A dilatometric analysis of inverse bainite transformation

Abstract

The unique two-stage dilatation curve observed during the inverse bainite transformation of a hypereutectoid low alloy steel is analyzed to understand the transformation kinetics. A new algorithm is proposed to extract the bainitic phase fractions from the raw dilatometry data. The proposed data extraction algorithm is generic that relies only on the density of phases involved in the transformation. To verify the extracted phase fraction, a kinetics model is developed using the principles of diffusion and Johnson–Mehl–Avrami–Kolmogorov kinetics. The predicted phase fractions by the kinetics model agree fairly well with the experimental phase fraction results from dilatometry, metallography, and XRD. The two-stage transformation can be explained by the kinetics of inverse bainite as a diffusion-controlled transformation product. The transformation proceeds in a para-equilibrium mode, involving only the diffusion of carbon at the inverse bainite/parent austenite interface

Appendix

Detailed derivation of Eq. 4 in “Algorithm to extract phase fraction from raw dilatometry data” section

Equation 3 is with the assumption that density of cementite and hence $\lambda_{\theta }$ does not vary with carbon concentration. In Eq. 3, the ratio of sample density to the density of austenite is taken as $\lambda_{\gamma }$ $\left( {\lambda_{\gamma } = \frac{{\rho_{o} }}{{\rho_{{\gamma \left( {T,C} \right)}} }}} \right)$.

$$ \frac{{\partial \lambda_{\gamma } }}{\partial t} = \frac{{ -\, \rho_{o} }}{{\rho {}_{{\gamma \left( {T,C} \right)}}}}\frac{1}{{\rho {}_{{\gamma \left( {T,C} \right)}}}}\frac{{\partial \rho_{\gamma } }}{\partial t} = \frac{{ - \rho_{o} }}{{\rho {}_{{\gamma \left( {T,C} \right)}}}}\frac{1}{{\rho {}_{{\gamma \left( {T,C} \right)}}}}\frac{{\partial \rho_{\gamma } }}{{\partial C_{\gamma } }}\frac{{\partial C_{\gamma } }}{\partial t} $$

(31)

Jablonka [31] fitted the variation in density of austenite with carbon content at constant temperature and found that the density of austenite varies linearly with carbon content. The density function of austenite at constant temperature is given by

$$ \rho_{{\gamma \left( {T,C} \right)}} = \rho_{{\gamma \left( {T,C = 0} \right)}} \left( {1 - 0.0146\% C_{\gamma } } \right) $$

(32)

$$ \frac{{\partial \rho_{{\gamma \left( {T,C} \right)}} }}{{\partial C_{\gamma } }} = - \,\rho_{{\gamma \left( {T,C = 0} \right)}} 0.0146 $$

(33)

Substituting Eqs. 32 and 33 in Eq. 31,

$$ \frac{{\partial \lambda_{\gamma } }}{\partial t} = \frac{{\rho_{o} }}{{\rho_{{\gamma \left( {T,C = 0} \right)}} }}\frac{0.0146}{{\left( {1 - 0.0146\% C} \right)^{2} }}\frac{{\partial C_{\gamma } }}{\partial t} $$

(34)

Substituting Eq. 34 in Eq. 3, the volume change rate can now be described by the following differential equation,

$$ \frac{1}{{V_{o} }}\frac{{{\text{d}}V}}{{{\text{d}}t}} = \frac{{\rho_{o} }}{{\rho_{{\gamma \left( {T,C = 0} \right)}} }}\frac{0.0146}{{\left( {1 - 0.0146\% C} \right)^{2} }}\frac{{\partial C_{\gamma } }}{\partial t}\left( {1 - f_{\theta } } \right) + \frac{{\partial f_{\theta } }}{\partial t}\rho_{o} \left[ {\frac{{\rho_{{\gamma \left( {T,C} \right)}} - \rho_{\theta } }}{{\rho_{{\gamma \left( {T,C} \right)}} \rho_{\theta } }}} \right] $$

(35)

The second term on the right-hand side in the above equation is negligible and is assumed zero because the fraction of cementite that nucleates and grows at the start of transformation is small, and the rate of change in fraction of cementite formed is also small. The justification for the approximation is given in “Error caused by the approximations used in the derivation of the algorithm” section. Therefore, Eq. 35 can be simplified as

$$ \frac{1}{{V_{o} }}\frac{{{\text{d}}V}}{{{\text{d}}t}} = \frac{{\rho_{o} }}{{\rho_{{\gamma \left( {T,C = 0} \right)}} }}\frac{0.0146}{{\left( {1 - 0.0146\% C} \right)^{2} }}\frac{{\partial C_{\gamma } }}{\partial t}\left( {1 - f_{\theta } } \right) $$

(36)

Writing mass balance with respect to carbon,

$$ C_{\gamma } f_{\gamma } + C_{\theta } f_{\theta } = 0.84 $$

$$ C_{\gamma } = \frac{{0.84 - 6.67f_{\theta } }}{{1 - f_{\theta } }} $$

$$ \frac{{\partial C_{\gamma } }}{\partial t} = -\, \frac{5.83}{{\left( {1 - f_{\theta } } \right)^{2} }}\frac{{\partial f_{\theta } }}{\partial t} $$

(37)

Substituting Eq. 37 in Eq. 36,

$$ \frac{1}{{V_{o} }}\frac{{{\text{d}}V}}{{{\text{d}}t}} = -\, \frac{{\rho_{o} }}{{\rho_{{\gamma \left( {T,C = 0} \right)}} }}\frac{0.0146}{{\left[ {1 - 0.0146\left( {\frac{{0.84 - 6.67f_{\theta } }}{{1 - f_{\theta } }}} \right)} \right]^{2} }}\frac{5.83}{{(1 - f_{\theta } )}}\frac{{\partial f_{\theta } }}{\partial t} $$

(38)

The term $\left[ {1 - 0.0146\left( {\frac{{0.84 - 6.67f_{\theta } }}{{1 - f_{\theta } }}} \right)} \right]^{2}$ is approximately equal to 1. The error caused by making this assumption is given in “Error caused by the approximations used in the derivation of the algorithm” section. Therefore, Eq. 38 can be simplified as

$$ \frac{3}{{L_{o} }}\frac{{{\text{d}}L}}{{{\text{d}}t}} = -\, \frac{{\rho_{o} }}{{\rho_{{\gamma \left( {T,C = 0} \right)}} }}0.0146\frac{5.83}{{(1 - f_{\theta } )}}\frac{{\partial f_{\theta } }}{\partial t} $$

$$ \frac{{{\text{d}}L}}{{{\text{d}}t}} = \frac{{\rho_{o} L_{o} 0.08511}}{{3\rho_{{\gamma \left( {T,C = 0} \right)}} }}\frac{{\partial \left( {{ \ln }\left( {1 - f_{\theta } } \right)} \right)}}{\partial t} $$

which leaves an Euler-type integration scheme as below,

$$ \frac{{L_{2} - L_{1} }}{K} = { \ln }\left[ {\frac{{1 - f_{2} }}{{1 - f_{1} }}} \right] $$

that calculates the fraction of cementite as a function of dilatation obtained from raw dilatometry data.

$$ f_{i + 1} = 1 - \left[ {\left( {1 - f_{i} } \right){ \exp }\left( {\frac{{L_{i + 1} - L_{i} }}{K}} \right)} \right] $$

where

$$ K = \frac{{\rho_{o} L_{o} 0.08511}}{{3\rho_{{\gamma \left( {T,C = 0} \right)}} }} $$

Error caused by the approximations used in the derivation of the algorithm

During the derivation of phase fraction of cementite, in Eq. 35, it is assumed that the contribution by term $\frac{{\partial f_{\theta } }}{\partial t}\rho_{o} \left[ {\frac{{\rho_{{\gamma \left( {T,C} \right)}} - \rho_{\theta } }}{{\rho_{{\gamma \left( {T,C} \right)}} \rho_{\theta } }}} \right]$ to the phase fraction calculation is negligible. Once, the phase fraction were calculated, the value of the term $\frac{{\partial f_{\theta } }}{\partial t}\rho_{o} \left[ {\frac{{\rho_{{\gamma \left( {T,C} \right)}} - \rho_{\theta } }}{{\rho_{{\gamma \left( {T,C} \right)}} \rho_{\theta } }}} \right]$ was determined to be 2 × $10^{ - 6}$. Error is calculated in the 7-min isothermal hold sample by accounting this term into phase fraction calculation, and it was found that the average error caused by not accounting this term is 7.6 ×  $10^{ - 6}$, i.e., 0.076%. The error caused is negligible, and therefore the approximation is valid. During the derivation of phase fraction of cementite, in Eq. 38, it is assumed that the term $\left[ {1 - 0.0146\left( {\frac{{0.84 - 6.67f_{\theta } }}{{1 - f_{\theta } }}} \right)} \right]^{2}$ is close to unity. Once, the phase fraction was calculated, the value of the term $\left[ {1 - 0.0146\left( {\frac{{0.84 - 6.67f_{\theta } }}{{1 - f_{\theta } }}} \right)} \right]^{2}$ was calculated and the average value was found to be 0.97653. Error in the phase fraction is calculated by accounting this term, and the absolute average error caused by the approximation is 6.13 × $10^{ - 4}$, i.e., 0.061%, which is negligible. Therefore, the approximation is justified.