*4.1. Cooling Rate and Continuous Cooling Transformation Curves*

Although the microstructures after quenching can be considered as a direct evidence of dynamic transformation of austenite to ferrite above the Ae3 temperature, the cooling rate during quenching is a critical data that should be obtained to verify the validity of the observations. This is due to the fact that ferrite can also form during cooling of austenite. In the present experiments, the quenching rate was measured to be above 1200 ◦C/s. For example, water quenching from 1020 ◦C to 200 ◦C (well below the Ae3) took 0.7 s (see Figure 5). These measurements were taken using a welded thermocouple attached to the surface of the sample and it remained connected even after deformation. Although the cooling rate of the cylindrical sample may vary from the surface going to its center, the microstructures shown in the previous section were obtained from near the surface of the samples, which can accurately represent the microstructures with cooling rates of more than 1200 ◦C/s.

**Figure 5.** Setpoint (red line) and measured temperature (black line) by the thermocouple on the surface of the samples. It is observed that took less than 1 s for the temperature to drop well below the Ae3.

The continuous cooling transformation (CCT) curves of the present material with austenite grain sizes of 10 μm and 26 μm are displayed in Figure 6a,b, respectively. These were calculated using the Fe Alloys module of JMatPro software 10.2 (Guildford, UK). Based on these plots, it can be seen that it requires approximately 4 s to obtain 1% ferrite. This required cooling time of austenite is applicable to both coarse and fine grain austenite structure. In the present work, the cooling time is always below 1 s. This means that the ferrite phase observed and measured from the previous section was solely due to the applied deformation at elevated temperature.

**Figure 6.** Calculated continuous cooling transformation (CCT) of the material quenched from: (**a**) 1020 ◦C and grain size of 10 μm and; (**b**) 1100 ◦C and grain size of 26 μm.

#### *4.2. Critical Stresses and Strains*

The stress–strain curves associated with 5-pass rolling simulation of Figure 2a are fitted with a 9th order polynomial using a Matlab software R2018b (The MathWorks, Natick, MA, USA) followed by application of the double differentiation method [18]. In this process, the critical stresses and strains were determined by initially plotting the *θ* versus *σ* curves where *θ* is strain hardening rate calculated

by *δσ*/*δε* at a fixed strain rate (first derivative). The critical stresses are characterized by inflection points of the *θ*-*σ* plot. This can be described by the equation:

$$\frac{\delta}{\delta \sigma} \left( \frac{\delta \theta}{\delta \sigma} \right) = \,\, 0,\tag{1}$$

The minima in the plot of *δθ*/*δσ* versus *σ* (see Figure 7) are related to the softening mechanisms during deformation. The lower critical stresses of each curve are associated with dynamic transformation while the higher critical stresses are for dynamic recrystallization [20]. The critical strains can also be identified from critical stresses of Figure 7.

**Figure 7.** Plots of−(d*θ*/d*σ*) versus *σ* used to identify the minima related to the initiation of dynamic transformation (DT) and dynamic recrystallization (DRX).

The dependence of critical strains on pass number is displayed in Figure 8. The critical strains for DT are estimated to be around 0.06 while the critical strain for DRX is 0.12. These values are consistent with previous investigations of the present authors in the present material [21]. Moreover, the critical strains for DRX falls within the range of values shown in the literature [22] using the same method employed in the current work [19].

**Figure 8.** Critical strains for dynamic transformation (DT) and dynamic recrystallization (DRX) as a function of roughing pass number.

The critical strains for both DT and DRX display a slight decrease with increasing pass number. Note that for DRX, the critical strains should increase during cooling from one pass to another. The discrepancy in the trend of critical strains can be attributed to retained work hardening from previous pass. This is quite noticeable in the flow curves displayed in Figure 2. Since it is expected that NbC precipitates can form and pin down the dislocations, there is less recovery in-between passes. This leads to higher retained work hardening, which provides additional energy on top of the applied stress. A lower retained work hardening is expected if the material is low-alloyed steel or if the interpass time is increased [16]. In the present work, the attention will be focused on DT and the energies associated with this phenomenon.

#### *4.3. Total Energy Obstacles and Driving Force*

The occurrence of dynamic transformation can be explained in terms of transformation softening model [21]. In this model, the driving force to transformation (*DF*) is the flow stress difference between the critical stress of the parent phase (*σC*) and the yield stress of the product phase (*σYS*) defined by Equation (2) below:

$$DF = \sigma\_{\mathbb{C}} - \sigma\_{\text{YS}} \tag{2}$$

The critical stresses from austenite phase were obtained from the previous section while the yield stress of ferrite was estimated in a previous work of the present authors [23]. The dependence of the calculated driving force with temperature is shown in Figure 9 (see solid circles). The work per unit volume are converted into thermodynamic quantities using the conversion factor 1 MPa = 7.2 J/mol [10].

**Figure 9.** Calculated driving force (red line) over the total energy obstacle with the temperature.

The *total energy obstacles* consist of the work of dilatation (*WD*) and shear accommodation (*WSA*) associated with the phase transformation of austenite to ferrite, and the free energy difference between the phases (**Δ***Gγ-α*). This is defined by Equations (3)–(5) below:

*Total Energy Obstacles* <sup>=</sup> *WSA* <sup>+</sup> *WD* <sup>+</sup> **<sup>Δ</sup>***Gγ-α*, (3)

$$\mathbf{W\_{SA}} = \sigma\_{\mathbf{C}} \times 0.36 \times m,\tag{4}$$

$$\mathbf{W\_D} = \sigma\_{\mathbf{C}} \times 0.03 \times m\_{\prime} \tag{5}$$

The work of dilatation and shear accommodation are dependent on the critical stresses, which were obtained from the previous section. Since the dilatation and shear accommodation strains are difficult to measure experimentally, the present work assumed values of 0.03 and 0.36, respectively [24]. These values are based on the theoretical required deformation strains to transform austenite into ferrite. The Schmid factor (*m*) is based on the transformation habit plane and the shear direction when austenite transforms into ferrite, which are (0.506, 0.452, 0.735) and (−0.867, 0.414, 0.277), respectively. The present calculation employs the highest Schmid factor of 0.5 since the most oriented grains (with respect to the applied stress) are expected to transform first. Note that the critical strains are normally associated with the transformation of grains that are well oriented with the direction of applied stress [21,23,25].

The free energy difference between the austenite and ferrite was calculated using the FSstel database of the FactSage thermodynamic software. The sum of the components of the energy obstacles was calculated and are presented in Figure 9 (see solid triangle). Since the driving force is greater than the total energy obstacles for the present temperature range of 1020 ◦C to 1100 ◦C, it is expected that transformation of austenite to ferrite can take place. Although the calculated values may not represent the exact driving force and total energy obstacles due to the assumptions specified above, the difference between the curves look reasonable. This is supported by the microstructures which show that austenite transforms into ferrite. The kinetics of transformation may possibly be hindered by niobium which can delay the progression of transformation due to pinning and/or solute drag effects [16]. For this reason, the present material only obtained less than 10% ferrite at total applied torsional strain of 1.5. This may also be the reason for lower difference between the calculated driving forces and total energy obstacle.

It is also important to note that the free energy difference curve for the present material shows a peak, see Figure 10. This behavior indicates that austenite can easily transform if the temperatures are either close to the Ae3 or near the delta-ferrite formation temperature. The present temperature range covers the peak free energy difference; thus, these temperatures requires higher amount of driving force to initiate the occurrence of dynamic phase transformation. Based on the present results at the selected temperature range, it is expected that austenite can be transformed into ferrite at any temperature above the Ae3 for the present material [1].

**Figure 10. <sup>Δ</sup>***G(α-γ)* vs **<sup>Δ</sup>***<sup>T</sup>* for the present Nb-microalloyed steel showing the Gibbs energy obstacle opposing dynamic transformation.

#### **5. Conclusions**

1. Dynamic transformation of austenite to ferrite can take place during roughing passes of the plate rolling process and its volume fractions formed and retained in the simulations increase as the pass number increases. The present work showed that this can take place on a Nb-microalloyed steel subjected to roughing passes at temperature range 1020 to 1100 ◦C.

2. The calculated CCT diagrams confirmed that the measured cooling rate of 1200 ◦C/s is enough to prevent the formation of ferrite by cooling. Therefore, the volume fraction of ferrite measured in present work is only attributed to the applied deformation.

3. The thermodynamic calculations show that the driving force for DT is higher than the total barrier in the present investigation. Assumptions were made on the values of dilatation and shear accommodation strains.

4. The critical strains to dynamic transformation were shown to be in the range 0.04 to 0.07. This value decreases from pass to pass due to retained work hardening from the previous pass. The highest critical strain pertains to the first pass (where there is no prior deformation) during multi-pass high-temperature deformation experiments.

**Author Contributions:** All the authors contributed to this research work: Conceptualization, S.F.R., F.S., C.A.J., M.J. and J.J.J.; Formal analysis, S.F.R., C.A.J. and J.J.J.; Data curation, S.F.R., F.S., G.S.R. and E.S.S.; Methodology, S.F.R., F.S. and J.J.J.; Software, S.F.R., C.A.J. and E.S.S.; Validation, S.F.R., M.Z. and J.J.J.; Investigation, S.F.R., C.A.J., G.S.R. and J.J.J.; Resources, S.F.R., F.S., G.S.R., and E.S.S.; Writing—original draft preparation, S.F.R., C.A.J., F.S. and J.J.J.; Writing—review and editing, C.A.J., F.S. and J.J.J.; Visualization, C.A.J., F.S. and E.S.S.; Supervision, S.F.R., M.Z., J.J.J. and F.S.; Project administration, S.F. and J.J.J.; Funding acquisition, S.F.R., F.S. and C.A.J.

**Acknowledgments:** The authors acknowledge with gratitude funding received from the Brazilian National Council for Scientific and Technological Development (CNPq), the Industrial Research Chair in Forming Technologies of High Strength Materials of the École de Technologie Supérieure of Montréal, the New Brunswick Innovation Foundation (NBIF) and the Natural Sciences and Engineering Research Council of Canada. They also thank Jon Jackson and Laurie Collins from the EVRAZ North America Research and Development Centre (Regina) for providing the steel investigated in this research.

**Conflicts of Interest:** The authors declare no conflict of interest.
