2.1.1. Phase I: Theoretical Study

The assumptions made here were that with the addition of Nb into the steel, the *T*5% and *T*95% temperatures will both be increased [18], and subsequentially the processing window above *T*95% will be reduced. In addition, the *T*5% temperature might be increased to where it enters the later portion of the rough rolling temperature range. Figure 4 shows a schematic of two Nb steels, namely, 0.1Nb and 0.04Nb, in wt.%. For a given strain (i.e., Σ*nPasses*) and temperature, and as the temperature decreases, the processing window at the end of roughing for a uniform austenite grain size distribution is reduced in the 0.1Nb steel, below which only partial recrystallization and grain refinement may occur; this is shown in the hatched area above 0.1Nb *T*95%. On the other hand, the processing window for the 0.04Nb steel to produce a uniform grain size distribution is higher even below 1050 ◦C, which is here designated as the end of rough rolling temperature.

**Figure 4.** Schematic showing the processing window of high and low Nb steels; hatched areas show the window for which a uniform grain refined austenite microstructure can be obtained by rolling above the respective *T*95% temperature.

In the context of hot rolling, the typical value of the recrystallization driving force of austenite subjected to a single low temperature rolling pass is found to be about 22 MPa [12]. This was for 900 ◦C; at 1100 ◦C it would be much lower, near 10 MPa. Thus, to retard recrystallization, a necessary pinning force is needed and several models were proposed to calculate the pinning force of precipitates in microalloyed steels as a result of the interaction between precipitates and austenite grain boundaries [18,23]. However, in this study, as mentioned earlier, the subgrain boundary model [3] was employed to estimate, theoretically, the pinning forces at different deformation temperatures and specifically during roughing passes. The reason for using this model is mainly that NbC particles are not randomly distributed but isolated on the subgrain boundaries of deformed austenite [24,25].

The estimations of pinning forces caused by strain induced precipitation of NbC were calculated using (i) the subgrain boundary model, together with (ii) the volume fractions of NbC estimated for the compositions and temperatures using the equilibrium solubility relation mentioned previously; (iii) particle sizes of NbC observed; and (iv) the dislocation cell sizes observed in the TEM in recovered austenite in earlier similar studies [17,20]. The subgrain boundary pinning force model used was [3]:

$$F = \frac{3\gamma f l l}{2\pi r^2} \tag{2}$$

where γ is the high angle austenite grain boundary energy, *fv* is the volume fraction of precipitates, *l* is the subgrain size and *r* is the particle size. The parameters used in this investigation were γ = 0.8 J/m2, particle size = 2 nm and subgrain size = 0.5 μm; and *fv* can be calculated from the solubility product.

It should be mentioned that the measured volume fractions of NbC for subgrain and grain boundaries were higher than predicted by equilibrium considerations [20]. It appeared that there was a segregation of Nb to the boundaries leading to higher than expected local volume fractions of NbC. This means that the local pinning forces are actually larger than those which would be calculated from equilibrium considerations

Table 1 shows the estimated pinning forces for two alloys, during roughing passes. The results indicate that higher Nb level gives the higher pinning force magnitudes at the end of roughing passes (i.e., 41 MPa at 1050 ◦C), due to higher volume fraction of NbC to precipitate at lower roughing temperature.


**Table 1.** NbC pinning force at the respective deformation temperature, MPa.
