Model for Strain-Induced Precipitation Kinetics in Microalloyed Steels

Abstract

Based on Dutta and Sellars’s expression for the start of strain-induced precipitation in microalloyed steels, a new model has been constructed which takes into account the influence of variables such as microalloying element percentages, strain, temperature, strain rate, and grain size. Although the equation given by these authors reproduces the typical “C” shape of the precipitation start time (P _s) curve well, the expression is not reliable for all cases. Recrystallization–precipitation–time–temperature diagrams have been plotted thanks to a new experimental study carried out by means of hot torsion tests on approximately twenty microalloyed steels with different Nb, V, and Ti contents. Mathematical analysis of the results recommends the modification of some parameters such as the supersaturation ratio (K _s) and constant B, which is no longer a constant, but a function of K _s when the latter is calculated at the nose temperature (T _N) of the P _s curve. The value of parameter B is deduced from the minimum point or nose of the P _s curve, where ∂t _0.05/∂T is equal to zero, and it can be demonstrated that B cannot be a constant. The new expressions for these parameters are derived from the latest studies undertaken by the authors and this work represents an attempt to improve the model. The expressions are now more consistent and predict the precipitation–time–temperature curves with remarkable accuracy. The model for strain-induced precipitation kinetics is completed by means of Avrami’s equation.

Appendix: Model Expressions for the Start of Precipitation

$$ t_{0.05} = A\varepsilon^{ - \beta } D^{\text{s}} Z^{\text{r}} \exp \left( {\frac{{Q_{\text{diff}} }}{\text{RT}}} \right)\exp \left[ {\frac{B}{{T^{3} \left( {\ln k_{\text{s}} } \right)^{2} }}} \right] $$

$$ T_{\text{N}} = T_{\text{s}} - T_{\text{N}} . $$

The expressions for T _s are as follows:

V-Steels: $T_{\text{s}} = \frac{7700}{{2.86 - \log \left( {{\text{V}}\,{\text{pct}}} \right)\left( {{\text{N}}\,{\text{pct}}} \right)}}.$

Nb-Steels: $T_{\text{s}} = \frac{9450}{{4.12 - \log \left( {{\text{Nb}}\,{\text{pct}}} \right)\left( {{\text{C}}\,{\text{pct}}} \right)^{0.7} \left( {{\text{N}}\,{\text{pct}}} \right)^{0.2} .}}$

V-Steels: $\Updelta T_{\text{N}} = 176.1\left[ {X_{\text{i}} } \right]^{0.382}.$

Nb-Steels: $\Updelta T_{\text{N}} = 139\left[ {X_{\text{i}} } \right]^{0.283} .$

V-Ti and Nb-Ti Steels: $\Updelta T_{\text{N}} = 118\left[ {X_{\text{i}} } \right]^{0.293} .$

V-Steels: $A({\text{s}}^{ - 1} ) = 3.142 \times 10^{ - 10} \exp \left( { - 1.20\ln K_{\text{s}} } \right).$

Nb-Steels: $A({\text{s}}^{ - 1} ) = 4.766 \times 10^{ - 11} \exp \left( { - 0.07\ln K_{\text{s}} } \right).$

V-Ti and Nb-Ti Steels: $A({\text{s}}^{ - 1} ) = 4.13 \times 10^{ - 10} \exp \left( { - 0.47\ln K_{\text{s}} } \right).$

$$ \beta = 1.96\left[ {1 - \exp \left( { - 3.994 \times 10^{ - 2} \left( \frac{1}{w} \right)^{0.813} } \right)} \right] $$

w is the microalloying element content (Nb, V) (wt. pct).

$$ r = - 0.20 $$

$$ s = 0.5 $$

$$ Z({\text{s}}^{ - 1} ) = \dot{\varepsilon }\exp \left( {\frac{{Q_{\text{def}} }}{\text{RT}}} \right) $$

$$ Q_{\text{def}} ({\text{J}}\,{\text{mol}}^{ - 1} ) = 267,000 - 2535.52\left[ {\text{C}} \right] + 1010\left[ {\text{Mn}} \right] + 33,620.76\left[ {\text{Si}} \right] + 70,729.85\left[ {\text{Nb}} \right]^{0.565} + 31,673.46\left[ {\text{V}} \right] + 93,680.52\left[ {\text{Ti}} \right]^{0.5919} . $$

Nb-Steels:$K_{\text{s}} = \frac{{\left[ {\text{Nb}} \right]\left[ {\text{C}} \right]^{0.7} \left[ N \right]^{0.2} }}{{\left[ {10^{{4.12 - \frac{9450}{T}}} } \right]}}.$

V-Steels:$K_{\text{s}} = \frac{{\left[ {\text{V}} \right]\left[ {\text{N}} \right]}}{{\left[ {10^{{2.86 - \frac{7700}{T}}} } \right]}}.$

$$ B = 4.238 \times 10^{8} \exp \left( {1.689\ln K_{\text{s}} } \right). $$

To draw curve P _s for any steel, the parameter t _0.05 is calculated as a function of the temperature, with fixed A and B values, calculated as a function of K _s (for the corresponding T _N temperature), ε ^−β, and D ^0.5. The other terms Z ^−0.2, exp(Q _diff/RT), and exp(B/T ³(lnK _s)²) will be calculated as a function of the temperature including K _s.