)

    Article  Google Scholar 

  11. Lemaitre, J.: A continuous damage mechanics model for ductile fracture. J. Eng. Mater. Technol. 107(1), 83–89 (1985)

    Article  Google Scholar 

  12. Hall, F.R., Hayhurst, D.R.: Continuum damage mechanics modelling of high temperature deformation and failure in a pipe weldment. Proc. R. Soc. Lond. A433, 383–403 (1991)

    ADS  Google Scholar 

  13. Altenbach, H., Kushnevsky, V., Naumenko, K.: On the use of solid- and shell-type finite elements in creep-damage predictions of thinwalled structures. Arch. Appl. Mech. 71, 164–181 (2001)

    ADS  Article  Google Scholar 

  14. Naumenko, K., Gariboldi, E.: Experimental analysis and constitutive modeling of anisotropic creep damage in a wrought age-hardenable Al alloy. Eng. Fract. Mech. 259, 108–119 (2022)

    Article  Google Scholar 

  15. Naumenko, K., Altenbach, H., Ievdokymov, M.: A constitutive model for inelastic behavior of casting materials under thermo-mechanical loading. J. Strain Anal. Eng. Design. 49(6), 421–428 (2014)

    Article  Google Scholar 

  16. Volkov, I.A., Igumnov, L.A., Korotkikh, Yu.G.: Applied Theory of Viscoplasticity. N. Novgorod, NNGU (2015). (in Russian)

  17. Volkov, I.A., Igumnov, L.A., Kazakov, D.A., Shishulin, D.N., Smetanin, I.V.: Defining relations of transient creep under complex stress state. Probl strength Plast 78(4), 436–451 (2016) (in Russian)

  18. Chaboche, J.L.: Continuum damage mechanics: part I-general concepts. ASME. J. Appl. Mech. 55(1), 59–64 (1988)

    ADS  Article  Google Scholar 

  19. Chaboche, J.L.: A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plast 24(10), 1642–1693 (2008)

    Article  Google Scholar 

  20. Frederick, C.O., Armstrong, P.J.: A mathematical representation of the multiaxial Bauschinger effect. Mater. High Temp. 24(1), 1–26 (2007)

    Article  Google Scholar 

  21. Malinin, N.N., Khadjinsky, G.M.: Theory of creep with anisotropic hardening. Int. J. Mech. Sci. 14(4), 235–246 (1972)

    Article  Google Scholar 

  22. Bodner, S.R., Lindholm, U.S.: An incremental criterion for time-dependent failure of materials. J. Eng. Mater. Technol. 98(2), 140–145 (1976)

    Article  Google Scholar 

  23. Perzyna, P.: Constitutive modeling of dissipative solids for post-critical behavior and fracture ASME. J. Eng. Mater. Technol. 106(4), 410–419 (1984)

    Article  Google Scholar 

  24. MacKenzie, J.K.: The elastic constants of a solids containing spherical holes. Proc. Phys. Soc. B63, 2–11 (1950)

    ADS  Article  Google Scholar 

  25. Kachanov, L.M.: Introduction to Continuum Damage Mechanics. M. Nijhoff, Boston (1986)

    Book  Google Scholar 

  26. Rabotnov, Y.N.: Creep Problems in Structural Members. North-Holland, Amsterdam (1969)

    MATH  Google Scholar 

  27. Murakami, S.: Continuum Damage Mechanics Book Subtitle A Continuum Mechanics Approach to the Analysis of Damage and Fracture. Springer, Cham (2012)

    Google Scholar 

  28. Lokoshchenko, A.M.: Criteria for determining the long-term strength under conditions of complex loading. Strength Mater. 21(9), 1121–1124 (1989)

    Article  Google Scholar 

  29. Bantahya, V., Mukeredzhi, S.: On an improved time integration scheme for stiff constitutive models of inelastic deformation. J. Eng. Mater. Technol. 107(4), 282–285 (1985)

    Article  Google Scholar 

  30. Kapustin, S.A., Kazakov, D.A., Churilov, Yu.A., Galushchenko, A.I., Vakhterov, A.M.: Experimental-theoretical study of the behavior of structural parts of heat-resistant alloy under high-temperature creep. Probl. Strength Plast. 70, 100–111 (2008). (in Russian)

    Google Scholar 

  31. Volkov, I.A., Igumnov, L.A., Kazakov, D.A., Emelyanov, A.A., Tarasov, I.S., Guseva, M.A.: Software implementation of viscoplastic deformation and damage accumulation processes in structural alloys under thermal-mechanical loading. Probl. Strength Plast. 78(2), 188–207 (2016). (in Russian)

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The work was carried out with the financial support of Russian Foundation for Basic Research (task 20-08-00450) in terms of experimental research and Scientific and Education Mathematical Center \(\ll \)Mathematics for Future Technologies\(\gg \) (Project No. 075-02-2021-1394) in terms of numerical calculations.

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Correspondence to Leonid A. Igumnov.

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Igumnov, L.A., Volkov, I.A., Boev, E.V. et al. A model of damaged media used for describing the process of non-stationary creep and long-term strength of polycrystalline structural alloys. Continuum Mech. Thermodyn. 34, 841–853 (2022).

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  • Nonstationary creep
  • Long-term strength
  • Numerical modeling
  • Constitutive relations
  • Mechanics of damaged medial
  • Temperature
  • Damage
  • Viscoplastic deformation
  • Material parameters