Mathematical modeling of strength characteristics of chromium ferritic-martensitic steels

To predict the chemical composition of heat-resistant high-chromium steels with ferritic-martensitic structure (HFMS) (with the number of alloying elements up to 10 and the same number of parameters of production and heat treatment technology), a mathematical model is needed. In this work, I searched for the dependences of the yield strength and ultimate strength of HFMS on the content of alloying elements and test temperature without analyzing technological factors due to their uniformity. Analysis of the samples from ten steel grades was carried out on the basis of the experimental data including 63 tensile tests at 20 – 720 °С. Regression multiplicative dependencies are proposed to take into account exponential and power-law form through the corresponding factors: solid solution and dispersion hardening, total temperature softening of the steel, carbon content, total molybdenum and tungsten content, and strengthening effect of manganese. Estimates of the effect of nitrogen and silicon on the predicted strength characteristics have shown that a factor that takes into account the effect of nitrogen improves the model and is necessary in the general formula, and introduction of a factor that takes into account silicon content, worsens the model. Introduction of a silicon factor in the formula may be necessary in analysis of steels with high silicon content (type EP-823). The experimental fact of a close relationship between yield strengths and tensile strengths for the studied HFMS steels made it possible to use for the yield strength the form of equation and forecast for the ultimate strength, which differs only by coefficients in variables. Deviation of the calculated model yield strengths and tensile strengths from experimental is 13 – 18 %. An example of analysis of the yield strength behavior of steels with experimental chemical compositions is given. It is shown that the dependences found for the yield strength and tensile strength are stable with respect to the increase of experimental data matrix: with an increase in the number of experiments from 94 and higher, the coefficient of variation V monotonously decreases up to a maximum array size of 299 experiments.

1. Lanskaya K.A. Vysokokhromistye zharoprochnye stali [High-chromium heat-resistant steels]. Moscow: Metallurgiya, 1976, 216 p. (In Russ.).

2. High chromium ferritic and martensitic steels for nuclear applications. Klueh R., Harries D. eds. West Conshohocken, PA: ASTM International, 2001, 217 p.

3. Yan Wei, Wang Wei, Shan Yiyin, Yang Ke, Sha Wei. 9-12Cr Heat-Resistant Steels. Springer, Ser.: Engineering Materials. 2015, 223 p.

4. Structural Materials for Liquid Metal Cooled Fast Reactor Fuel Assemblies: Operational Behaviour. Vienna: International Atomic Energy Agency, 2012, 87 p.

5. Gorskii V.G. Applied mathematical statistics – our profile. Zavodskaya laboratoriya. Diagnostika materialov. 1997, no. 1, pp. 96–100. (In Russ.).

6. Dubrov A.M. Obrabotka statisticheskikh dannykh metodom glavnykh component [Statistical data processing by a method of principle components]. Moscow: Statistika, 1978, 135 p. (In Russ.).

7. Rachkov V.I., Belomyttsev M.Yu., Konobeev Yu.V., Obraztsov S.M., Pyshin I.V. Yield strength research of ferritic-martensitic steel with the help of neural-net and principal component analysis. Voprosy materialovedeniya. 2014, no. 1(77), pp. 11–19. (In Russ.).

8. Obraztsov S.M., Birzhevoi G.A., Konobeev Yu.V., Solov’ev V.A., Rachkov V.I. Neural-net improvement of ferritic-martensitic steel of EP-450 type by maximum of strength and plasticity. Perspektivnye materialy. 2005, no. 4 pp. 14–19. (In Russ.).

9. Obraztsov S.M., Birzhevoi G.A., Konobeev Yu.V., Rachkov V.I., Solov’ev V.A. Neural-net experiments on mutual influence of alloying elements on mechanical properties of 12 % Cr ferritic-martensitic steels. Izv. vuz. Yadernaya energetika. 2008, no. 3 pp. 119–124. (In Russ.).

10. Sangiovanni D.G., Hultman L., Chirita V. Supertoughening in B1 transition metal nitride alloys by increased valence electron concentration. Acta Materialia. 2011, vol. 59, no. 5, pp. 2121–2134.

11. Brachet J.-C., Gavard L., Boussidan C., Lepoittevin C., Denis S., Servant C. Modelling of phase transformations occurring in low activation martensitic steels. Journal of Nuclear Materials. 1998, vol. 258-263, part B, pp. 1307–1311.

12. Ke Jia-Hong, Ke Huibin, Odette G.R., Morgan D. Cluster dynamics modeling of Mn-Ni-Si precipitates in ferriticmartensitic steel under irradiation. Journal of Nuclear Materials. 2018, vol. 498, pp. 83–88.

13. Belomyttsev M.Yu., Mikhailov M.A., Obraztsov S.M., Solov’ev V.A., Rachkov V.I. Influence of manganese on strength properties of ferritic-martensitic steels with 12 % of chromium. Izvestiya. Ferrous Metallurgy. 2011, no. 3, pp. 45-47. (In Russ.).

14. Rachkov V.I., Obraztsov S.M., Solov’ev V.A., Belomyttsev M.Yu., Mikhailov M.A., Chizhikov K.E. Optimization of the chemical composition of ferrite-martensite steel to increase short-time mechanical properties. Atomic Energy. 2013, vol. 115, no. 1, pp. 26–31.

15. Lanskaya K.A. Zharoprochnye stali [Heat-resistant steels]. Moscow: Metallurgiya, 1967, 247 p. (In Russ.).

16. Khimushin F.F. Nerzhaveyushchie stali [Stainless steels]. Moscow: Metallurgiya, 1967, 799 p. (In Russ.).

17. Khimushin F.F. Zharoprochnye stali i splavy [Heat-resistant steels and alloys]. Moscow: Metallurgiya, 1969, 752 p. (In Russ.).

18. Klueh R. L. Elevated–temperature ferritic and martensitic steels and their application to future nuclear reactors. International Materials Reviews. 2005, vol. 50, no. 5, no. 287–310.

19. Roy A. K., Kukatla S. R., Yarlagadda B., Potluri V. N., Lewis M., O’Toole B. Tensile properties of martensitic stainless steels at elevated temperatures. Journal of Materials Engineering and Performance. 2005, vol. 14, no. 2, pp. 212–218.

20. Zhongfei Ye, Wang Pei, Li Dianzhong, Zhang Yutuo, Li Yiyi. Effect of carbon and niobium on the microstructure and impact toughness of a high silicon 12% Cr ferritic/martensitic heat resistant steel. Materials Science and Engineering: A. 2014, vol. 616, pp. 12–19.

21. Ye Zhongfei, Wang Pei, Li Dianzhong, Li Yiyi. M 23 C 6 precipitates induced inhomogeneous distribution of silicon in the oxide formed on a high-silicon ferritic/martensitic steel. Scripta Materialia. 2015, vol. 97, pp. 45–48.

22. Fujita S.N., Ohmura K., Kikuchi M., Suzuki T., Funaki S., Hiroshige I. Effect of Nb on high-temperature properties for ferritic stainless steel. Scripta Materialia. 1996, vol. 35, no. 6, pp. 705–710.

23. Ule B., Nagode A. The improved power-law, stress-dependent, energy-barrier model of 9Cr–1Mo–0.2V steel using short-term creep data. Scripta Materialia. 2007, vol. 57, no. 5, pp. 405–408.

24. Sawada K., Takeda M., Maruyama K., Ishii R., Yamada M., Nagae Y., Komine R. Effect of W on recovery of lath structure during creep of high chromium martensitic steels. Materials Science and Engineering: A. 1999, vol. 267, no. 1, pp. 19–25.

25. Helisa L., Todaa Y., Harab T., Miyazakic H., Abe F. Effect of cobalt on the microstructure of tempered martensitic 9Cr steel for ultrasupercritical power plants. Materials Science and Engineering: A. 2009, vol. 510 – 511, pp. 88–94.

26. Li S., Zhou Z., Jang J., Wang M., Hu H., Sun H., Zou L., Zhanga G., Zhanga L. The influence of Cr content on the mechanical properties of ODS ferritic steels. Journal of Nuclear Materials. 2014, vol. 455, no. 1-3, pp. 194–200.

27. Klueh R.L., Alexander D.J., Sokolov M.A. Effect of chromium, tungsten, tantalum, and boron on mechanical properties of 5–9CrWVTaB steels. Journal of Nuclear Materials. 2002, vol. 304, no. 2-3, pp. 139–152.

28. Belomyttsev Yu.S., Lyashenko V.S., Abramovich M.D. Effect of alloyed elements on the heat resistance of chromium-silicon low-carbon steel. Metal Science and Heat Treatment. 1964, no. 7, pp. 427–429.

29. Rachkov V.I., Obraztsov S.M., Birzhevoi G.A., Konobeev Yu.V., Solov’ev V.A., Silkina O.S. Neural-net analysis of the plasticity of ÉP-450 ferrite–martensite steel with different alloying element concentrations. Atomic Energy. 2004, vol. 96, no. 2, pp. 111–116.

30. Chen S., Rong L. Effect of silicon on the microstructure and mechanical properties of reduced activation ferritic/martensitic steel. Journal of Nuclear Materials. 2015, vol. 459, pp. 13–19.

31. Das C.R., Albert S.K., Bhaduri A.K., Murty B.S. Effect of boron and Ni addition and initial heat-treatment temperature on microstructure and mechanical properties of modified 9Cr-1Mo steels under different heat-treatment conditions. Metallurgical and materials transactions A. 2013, vol. 44A, pp. 2171–2186.

32. Shtremel’ M.A. Prochnost’ splavov. Ch. 2: Deformatsiya [Alloy strength. Part 2. Deformation]. Moscow: Izd. MISIS, 1997, 527 p. (In Russ.).

33. He M.Y., Odette G.R., Yamamoto T., Klingensmith D. A universal relationship between indentation hardness and flow stress. Journal of Nuclear Materials. 2007, vol. 367–370, part A, pp. 556–560.

34. Gol’dshtein M.I., Grachev S.V., Veksler Yu.G. Spetsial’nye stali [Special steels]. Moscow: Metallurgiya, 1985, 408 p. (In Russ.).

35. Gol’tdshtein M.I., Litvinov V.S, Bronfin B.M. Metallofizika vysokoprochnykh splavov [Metal physics of high-strength alloys]. Moscow: Metallurgiya, 1986, 312 p. (In Russ.).

36. Superalloys II: High-Temperature Materials for Aerospace and Industrial Power. Chester T. Sims, Stoloff N.S., Wolliam C. Hagel eds. New York: Wiley, 1987, 615 p. (Russ.ed.: Supersplavy II: Zharoprochnye materialy dlya aerokosmicheskikh i promyshlennykh energoustanovok. Sims Ch.T. Stoloff N.S., Hagel W.C. eds. Moscow: Metallurgiya, 1995, 384 p.).

37. Logunov A.V. Zharoprochnye nikelevye splavy dlya lopatok i diskov gazovykh turbin [Heat-resistant nickel alloys for gas turbine blades and discs]. Rybinsk: ID Gazoturbinnye tekhnologii, 2017, 854 p. (In Russ.).

38. Pauling L. The nature of the interatomic forces in metals. Physical Review II. December 1938, vol. 54, pp. 899–904.

39. Mel’nichenko A.S. Statisticheskii analiz v metallurgii i materialovedenii [Statistical analysis in metallurgy and material science]. Moscow: ID MISiS, 2009, 268 p. (In Russ.).

40. Hong S.G., Lee W.B., Park C.G. The effects of tungsten addition on the microstructural stability of 9Cr-Mo steels. Journal of Nuclear Materials. 2001, vol. 288, no. 2-3, pp. 202–207.

41. Hamilton M.L., Gelles D.S. Tensile response of low activation ferritic steels irradiated in ORR at temperatures in the range 60–400 °C. Journal of Nuclear Materials. 2002, vol. 307, part A, pp. 256–259.

42. Yan W., Hu P., Deng L., Wang W., Sha W., Shan Y., Yang K,. Effect of carbon reduction on the toughness of 9CrWVTaN steels. Metallurgical and materials transactions A. 2012, vol. 43A, no. 6, pp. 1921–1933.