Skip to main content Accessibility help

A modified constitutive model based on Arrhenius-type equation to predict the flow behavior of Fe–36%Ni Invar alloy

  • Shuai He (a1), Chang-sheng Li (a1), Zhen-yi Huang (a2) and Jian-jun Zheng (a1)


The predictability of modified constitutive model, based on Arrhenius type equation, for illustrating the flow behavior of Fe–36%Ni Invar alloy was investigated via isothermal hot compression tests. The hot deformation tests were carried out in a temperature range of 850–1100 °C and strain rates from 0.01 to 10 s−1. True stress-true strain curves exhibited the dependence of the flow stress on deformation temperatures and strain rates, which then described in Arrhenius-type equation by Zener–Holloman parameter. Moreover, the related material constants and hot deformation activation energy (Q) in the constitutive model were calculated by considering the effect of strain as independent function on them and employing sixth polynomial fitting. Subsequently, the performance of the modified constitutive equation was verified by correlation coefficient and average absolute relative error which were estimated in accordance with experimental and predicted data. The results showed that the modified constitutive equation possess reliable and stable ability to predict the hot flow behavior of studied material under different deformation conditions. Meanwhile, Zener–Holloman parameter map was established according to the modified constitutive equation and used to estimate the extent of dynamic recrystallization.


Corresponding author

a) Address all correspondence to this author. e-mail:


Hide All

Contributing Editor: Michael E. McHenry



Hide All
1. Lentz, M., Gall, S., Schmack, F., Mayer, H.M., and Reimers, W.: Hot working behavior of a WE54 magnesium alloy. J. Mater. Sci. 49, 1121 (2013).
2. Soliman, M. and Palkowski, H.: Influence of hot working parameters on microstructure evolution, tensile behavior and strain aging potential of bainitic pipeline steel. Mater. Des. 88, 759 (2015).
3. Sun, C.Y., Guo, N., Fu, M.W., and Liu, C.: Experimental investigation and modeling of ductile fracture behavior of TRIP780 steel in hot working conditions. Int. J. Mech. Sci. 110, 108 (2016).
4. Mehtedi, M.E., Musharavati, F., and Spigarelli, S.: Modelling of the flow behaviour of wrought aluminium alloys at elevated temperatures by a new constitutive equation. Mater. Des. 54, 869 (2014).
5. Lin, Y.C., Li, Q.F., Xia, Y.C., and Li, L.T.: A phenomenological constitutive model for high temperature flow stress prediction of Al–Cu–Mg alloy. Mater. Sci. Eng., A 534, 654 (2012).
6. Li, H.Y., Liu, Y., Lu, X.C., and Su, X.J.: Constitutive modeling for hot deformation behavior of ZA27 alloy. J. Mater. Sci. 47, 5411 (2012).
7. Wang, X., Chandrashekhara, K., Rummel, S.A., Lekakh, S., Van Aken, D.C., and O’Malley, R.J.: Modeling of mass flow behavior of hot rolled low alloy steel based on combined Johnson–Cook and Zerilli–Armstrong model. J. Mater. Sci. 52, 2800 (2016).
8. Samantaray, D., Mandal, S., and Bhaduri, A.K.: A comparative study on Johnson Cook, modified Zerilli–Armstrong and Arrhenius-type constitutive models to predict elevated temperature flow behaviour in modified 9Cr–1Mo steel. Comput. Mater. Sci. 47, 568 (2009).
9. Abbasi-Bani, A., Zarei-Hanzaki, A., Pishbin, M.H., and Haghdadi, N.: A comparative study on the capability of Johnson–Cook and Arrhenius-type constitutive equations to describe the flow behavior of Mg–6Al–1Zn alloy. Mech. Mater. 71, 52 (2014).
10. Sun, Y., Ye, W.H., and Hu, L.X.: Constitutive modeling of high-temperature flow behavior of Al–0.62Mg–0.73Si aluminum alloy. J. Mater. Eng. Perform. 25, 1621 (2016).
11. Zhao, J., Jiang, Z., Zu, G., Du, W., Zhang, X., and Jiang, L.: Flow behaviour and constitutive modelling of a ferritic stainless steel at elevated temperatures. Met. Mater. Int. 22, 474 (2016).
12. Tan, Y.B., Duan, J.L., Yang, L.H., Liu, W.C., Zhang, J.W., and Liu, R.P.: Hot deformation behavior of Ti–20Zr–6.5Al–4V alloy in the α + β and single β phase field. Mater. Sci. Eng., A 609, 226 (2014).
13. Ashtiani, H.R.R. and Shahsavari, P.: Strain-dependent constitutive equations to predict high temperature flow behavior of AA2030 aluminum alloy. Mech. Mater. 100, 209 (2016).
14. Zhou, M., Lin, Y.C., Deng, J., and Jiang, Y.Q.: Hot tensile deformation behaviors and constitutive model of an Al–Zn–Mg–Cu alloy. Mater. Des. 59, 141 (2014).
15. Zhang, C., Ding, J., Dong, Y., Zhao, G., Gao, A., and Wang, L.: Identification of friction coefficients and strain-compensated Arrhenius-type constitutive model by a two-stage inverse analysis technique. Int. J. Mech. Sci. 98, 195 (2015).
16. Changizian, P., Zarei-Hanzaki, A., and Roostaei, A.A.: The high temperature flow behavior modeling of AZ81 magnesium alloy considering strain effects. Mater. Des. 39, 384 (2012).
17. Yu, D.H.: Modeling high-temperature tensile deformation behavior of AZ31B magnesium alloy considering strain effects. Mater. Des. 51, 323 (2013).
18. Askariani, S.A. and Hasan Pishbin, S.M.: Hot deformation behavior of Mg–4Li–1Al alloy via hot compression tests. J. Alloys Compd. 688, 1058 (2016).
19. Gao, F., Liu, Z., Misra, R.D.K., Liu, H., and Yu, F.: Constitutive modeling and dynamic softening mechanism during hot deformation of an ultra-pure 17%Cr ferritic stainless steel stabilized with Nb. Met. Mater. Int. 20, 939 (2014).
20. Cao, Y., Di, H., Misra, R.D.K., Yi, X., Zhang, J., and Ma, T.: On the hot deformation behavior of AISI 420 stainless steel based on constitutive analysis and CSL model. Mater. Sci. Eng., A 593, 111 (2014).
21. Marandi, A., Zarei-Hanzaki, A., Haghdadi, N., and Eskandari, M.: The prediction of hot deformation behavior in Fe–21Mn–2.5Si–1.5Al transformation-twinning induced plasticity steel. Mater. Sci. Eng., A 554, 72 (2012).
22. Wu, S.W., Zhou, X.G., Cao, G.M., Liu, Z.Y., and Wang, G.D.: The improvement on constitutive modeling of Nb–Ti micro alloyed steel by using intelligent algorithms. Mater. Des. 116, 676 (2017).
23. Wang, L., Liu, F., Cheng, J.J., Zuo, Q., and Chen, C.F.: Arrhenius-type constitutive model for high temperature flow stress in a Nickel-based corrosion-resistant alloy. J. Mater. Eng. Perform. 25, 1394 (2016).
24. Samantaray, D., Patel, A., Borah, U., Albert, S.K., and Bhaduri, A.K.: Constitutive flow behavior of IFAC-1 austenitic stainless steel depicting strain saturation over a wide range of strain rates and temperatures. Mater. Des. 56, 565 (2014).
25. Cai, J., Lei, Y., Wang, K., Zhang, X., Miao, C., and Li, W.: A comparative investigation on the capability of modified Zerilli–Armstrong and Arrhenius-type constitutive models to describe flow behavior of BFe10-1-2 cupronickel alloy at elevated temperature. J. Mater. Eng. Perform. 25, 1952 (2016).
26. Guan, Z., Ren, M., Zhao, P., Ma, P., and Wang, Q.: Constitutive equations with varying parameters for superplastic flow behavior of Al–Zn–Mg–Zr alloy. Mater. Des. 54, 906 (2014).
27. Jia, W., Xu, S., Le, Q., Fu, L., Ma, L., and Tang, Y.: Modified Fields–Backofen model for constitutive behavior of as-cast AZ31B magnesium alloy during hot deformation. Mater. Des. 106, 120 (2016).
28. Tsao, L.C., Huang, Y.T., and Fan, K.H.: Flow stress behavior of AZ61 magnesium alloy during hot compression deformation. Mater. Des. 53, 865 (2014).
29. Zhu, Y., Zeng, W., Sun, Y., Feng, F., and Zhou, Y.: Artificial neural network approach to predict the flow stress in the isothermal compression of as-cast TC21 titanium alloy. Comput. Mater. Sci. 50, 1785 (2011).
30. Han, Y., Zeng, W., Zhao, Y., Qi, Y., and Sun, Y.: An ANFIS model for the prediction of flow stress of Ti600 alloy during hot deformation process. Comput. Mater. Sci. 50, 2273 (2011).
31. Qin, Y.J., Pan, Q.L., He, Y.B., Li, W.B., Liu, X.Y., and Fan, X.: Artificial neural network modeling to evaluate and predict the deformation behavior of ZK60 magnesium alloy during hot compression. Mater. Manuf. Processes 25, 539 (2010).
32. Sabokpa, O., Zarei-Hanzaki, A., Abedi, H.R., and Haghdadi, N.: Artificial neural network modeling to predict the high temperature flow behavior of an AZ81 magnesium alloy. Mater. Des. 39, 390 (2012).
33. Quan, G.Z., Wang, T., Li, Y.L., Zhan, Z.Y., and Xia, Y.F.: Artificial neural network modeling to evaluate the dynamic flow stress of 7050 aluminum alloy. J. Mater. Eng. Perform. 25, 553 (2016).
34. Wang, M.H., Wang, G.T., and Wang, R.: Flow stress behavior and constitutive modeling of 20MnNiMo low carbon alloy. J. Cent. South Univ. 23, 1863 (2016).
35. Ji, G., Yang, G., Li, L., and Li, Q.: Modeling constitutive relationship of Cu–0.4Mg alloy during hot deformation. J. Mater. Eng. Perform. 23, 1770 (2014).
36. Peng, W., Zeng, W., Wang, Q., Zhao, Q., and Yu, H.: Effect of processing parameters on hot deformation behavior and microstructural evolution during hot compression of as-cast Ti60 titanium alloy. Mater. Sci. Eng., A 593, 16 (2014).
37. Samantaray, D., Mandal, S., Phaniraj, C., and Bhaduri, A.K.: Flow behavior and microstructural evolution during hot deformation of AISI type 316 L(N) austenitic stainless steel. Mater. Sci. Eng., A 528, 8565 (2011).
38. Guo, L., Fan, X., Yu, G., and Yang, H.: Microstructure control techniques in primary hot working of titanium alloy bars: A review. Chin. J. Aeronaut. 29, 30 (2016).
39. Wang, M.H., Wang, W.H., Zhou, J., Dong, X.G., and Jia, Y.J.: Strain effects on microstructure behavior of 7050-H112 aluminum alloy during hot compression. J. Mater. Sci. 47, 3131 (2011).
40. Wu, B., Li, M.Q., and Ma, D.W.: The flow behavior and constitutive equations in isothermal compression of 7050 aluminum alloy. Mater. Sci. Eng., A 542, 79 (2012).
41. Liu, Y., Yao, Z., Ning, Y., Nan, Y., Guo, H., Qin, C., and Shi, Z.: The flow behavior and constitutive equation in isothermal compression of FGH4096–GH4133B dual alloy. Mater. Des. 63, 829 (2014).
42. Feng, D., Zhang, X.M., Liu, S.D., and Deng, Y.L.: Constitutive equation and hot deformation behavior of homogenized Al–7.68Zn–2.12Mg–1.98Cu–0.12Zr alloy during compression at elevated temperature. Mater. Sci. Eng., A 608, 63 (2014).
43. Bobbili, R. and Madhu, V.: Dynamic recrystallization behavior of a biomedical Ti–13Nb–13Zr alloy. J. Mech. Behav. Biomed. Mater. 59, 146 (2016).
44. Chen, X.M., Lin, Y.C., Wen, D.X., Zhang, J.L., and He, M.: Dynamic recrystallization behavior of a typical nickel-based superalloy during hot deformation. Mater. Des. 57, 568 (2014).
45. Kai, X., Chen, C., Sun, X., Wang, C., and Zhao, Y.: Hot deformation behavior and optimization of processing parameters of a typical high-strength Al–Mg–Si alloy. Mater. Des. 90, 1151 (2016).
46. Mirzadeh, H., Cabrera, J.M., Najafizadeh, A., and Calvillo, P.R.: EBSD study of a hot deformed austenitic stainless steel. Mater. Sci. Eng., A 538, 236 (2012).
47. Haghdadi, N., Martin, D., and Hodgson, P.: Physically-based constitutive modelling of hot deformation behavior in a LDX 2101 duplex stainless steel. Mater. Des. 106, 420 (2016).
48. Gall, S., Huppmann, M., Mayer, H.M., Müller, S., and Reimers, W.: Hot working behavior of AZ31 and ME21 magnesium alloys. J. Mater. Sci. 48, 473 (2012).
49. Cao, Y., Di, H., Zhang, J., Zhang, J., Ma, T., and Misra, R.D.K.: An electron backscattered diffraction study on the dynamic recrystallization behavior of a nickel–chromium alloy (800H) during hot deformation. Mater. Sci. Eng., A 585, 71 (2013).
50. Shiga, M.: Invar alloys. Crit. Rev. Solid State Mater. Sci. 1, 340 (1996).
51. Park, W.S., Chun, M.S., Han, M.S., Kim, M.H., and Lee, J.M.: Comparative study on mechanical behavior of low temperature application materials for ships and offshore structures: Part I—Experimental investigations. Mater. Sci. Eng., A 528, 5790 (2011).
52. Xiong, W., Zhang, H., Vitos, L., and Selleby, M.: Magnetic phase diagram of the Fe–Ni system. Acta Mater. 59, 521 (2011).
53. He, Y., Wang, F., Li, C., Yang, Z., Zhang, J., and Li, Y.: Effect of Mg content on the hot ductility of wrought Fe–36Ni alloy with Ti addition. Mater. Sci. Eng., A 673, 99 (2016).
54. Valenzuela, J.L., Valderruten, J.F., Pérez Alcázar, G.A., Colorado, H.D., Romero, J.J., González, J.M., Greneche, J.M., and Marco, J.F.: Low temperature study of mechanically alloyed Fe67.5Ni32.5 Invar sample. J. Magn. Magn. Mater. 385, 83 (2015).
55. Michler, T.: Influence of gaseous hydrogen on the tensile properties of Fe–36Ni Invar alloy. Int. J. Hydrogen Energy 39, 11807 (2014).
56. Zheng, J.J., Li, C.S., He, S., Cai, B., and Song, Y.L.: Microstructural and tensile behavior of Fe–36%Ni alloy after cryorolling and subsequent annealing. Mater. Sci. Eng., A 670, 275 (2016).
57. Zener, C. and Hollomon, J.H.: Effect of strain rate upon plastic flow of steel. J. Appl. Phys. 15, 22 (1944).
58. Yin, X.Q., Park, C.H., Li, Y.F., Ye, W.J., Zuo, Y.T., Lee, S.W., Yeom, J.T., and Mi, X.J.: Mechanism of continuous dynamic recrystallization in a 50Ti–47Ni–3Fe shape memory alloy during hot compressive deformation. J. Alloys Compd. 693, 426 (2017).
59. Sakai, T., Belyakov, A., Kaibyshev, R., Miura, H., and Jonas, J.J.: Dynamic and post-dynamic recrystallization under hot, cold and severe plastic deformation conditions. Prog. Mater. Sci. 60, 130 (2014).



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed