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Modeling the Double-Pass Flow Curve of Nb Micro-Alloyed Steel by Machine Learning and its Extrapolation to Static Softening Kinetics

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Abstract

This paper presents a method to predict the double-pass flow stress behavior of hot-rolled niobium (Nb) micro-alloyed steel by combining the constitutive model and back-propagation neural network (BPNN) model (the machine learning (ML) model). The constitutive model is used to predict the first-pass flow stress behavior, and the BPNN model is used to predict the second-pass flow stress behavior. The results show that the ML model can accurately predict the double-pass flow stress behavior at different temperatures and holding time. The second-pass flow stress predicted by the BPNN model is compared with those predicted by the mathematical model in the previous literature. The results show that the BPNN model can more accurately predict the second-pass flow stress than the mathematical model in the previous literature. In addition, the static softening fraction during the interval time is calculated by using the predicted double-pass flow stress curve in this paper. The results show that the precision of the static softening fraction calculated in this paper is not only higher than that of the traditional Avrami kinetic model but also higher than that of the traditional model which couples recovery, precipitation and recrystallization. Therefore, the ML model for predicting the double-pass flow stress behavior has high accuracy and provides a theoretical basis for predicting the double-pass flow stress behavior of other alloys.

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Abbreviations

σ 1 :

The flow stress of the work hardened (Eq B1)

σ 0 :

The flow stress of completely softened material (Eq B1)

σ 2 :

The flow stress of partially softened (Eq B1)

ε :

Strain (Eq B1)

\({\dot{\varepsilon }}\) :

Strain rate (Eq B1)

\(D_{\gamma }\) :

Initial austenite grain size (Eq B2)

T :

Deformation temperature (Eq B2)

C Nb :

Nb concentration in steel (Eq B2)

X :

Recrystallization fraction (Eq B3)

σ Rex :

5% Flow stress of the recrystallized portions of the material (Eq B3)

σ Non -Rex :

5% Flow stress of the unrecrystallized portions of the material (Eq B3)

σ Rex -Matrix :

The 5% flow stress evaluated upon reloading using Eq (B2), with the current Nb content of the matrix (Eq B3)

σ UnRex -Matrix :

The contribution from the matrix including dislocation strengthening (Eq B3)

σ ppt :

Precipitation strengthening strength (Eq B4-1)

k :

Shearing constant (Eq. B4-1)

b :

Burgers vector (Eq B4-1)

f V :

Volume fraction of precipitates (Eq B4-1)

R :

The radius of precipitates (Eq B4-1)

k p :

0.8 (Eq B4-2)

k d :

1.1 (Eq B4-2

N(t):

Number of precipitate particles (Eq B5)

M :

Taylor factor (~ 3.1 for FCC) (Eq B5)

α r :

A constant of the order of 0.15 (Eq B5)

U a :

Recovery activation energy, 286 kJ/mol (Eq B5)

V a :

Recovery activation volume (Eq B5)

v d :

Debye frequency, 2 × 1012 s−1(Eq B5)

E :

Young’s modulus, 2.06 × 105 MPa (Eq B5)

k B :

Boltzman’s constant, 1.38 × 10−23 J/K (Eq B5)

Nc(t) :

Number of dislocation nodes, 0.5ρ(t)1.5(Eq B5)

G(t) :

Net driving force for recrystallization (Eq B6)

N rex :

Number of recrystallization nuclei (Eq B6)

\(\rho \left( t \right)\) :

Dislocation density (Eq B6)

γ gb :

Grain-boundary energy (Eq B6), γgb = 1.3115-0.0005 T

F v (t) :

Volume fraction of precipitated phase (B6)

R(t) :

Average radius of precipitated phase (B6)

M pure :

Grain-boundary mobility of pure material (Eq B6)

δ :

Grain-boundary thickness (B6)

N v :

Number of atoms per unit volume (Eq B6)

E b :

Binding energy of Nb to grain boundaries (Eq B6)

\(D_{gb}^{x}\) :

Average diffusion coefficient of Nb across the Fe grain boundary (Eq B6)

R :

The precipitate radius after a certain time t (Eq B7-1)

R 0 :

The average critical radius of the precipitates that have nucleated during the incubation period, R0≈1.05Rc (Eq B7-1)

\(X_{Nb}^{p}\) :

The concentration of Nb in the precipitate (Eq B7-2)

f :

The equilibrium volume fraction of precipitated phase of composite carbonitride (Eq B8)

M :

The solution concentration of M element in austenite at reheating temperature (Eq B8)

[M]:

The equilibrium concentration of M element in austenite at temperature T (Eq B8)

A M :

The atomic weights of elements M (Eq B8)

A X :

The atomic weights of X (Eq B8)

d Fe :

The densities of iron matrix (Eq B8)

d MXx :

The densities of MXx phase (Eq B8)

σ y :

Yield stress (Eq B9)

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Acknowledgments

Authors would like to acknowledge the financial supports from Ministry of Science and Technology, China (No. 2021YFB3702404), the project from Liaoning Province (No. XLYC1902034) and directional items of Liaoning National Science and Technology Award (2022JH25/10200001), Postdoctoral Science Foundation of China (2022T150205) and Postdoctoral Research Fund for Northeastern University (20210203), and National Natural Science Foundation of China (52104370).

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Authors and Affiliations

  1. State Key Laboratory of Rolling and Automation, Northeastern University, Shenyang, 110819, China

    Xin Li, Qiming Jiang, Xiaoguang Zhou, Siwei Wu, Guangming Cao & Zhenyu Liu

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Appendices

Appendix A Hernandez’s Model for Calculating Flow Stress

$$\left\{ \begin{gathered} \sigma = \sigma^{\prime} - \Delta \sigma \hfill \\ \sigma^{\prime} = B\left[ {1 - \exp \left( { - C\varepsilon } \right)} \right]^{m} \hfill \\ \Delta \sigma = B^{\prime}\left( {1 - \exp \left[ { - k\left( {\frac{{\varepsilon - a\varepsilon_{p} }}{{\varepsilon_{p} }}} \right)^{{m^{\prime}}} } \right]} \right) \hfill \\ B = \left( {9.5326 + 0.6196\ln \left( \frac{Z}{A} \right)} \right)^{2} \left( {{\text{MPa}}} \right) \hfill \\ C = 3.9202\left( \frac{Z}{A} \right)^{0.0592} \hfill \\ m = 0.3449\exp \left( {0.0139\sqrt{\frac{Z}{A}} } \right) \hfill \\ B^{\prime} = 26.0310\left( \frac{Z}{A} \right)^{0.1351} \left( {{\text{MPa}}} \right) \hfill \\ k = 0.5974\exp \left[ {1.2333\left( \frac{Z}{A} \right)^{{ - \frac{1}{2}}} } \right] \hfill \\ m^{\prime} = 1.0901\exp \left( {0.0264\sqrt{\frac{Z}{A}} } \right) \hfill \\ \varepsilon_{p} = 0.0952D_{0}^{0.195} \left( {\sqrt{\frac{Z}{A}} } \right)^{0.159} \hfill \\ Q = 267000 - 2535.52 \times {\text{C}}\% + 1010 \times {\text{Mn}}\% + 33620.76 \times {\text{Si}}\% + 36651.28 \times {\text{Mo}}\% \hfill \\ \quad \quad + 93680.52 \times {\text{Ti}}\%^{0.05919} + 31673.46 \times {\text{V}}\% + 70729.85 \times {\text{Nb}}\%^{0.5679} \left( {{\text{J}}/{\text{mol}}} \right) \hfill \\ A = \left[ {12.197 + 65.590 \times {\text{C}}\% - 49.052 \times {\text{Nb}}\% } \right]\exp \left( {7.076 \times 10^{ - 5} Q} \right)\left( {{\text{s}}^{ - 1} } \right) \hfill \\ \end{gathered} \right.$$
(A1)

Appendix B The Conventional Model for Calculating the Static Softening Fraction Coupling Recovery, Precipitation and Recrystallization

See in Table

Table 2 The conventional model coupling recovery, precipitation and recrystallization (Ref 16)
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2

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Li, X., Jiang, Q., Zhou, X. et al. Modeling the Double-Pass Flow Curve of Nb Micro-Alloyed Steel by Machine Learning and its Extrapolation to Static Softening Kinetics. J. of Materi Eng and Perform 33, 3669–3679 (2024). https://doi.org/10.1007/s11665-023-08221-9

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