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


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.

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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:

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

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:

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

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

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

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

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

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

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

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:

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

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 \(n_{TTT}^{Max} = 5\)


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.

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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).

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