**3. Validation of the Numerical Model**

The accuracy of the numerical model was validated prior to evaluating the results of the numerical simulation by comparing the numerically simulated geometries with the measured geometries of the flat-rolled TWIP steel wire. The analyzed chemical composition of TWIP steel is Fe-19.94Mn-0.60C-1.03Al (wt.%), which was fabricated by vacuum induction method. Prior to hot rolling, the ingot measuring 125 mm in thickness was homogenized at 1200 ◦C for 3 h. Then, the ingot was directly rolled onto a plate measuring 20 mm in thickness using multipass rolling at temperatures above 950 ◦C, followed by air cooling to simulate the hot rod rolling process. The hot-rolled plate was machined into several round bars with a diameter of 13 mm for the flat wire rolling test. The bar was rolled into the flat-rolled wire using flat rolls with a diameter of 400 mm at a rolling speed of 5 RPM. The other process conditions were kept the same as those of the numerical simulation.

Figure 3a shows the comparison of the cross-sectional shapes of flat-rolled wires based on the experimental and numerical simulation, as well as the used terminologies in this study, while Figure 3b compares the measured and numerically simulated width of the contact area (*b*) and the lateral spread (*W*) values with the total *Rh*. As expected, the *W* and *b* values increased when increasing the total *Rh*. Overall, the *W* and *b* values obtained by the numerical simulation were in good agreement with the experimental values. However, it was found that the deviation of the *W* value between the two results slightly increased with the total *Rh*. This inconsistency was related to the hardening model applied in this study [5] and the selected friction coefficient of 0.3.

**Figure 3.** (**a**) Photograph of the caliber-rolled TWIP steel wire in this experiment and the shape of the deformed wire based on the numerical simulation. (**b**) Comparison of the measured, simulated, theoretically derived *W* and *b* values as a function of the total reduction in height.

Meanwhile, Kazeminezhad and Karimi Taheri [1,3] suggested the prediction of *b* and *W* values as a function of the total reduction in the height (Δ*h*), *h*0, and *h*<sup>1</sup> from the plain carbon steels as the following equations:

$$\mathbf{b} = \sqrt{2\Delta l h\_0} \tag{4}$$

$$\frac{W\_1}{W\_0} = 1.02 \left(\frac{h\_0}{h\_1}\right)^{0.45} \tag{5}$$

where the subscripts of 0 and 1 indicate the initial and final values during the flat rolling process, respectively. As shown in Figure 3b, the *W* and *b* values of TWIP steel were lower compared to those of the plain carbon steel theoretically suggested in the relationships. Hwang [14] suggested that this result is highly related to the different strain hardening behavior between plain carbon steels and TWIP steel. Namely, the strain hardening exponents of TWIP steel are much higher than those of plain carbon steels [23]. Overall, it can be concluded from an engineering application point of view that the proposed FE analysis for the flat wire rolling process can be used to evaluate the characteristics of the shape and strain distribution with the process conditions.

### **4. Results and Discussion**

#### *4.1. Strain Distribution with Roll Design*

Figure 4 shows a comparison of the contours of the von Mises strain (effective strain) and normal pressure of the flat-rolled wire with the representative roll designs (i.e., flat roll design, oval-grooved and cambered rolls with a radius of 10 mm). Clearly, the strain distribution in the flat-rolled wire was complex—it had two MSBs with high effective strain [8]. The center area tended to have a maximum strain, while the free surface area tended to have a minimum strain. However, the distribution of the effective strain of the wire was different with the roll design. In particular, the shapes of the MSBs changed with the roll design. During the first pass, the total width of the MSBs increased with the oval-grooved roll, while that of the MSBs decreased with the cambered roll. The different behavior of MSBs with the roll design was related to the normal pressure on the wire surface [24,25], as shown in Figure 4, indicating that the MSBs can be controlled by tailoring the contact pressure on the specimen during the rolling process. It should be noted that it is necessary to increase the strength on the flat surface of a wire, because the external stress was mainly imposed on the flat surface of the wires under service. For a better understanding of the strain distribution of the flat-rolled wire, the effective strain was extracted from the contour maps in both the cross-section and the flat surface, as shown in Figure 5 for the final product (i.e., after the second pass). The maximum effective strain in the center area decreased with the cambered roll, whereas it increased with the oval-grooved roll. Meanwhile, the minimum effective strain in the free surface area was similar regardless of the roll design, indicating that the cambered roll reduced the overall strain inhomogeneity of the flat-rolled wire.

**Figure 4.** Contour maps of the effective strain and normal pressure of flat-rolled wires with a roll design and pass.

Interestingly, the effective strain on the flat surface of the wire increased when using the cambered roll, as shown in Figure 5c, whereas the oval-grooved roll decreased the effective strain on the flat surface of the wire. To deduce a general conclusion, a new non-dimensional indicator for the roll design (*IRD*) was defined as follows:

$$I\_{RD} = \frac{D\_{\text{wire}}}{R\_R} \tag{6}$$

where *RR* and *Dwire* are the surface radius of the roll (Figure 1) and the diameter of the initial round wire (13 mm), respectively. To calculate the *RR* value, the surface radius of the cambered roll was taken to be negative due to the reverse roll shape compared to the oval-grooved roll, as listed in Table 1. Figure 6a shows the maximum and minimum effective strains in the cross-section of the flat-rolled wire at the second pass. The maximum strain at the center area of the wire slightly increased with *IRD*, meaning that the maximum strain increased with the decrease in the radius of the oval-grooved roll. In other words, the oval-grooved roll increased the stress concentration at the center area of the specimen due to the restriction of the plastic deformation according to roll shape, as compared to the flat and cambered rolls, as shown in Figure 2. In addition, it is well known that the strain is highly concentrated in the center area during bar and rod rolling with the oval-round roll pass sequence [26–28]. The minimum effective strain had a constant value with *IRD*, indicating that the oval-grooved roll and camber roll did not affect the level of strain in the free surface area of a wire. In summary, the cambered roll slightly reduced the strain inhomogeneity of the specimen during the flat rolling process.

**Figure 5.** (**a**) Schematic showing the terminologies used in this study. Comparison of the effective strain profiles in (**b**) cross-sections of flat-rolled wires along the horizontal and vertical directions, and (**c**) the flat surface of a flat-rolled wire with the representative roll design.

**Table 1.** Comparison of the seven roll designs and related values during the flat rolling process used in this study.


**Figure 6.** Variations in (**a**) the maximum and minimum effective strains in the cross-sections of the flat-rolled wires and (**b**) the maximum and centered effective strains on the flat surface of a wire with *IRD*.

The maximum effective strain on the flat surface of the wire decreased with *IRD*, as shown in Figure 6b, meaning that the camber roll with the small radius increased the strength on the flat surface of the wire. This was due to the strong concentration of the normal pressure on the wire surface, as shown in Figure 4c. Namely, the strain distribution on the flat surface of the wire was highly dependent on the normal contact pressure. Based on a similar mechanism, the strain of the center area on the flat surface of the wire was improved when using the cambered roll with a small radius. This result is attractive for industrial plants because many process designers want to improve the hardness of the flat surfaces in a wire owing to the strict customer demands.
