Inﬂuence of Roll Diameter on Material Deformation and Properties during Wire Flat Rolling

: The inﬂuence of roll diameter on the strain distribution, shape change, contact pressure, and damage value of a workpiece was investigated during wire ﬂat rolling to control the material properties of the ﬂattened wire. The ﬂattened wires fabricated by the four different rolls were compared using ﬁnite element analysis. The strain inhomogeneity of the ﬂat-rolled wire increased with the roll diameter; thus, the macroscopic shear bands were strengthened as the roll diameter increased during wire ﬂat rolling. The contact width and lateral spreading of the ﬂattened wire increased with the roll diameter; therefore, the reduction in area decreased with the roll diameter. The contour of the normal contact pressure on the wire surface exhibited a similar pattern regardless of the roll diameter. The contact pressure showed higher values at the entry, edge, and exit zones in the contact area. The distribution of the damage value varied with the roll diameter. The free surface region tended to have the peak damage value during the process; however, the center region exhibited the maximum damage value with the roll diameter. From the perspective of the damage value, the optimum roll diameter was in existence during wire ﬂat rolling. The underlying cause of the different strain distributions, shape changes, and damage values of the ﬂat-rolled wire was the different contact lengths originating from the different roll diameters during wire ﬂat rolling.


Introduction
Wire flat rolling is the process of reducing the height of a round wire by passing it through two counter-rotating flat rolls. Despite the use of flat rolls, the deformation behavior of a workpiece is relatively complex during wire flat rolling because a round wire is used as the initial workpiece, which is different from the strip or plate flat rolling. The issue of strain inhomogeneity in a flat-rolled wire is inevitable because stress is concentrated in the central regions of the wire owing to the geometric forming and loading conditions during the process [1]. The more essential reason is that the wire flat rolling process belongs to the compression-type cold forming process. The strain inhomogeneity in the compression-type forming processes is attributed to the restricted metal flow in the workpiece and die interface owing to the resistant shear stress generated by friction. In this manner, macroscopic shear bands (MSBs) commonly occur during compression-type forming [2,3]. Meanwhile, the issue of strain inhomogeneity of the flattened wire is becoming critical because the demand for nonheat-treated products increases continuously from metal forming and its downstream industries. In addition, the elimination of heat treatments is a major trend in the metal manufacturing industry to reduce production costs, production cycles, and greenhouse gas emissions. For instance, Hwang [4,5] recommended twinning-induced plasticity (TWIP) steel as an innovative metal for use in high-performance flattened wire products. He suggested the use of TWIP steel as flat-rolled wire products without the application of any heat treatments during the entire manufacturing process owing to its

FE Analysis and Boundary Conditions
During wire flat rolling, the workpiece exhibits three-dimensional (3D) deformation behavior owing to the initial round shaped wire [15]. Accordingly, FE analysis was conducted using the commercial software, DEFORM 3D version 11.0(Scientific Forming Technologies Corporation, Columbus, OH, USA) to evaluate the deformation behavior of the workpiece during wire flat rolling.
The 13 mm-diameter wire was reduced to a 9.1 mm-thick flattened wire by flat rolls with diameters of 150, 400, 800, and 1600 mm at room temperature (RT, 24 • C), as schematically shown in Figure 1. Hereafter, the flat-rolled wires fabricated using rolls with diameter of 150, 400, 800, and 1600 mm are referred to as "150-wire", "400-wire", "800-wire", and "1600-wire", respectively. During the process, the reduction in height (R h ) and reduction in area (R A ) of the workpiece were calculated using the following equations: where h o and h 1 represent the initial and final heights of the workpiece; and A o and A 1 indicate the initial and final areas of the workpiece, respectively. The nominal strain (ε n ) of the workpiece was calculated using the height of the workpiece as follows: ing using the friction coefficient of 0.17 although it is known that the friction coefficien in the roll bite varies with the rolling [23] and width [24] directions of the workpiec during rolling. In addition, the friction coefficient of 0.15 was a reasonable value for th cold rolling process without lubrication. The influences of temperature increase and rolling speed were not considered dur ing the process because the rolling velocity was set to 5 rpm, that is, a low strain rate o forming speed. The roll was considered to be a rigid body because the plastic defor mation of the workpiece was relatively large, i.e., Rh of 30%. In such a case, the elasti deformation of roll can be ignored.
Owing to the double symmetry condition in both the reduction and spreading di rections during wire flat rolling, only a quarter of the full geometry of both the wire and roll were modeled. Hence, a significant reduction in computational costs was achieved.

Experimental Procedures
Experimental tests were conducted to validate the present FE model. A 50 kg ingo with a chemical composition of Fe-19.94Mn-0.6C-1.03Al in weight percent was fabricated using a vacuum induction method under inert gas. After homogenization in a box fur nace at 1200 °C for 12 h, the ingot with 125 mm in thickness was directly rolled to a 2 According to the equations above, the R h and ε n values in the present study were 30% and 0.36, respectively.
Friction plays a crucial role in determining the deformation behavior of a workpiece during the rolling process because friction is the main reason for the rolling process to take place. Although a Tresca model (shear friction factor) is preferable in high-pressure bulk forming processes [20], the Coulomb friction model was applied at the roll and wire interfaces because it is known that the shear friction model cannot predict the sticking region near the neutral point during plat flat rolling [21]. Accordingly, in this study, the Coulomb friction model was applied as follows: where, τ, µ, and p are the frictional stress, coefficient of friction, and normal pressure, respectively. In the present study, a constant friction coefficient of 0.15 was used because Kazeminezhad and Karimi Taheri [22] well predicted the roll force during wire flat rolling using the friction coefficient of 0.17 although it is known that the friction coefficient in the roll bite varies with the rolling [23] and width [24] directions of the workpiece during rolling. In addition, the friction coefficient of 0.15 was a reasonable value for the cold rolling process without lubrication. The influences of temperature increase and rolling speed were not considered during the process because the rolling velocity was set to 5 rpm, that is, a low strain rate or forming speed. The roll was considered to be a rigid body because the plastic deformation of the workpiece was relatively large, i.e., R h of 30%. In such a case, the elastic deformation of roll can be ignored.
Owing to the double symmetry condition in both the reduction and spreading directions during wire flat rolling, only a quarter of the full geometry of both the wire and roll were modeled. Hence, a significant reduction in computational costs was achieved.

Experimental Procedures
Experimental tests were conducted to validate the present FE model. A 50 kg ingot with a chemical composition of Fe-19.94Mn-0.6C-1.03Al in weight percent was fabricated using a vacuum induction method under inert gas. After homogenization in a box furnace at 1200 • C for 12 h, the ingot with 125 mm in thickness was directly rolled to a 20 mm-thick plate in the temperature range of hot deformation in steels (from 950 • C to 1150 • C) using Appl. Sci. 2021, 11, 8381 4 of 15 several rolling machines. Subsequently, it was naturally cooled in air to RT in order to simulate the hot rod rolling process. Tensile specimens (5 mm-diameter and 25 mm-long in gauge) machined from the hot-rolled plate were pulled at a strain rate of 10 −3 s −1 at RT. Figure 2a presents the true stress-strain curve of the hot-rolled steel. This steel exhibited high strength and excellent elongation owing to the deformation twinning during plastic deformation [6,7]. For a better evaluation of the strain hardening behavior of this steel, Holloman's law was used to evaluate the strain hardening exponent (n) as follows: where K is the strain hardening coefficient. Based on the above equation, the instantaneous n was calculated with true strain using the following equation: ε, T=const. (6) exhibited high strength and excellent elongation owing to the deformation twin during plastic deformation [6,7]. For a better evaluation of the strain hardening beha of this steel, Holloman's law was used to evaluate the strain hardening exponent ( follows: where K is the strain hardening coefficient. Based on the above equation, the insta neous n was calculated with true strain using the following equation: The instantaneous n increased gradually with the true strain as shown in Figure  Several 13 mm-diameter wires were machined from the hot-rolled plate along rolling direction. In addition, the initial wires were reduced at an Rh of 10%, 20%, and 40% using a single pass-type rolling simulator with 400 mm-diameter flat rolls a No lubricant was applied to the wire prior to the wire flat rolling process. The ro speed was set to 5 rpm to ignore the temperature rise or strain rate effects during forming process.

Flow Stress Model
TWIP steel was used as the model material to simulate the wire flat rolling pro The workpiece was assumed to be a rigid-plastic material because the plastic d mation occurred primarily during rolling, and the elastic deformation was relati small. Hence, the elastic deformation part is negligible. In addition, the workpiece assumed to be an isotropic material with strain hardening effect. The strain rate e was not considered owing to the cold forming process and the slow rolling velocity rpm. In such a case, Hollomon's law in Equation (5) can describe the constitutive be ior of a workpiece. From the curve fitting of the stress-strain curve in the TWIP (Figure 2), K and n values were determined to be 1980 MPa and 0.54, respectively.

Convergence Analysis of FE Mesh
To determine the appropriate number of elements, the deformation behavior o wire was evaluated with the number of elements at the roll diameter of 400 mm. number of elements in the cross section of the wire was changed from 100 to 500, and  The instantaneous n increased gradually with the true strain as shown in Figure 2b. Several 13 mm-diameter wires were machined from the hot-rolled plate along the rolling direction. In addition, the initial wires were reduced at an R h of 10%, 20%, 30%, and 40% using a single pass-type rolling simulator with 400 mm-diameter flat rolls at RT. No lubricant was applied to the wire prior to the wire flat rolling process. The rolling speed was set to 5 rpm to ignore the temperature rise or strain rate effects during the forming process.

Flow Stress Model
TWIP steel was used as the model material to simulate the wire flat rolling process. The workpiece was assumed to be a rigid-plastic material because the plastic deformation occurred primarily during rolling, and the elastic deformation was relatively small. Hence, the elastic deformation part is negligible. In addition, the workpiece was assumed to be an isotropic material with strain hardening effect. The strain rate effect was not considered owing to the cold forming process and the slow rolling velocity of 5 rpm. In such a case, Hollomon's law in Equation (5) can describe the constitutive behavior of a workpiece. From the curve fitting of the stress-strain curve in the TWIP steel (Figure 2), K and n values were determined to be 1980 MPa and 0.54, respectively.

Convergence Analysis of FE Mesh
To determine the appropriate number of elements, the deformation behavior of the wire was evaluated with the number of elements at the roll diameter of 400 mm. The number of elements in the cross section of the wire was changed from 100 to 500, and Appl. Sci. 2021, 11, 8381 5 of 15 that in the longitudinal direction of the wire was fixed at 60. The shape change and strain distribution were used as the parameters for the convergence analysis of the FE model. Figure 3a shows the shape of the deformed flat-rolled wire and contour of the von Mises equivalent strain (effective strain) with the number of elements. Figure 3b compares the contact width (b) and lateral spreading (W), and Figure 3c shows the maximum effective strain (ε max ) of the flat-rolled wire against the number of elements. The values of b, W, and ε max increased gradually with the number of elements. As expected, the difference in the variables between the two closest decreased with an increase in the number of elements. From an engineering perspective, the number of elements was selected as 300 because it adequately reflected the deformation behavior of the wire. Therefore, approximately 18,000 only hexahedral elements with eight nodes were employed in this study to obtain appropriate results regarding the deformation behavior of the workpiece during wire flat rolling.
l. Sci. 2021, 11, 8381 5 of in the longitudinal direction of the wire was fixed at 60. The shape change and str distribution were used as the parameters for the convergence analysis of the FE mod Figure 3a shows the shape of the deformed flat-rolled wire and contour of the von Mi equivalent strain (effective strain) with the number of elements. Figure 3b compares contact width (b) and lateral spreading (W), and Figure 3c shows the maximum effect strain (εmax) of the flat-rolled wire against the number of elements. The values of b, W, a εmax increased gradually with the number of elements. As expected, the difference in variables between the two closest decreased with an increase in the number of elemen From an engineering perspective, the number of elements was selected as 300 becaus adequately reflected the deformation behavior of the wire. Therefore, approximat 18,000 only hexahedral elements with eight nodes were employed in this study to obt appropriate results regarding the deformation behavior of the workpiece during wire f rolling.

Model Validation
During the general forming process, the FE model is typically validated by co paring the external imposed force for the deformation. In this study, since shape cont is a large issue in the wire flat rolling industry, the shape changes between the expe mental and simulated results were compared to validate the results of the FE analy Figure 4a presents the b and W of the simulated results with Rh, that is, 10%, 20%, 30 and 40%, at a roll diameter of 400 mm. Both b and W increased with Rh. Figure 4b co pares W and b from the FE analysis with the experimental results. The values from the analysis were slightly lower than those of the experiment. The authors believe that deviation in the numerical and experimental values was primarily associated with friction coefficient or model selected for the FE analysis in this study. For example, Jia et al. [24] revealed that the friction conditions along both the rolling and width directio of a workpiece significantly affected the strip shape and profile during the strip flat ro ing process. In this study, a constant friction coefficient of 0.15 was used from the p vious study [22] and the experiences of the authors. Based on the comparison of shape change of the flattened wire with Rh, a higher friction coefficient was necessary

Model Validation
During the general forming process, the FE model is typically validated by comparing the external imposed force for the deformation. In this study, since shape control is a large issue in the wire flat rolling industry, the shape changes between the experimental and simulated results were compared to validate the results of the FE analysis. Figure 4a presents the b and W of the simulated results with R h , that is, 10%, 20%, 30%, and 40%, at a roll diameter of 400 mm. Both b and W increased with R h . Figure 4b compares W and b from the FE analysis with the experimental results. The values from the FE analysis were slightly lower than those of the experiment. The authors believe that the deviation in the numerical and experimental values was primarily associated with the friction coefficient or model selected for the FE analysis in this study. For example, Jiang et al. [24] revealed that the friction conditions along both the rolling and width directions of a workpiece significantly affected the strip shape and profile during the strip flat rolling process. In this study, a constant friction coefficient of 0.15 was used from the previous study [22] and the experiences of the authors. Based on the comparison of the shape change of the flattened wire with R h , a higher friction coefficient was necessary to more accurately predict the shape change of the flat-rolled wire because it is known that the lateral spreading of the flat-rolled wire increases with the friction coefficient [25]. In addition, roll flattening also affected the wire shape because the elastic deformation of rolls becomes larger and more important as the strength of the workpiece increases [26]. That is, the effective roll diameter increases during the cold rolling process owing to the roll flattening. In this FE analysis, the roll was considered to be a rigid body, which can lead to a decrease in the b and W values of the flat-rolled wire during FE analysis.
addition, roll flattening also affected the wire shape because the elastic deformatio rolls becomes larger and more important as the strength of the workpiece increases That is, the effective roll diameter increases during the cold rolling process owing to roll flattening. In this FE analysis, the roll was considered to be a rigid body, which lead to a decrease in the b and W values of the flat-rolled wire during FE analysis.
Overall, although minor errors existed between the results of the experiment the FE analysis, it can be concluded that the FE model used in this study was adeq for analyzing the flat-rolled wire during the process because the errors of the W a values between the experiments and simulations were relatively small, namely, w 5%.  Figure 5a shows a comparison of the effective strains of the wire against the diameter. The trends in the strain distribution were similar regardless of the roll dia ter. The center region had the maximum effective strain, and the free surface region hibited the minimum strain. Additionally, the contact surface region exhibited a hi effective strain than the free surface region. MSBs were also observed in all the wires However, the density of strain distribution was different with the roll diameter. strain distribution of the flattened wire was more inhomogeneous with the roll diam thus the shape of the MSBs was strengthened with the roll diameter.

Strain Distribution with Roll Diameter
To compare the strain distributions of the flattened wire with the roll diam comprehensively, the effective strains were extracted based on the strain contour m Figure 5a,b compare the extracted effective strain along the horizontal and vertica rections of the wire against the roll diameter. The effective strains at the center re increased with the roll diameter, and those of the free and contact surface regions  Overall, although minor errors existed between the results of the experiment and the FE analysis, it can be concluded that the FE model used in this study was adequate for analyzing the flat-rolled wire during the process because the errors of the W and b values between the experiments and simulations were relatively small, namely, within 5%. Figure 5a shows a comparison of the effective strains of the wire against the roll diameter. The trends in the strain distribution were similar regardless of the roll diameter. The center region had the maximum effective strain, and the free surface region exhibited the minimum strain. Additionally, the contact surface region exhibited a higher effective strain than the free surface region. MSBs were also observed in all the wires [10]. However, the density of strain distribution was different with the roll diameter. The strain distribution of the flattened wire was more inhomogeneous with the roll diameter, thus the shape of the MSBs was strengthened with the roll diameter.

Strain Distribution with Roll Diameter
increased with the roll diameter, whereas the εmin decreased along both the horizon and vertical directions of the wire with the roll diameter. Figure 6b compares the str inhomogeneity factor along both the horizontal direction (SIFh) and vertical direct (SIFv) against the roll diameter. Regardless of the roll diameter, SIFh was higher than S which is consistent with previous results [1,4]. It is evident that the SIF increased with roll diameter.   To compare the strain distributions of the flattened wire with the roll diameter comprehensively, the effective strains were extracted based on the strain contour maps. Figure 5a,b compare the extracted effective strain along the horizontal and vertical directions of the wire against the roll diameter. The effective strains at the center region increased with the roll diameter, and those of the free and contact surface regions decreased with the roll diameter, resulting in a larger strain inhomogeneity of the wire with increasing roll diameter. For a better comparison of the inhomogeneity level, the strain inhomogeneity factor (SIF), defined as follows, was introduced: where ε ave and ε min indicate the average and minimum effective strains, respectively. Figure 6a shows the ε max , ε min , and ε ave of the wire against the roll diameter. Regardless of the roll diameter, all the wires had a similar ε ave in both directions. The ε max of the wires increased with the roll diameter, whereas the ε min decreased along both the horizontal and vertical directions of the wire with the roll diameter. Figure 6b compares the strain inhomogeneity factor along both the horizontal direction (SIF h ) and vertical direction (SIF v ) against the roll diameter. Regardless of the roll diameter, SIF h was higher than SIF v , which is consistent with previous results [1,4]. It is evident that the SIF increased with the roll diameter.    Figure 7a shows a comparison of the wire shape during wire flat rolling, and Figure 7b shows the variations in b and W with respect to the roll diameter. The b and W of the flattened wires increased as the roll diameter increased. Lambiase and Ilio [27] reported that W increases with the roll diameter during wire drawing with a flat roll, which is consistent with the present result. As the spreading of the wire increased with the roll diameter, R A decreased with the roll diameter, as shown in Figure 7c. Meanwhile, it is noteworthy that R A was approximately half the value of R h owing to the significant spreading effect of the workpiece during wire flat rolling.  Figure 7a shows a comparison of the wire shape during wire flat rolling, and Figure  7b shows the variations in b and W with respect to the roll diameter. The b and W of the flattened wires increased as the roll diameter increased. Lambiase and Ilio [27] reported that W increases with the roll diameter during wire drawing with a flat roll, which is consistent with the present result. As the spreading of the wire increased with the roll diameter, RA decreased with the roll diameter, as shown in Figure 7c. Meanwhile, it is noteworthy that RA was approximately half the value of Rh owing to the significant spreading effect of the workpiece during wire flat rolling.

Contact Area and Pressure Distribution with Roll Diameter
The contact area and its pressure distribution in the workpiece and roll interface have a great effect on the product shape, roll force, temperature drop, surface properties production costs, and roll wear during the rolling process. Therefore, the evaluation of the contact pressure in the roll bite is essential to gain a better understanding of the wire

Contact Area and Pressure Distribution with Roll Diameter
The contact area and its pressure distribution in the workpiece and roll interface have a great effect on the product shape, roll force, temperature drop, surface properties, production costs, and roll wear during the rolling process. Therefore, the evaluation of the contact pressure in the roll bite is essential to gain a better understanding of the wire flat rolling process. It is known that the distribution of the contact pressure during the rolling process depends on R h , friction at the workpiece and roll interface, and material properties [5,[27][28][29][30][31]. Figure 8 shows the distribution of the normal contact pressure on the wire surface with the roll diameter. The contour of the normal contact pressure on the wire exhibited similar patterns regardless of the roll diameter. The contact pressure showed higher values at the entry, edge, and exit zones of the contact area. This shape of the contact pressure distribution was reported in earlier studies [32,33]. Summarizing the literature above, the higher contact pressures in the entry and edge zones of the contact area are explained by the restrictions caused by the more rigid surrounding material that deforms elastically. In addition, the higher contact pressure on the exit zone of the contact area is attributable to the strain hardening effect of the metal during the process. Figure 9 shows a comparison of the effective stress distribution of the wire along the rolling direction. The similar effective stress distributions of the wire, regardless of the roll diameter, partially supported the present similar contact pressure distributions.
where Δh indicates the height reduction, i.e., h0 − h1. Figure 10b compares L with the roll diameter. Although α decreased with the roll diameter, L increased gradually with the roll diameter owing to the increased roll size. During the plate or strip flat rolling process, it was reported that L increases with the roll diameter and Rh as follows [18]: Figure 10b also shows a comparison between the geometrically derived L and the L value obtained using Equation (9). The equation above predicted the L of the wire well during wire flat rolling. Obviously, L increased with increasing roll diameter during wire flat rolling.   Appl. Sci. 2021, 11, 8381 the roll diameter. The value of α was derived based on the geometrical cond wire flat rolling process (Figure 1) as follows: where Δh indicates the height reduction, i.e., h0 − h1. Figure 10b compares L w diameter. Although α decreased with the roll diameter, L increased gradual roll diameter owing to the increased roll size. During the plate or strip flat r cess, it was reported that L increases with the roll diameter and Rh as follows [

= •
Figure 10b also shows a comparison between the geometrically derived L value obtained using Equation (9). The equation above predicted the L of the during wire flat rolling. Obviously, L increased with increasing roll diameter d flat rolling.   Meanwhile, the contact area increased with the roll diameter owing to the increased contact length (L) of the workpiece and roll. Figure 10a shows the contact angle (α) with the roll diameter. The value of α was derived based on the geometrical condition of the wire flat rolling process (Figure 1) as follows: where ∆h indicates the height reduction, i.e., h 0 − h 1 . Figure 10b compares L with the roll diameter. Although α decreased with the roll diameter, L increased gradually with the roll diameter owing to the increased roll size. During the plate or strip flat rolling process, it was reported that L increases with the roll diameter and R h as follows [18]:

Damage Value with Roll Diameter
The damage value of the wire was analyzed during the process to evaluate formability with the roll diameter. In this study, the normalized Cockcroft and Lath fracture criterion [34,35] was applied as follows: where σ* is the maximum principal tensile stress and is the von Mises equiva stress. Figure 11a compares the contour of the damage value in the wire with the diameter. The distribution of the damage value varied with the roll diameter. The surface region tended to have the peak damage value during wire flat rolling, whic consistent with previous results using hardening material by Cao et al. [36] and Mass al. [12]. However, the center region exhibited the maximum damage value as the diameter increased, as shown in Figure 11b Figure 10b also shows a comparison between the geometrically derived L and the L value obtained using Equation (9). The equation above predicted the L of the wire well during wire flat rolling. Obviously, L increased with increasing roll diameter during wire flat rolling.

Damage Value with Roll Diameter
The damage value of the wire was analyzed during the process to evaluate the formability with the roll diameter. In this study, the normalized Cockcroft and Latham fracture criterion [34,35] was applied as follows: where σ* is the maximum principal tensile stress and σ is the von Mises equivalent stress. Figure 11a compares the contour of the damage value in the wire with the roll diameter. The distribution of the damage value varied with the roll diameter. The free surface region tended to have the peak damage value during wire flat rolling, which is consistent with previous results using hardening material by Cao et al.
[36] and Masse et al. [12]. However, the center region exhibited the maximum damage value as the roll diameter increased, as shown in Figure 11b,c.
stress. Figure 11a compares the contour of the damage value in the wire with the roll diameter. The distribution of the damage value varied with the roll diameter. The free surface region tended to have the peak damage value during wire flat rolling, which is consistent with previous results using hardening material by Cao et al.
[36] and Masse et al. [12]. However, the center region exhibited the maximum damage value as the roll diameter increased, as shown in Figure 11b,c.

Discussion
The different behaviors of shape change, strain distribution, and damage value of the wire with the roll diameter during wire flat rolling must be evaluated. First, each strain imposed on the flat-rolled wire was compared with the direction. During the rolling process, the compressive stress imposed on the workpiece by the roll mainly deformed the workpiece in three directions (Figure 1), that is, the height, longitudinal, and transverse directions of the workpiece, which are typically known as reduction, elongation, and spreading, respectively. Figure 12 compares the strain contours in the three directions of the wire with the roll diameter. The reduction exhibited similar contours regardless of the roll diameter, and the spreading tended to concentrate on the center region with increasing roll diameter. Interestingly, the elongation decreased with increasing roll diameter. Figure 13 shows a comparison of the average elongation with the roll diameter. The decreased elongation with the roll diameter increased b and W, resulting in a decrease in R A with the roll diameter (Figure 7). The decreased elongation with the roll diameter also decreased the ε min of the wire at the free and contact surface regions, leading to an increase in SIF h and SIF v with the roll diameter ( Figure 6). Accordingly, both the shape change and strain distribution of the specimen during wire flat rolling were associated significantly with the elongation of the wire during the process.
The decreased elongation of the wire with the roll diameter was highly dependent on L. The increased L with the roll diameter ( Figure 10) rendered the frictional effect more prominent owing to the higher contact surface. That is, the increased contact area between the wire and roll resulting from the increased L increased the frictional effect along the longitudinal direction of the wire (F µlength ) compared with the transverse direction of the wire (F µwidth ), as shown in Figure 14. The increased F µlength decreased the elongation, leading to an increase in the spreading, including b and W owing to the conservation of volume during the plastic deformation. Esteban et al. [37] reported that lateral spreading increases as the friction coefficient increases during the plate flat rolling process. stronger as the roll diameter increased, this is attributable to the high L durin rolling. Kazeminezhad and Taheri [1] revealed that the strain inhomoge flat-rolled wire increases as the friction coefficient at the interface between th roll increases. As a result, the underlying cause of the strain distribution change of the flattened wire was highly related to L during deformation.  The damage value analysis showed that the wire fabricated using the small roll was easy to fracture in the free surface region; whereas, the wire manufactured by the large roll tended to be fractured in the center region ( Figure 11). For the wires with a small roll, the high elongation (σ*) and small in the free surface region were related to the fracture in this region. By contrast, the high spreading (σ*) in the center region of the wire fabricated using large roll ( Figure 12) yielded a high damage value in this region. Accordingly, the wire can be fractured with the different manner depending on the roll diameter during wire flat rolling. For instance, cracks occurred in the free surface region along the rolling direction in the 150-wire; whereas, cracks were generated in the center region along the spreading direction in the 1600-wire. It is noteworthy that the optimum roll diameter was in existence during wire flat rolling from the perspective of the damage value. In this study, the 800-wire exhibited the minimum damage value, as shown in Figure 11c. In addition, the damage value can vary with materials because σ* and in Equation (10) varied with n value of the material. In other words, the optimum roll size was different with materials during wire flat rolling.

Conclusions
The influence of the roll diameter on the strain distribution, shape change, contact pressure, and damage value of the workpiece was comprehensively investigated during wire flat rolling. The conclusions obtained are as follows: 1 The strain inhomogeneity of the flat-rolled wire increased with the roll diameter; therefore, the MSBs were strengthened with the roll diameter during wire flat rolling. 2 The b and W of the wire increased as the roll diameter increased, leading to a decrease in RA with the roll diameter. The damage value analysis showed that the wire fabricated using the small roll wa easy to fracture in the free surface region; whereas, the wire manufactured by the larg roll tended to be fractured in the center region ( Figure 11). For the wires with a small rol the high elongation (σ*) and small in the free surface region were related to the fra ture in this region. By contrast, the high spreading (σ*) in the center region of the wir fabricated using large roll ( Figure 12) yielded a high damage value in this region. A cordingly, the wire can be fractured with the different manner depending on the ro diameter during wire flat rolling. For instance, cracks occurred in the free surface regio along the rolling direction in the 150-wire; whereas, cracks were generated in the cente region along the spreading direction in the 1600-wire. It is noteworthy that the optimum roll diameter was in existence during wire flat rolling from the perspective of the damag value. In this study, the 800-wire exhibited the minimum damage value, as shown i Figure 11c. In addition, the damage value can vary with materials because σ* and i Equation (10) varied with n value of the material. In other words, the optimum roll siz was different with materials during wire flat rolling.

Conclusions
The influence of the roll diameter on the strain distribution, shape change, conta pressure, and damage value of the workpiece was comprehensively investigated durin wire flat rolling. The conclusions obtained are as follows: 1 The strain inhomogeneity of the flat-rolled wire increased with the roll diamete therefore, the MSBs were strengthened with the roll diameter during wire flat rol ing. 2 The b and W of the wire increased as the roll diameter increased, leading to a de crease in RA with the roll diameter. Meanwhile, it is known that a higher frictional effect concentrates the stress in the center region during the compression-type forming process. The rolling process mainly deformed the material under the compression stress state; therefore, the higher frictional effect with increasing roll diameter due to the higher L led to an increase in the ε max at the center region of the wire. In other words, the strain concentration in the center region was stronger as the roll diameter increased, this is attributable to the high L during wire flat rolling. Kazeminezhad and Taheri [1] revealed that the strain inhomogeneity of a flat-rolled wire increases as the friction coefficient at the interface between the wire and roll increases. As a result, the underlying cause of the strain distribution and shape change of the flattened wire was highly related to L during deformation.
The damage value analysis showed that the wire fabricated using the small roll was easy to fracture in the free surface region; whereas, the wire manufactured by the large roll tended to be fractured in the center region ( Figure 11). For the wires with a small roll, the high elongation (σ*) and small σ in the free surface region were related to the fracture in this region. By contrast, the high spreading (σ*) in the center region of the wire fabricated using large roll ( Figure 12) yielded a high damage value in this region. Accordingly, the wire can be fractured with the different manner depending on the roll diameter during wire flat rolling. For instance, cracks occurred in the free surface region along the rolling direction in the 150-wire; whereas, cracks were generated in the center region along the spreading direction in the 1600-wire. It is noteworthy that the optimum roll diameter was in existence during wire flat rolling from the perspective of the damage value. In this study, the 800-wire exhibited the minimum damage value, as shown in Figure 11c. In addition, the damage value can vary with materials because σ* and σ in Equation (10) varied with n value of the material. In other words, the optimum roll size was different with materials during wire flat rolling.

Conclusions
The influence of the roll diameter on the strain distribution, shape change, contact pressure, and damage value of the workpiece was comprehensively investigated during wire flat rolling. The conclusions obtained are as follows: 1.
The strain inhomogeneity of the flat-rolled wire increased with the roll diameter; therefore, the MSBs were strengthened with the roll diameter during wire flat rolling.

2.
The b and W of the wire increased as the roll diameter increased, leading to a decrease in R A with the roll diameter.

3.
The contour of the normal contact pressure on the wire surface exhibited similar patterns regardless of the roll diameter. The contact pressure of the wire was higher at the entry, edge, and exit zones in the contact area. 4.
The distribution of the damage value varied with the roll diameter. The free surface region tended to have the peak damage value during wire flat rolling; however, the center region exhibited the maximum damage value as the roll diameter increased. The optimum roll diameter was in existence during wire flat rolling from the perspective of the damage value.

5.
The underlying cause of the different strain inhomogeneities, shape changes, and damage values of the flattened wire was the different contact lengths originating from the different roll diameters during wire flat rolling.