High-Order Groove-Shape Curve Roll Design for Aluminum Alloy 7075 Wire Rolling

Abstract

A Bézier curve groove-shape roll design method was proposed and compared to round-oval-round designs for the wire rolling process. CAE simulations were adopted to predict the rolling forces and torques required for the rolling process. The rolling torque required for the case of 28% vertical compression rate is 73% higher than the case of 12% compression rate. The curve fittings of the rolling torque and the compression force with respect to the compression rate were obtained with very high R-square values (0.99 and 0.98) in the first rolling pass. The rolling force required for the Bézier curve groove-shape design is no different compared to the oval design with the same compression rate, but the rolling torque requirement for the next rolling station is 6% less than the oval groove design. Furthermore, the equivalent strain distribution on the cross section of the product was uniform, and no fin flash defects occurred. The proposed Bézier curve groove-shape is a better design from the viewpoint of uniform product mechanical property requirement, and a larger compression rate per pass could be achieved.

1. Introduction

The wire rolling process uses a multi-station of roll pairs to reduce the wire diameter to the product dimensions. The rolling contact of the roll to the wire gives smaller surface traction than the conventional wire drawing process. However, the roundness and precise dimensions of the wire product should be obtained via the final wire drawing process.

The wire drawing process uses a drawing die to reduce the wire diameter according to a specific area reduction ratio in order to prevent defects, such as wire break and surface cracking. The cross section profile of the wire drawing dies are usually round in shape with different inlet and outlet diameters. The wire rolling process uses roll pairs with grooves of round or oval shapes. The cross section area of the roll groove from station to station was reduced according to the area reduction design. The larger area reduction was able to reduce the station numbers required for the wire rolling process.

Kim and Im [1] used five groove shapes (round, ellipse, square, star, and trapezoid) to establish an expert system for the roll pass design. The rolling force was predicted via empirical equations. The high rolling force could cause rolling defects, such as laps and flash fins. Behzadipour et al. [2] proposed a power consumption estimation method for the flat- and the shape-rolling processes. The power consumptions obtained by empirical equations were used to build an ANN model. The power requirement for the different geometry and thicknesses of the rolling process could be estimated and then used to design the rolling machine. Bontcheva and Petzov [3] used the FEM method to simulate a 12-pass square and round wire rolling process. The roll grooves of the breakdown, the shaping, and the finishing stations were designed with different groove shapes. The distributions of the rolling temperature, the yield stress, and the longitudinal strains were calculated and compared with the experimental results. The FEM simulation results were used to evaluate the feasibility of a groove-shape roll design.

Sakhaei et al. [4] used a constant area reduction ratio to determine the roll pass number and design the cross section of the roll groove to make a u-shaped product using a square billet. The minimum pass number of the rolling process was obtained by increasing the material flow from the billet center to the edge of the ribs. The FEM method was used to evaluate the tendency of the material flow into the rib and the web regions. The simulation results confirmed that the proposed design method was able to reduce the pass numbers. Lambiase [5] introduced an expert system for the selection of the rolling pass number. FEM simulation was used to support the expert system for the pass sequence selection of the square and the round groove shapes. The pass number could be reduced by using the simulated maximum rolling force to efficiently utilize the maximum power capacity of the rolling machine. Fu and Yu [6] established a pass design method to obtain no-recrystallization temperature rolling; the temperature of the recrystallization was obtained via cold compression-and-tension experiments. A fine grain size of the roll-formed material could be obtained via this method.

Overhagen et al. [7] modified the two-roll machine configuration to design a three-roll forming machine. A process prediction model was proposed to set up the rolling speed, calculate the rolling force and the rolling torque, and predict the material flow. The rolling force could be reduced with a proper setup of the rolling speed. Hwang [8] used FEM simulation to predict the hot rolling processes of the round and the plate billets. The temperature variation and the rolling torque were obtained. The high temperature variation in the radial direction and the longitudinal direction (the rolling direction) could increase the surface defects for the round billet rolling. Ksenov et al. [9] proposed the round-oval-round and the round-flat-round methods for the bar rolling process design. FEM simulations were carried out to adjust the roll gaps in order to obtain more uniform rolling force for each stand. Proper gap adjustments were able to achieve the requirement of the bar geometry and the even-forming load for each stand.

Li et al. [10] used the round-oval-round design to study the 16MnCrS5 wire hot rolling process. The rolling temperature was 1100 degree Celsius. Different major–minor axis ratios of the elliptical orifice were evaluated using Oyane damage criteria. The high-friction coefficient conditions required high rolling force, while the low-friction coefficient conditions were not able to supply enough dragging force for the rolling process. Altan et al. [11] suggested the ranges of the constant shear friction factor for different forming processes. The constant shear friction factor suggested for the hot steel rolling process and the aluminum alloy hot extrusion are 0.6 to 1.0. Bayoumi [12] studied the round-oval-round rolling process and found that a small change in the roll gap has no significant effect on the rolling force and the rolling torque, while an increase of the friction could increase the rolling force and the rolling torque significantly. Zhao et al. [13] used the finite element method to study the heat transfer effects on the plate hot rolling of 7075 aluminum alloy. The distributions of the temperature and the effective stress and the effective strain distribution of the plate were investigated. The results showed that a lower heat transfer coefficient can help to reduce the heat loss, reduce the rolling stress, and improve the rolling deformation.

The groove-shape design, the rolling process simulation, and the defects of the wire rolling process are still significant issues of the wire rolling industry. In this paper, a new groove-shape design method was proposed to reduce the rolling force and improve the material flow of the wire rolling process. FEM simulations were carried out to predict the defects of the rolling process and evaluate the proposed groove-shape design method. This study used the Oyane damage criteria [14], regarding the various strain paths and stress paths, along with the FEM simulations to evaluate the possibility of defects during the wire rolling process. This made it possible to deal with the complex combinations of the tension and the compression strains for the multi-pass wire rolling process.

2. Design of the Wire Rolling Process and the Groove Shape of the Roll

2.1. Wire Rolling Process Design

The multi-pass rolling process is applied to make sheets, rods, bars, and plates by using different roller designs, with or without grooves on the roller surface. Coils of wire or sheets are fed into the rolling machine and pass through the forming stations to change the dimensions and the cross-section geometry of the wire or the sheet. The wire rolling process uses a series of two-roll or three-roll stands to reduce the diameter of the rods or the wires until the required diameter is achieved. The rolling surface contact of the wire rolling process avoids the high surface friction that occurs in the conventional wire drawing process. The main process parameters of the wire rolling process include the area reduction ratio, the rolling temperature, the rolling speed, and the lubrication of the rolling stands. The groove geometry on the roller surface is designed to obtain the area reduction ratio and control the cross section of the billet deformation. The round-oval-round (R-O-R) and the round-Bézier-round (R-B-R) wire rolling processes were compared. The original wire was 20 mm in diameter and the product was 16 mm in diameter. The area reduction ratio for each stand was 20%. The round wire was fed into the first rolling stand with oval or Bézier curve groove-shapes and went through the second rolling stand with a round groove-shape. The calculation of the area reduction ratio is given in Equation (1):

AR_n = (A_n − A_n−1)/(A_n−1)(1)

where $A_{\mathrm{n}}$ and $A_{\mathrm{n}-1}$ are the cross-section area of the nth and the (n − 1)th stand, respectively, and $A{R}_{\mathrm{n}}$ is the area reduction ratio of the nth stand. The constant area reduction method was adopted for this two-step wire rolling process study.

2.2. Roll Groove-Shape Design

In this paper, a two-pass hot wire rolling process of aluminum alloy 7075 was designed to make a 16 mm round wire. The selected area reduction ratio was 20% for each pass. The groove cross section of the roller was designed using elliptic and Bézier curve profiles. The aspect ratios of the minor axis (in the vertical direction) length to the original wire diameter for the proposed ellipse groove-shape roll were set to 1.1, 1.0, and 0.9 times the product diameter, which were 17.6 mm, 16.0 mm, and 14.4 mm, respectively. The compression rate of the wire rolling process was calculated using the vertical compression increment divided by the wire diameter of 20 mm; the compression rates for cases A, B, C, and D were 28%, 20%, 12%, and 28%, respectively. A large compression rate means a higher compression increment and more severe deformation when wire passes through the roll groove. The Bézier curve groove-shape design (case D) used a 28% compress rate (same as case A) to obtain the large diameter decrease, which is helpful in reducing the roll pass number required. The dimensions of the roller groove-shape design of the elliptical curve (case A, B, C) and the Bézier curve (case D) are shown in Figure 1. The major (horizontal) axes of the elliptical groove profiles were 18.2 mm, 20 mm, and 22.2 mm, calculated by the constraint of the 20% area reduction requirement. A schematic diagram of the control points of the proposed Bézier curve groove-shape design is shown in Figure 2. The y coordinate (vertical direction) of the control points P₀ and P₁ of the half model of the Bézier curve groove is 7.2. The x coordinates (horizontal direction) of the control points P₂ and P₃ are set to one-third and two-thirds of the half groove width, respectively.

2.3. Bézier Curved Groove Design

The Bézier curve groove-shape was designed using four control points to define the half groove; the parametric form of the Bézier curve is given in Equation (2):

(2)$P_r\left(v\right)={\left(1-v\right)}^3{P}_{0r}+3v{\left(1-v\right)}^2{P}_{1r}+3v^2\left(1-v\right)P_{2r}+v^3{P}_{3r}$

where $v$ is the normalized parameter of the Bézier curve, r is the index of the x and the y coordinates, and $p^r$ defines the coordinates (x, y) of the Bézier curve continuously. The Bézier curve groove-shape was designed via the following steps:

Step 1: Define the groove height, the y coordinates of the control points P₀, P₁, and P₂.

The groove height (y coordinates of the control points P₀, P₁) is calculated according to the compression rate. The y coordinates of control point P₁ are equal to the value of control point P₀ in order to guarantee a zero slope at the top of the groove, which is the symmetric boundary condition for the left- and the right-hand side Bézier curve segment. To obtain a smooth groove shape, the y coordinate of control point P₂ is equal to one-third of control point P₁.

Step 2: Calculate the width of the groove (the x coordinate of control point P₃).

The x coordinate of control point P₃ is the groove width, calculated according to the area reduction ratio design. In order to guarantee the vertical slope to control the vertical material flow during the rolling process, the x coordinate of control point P₂ is equal to the x coordinate of control point P₃. The x coordinate of control point P₁ is set to one-third of the x coordinate of P₂ to smoothly control the lateral material flow.

Step 3: Calculate the area of the groove cross section bounded by the Bézier curve.

The groove-bounded area, Ag, of the Bézier curve was integrated from the control points P₀ to P₃ and given by Equation (3). The total cross section area of the Bézier groove-shape was four times the calculated vale because only one-fourth of the Bézier curve is taken into consideration for the symmetric boundary conditions.

(3)$Ag={{\displaystyle \int }}_{p_3\left(\mathrm{x}\right)}^{p_0\left(\mathrm{x}\right)}[{\left(1-v\right)}^3{P}_0\left(\mathrm{y}\right)+3v{\left(1-v\right)}^2{P}_1\left(\mathrm{y}\right)+3v^2\left(1-v\right)P_2\left(\mathrm{y}\right)+v^3{P}_3\left(\mathrm{y}\right)]d\mathrm{x}$

2.4. Rotation Speed of Roll for Each Stand

The flow rate of the material passing through the roll groove at each stand should be kept equal in order to avoid unbalanced material flow in the rolling direction. Unbalanced material flow between the neighboring stands could cause defects of wire buckle (under compression force condition) or wire break (under tension force condition). In order to avoid unexpected compression or tension force, the volume rate of the material flow was calculated using the tangential rolling speed multiplied by the groove-bounded area given in Equation (3). The average tangential rolling speed, V_i, at the ith stand is given in Equation (4):

V_i = πD_rN/60(4)

where N is the rotational speed per minute (rpm) of the rollers, and $D_{\mathrm{r}}$ is the cylindrical diameter of the roller.

2.5. CAE Simulation of the Wire Rolling Process

The FEM software Simufact was adopted for the rolling process simulation. The workpiece, initial velocity, the symmetric boundary conditions, and the rolling stands are shown in Figure 3. The parameters for the CAE simulation are given in Table 1. The initial temperatures of the workpiece and the rolls were 400 °C and 20 °C, respectively. A hexahedral mesh with an element size of 0.5 mm (determined by convergence test) was adopted for the modeling of workpiece. Two symmetry boundary conditions were applied to the vertical and the horizontal planes of the workpiece. The workpiece was given an initial velocity in the rolling direction to lead into the first rolling stand and was pulled into the rollers by the friction force of the rotating rollers. The constant friction shear factor was set to 0.6 according to the suggestion of reference [11] to give enough friction force for the requirement of the natural rolling. The material of the wire is typical aluminum alloy 7075. The flow stress model of aluminum alloy 7075 is given in Figure 4 for different temperatures and strain rates. The Oyane damage model was applied to predict the defects of the workpiece during the wire rolling process.

3. Results and Discussion

3.1. Torque Requirement for Different Process and Groove-Shape Design

The characteristics of the wire rolling process can be observed via the history of rolling torque and the maximum principal stress distributions, as shown in Figure 5, which shows the simulation results of the first rolling stand using a Bézier curve groove-shape. The rolling torque was increased gradually at the start of the wire rolling, and a steady rolling state was achieved when the deformed materials left the forming zone. The rolling torque was gradually increased as the strain-hardened rear part of the wire entered the deformation zone. The largest torque requirement was observed when all of the strain-hardened rear parts of the wire entered the deformation zone. After that, the rolling torque decreased with the decreasing of the wire length under the deformation zone. The maximum principal stress distribution had revealed the entrance, the neutral, and the exit points of the deformation zone. The highest maximum principal stress distribution was developed from the wire surface to the wire center. The contact region at the entrance applied downward compression and forward dragging forces simultaneously; the maximum principal stress was developed from the workpiece surface to the center. The compression and dragging force gave the surface region higher material flow and a concave wire front end.

3.2. Equivalent Stress Distributions

The equivalent stress distributions on the cross sections cut at the roller center along the vertical and the horizontal planes are shown in Figure 6. The maximum equivalent stress levels of the largest compress rate (28%, case A and case D) were larger than those of case B and case C. The design of case B uses the same compression rate (20%) for stands 1 and 2; that is, the workpiece was compressed in the vertical direction and the horizontal material flow was constrained with the groove boundary. For case B, the maximum equivalent stress was the highest, and the deformation zone was the shorter in the rolling direction than in the other cases. The compression rate of case A and case D was equal, but the maximum equivalent stress of case D was higher, and the lateral material flow of case D was obviously constrained. In stand 2, the maximum stress of case C is higher than that of case D, and the flash fin defect was observed in case C but not in case D. For stand 2, the equivalent stress distribution of case D at the exit cross section was more uniform than the other three cases. These result comparisons revealed that uniform deformation could be obtained using the proposed Bézier curve groove-shape design.

3.3. The Equivalent Strain Distributions

The equivalent strain distributions on the cross sections cut through the roller center along the vertical and the horizontal planes are shown in Figure 7. The high-strain concentration zones occurred at the upper and the lower surfaces for pass 1 rolling and occurred at the left and the right surfaces for pass 2 rolling. These results were due to a combination of the horizontal roll arrangement in pass 1 and the vertical roll layout in pass 2. The compression deformation of the workpiece was mainly applied by the roll groove in a direction vertical to the roll shaft. For cases B and C, the high equivalent strain concentration was generated at the region of flash extrusions on the upper and the lower surfaces of the product at the exit of pass 2. For the Bézier curve groove-shape design (case D), the equivalent strain of the final product (exit of pass 2) was distributed more uniformly on the cross section of the product, except for the top and the bottom surface regions, and no flash occurred on the circumference of the product.

3.4. Defects of Wire Rolling Process

The damage distributions at the cross sections along the vertical direction and the rolling direction are shown in Figure 8. High-damage zones appeared on the outer surfaces of the wires, which revealed that the deformations were not uniform in the radial direction. The damage distribution of pass 2 for the Bézier curve groove-shape design (case D) was more uniform than the other cases; the maximum damage value of case D was 0.5, which is less than the others. The more uniform damage distribution means that more uniform deformation was obtained, and the formability of the roll-formed wire should be better for further roll-forming sequences, if necessary. A smaller damage value also means less chance of cracking defect occurrence.

3.5. Friction Stress Ratio Distribution

The friction stress ratios on the surface of the deformation zone for all cases are shown in Figure 9. The friction ratio is the result of the friction stress divided by the shear yield stress. The regions of high-friction stress ratio indicate surfaces where there is high dragging force. A larger area of high-friction stress ratio could reduce the maximum friction shear stress and prevent surface defects. A larger contact length in the rolling direction is preferred for better dragging force control of the roller. From the viewpoint of wire buckling or breaking prevention, the regions of high-friction stress ratio and the contact length in the rolling direction in pass 1 and pass 2 for the Bézier curve groove-shape design (case D) were more even, and therefore the above-mentioned defects are unlikely to occur. The regions of high-friction stress ratio and the contact length in the rolling direction of pass 1 and pass 2 for case A and case C were more uneven, and buckling or breaking defects are therefore very likely to occur.

3.6. The Section Profile Deviation of the Roll Formed Wire

The section profile deviations of the wire after the two-pass rolling are shown in Figure 10. Both of the section profiles of case B and case C had defects of flash spikes at the roll gaps; the convex profile deviations were 0.72 mm and 3.11 mm, respectively. These two cases used smaller vertical compression ratios in pass 1 than case A; the material flow along the vertical direction at stand 2 caused the flash spikes on the top and the bottom regions of the wire. For the designs of case A and case D, no flash spikes occurred, and the concave profile deviations were 0.28 mm and 0.20 mm, respectively.

3.7. The Rolling Torque and the Compression Force Prediction

The rolling torques of pass 1 and pass 2 are shown in Figure 11 for the four groove-shape designs with different compression rates for pass 1. The rolling torque increased with the increase of the compression rate. The higher the compression rate adopted, the higher the rolling torque required at pass 1. Case C (compression rate 12%) had the smallest rolling torque in pass 1, but the rolling torque required in pass 2 was the highest and was almost 1.5 times the other cases.

The compression forces (in the direction vertical to the roll shaft) of pass 1 and pass 2 are shown in Figure 12 for the four groove-shape designs with the different compression rates of pass 1. The rolling force increased with the increasing compression rates. The higher the compression rate adopted, the higher the rolling force required at pass 1. Case C (compression rate 12%) had the smallest rolling force in pass 1, but the rolling force required in pass 2 was the highest, and almost two times the other cases.

In pass 1, the increase of the rolling torques seems to be proportional to the increase of the compression rate. The linear curve fittings for the rolling torque and the compression force with respect to the compression rate are shown in Figure 13; the fitting results have a very high R-square value in the first rolling pass. The R-square values for the curve fitting of the rolling torque and the compression force were 0.99 and 0.98, respectively. The Bézier curve groove-shape design adopted the highest compression rate of 28% in order to reduce the rolling pass of the future application. In pass 1 and pass 2, the rolling force required using the Bézier curve groove-shape design makes no obvious difference compared to the oval design with the same compression rate, but the rolling torque requirement of pass 2 is 6% less than the oval groove design in the case of using the Bézier curve groove for pass 1 pre-forming.

3.8. The Wire Dragging Force (Z-Direction Force) in the Rolling Direction

A minimum dragging force is preferred for a successful wire rolling process from the viewpoints of energy saving and smooth deformation. The dragging forces required for the wire rolling of pass 1 and pass 2 are shown in Figure 14 for all of the groove-shape designs. For pass 1, the higher compression rate requires higher dragging force (case A > case B > case D > case C), except for case D. Case D (Bézier curve groove-shape) has the same compression rates as case A, but the dragging force required is smaller than case A and case B. The reason for this is that a smooth material flow and uniform deformation conditions were achieved for the Bézier curve groove-shape design. For pass 2, the dragging force is higher, with a higher second pass compression rate (case C > case B = case A > case D). Case D had the smallest dragging force required; this is proof of the smooth material flow and the uniform deformation (small damage value) achieved in the first pass using the Bézier curve groove-shape design.

4. Conclusions

• The maximum equivalent stress level of the largest compress rate (28%, case A and case D) was larger than that of case B (compress rate 20%) and case C (compress rate 12%).

• For the Bézier curve groove-shape design (case D), both the equivalent stress and the equivalent strain were distributed more uniformly on the vertical cross section of the product, and no flash occurred on the circumference of the product. This design could obtain a wire product with a more uniform mechanical property because the equivalent stress and the equivalent strain were distributed uniformly.

• In pass 1, the increasing of the rolling torques and the rolling force seemed to be proportional to the increase of the compression rate. The linear curve fittings for the rolling torque and the compression force with respect to the compression rate were obtained with a very high R-square value. These curve fitting equations could be used for the choice of the rolling machine and the design of the shaft of the rolls.

• The damage distribution of pass 2 for the Bézier curve groove-shape design (case D) was more uniform than the other cases. The more uniform damage distribution means a more uniform deformation was obtained, and the formability of the roll-formed wire should be better for further roll-forming sequences if necessary.

• The curve fittings of the rolling torque and the compression force requirement in the first pass were able to give empirical estimation of the torque capacity for the rolling machine design.

Author Contributions

Conceptualization, J.-J.S. and C.-J.H.; methodology, J.-J.S. and C.-J.H.; validation, C.-H.Y.; formal analysis, C.-Y.K.; investigation, and C.-J.H. and C.-H.Y.; resources, C.-J.H.; data curation, C.-Y.K.; writing—original draft preparation, C.-Y.K.; writing—review and editing, J.-J.S.; visualization, C.-Y.K.; supervision, C.-H.Y.; project administration, J.-J.S.; funding acquisition, J.-J.S. All authors have read and agreed to the published version of the manuscript.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Acknowledgments

The authors would like to acknowledge the financial support by the Taiwan Ministry of Science and Technology (MOST 110-2221-E-992-074).

Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Table

Figure 1. Roller groove-shape designs for round-oval-round (case A, B, C) and round-Bézier-round (case D).

Figure 2. Bézier curve groove-shape design and the control points layout.

Figure 3. FEM model of the wire billet and the boundary conditions.

Figure 4. Flow stress curves of aluminum alloy 7075 for different temperatures.

Figure 5. The history of rolling torque and the maximum principal stress distribution for the first stand using a Bézier curve groove-shape.

Figure 6. The equivalent stress distributions at the cross sections along the vertical direction and the rolling direction.

Figure 7. The equivalent strain distributions at the cross sections along the vertical direction and the rolling direction.

Figure 8. The damage value distributions at the cross sections along the vertical direction and the rolling direction.

Figure 9. The friction stress ratio and the contact pressure distributions at the cross sections along the vertical direction and the rolling direction.

Figure 10. The section profile deviations of the roll-formed wire.

Figure 11. Rolling torques of passes 1 and 2 for different compression rates (C. rate) at pass 1.

Figure 12. Compression loads of passes 1 and 2 for different compression rates (C. rate) in pass 1.

Figure 13. Linear curve fitting of rolling torque and compression load in pass 1.

Figure 14. The dragging (z-direction) forces of pass 1 and pass 2.

Table 1

CAE simulation parameters.

Process Conditions	Values
Initial wire diameter (mm)	20
Initial wire length (mm)	180
Maximum roll diameter (mm)	80
Number of passes	2
Stand 1 (pass 1) speed (rpm)	14.12
Stand 2 (pass 2) speed (rpm)	17.77
Shear friction factor	0.6
Element type	Hexahedral
Element size(mm)	0.5

References

1. Kim, S.H.; Im, Y.T. A knowledge-based expert system for roll pass and profile design for shape rolling of round and square bars. J. Mater. Proc. Tech.; 1999; 89, pp. 145-151. [DOI: https://dx.doi.org/10.1016/S0924-0136(99)00055-2]

2. Behzadipour, S.; Khajepour, A.; Lenard, J.G.; Biglou, J. A new shape change quantification method for estimation of power in shape rolling. J. Mater. Proc. Tech.; 2004; 148, pp. 353-361. [DOI: https://dx.doi.org/10.1016/j.jmatprotec.2004.02.047]

3. Bontcheva, N.; Petzov, G. Total simulation model of the thermo-mechanical process in shape rolling of steel rods. Comput. Mater. Sci.; 2005; 34, pp. 377-388. [DOI: https://dx.doi.org/10.1016/j.commatsci.2005.01.009]

4. Sakhaei, A.H.; Salimi, M.; Kadkhodaei, M. New multi-pass hot channel section rolling design by the finite element method. J. Mech. Eng. Sci.; 2013; 227, pp. 2742-2750. [DOI: https://dx.doi.org/10.1177/0954406213475402]

5. Lambiase, F. Optimization of shape rolling sequences by integrated artificial intelligent techniques. J. Adv. Manuf. Tech.; 2013; 68, pp. 443-452. [DOI: https://dx.doi.org/10.1007/s00170-013-4742-2]

6. Kuroda, K.; Kuboki, T.; Imamura, Y.; Hayashi, C. Design evaluation of multiroll mills for small-diameter wire rolling. J. Mech. Eng. Sci.; 2001; 215, pp. 77-86. [DOI: https://dx.doi.org/10.1243/0954406011520535]

7. Overhagen, C.; Mauk, P.J. A new rolling model for three-roll rolling mills. Key Eng. Mater.; 2014; 622, pp. 879-886. [DOI: https://dx.doi.org/10.4028/www.scientific.net/KEM.622-623.879]

8. Hwang, J.K. Thermal behavior of a rod during hot shape rolling and its comparison with a plate during flat rolling. Processes; 2020; 8, 327. [DOI: https://dx.doi.org/10.3390/pr8030327]

9. Aksenov, S.A.; Chumachenko, E.N.; Logashina, I.V.; Kubina, T. Numerical simulation in roll pass design for bar rolling. Metalurgija; 2015; 54, pp. 75-78.

10. Li, X.; Wang, S.; Lohmar, J.; Hirt, G. Design of caliber rolls incorporating load path dependent damage evolution. Proc. Manuf.; 1990; 47, pp. 643-648. [DOI: https://dx.doi.org/10.1016/j.promfg.2020.04.198]

11. Kobayashi, S.; Oh, S.I.; Altan, T.; Chaudhary, A. Metal forming and the finite-element method. J. Mater. Shap. Tech.; 1989; 8, 65. [DOI: https://dx.doi.org/10.1007/BF02834794]

12. Bayoumi, L.S. Flow and stresses in round–oval–round roll pass sequence. Int. J. Mech. Sci.; 1998; 40, pp. 1223-1234. [DOI: https://dx.doi.org/10.1016/S0020-7403(98)00005-8]

13. Zhao, X.; Chen, L.; He, K.; Wu, N.; Zeng, J. Effect of contactheat transfer on hot rolling of aluminum alloy. Proc. Manuf.; 2019; 37, pp. 91-96.

14. Oyane, M.; Sato, T.; Okimoto, K.; Shima, S. Criteria for ductile fracture and their applications. J. Mech. Work. Tech.; 1980; 4, pp. 65-81. [DOI: https://dx.doi.org/10.1016/0378-3804(80)90006-6]