Skip to main content

Improved Method to Verify the Additivity Rule for Pearlite Transformation in Eutectoid Steel

Abstract

The additivity rule is beneficial when kinetics between the isothermal and non-isothermal (usually continuous cooling) phase transformation is interrelated. When isothermal transformation (Time Temperature Transformation (TTT)) follows the Johnson–Mehl–Avrami equation \(\left( {X = 1 - \exp \left( { - k_{TTT} \tau_{TTT}^{{n_{TTT} }} } \right)} \right)\), even though the reaction exponent is a function of temperature, if \(n_{TTT}\) is larger than \(1.0\), as happens in pearlite transformation, it is confirmed that isothermal kinetics is derived from continuous cooling kinetics by the same approach proposed by Ozawa, if the additivity rule is applicable. In this case, the Continuous Cooling Transformation (CCT) with cooling rate q, is described by \(X = 1 - \exp \left\{ { - [J/q]^{{n_{TTT} }} } \right\}\), where \(J = \int_{T}^{{T_{eq} }} {k_{TTT}^{{1/n_{TTT} }} dT}\). It is suggested that the additivity rule is applicable when the difference between \(J\left( T \right)\) from TTT (i.e. \(J_{TTT} (T)\)) and \(J(T)\) from CCT (i.e. \(J_{TTT \leftarrow CCT} (T)\)) is within experimental error. Both TTT and CCT are measured for austenite decomposition to pearlite in eutectoid steel. Considering the error of both CCT and TTT experiments, pearlite transformation is proved to be partially additive. When temperatures are from 610 to 640 °C, pearlite transformation is additive. However, when the temperature is 650 °C or above, pearlite transformation is non-additive due to a change of the transformation mechanism, which is confirmed by the isokinetic condition using the Master Curve Method.

Graphic Abstract

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

References

  1. E. Scheil, Arch. Eisenhüttenwes. 8, 565 (1935)

    CAS  Article  Google Scholar 

  2. J.-O. Andersson, T. Helander, L. Höglund, P.F. Shi, B. Sundman, Calphad 26, 273 (2002)

    CAS  Article  Google Scholar 

  3. B. Pawłowski, J. Achiev. Mater. Manuf. Eng. 49, 331 (2011)

    Google Scholar 

  4. T. Ozawa, Polymer 12, 150 (1971)

    CAS  Article  Google Scholar 

  5. P.R. Rios, Acta Mater. 53, 4893 (2005)

    CAS  Article  Google Scholar 

  6. T. Jia, M. Militzer, Z.Y. Liu, ISIJ Int 50, 583 (2010)

    CAS  Article  Google Scholar 

  7. M. Fontana, M.A. Urena, B. Arcondo, M.T. Clavaguera-Mora, Int. J. Therm. Sci 109, 33 (2016)

    CAS  Article  Google Scholar 

  8. M. Umemoto, K. Horiuchi, I. Tamura, Trans. ISIJ. 23, 690 (1983)

    CAS  Article  Google Scholar 

  9. T. Réti, L. Horvath, I. Felde, J. Mater. Eng. Perform 6, 433 (1997)

    Article  Google Scholar 

  10. F.X. Bai, J.H. Yao, Y. Li, Intermetallics 86, 73 (2017)

    CAS  Article  Google Scholar 

  11. C.-W. Lee, S.-H. Uhm, K.-M. Kim, K.-J. Lee, C.-H. Lee, ISIJ Int. 41, 1383 (2001)

    CAS  Article  Google Scholar 

  12. W. Yu, L. Xu, G. Feng, C. Wu, C. Zhou, H. Wang, Int. J. Min. Met. Mater. 17, 558 (2010)

    CAS  Article  Google Scholar 

  13. S.E. Offerman, L.J.G.W. van Wilderen, N.H. van Dijk, J. Sistsma, M.T. Rekveldt, S. van der Zwaag, Acta Mater. 51, 3927 (2003)

    CAS  Article  Google Scholar 

  14. C. Zener, T. Am. I. Min. Met. Eng. 167, 550 (1946)

    Google Scholar 

  15. M. Hillert, Jernkontorets Annaler. 141, 757 (1957)

    CAS  Google Scholar 

  16. M.P. Puls, J.S. Kirkaldy, Metall. Mater. Trans. B 3, 2777 (1972)

    CAS  Article  Google Scholar 

  17. N.A. Razik, G.W. Lorimer, N. Ridley, Metall. Trans. A 7, 209 (1976)

    Article  Google Scholar 

  18. M. Hillert, Metall. Mater. Trans. B 3, 2729 (1972)

    CAS  Article  Google Scholar 

  19. A.S. Pandit, H.K.D.H. Bhadeshia, Proc. R. Soc. A Math. Phys. 467, 2948 (2011)

    CAS  Article  Google Scholar 

  20. S.-W. Seo, H.K.D.H. Bhadeshia, D.-W. Suh, Mater. Sci. Technol. 31, 487 (2015)

    CAS  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kyung Jong Lee.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Validity of Approximation of Eq. (7)

Appendix A: Validity of Approximation of Eq. (7)

If \(n_{TTT} \left( T \right)\) exists between \(n_{TTT}^{Min}\) to \(n_{TTT}^{Max}\), the upper and lower boundaries of the exact solution are expressed as:

$$\left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT}^{Max} }}}} *J_{TTT} \left( T \right) < \mathop \int \limits_{T}^{{T_{eq} }} \left( {\frac{{k_{TTT} \left( T \right)}}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT} \left( T \right)}}}} dT < \left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT}^{Min} }}}} *J_{TTT} \left( T \right)$$
(21)

where \(J_{TTT} \left( T \right) = \mathop \int \limits_{T}^{{T_{eq} }} k_{TTT}^{{1/n_{TTT} }} dT\). Since the exact solution in Eq. (21) is approximated as below:

$$\mathop \int \limits_{T}^{{T_{eq} }} \left( {\frac{{k_{TTT} \left( T \right)}}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT} \left( T \right)}}}} dT \cong \left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT} \left( T \right)}}}} *J_{TTT} \left( T \right)$$
(22)

The error of approximation \(\left( {\varepsilon \left( {n_{TTT}^{Max} , n_{TTT}^{Min} ,T} \right)} \right)\) is written as:

$$\begin{aligned} & \left[ {\left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT}^{Max} }}}} - \left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT} \left( T \right)}}}} } \right]*J_{TTT} \left( T \right) < \varepsilon \left( {n_{TTT}^{Max} , n_{TTT}^{Min} ,T} \right) \\ & \quad < \left[ {\left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT}^{Min} }}}} - \left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT} \left( T \right)}}}} } \right]*J_{TTT} \left( T \right) \\ \end{aligned}$$
(23)

Therefore, the relative error \(\varepsilon_{f} \left( {n_{TTT}^{Max} , n_{TTT}^{Min} ,T} \right)\) equals Eq. (23) divided by RHS of Eq. (22):

$$\frac{{\left[ {\left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT}^{Max} }}}} - \left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT} \left( T \right)}}}} } \right]}}{{\left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT} \left( T \right)}}}} }} < \varepsilon_{f} < \frac{{\left[ {\left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT}^{Min} }}}} - \left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT} \left( T \right)}}}} } \right]}}{{\left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT} \left( T \right)}}}} }}$$
(24)

Tolerance \({\rm O}\left( T \right)\), calculated from the error of TTT, is calculated as below:

$${\rm O}\left( T \right) = \left| {\frac{{\mathop \int \nolimits_{T}^{{T_{eq} }} \left( {\frac{{k_{TTT}^{Upper} \left( T \right)}}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT}^{Upper} \left( T \right)}}}} - \mathop \int \nolimits_{T}^{{T_{eq} }} \left( {\frac{{k_{TTT}^{Lower} \left( T \right)}}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT}^{Lower} \left( T \right)}}}} }}{{\left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT} \left( T \right)}}}} *J_{TTT} \left( T \right)}}} \right|$$
(25)

The average tolerance \({\rm O}_{avg} = 0.613\) due to experimental error (\(\pm 0.3*\tau_{TTT}\)), as mentioned in Sect. 2. The condition \(\varepsilon_{f} \left( {n_{TTT}^{Max} , n_{TTT}^{Min} ,T} \right) < 0.613\) is checked to confirm the validity of Eq. (7).

Since the \(n_{TTT} \left( T \right)\) exists from \(n_{TTT}^{Min}\) to \(n_{TTT}^{Max}\), the maximum relative error \(\varepsilon_{f}^{Max} \left( {n_{TTT}^{Max} , n_{TTT}^{Min} ,T} \right)\) equals:

$$\varepsilon_{f}^{Max} \left( {n_{TTT}^{Max} , n_{TTT}^{Min} ,T} \right) = \left| {\frac{{\left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT}^{Min} }}}} - \left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT}^{Max} }}}} }}{{\left( {\frac{1}{{ - \ln \left( {1 - X} \right)}}} \right)^{{\frac{1}{{n_{TTT}^{Max} }}}} }}} \right|$$
(26)

Normally, \(n_{TTT}^{Max}\) does not exceed 5 for most of the transformation. The calculated \(n_{TTT}^{Min}\), which satisfies \(\varepsilon_{f}^{Max} \left( {n_{TTT}^{Max} , n_{TTT}^{Min} ,X,T} \right) = 0.613\) when \(n_{TTT}^{Max} = 5\) at different fractions \(X,\) is shown in Fig. 

Fig. 11
figure 11

The calculated \(n_{TTT}^{Min}\) which satisfies \(\varepsilon_{f}^{Max} \left( {n_{TTT}^{Max} , n_{TTT}^{Min} ,T} \right) = 0.613\) when \(n_{TTT}^{Max} = 5\)

11.

The average value of \(n_{TTT}^{Min} = 1.02083\). This is the reason why Eq. (7) is valid for pearlite transformation, since the reaction exponent is larger than 1 at the transformation range (Fig. 2a). Figure 

Fig. 12
figure 12

The comparison between exact solution (solid line) and approximation (dashed line) for studied steel

12 shows the comparison between the exact solution and the approximation for studied steel. As shown in Fig. 12, the approximation is well established at all cooling rates, which proves the approximation is valid for pearlite transformation.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kim, J.M., Hong, S. & Lee, K.J. Improved Method to Verify the Additivity Rule for Pearlite Transformation in Eutectoid Steel. Met. Mater. Int. (2022). https://doi.org/10.1007/s12540-022-01228-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12540-022-01228-2

Keywords

  • Phase transformation kinetics
  • Pearlitic steels
  • Analytical method
  • Non-isothermal
  • Additivity rule