**1. Introduction**

Flat-rolled wires are widely used in windshield wipers, springs, guide rails, and saw blades [1]. There are two main issues in the flat rolling of wire. The first issue is shape control of the flat-rolled wires, because direct shape control is impossible during the flat rolling of wire; that is, lateral spreading occurs in the free surface of the wire [2]. Consequently, several studies have been conducted regarding the influence of process conditions, such as reductions in the height, initial wire size, friction, and rolling speed on the lateral spreading of a wire using empirical or numerical methods [1–7]. The studies reported that the lateral spreading of a wire increased with reductions in the height and friction, which was independent of the rolling velocity. The second issue is the inhomogeneity of the mechanical properties of flat-rolled wires. In particular, the low hardness on the flat surfaces of a wire is a crucial issue in the industry. Therefore, the strain distribution of flat-rolled wire has been investigated using finite element (FE) analysis and hardness tests [8–14]. Kazeminezhad and Karimi Taheri [9] reported that the strain inhomogeneity of a wire increased with decreasing the reduction in height and increasing the friction coefficient. Vallellano et al. [12] reported that the maximum contact pressure occurred in the roll entry zone due to the local inhomogeneity of deformation. Hwang [14] reported that the difference of the effective strain along the horizontal direction was much higher than that of the vertical direction, and the maximum difference of the effective strain occurred in between the center area and the free surface area of the flat-rolled wire. It is well known that the occurrence of macroscopic shear bands (MSBs) is highly related to the strain inhomogeneity of flat-rolled wire [8], because the occurrence of MSBs indicates that the deformation is highly inhomogeneous during the

forming process. The occurrence of MSBs has been reported in several compression-type metal-forming processes, such as plain strain compression [15], uniaxial compression [16], flat rolling of wire [8], and flat roll drawing [17]. The restricted metal flow at the interface between the specimen and tool is the main reason for the occurrence of MSBs in specimens during the compression-type forming process. Therefore, the strain inhomogeneity can be improved by controlling the behavior of MSBs during the rolling process.

Over the past three decades, although several studies have reported the influence of process conditions on the strain inhomogeneity of flat-rolled wires [8–11], most of the studies have been conducted based on external process conditions, such as the roll diameter, reduction in thickness, rolling speed, and friction coefficient. Meanwhile, it may be inferred from experience that the roll shape is an important design parameter in shape rolling processes, such as caliber rolling [18,19] and flat rolling of wire. However, no studies have reported on the effect of the roll design on the strain distribution and shape control of a flat-rolled wire.

Therefore, the present study focuses on the effect of the roll design on the strain distribution in a flat-rolled wire in order to improve the homogeneity of mechanical properties and to increase the hardness on the flat surface of a wire. Oval-grooved roll and cambered roll techniques with various radii were applied to the flat rolling process based on numerical simulation, and then a general strategy for fabricating high-quality flat-rolled wire products was deduced, considering the process and working conditions in industries.

#### **2. Numerical Procedures**

The DEFORM FE commercial software developed by Scientific Forming Technologies Corporation in Ohio, USA, version 11.0 with a three-dimensional (3D) module was used to analyze the flat wire rolling process, because a flat-rolled wire experiences 3D inhomogeneous deformation during the process [14]. The workpiece was assumed to be an isotropic and rigid plastic material, while the effect of the strain rate was not considered in this study. In this case, the constitutive behavior was generally described by Hollomon's law [20,21], as follows:

$$
\sigma = \mathbb{K} \mathfrak{e}^{\mathfrak{n}} \tag{1}
$$

where *K* means the strength coefficient and *n* refers to the strain hardening exponent. The *K* and *n* values of the present material are chosen as 1980 and 0.54, respectively, based on the curve fitting of the tensile test in twinning-induced plasticity (TWIP) steel in [22]. Each value was inserted into the DEFORM commercial software model as follows:

$$
\sigma = 1980 \epsilon^{0.54} \tag{2}
$$

All rolls with 400 mm diameter were assumed to be rigid bodies. The rolling speed was set to 5 revolutions per minute (RPM) to ignore the effect of temperature rise. Oval-grooved rolls with radii of 10, 20, and 30 mm were applied to the flat rolling process to tailor the strain distribution, as shown in Figure 1b. A cambered roll was also applied with various radii to understand the effect of the roll shape on the strain distribution of a flat-rolled wire, as shown in Figure 1c. The oval-grooved roll and cambered roll were only used for the first rolling pass, while the same flat roll was applied to the second pass, as shown in Figure 1, to fabricate a flat-rolled wire with the desired shape for customers. The reduction in height (*Rh*) was calculated using the following equation:

$$R\_h = \frac{h\_0 - h\_1}{h\_0} \times 100 \, (\%) \tag{3}$$

where *h*<sup>0</sup> and *h*<sup>1</sup> are the initial and final heights of the wire, respectively. *Rh* was about 22% at the first pass, regardless of the roll design. At the second pass, the *Rh* was approximately 18%, and thus the total *Rh* was 35%. The shear friction coefficient between the wire and the roll interface was selected as 0.3 [14].

**Figure 1.** Schematic of the cross-sectional shape of the wire and the roll design during the flat rolling process with a (**a**) flat roll, (**b**) oval-grooved roll, and (**c**) cambered roll.

To reduce the computational cost, only one-quarter of the full geometry was calculated, owing to the symmetric condition of the flat-rolled wire. Approximately 15,000 brick-type elements were used, while 300 elements were used in the cross-section of the wire, as shown in Figure 2.

**Figure 2.** Detailed geometry and elements of the numerical modeling with a (**a**) flat roll, (**b**) oval-grooved roll with a radius of 10 mm, and (**c**) cambered roll with a radius of 10 mm.
