# A Rapid, Open-Source CCT Predictor for Low-Alloy Steels, and Its Application to Compositionally Heterogeneous Material

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## Abstract

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## 1. Introduction

## 2. Model Formulation

#### 2.1. Modelling Reaction Kinetics

#### 2.2. Converting to Non-Isothermal Behaviour

#### 2.3. Predicting Constituent Behaviour

#### 2.3.1. Predicting Ferrite

#### 2.3.2. Predicting Pearlite

#### 2.3.3. Predicting Bainite

#### 2.4. Model Modifications

#### 2.4.1. Modelling Carbon Partitioning

#### 2.4.2. Adjusting Boundary Conditions

#### 2.4.3. The Upper-to-Lower Bainite Transition

#### Predicting Ferrite Decarburisation

#### Predicting Cementite Formation Kinetics

#### Estimating ${L}_{\mathrm{s}}$

#### 2.4.4. Predicting Martensite

#### 2.5. Model Layout

#### 2.5.1. Model Implementation

#### 2.5.2. Model Assumptions

- Simple, homogeneous austenite grains—the basis of the model predicts steel cooling transformation behaviour within a single, homogeneous austenite grain of a set size. The rest of the material is assumed to comprise solely these homogeneous grains. As a consequence, constraints to constituent nucleation and growth due to the size and shape of the austenite grain, neighbouring grains and ongoing transformations within the grain are not explicitly considered, and any effect on transformation behaviour is assumed to be accounted for empirically. This also includes the impact of residual plastic strain as a result of specimen deformation. The partitioning of carbon is also assumed to occur homogeneously, and the remaining untransformed austenite is enriched equally.
- Additivity is satisfied—all transformations are assumed to be additive and a conversion to a ‘true’ TTT (before conversion to non-isothermal conditions) is deemed unnecessary.
- Consistent reaction rate—the time-dependent reaction rate of austenite decomposition is assumed to be consistent between each constituent, aside from martensite. This follows the original assumptions made in the Li model.
- Simplified transformations—austenite is assumed to only decompose into either ferrite ($\mathsf{\alpha}$), pearlite ($\mathsf{\alpha}$ + $\mathsf{\theta}$), upper bainite ($\mathsf{\alpha}$), lower bainite ($\mathsf{\alpha}$ + intralath $\mathsf{\theta}$) and martensite ($\mathsf{\alpha}$’). Other transformations, such as second-phase particles or interlath carbides, are not currently considered. It is assumed that no constituent can transform simultaneously with another and transformations obey an order. Predictions for ferritic transformation behaviour are assumed to encompass the formation of all ferrite morphologies (allotriomorphic, Widmanstätten, and idiomorphic), as the predictions made by both Kirkaldy and Li do not distinguish between them. There is debate as to whether some of these morphologies should be considered ferritic; however, the determination of this lies outside the scope of this study.
- Diffusionless bainite transformation—there is much debate regarding the mechanism of the bainitic transformation and whether it is diffusion- or displacive controlled. For this model, for simplicity, it is assumed that bainite initially transforms displacively in the form of supersaturated laths of ferrite with carbon diffusion occurring after transformation.

## 3. Validation of Model

#### 3.1. Experimental Measurements

#### 3.2. Evaluating Modelled CCTs

#### 3.2.1. EN3B

#### 3.2.2. EN8

#### 3.2.3. SA-540 B24

## 4. Discussion

#### 4.1. Model Advantages

#### 4.1.1. Improved CCT Predictions

#### 4.1.2. Expanded Predictive Capabilities

#### 4.1.3. Enhanced Usability and Accessibility

#### 4.2. Model Limitations

#### 4.2.1. Inherent Limitations

#### 4.2.2. Over Partitioning

#### 4.2.3. Material Heterogeneity

#### 4.3. Considering Chemical Heterogeneity

## 5. Conclusions

- Li model predictions of continuous cooling behaviour were improved by the modifications outlined in this study. More accurate predictions were observed for all steels examined; however, this was best demonstrated in the SA-540 prediction, where the Li model underestimated the martensitic behaviour succeeding a higher-temperature bainitic transformation.
- The predictive capabilities of the model were expanded to include a prediction of final constituent fraction. The accuracy of these estimations were determined using hardness predictions, which agreed well with the experimental measurements.
- Although improvements were made to the CCT predictions, the accuracy of the proposed model is restricted by the inherent, empirical limitations of the original semi-empirical expressions. Correcting these limitations would require developing new sets of constituent transformation expressions around reliable and carefully controlled TTT and CCT datasets.
- The extent of carbon partitioning during austenite decomposition appears to be over-predicted, likely due to the technique used when measuring the empirical data, resulting in larger suppressions of ${T}_{\mathrm{s}}$. Nevertheless, the inclusion of this model incorporates a more realistic CCT prediction, but a more accurate measurement of partitioning would be required if this model was to be improved.
- Many disparities between the measured and predicted CCT behaviour were considered to be a result of microstructural heterogeneity within the examined material. The proposed model was implemented into a more sophisticated model that considered the measured chemical segregation in SA-540. Results from this adapted model showed an improved prediction of the SA-540 cooling behaviour.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Deriving the ${\mathit{T}}_{\mathbf{0}}^{{}^{\prime}}$ Equation

**Table A1.**Approximate representations of the free-energy components for the $\gamma \to \alpha $ transformation in pure Fe [53].

Function | a | b | Temperature Range |
---|---|---|---|

$\Delta {G}_{\mathrm{NM}}^{\gamma \to \alpha}=a+bT$ (J mol${}^{-1}$) | $-6660$ | 7 | $900>T>300\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

$\Delta {G}_{\mathrm{M}}^{\gamma \to \alpha}=a+bT$ (J mol${}^{-1}$) | 650 | $-1$ | $900>T>620\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

$\Delta {G}_{\mathrm{M}}^{\gamma \to \alpha}=a+bT$ (J mol${}^{-1}$) | 0 | 0 | $T<620\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

**Table A2.**Values of $\Delta {T}_{\mathrm{M}}$ and $\Delta {T}_{\mathrm{NM}}$ for specific substitutional alloying additions [53].

Alloying Element | $\mathbf{\Delta}{\mathit{T}}_{\mathbf{M}}$ (K per at.%) | $\mathbf{\Delta}{\mathit{T}}_{\mathrm{NM}}$ (K per at.%) |
---|---|---|

Si | $-3$ | 0 |

Mn | $-37.5$ | $-39.5$ |

Ni | $-6$ | $-18$ |

Mo | $-26$ | $-17$ |

Cr | $-19$ | $-18$ |

V | $-44$ | $-32$ |

Co | 19.5 | 16 |

Al | 8 | 15 |

Cu | 4.5 | $-11.5$ |

## Appendix B. Calculating $\underline{\mathrm{D}}$

**Table A3.**Alloying element parameters (${k}_{1}$ and ${k}_{2}$) for calculating the diffusivity of carbon in austenite [80].

M | Mn | Si | Ni | Cr | Mo | Al |
---|---|---|---|---|---|---|

${k}_{1}$ | $-0.0315$ | 0.0509 | $-0.0085$ | 0.0 | 0.3031 | $-0.0520$ |

${k}_{2}$ | $-4.3663$ | 4.0507 | $-1.2407$ | 7.7260 | 12.1266 | $-6.7886$ |

## Appendix C. Predicting Hardness

## Appendix D. Chemical Compositional Ranges

**Table A4.**Chemical composition ranges (in wt.%) used to develop equations for the ${\mathit{Ae}}_{3}$, ${\mathit{Ae}}_{1}$, ${\mathit{Ae}}_{\mathrm{cm}}$ and ${M}_{\mathrm{s}}$ temperatures.

Eq. | C | Si | Mn | Ni | Cr | Mo | V | S | P | Ref. |
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{Ae}}_{3}$/ ${\mathit{Ae}}_{1}$ | 0.30–0.63 | 0.15–0.30 | 0.37–1.85 | 0.44–3.41 | 0.49–0.98 | 0.18–0.33 | [42] | |||

${\mathit{Ae}}_{\mathrm{cm}}$ | 0.2–0.7 | 0.0–0.3 | 0.0–1.5 | 0.0–2.8 | 0.0–1.5 | 0.0–0.6 | [55] | |||

${M}_{\mathrm{s}}$ | 0.09–0.55 | 0.11–1.74 | 0.20–1.67 | 0.15–5.04 | 0.07–3.34 | 0.0–1.0 | 0.01–0.20 | 0.004–0.043 | 0.005–0.038 | [61] |

## Appendix E. EPMA Setup and Parameters

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**Figure 1.**A schematic showing how cooling curves can be sectioned into discrete steps when using Scheil’s additivity rule.

**Figure 2.**A flowchart showing the general layout of the model. Constituent nomenclature include: ferrite—f, pearlite—p, bainite—b, and martensite—m.

**Figure 3.**Predicted CCT behaviour for EN3B steel for an ASTM PAG size of 5.6 using (

**a**) the Li model, and (

**b**) the modified, proposed model. Constituent nomenclature includes: f—ferrite, p—pearlite, bu—upper bainite, bl—lower bainite, and m—martensite. Experimental ${T}_{\mathrm{s}}$ measurements are plotted alongside the predicted results as a dashed black line.

**Figure 5.**(

**a**) A plot showing the EN3B experimental hardness with cooling rate and modelled hardness predicted by the results of the proposed model. (

**b**) The final constituent fractions for EN3B predicted by the proposed model at each cooling rate. Constituent nomenclature includes: f—ferrite, p—pearlite, bu—upper bainite, bl—lower bainite, and m—martensite.

**Figure 6.**Predicted CCT behaviour for EN8 steel for an ASTM PAG size of 11.0 using (

**a**) the Li model, and (

**b**) the modified, proposed model. Constituent nomenclature includes f—ferrite, p—pearlite, bu—upper bainite, bl—lower bainite, and m—martensite. Experimental ${T}_{\mathrm{s}}$ measurements are plotted alongside the predicted results as a dashed black line.

**Figure 8.**(

**a**) A plot showing the EN8 experimental hardness with cooling rate and modelled hardness predicted by the results of the proposed model. (

**b**) The final constituent fractions for EN8 predicted by the proposed model at each cooling rate. Constituent nomenclature includes: f—ferrite, p—pearlite, bu—upper bainite, bl—lower bainite, and m—martensite.

**Figure 9.**Predicted CCT behaviour for SA-540 B24 steel for an ASTM PAG size of 6.7 using (

**a**) the Li model, and (

**b**) the modified, proposed model. Constituent nomenclature includes: f—ferrite, p—pearlite, bu—upper bainite, bl—lower bainite, and m—martensite. Experimental ${T}_{\mathrm{s}}$ measurements are plotted alongside the predicted results as a dashed black line.

**Figure 11.**(

**a**) A plot showing the SA-540 experimental hardness with cooling rate and modelled hardness predicted by the results of the proposed model. (

**b**) The final constituent fractions for SA-540 predicted by the proposed model at each cooling rate. Constituent nomenclature includes: f—ferrite, p—pearlite, bu—upper bainite, bl—lower bainite, and m—martensite.

**Figure 12.**Continuous cooling transformation modelling of chemically heterogeneous SA-540, for a 6.7 ASTM PAG size, showing (

**a**) the Ni EPMA map used, (

**b**) the predicted constituent map for a 0.5 °C s${}^{-1}$ cool, and (

**c**) the resultant predicted CCT alongside experimentally measured ${T}_{\mathrm{s}}$ curves. Constituent nomenclature includes: f—ferrite, p—pearlite, bu—upper bainite, bl—lower bainite, m—martensite and a—retained austenite.

Alloy | C | Si | Mn | Ni | Cr | Mo | S | Al | Cu | Fe |
---|---|---|---|---|---|---|---|---|---|---|

EN3B | 0.18 | 0.16 | 0.73 | 0.04 | 0.06 | 0.01 | 0.008 | 0.000 | 0.11 | Bal. |

EN8 | 0.44 | 0.20 | 0.77 | 0.07 | 0.14 | 0.02 | 0.029 | 0.028 | 0.15 | Bal. |

SA-540 | 0.40 | 0.26 | 0.75 | 1.81 | 0.86 | 0.32 | 0.008 | 0.031 | 0.08 | Bal. |

Alloy | T${}_{\mathbf{A}}$ (°C) | Step 1 | Step 2 | Step 3 |
---|---|---|---|---|

EN3B | 900 | Heat to T${}_{\mathrm{A}}$ at 10 °C s${}^{-1}$. | Hold at T${}_{\mathrm{A}}$ for 10 min. | Cool at 0.1, 0.2, 0.5, 1, 2, 5, 10, 20 or 50 °C s${}^{-1}$. |

EN8 | 900 | Heat to T${}_{\mathrm{A}}$ at 10 °C s${}^{-1}$. | Hold at T${}_{\mathrm{A}}$ for 10 min. | Cool at 0.1, 0.2, 0.5, 1, 2, 5, 10, 20 or 50 °C s${}^{-1}$. |

SA-540 | 870 | Heat to T${}_{\mathrm{A}}$ at 10 °C s${}^{-1}$. | Hold at T${}_{\mathrm{A}}$ for 2 h. | Cool at 0.1, 0.2, 0.5, 1, 2, 5, 10, 20 or 50 °C s${}^{-1}$. |

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## Share and Cite

**MDPI and ACS Style**

Collins, J.; Piemonte, M.; Taylor, M.; Fellowes, J.; Pickering, E.
A Rapid, Open-Source CCT Predictor for Low-Alloy Steels, and Its Application to Compositionally Heterogeneous Material. *Metals* **2023**, *13*, 1168.
https://doi.org/10.3390/met13071168

**AMA Style**

Collins J, Piemonte M, Taylor M, Fellowes J, Pickering E.
A Rapid, Open-Source CCT Predictor for Low-Alloy Steels, and Its Application to Compositionally Heterogeneous Material. *Metals*. 2023; 13(7):1168.
https://doi.org/10.3390/met13071168

**Chicago/Turabian Style**

Collins, Joshua, Martina Piemonte, Mark Taylor, Jonathan Fellowes, and Ed Pickering.
2023. "A Rapid, Open-Source CCT Predictor for Low-Alloy Steels, and Its Application to Compositionally Heterogeneous Material" *Metals* 13, no. 7: 1168.
https://doi.org/10.3390/met13071168