**Process Parameters Decision to Optimization of Cold Rolling-Beating Forming Process through Experiment and Modelling**

### **Long Li, Yan Li \*, Mingshun Yang and Tong Tong**

School of Mechanical and Precision Instrument Engineering, Xi an University of Technology, Xi'an 710048, China; lilong678@stu.xaut.edu.cn (L.L.); yangmingshun@xaut.edu.cn (M.Y.); 2170221141@stu.xaut.edu.cn (T.T.) **\*** Correspondence: jyxy-ly@xaut.edu.cn; Tel.: +86-029-82312776

Received: 8 March 2019; Accepted: 31 March 2019; Published: 2 April 2019

**Abstract:** The cold roll-beating forming (CRBF) process is a particular cold plastic bulk forming technology for metals that is adequate for shaping the external teeth of important parts. The process parameters of the CRBF process were studied in this work to improve the process performance. Of the CRBF process characteristics, the forming forces, tooth profile angle, surface roughness, and forming efficiency were selected as the target indices to describe the process performance. Single tooth experimental tests of ASTM 1045 steel were conducted with different roll-beating modes, spindle rotation speeds, and feed speeds. Using analysis of variance (ANOVA) and regression analysis, the influence of the process parameters in each index was investigated, and regression models of each index were established. Then, the linear weighted sum method and compound entropy weight method were used to determine the process parameters for multi-objective optimization. The results show that the impact capacity and optimum value range of the process parameters vary in different indices, and that, to achieve the comprehensive optimum effect of a small forming force, high product quality, and high forming efficiency, the optimal process parameter combination is the up-beating mode, a spindle rotation speed of 801 r/min, and a feed speed of 960 mm/min.

**Keywords:** plasticity forming; cold roll-beating forming; process parameter; multi-objective optimization

### **1. Introduction**

As important mechanical products, gears, spline shafts, and other transmission parts are widely used in various mechanical products [1,2], the rapid development of major industries, such as automobile, aircraft, special engineering machinery, and wind and nuclear power equipment, has led to a huge demand for transmission parts with higher performance [3,4]. In the production of mechanical parts, objectives such as creating a lightweight design, establishing short process chains, and improving material and energy efficiency are difficult to meet using the current cutting methods to shape the external teeth of transmission parts [3,5,6]. Plastic forming, as a near-net shape forming technology, is a promising method to solve some of the problems in traditional cutting. As such, the new process and novel production equipment have received attention [7,8]. Among the varieties of plastic forming technologies, cold roll-beating forming (CRBF) is an incremental metal bulk forming technology for forming the external teeth of high-load transmission parts, and it has the advantages of environmental protection, flexibility, and lower cost. It has a wide range of application prospects in the automobile industry, the aerospace sector, and equipment manufacturing [9–11].

Some studies have been carried out on the CRBF process, and some research results have been published. On the basis of the kinematics of the CRBF process, Fengkui Cui et al. [12,13] analyzed and studied the forming process of involute spline cold roll-beating in which the workpiece was continuously indexed. In view of the forming error caused by the continuous indexing motion of workpiece, a design and modification method of the rolling wheel were put forward, and a technological scheme of roll wheel manufacturing was provided that improved the geometric accuracy of involute spline CRBF. Mingshun Yang et al. [14] established a simplified mathematical model for describing the residual height of the formed tooth bottom under the assumption of rigid plastic deformation, and they discussed the influence of process parameters on the residual height of the tooth bottom. These studies illustrate the basic kinematic characteristics of the CRBF process. On the basis of these studies, there have been some numerical simulation and experimental studies dedicated to the metal deformation characteristics and mechanism of the CRBF process. Fengkui Cui [15] and Xinqin Gao [16] established a simplified finite element model of the CRBF process and simulated the material behavior and changes in the principal stress, hydrostatic pressure, and principal strain of the deforming area. Then, the metal deformation characteristics of the CRBF process were given. On the basis of the deformation characteristics, a constitutive model of the material for the CRBF process was further studied [17–19]. The results of these studies provide guidance for simulating the CRBF process and explaining the forming mechanism. Through the simulation of a complete tooth groove CRBF process, Xiaoming Liang et al. [20] analyzed changes in the radial and tangential forming force throughout the whole forming process and discussed the influence of the roll-beating mode on forming forces. Fengkui Cui et al. [21] linked the plastic deformation of metal to the surface residual stress that occurs during the CRBF process by observing the grains changing in the spline tooth profile section, and they explained the residual stress generation in terms of the forming mechanism. Zhiqi Liu et al. [11] performed an experimental study on the CRBF mechanism from the microcosmic point of view. The depth of the microhardness layer and the model of grain evolution during the CRBF process were established. Fengkui Cui et al. [10,22,23] measured the residual stress, hardness, and roughness of a spline surface fabricated by CRBF, and a series of empirical expressions were obtained to describe the effects of the roll-beating speed and feed rate on these indices.

On the basis of clear characteristics and the forming mechanism of CRBF, there is a need for further quantitative discussion about the effect of process parameters on the CRBF process to promote the practical application of CRBF. In particular, to improve the process performance of CRBF, it is necessary to clarify the influence of process parameters on the forming force, forming quality, and forming efficiency. In view of this situation, ASTM 1045 steel was tested by a CRBF experiment on the outer teeth. The empirical formulas of the indices of the forming force, product quality, and forming efficiency under different process parameters were established according to the experimental results. The aim of this study is to determine the influence of different process parameters on CRBF and provide a method to realize the comprehensive optimization of the forming force, forming quality, and forming efficiency by judiciously selecting the appropriate process parameters.

### **2. Materials and Methods**

### *2.1. Description of CRB Process*

CRBF is a special intermittent free cold forging process. It is different from the quasi-static forming process of die forging, drawing, rolling, etc. During the CRBF process, the metal is subjected to intermittent impacts at a certain frequency, and the deformation produced by each impact is small. The deformation zone continuously migrates through the feed of the workpiece, and the amount of deformation gradually accumulates to accomplish the final forming purpose. For different types of parts, the process of forming the external tooth profile in CRBF can be described by the single tooth groove forming process. A schematic diagram is presented in Figure 1. The roller is assembled eccentrically on the spindle, and the geometric profile of the roller is designed according to the profile of the target tooth. The spindle rotation drives the roller to roll-beat the workpiece, and the roller can rotate around the roller shaft as it beats and rolls the workpiece. During the forming process, the workpiece is fed continuously to accumulate the plastic deformation caused by each roll-beating

of the roller. Finally, a tooth groove consistent with the profile of the target tooth is formed on the workpiece. When the feed direction of the workpiece is clear, we can change the rotation direction of the spindle to choose the mode of the CRBF process. In order to express this clearly in a way similar to the milling process, CRBF can be defined by a coordinate system in which the y-axis is the feed direction of the workpiece and the z-axis is perpendicular to the workpiece to be processed. As such, when the angular velocity direction of the spindle rotation is consistent with the x-axis, it is an up-beating mode; when it is inconsistent, it is a down-beating mode. Through the above description of the CRBF process, it can be understood that the main processing parameters are spindle rotation speed, feed speed, and roll-beating mode.

**Figure 1.** Schematic of the cold roll-beating (CRB) principle of an external tooth groove.

### *2.2. Material Characterization*

The material of the workpieces was ASTM 1045 steel for the CRBF experiment because it has good mechanical properties and manufacturability, and it is widely used to manufacture functional parts. The main chemical composition of the material is shown in Table 1.



The main deformation of materials during the forming process is compression deformation. After normalization, a compression test was carried out by a 100-kN mechanical testing machine (MTS Inc., Eden Prairie, MN, USA). The size of the sample was Φ6 mm × 10 mm, and the down-pressure velocity was 0.5 mm/min. Figure 2 illustrates the experimental result.

**Figure 2.** Stress-strain curve of ASTM 1045 steel.

For the CRBF experiment, the material was made into workpieces by wire-electrode cutting and milling. The length along the feed direction of the workpiece was 45 mm. The surface hardness and roughness of the workpieces were HV 180–220 and Ra 0.8–1.6, respectively.

### *2.3. Experimental Setup*

According to the principle of the external tooth groove CRBF, a CRBF device was designed and realized by a horizontal milling machine to carry out the forming experiment, as shown in Figure 3. In this CRBF device, the roller is eccentrically mounted on the spindle through a roller shaft and two bearings (INA NKXR20Z, INA Inc., Nuremberg, Germany), and the axial clearance is adjusted by two sets of gaskets. The radius of the trajectory of the roller edge is 74 mm, and the roller is made of 20CrMnTi. After quenching and tempering treatment, the surface of the roller is conditioned and Rockwell hardness number reaches 58–64 HRC. The dimensions and tooth detail of the roller are shown in Figure 4.

**Figure 3.** CRBF experimental equipment.

**Figure 4.** Dimensions and tooth detail of the roller (dimensions in mm).

The spindle rotation speed, feed speed, and roll-beating mode were chosen as the research factors. The spindle rotation speeds are 475 r/min, 950 r/min, and 1500 r/min. The feed speeds of the workpiece are 30 mm/min, 60 mm/min, 120 mm/min, 240 mm/min, 480 mm/min, and 960 mm/min. The different roll-beating modes of the CRBF process were realized by changing the rotation direction of the spindle. The experiments were carried out with all combinations of all process parameters at all levels. The experimental implementation scheme is shown in Table 3. The roll-beating depth was set at 2.5 mm, and the workpiece was coated with lubricating oil.

During the forming process, a piezoelectric three-direction force sensor PCB261A03 (PCB Inc., Buffalo, NY, USA) was used to obtain forming force data. After removing oil from the surface of the experimental tooth grooves with a metal cleaning agent, the three-dimensional macroscopic geometrical data of the surface geometry of the tooth grooves were measured and obtained by the super 3-D microscope system VHX-5000 (Keyence Inc., Osaka, Japan). The measuring accuracy of this instrument is 0.01 mm. The surface roughness of the tooth wall was measured by laser confocal microscopy using the DCM 3D (Leica Inc., Wetzlar, Germany).

### **3. Results**

For metal bulk forming technology, forming forces, forming quality, and forming efficiency are the important objectives that we are interested in optimizing.

In the coordinate system described in Figure 1, the forming force in the x-direction was low during the forming process because of the geometric symmetry of the tooth shape of the roller. The forming force in the z-direction *Fz* is the main forming force in the CRBF process, and the forming force in the y-direction *Fy* determines the main shaft torque load and the workpiece feed system load of the forming equipment. *Fz* and *Fy* directly affect the design of the forming equipment and the implementation of the forming process.

Figure 5 shows *Fz* and *Fy* measured in the up-beating and down-beating modes for a spindle rotation speed of 475 r/min and a feed speed of 240 mm/min. For the same forming equipment and workpiece, the total time of the CRBF process is mainly related to the feed speed. Figure 5 shows that the actual roll-beating lasts nearly 16.5 s when the feed speed is 240 mm/min and that the forming forces of CRBF form a pronounced sharp pulse. This is consistent with the characteristics of intermittent impact loading in the CRBF process. In either mode, the *Fz* peak of each roll-beating gradually increases in the early stage of the forming process; then, it reaches the maximum and remains stable, after which it decreases gradually until the end of the forming process. Therefore, *Fz*am—the average value of the *Fz* peak of each roll-beating in the stability region—is used to characterize *Fz*. The *Fy* peak of each roll-beating during the CRBF process also has increasing, stable, and decreasing regions. However, the *Fy* peak of each roll-beating in the stable region is not the maximum of the whole forming process, differing from the pattern for *Fz* and varying for different roll-beating modes: In the up-beating mode, the maximum *Fy* is obtained between the increasing and stable regions, and the maximum *Fy* in the down-beating mode occurs before the end of the stable region. For practical applications, the maximum

*Fy* of the whole forming process is worth more than the average value of the *Fy* peak of each roll-beating in the stable region. As such, to express the maximum *Fy* of the whole forming process, *Fy*max is used to characterize *Fy* in the CRBF process.

**Figure 5.** Forming forces of the CRBF forming process: (**a**) Up-beating; (**b**) down-beating.

For product quality attributes, this paper mainly considers geometric accuracy, surface roughness, and hardening. The geometry accuracy of the single tooth profile is usually described by profile error and lead error. The lead error is primarily determined by the feeding straightness of the feed system of the forming equipment and the axial positioning accuracy of the spindle rotation. The lead error is not sensitive to the process parameters analyzed in this paper. There is an obvious tooth profile error in the tooth groove formed by CRBF because, in the real forming process, the material has elastic recovery, which produces profile errors. These profile errors mainly appear as angle errors in the tooth profile, as shown in Figure 6. When the roller roll-beats the workpiece, the metal extends along the tooth profile. After the roller leaves the workpiece, the metal flowing along the tooth profile has elastic recovery, which ultimately makes the angle of the tooth groove wall smaller than the tooth profile angle of the roller. The above discrepancy in the tooth angle is defined as the angle error of the tooth profile and is represented by β. For a different roll-beating mode, roll-beating speed, and feed speed, the stress field and hardening of metals are different, and this variation affects the elastic recovery. As such, β is obviously affected by the process parameters. Therefore, β is used to characterize the geometry accuracy in this paper.

**Figure 6.** Schematic of profile error generation.

Transmission parts mainly transfer power through tooth wall meshing. Reducing the surface roughness of the tooth wall directly improves the service life and transfer efficiency of the parts and reduces the working vibration. Therefore, the surface roughness is an important standard of surface quality. To facilitate measurement, each groove formed in the experiment was divided into two parts by the middle of the tooth bottom using wire-electrode cutting. The length of the specimen was 10 mm along the tooth's lead direction, as shown in Figure 7. It can be seen that the desired surface finish of the tooth wall can be achieved by grinding.

**Figure 7.** Experimental forming of parts and specimens.

From the measurement of the surface roughness of the tooth wall, it was observed that the micro-morphology of the tooth wall formed by CRBF is an irregular fish-scale pattern, as shown in Figure 8. This phenomenon is caused by the multiple rolling of the surface of the workpiece and the rotation of the roller in the roll-beating process. Therefore, surface roughness Sa of the tooth wall is used to characterize the surface quality.

**Figure 8.** Surface morphology of tooth wall.

The metallographic structure of the metal after CRBF is shown in Figure 9. As a result of the strong plastic deformation of the metal during the forming process, the grain was refined and highly fibrous on the tooth wall and tooth bottom. Because of the impact loading of CRBF, the grain refinement and fibrosis are mainly concentrated in the surface layer of the metal. Comparing the metallographic structure of the tooth top, tooth wall, and tooth bottom, it can be found that the grain refinement degree of the tooth bottom near the tooth wall is the highest, and the grain refinement degree is the weakest at the tooth top.

**Figure 9.** Metallographic structure of different parts of formed tooth groove: (**a**) Tooth top; (**b**) tooth wall; (**c**) tooth bottom.

The hardness of each part was measured. The hardness at the position of the tooth top, tooth wall, and tooth bottom near the tooth wall is 256, 292, and 310 HV, respectively. The hardness of the internal metal, which does not change in structure, is 210 HV. This is consistent with the change in grain refinement. The hardness of the tooth wall resulting from the use of different process parameters was measured. The results show that the hardening rate is between 135% and 145%, and the hardening degree is not very sensitive to the change in process parameters. This is because the change in processing parameters has little effect on the metal deformation after final forming. Therefore, workpiece hardening is not considered as an index of forming quality in this study.

For the forming efficiency, the feed speed is directly used to characterize the forming efficiency. The research objectives and their corresponding indices are shown in Table 2.


**Table 2.** Research objectives and their corresponding indices.

The experimental results of the above indices and corresponding process parameters are listed in Table 3, in which the results for β and Sa are the average of the two sides of the tooth wall. In addition, in order to facilitate mathematical representation and analysis, the variable *m* is used to denote the roll-beating mode: *m* = 1 represents up-beating and *m* = −1 represents down-beating.


**Table 3.** CRBF experimental results.


**Table 3.** *Cont.*

### **4. Discussion**

### *4.1. Significance Analysis of Process Parameters to Objectives*

ANOVA was used to test the significance of each process parameter, and the results are shown in Table 4. The probability of *F*(0.05) > *F* is represented by the *P* value of the right-sided test. When the *F* value of the objective is greater than *F*(0.05) and the *P* value is less than 0.05, the factor has a significant effect on the objective [24,25].


**Table 4.** Significance analysis of process parameters to objectives.

The results of ANOVA show that for the ranges of the tested levels, the three process parameters have a significant influence on *Fz*am, for which the most important factor is feed speed, the second most important is the roll-beating mode, and the least important is the spindle rotation speed. The interaction item *m* × *w* is significant for *Fz*am, and *m* × *f* is considered statistically non-significant. The *F* value of *w* × *f* is larger than *F*(0.05), but the *P* value is bigger than 0.05. These values indicate that, although the influence of *w* × *f* on *Fz*am is very weak, it still has some impact.

All the tested sources of variation in *Fy*max are significant, and the order of importance of those sources is *w* > *m* > *f* > *m* × *f* > *m* × *w* > *w* × *f*.

Each process parameter has a significant influence on β, and *m* is the most important factor. The parameters *w* and *f* almost have the same effect on β. Of the interactive terms, *m* × *w* and *m* × *f* are significant. *P* > 0.05, the *w* × *f* interaction is considered statistically non-significant, but its *F* value is bigger than *F*(0.05), so its impact cannot be ignored.

For Sa, *m* and the interaction terms containing *m* are not significant. So, from a statistical view, it can be asserted that the change in roll-beating mode does not affect the roughness of the tooth wall. All the other sources are significant, and *w* is the most important.

### *4.2. Regression Models*

Because *m* is not a continuous variable—i.e., it only takes values of 1 or −1—a piecewise function, denoted by *T*(*w*, *f*), that consists of two polynomial models is used to characterize the response of *Fz*am, *Fy*max, and β to the influences of the different parameters. Meanwhile, given that *m* is irrelevant to Sa, *T*(*w*, *f*) is directly used to express Sa. The functional form of *T*(*w*, *f*) is a polynomial model which is widely used in regression analysis and empirical modeling. The function is shown in Equation (1). In order to ensure regression accuracy, the degrees of freedom of *w* and *f* in the function *T*(*w*, *f*) are 2 and 3, respectively, and the interaction between *w* and *f* is accounted for.

$$T(w, f) = a\_{00} + a\_{10}w + a\_{01}f + a\_{20}w^2 + a\_{11}wf + a\_{02}f^2 + a\_{21}w^2f + a\_{12}wf^2 + a\_{03}f^3 \tag{1}$$

Table 5 gives the final regression model of each index. The applicable range of these regression models is within the range of the experimental parameters. The term *R*<sup>2</sup> denotes the prediction capability of the regression equations. Generally, an *R*<sup>2</sup> value greater than 0.9 indicates that the regression equation is acceptable for fitting the experimental data, and an *R*<sup>2</sup> value greater than 0.95 indicates that all predicted values are reliable and quite close to the actual values. Apart from the *R*<sup>2</sup> value of β's regression model in the down-beating mode being slightly less than 0.95, the *R*<sup>2</sup> values of all the other regression models are higher than 0.95. Furthermore, the *P* values of all regression models are less than 10<sup>−</sup>4, as shown in Table 4.

**Table 5.** Regression models and validation results.


From the established regression models, the predicted values of the indices resulting from the process parameters in Table 3 were obtained. Then, we plotted these points to the coordinates that define the experimental values compared with the predicted values, as seen in Figure 10. The distribution of these points is not related to the form and coefficient of the regression equation but only to the fitting error. Figure 10 shows that the points are close to the straight line of y = x, and the error of the predicted values versus experimental values is distributed uniformly. These results indicate that the regression models have high fitting ability.

**Figure 10.** Correlation of predicted values versus experimental values: (**a**) *Fz*am; (**b**) *Fy*max; (**c**) β; and (**d**) Sa.

As such, according to the results of the above analysis, it is proved that the established regression models can effectively and reliably describe the influence of the tested process parameters on *Fz*am, *Fy*max, β, and Sa.

### *4.3. Influence of Process Parameters on Each Index*

In accordance with the regression equations listed in Table 5, Figures 11–14 illustrate the relation surface between each index and process parameter.

**Figure 11.** Surface graphs of *Fz*am regression model: (**a**) Up-beating (*m* = 1); (**b**) Down-beating (*m* = −1).

**Figure 12.** Surface graphs of *Fy*max regression model: (**a**) Up-beating (*m* = 1); (**b**) Down-beating (*m* = −1).

**Figure 13.** Surface graphs of β regression model: (**a**) Up-beating (*m* = 1); (**b**) Down-beating (*m* = −1).

**Figure 14.** Surface graphs of Sa regression model.

Figure 11 shows that in the different roll-beating modes, the change rules of *Fz*am with *w* and *f* are consistent, but the change in *Fz*am in the up-beating mode is larger. The value of *f* has a great influence on *Fz*am, where lower *f* means smaller *Fz*am. The degree of this effect is greater when *f* is less than 400 mm/min, below which *Fz*am decreases rapidly with decreasing *f*. The effect of *w* on *Fz*am varies in different *f* regions. When *f* is in the lower range, *Fz*am increases first and then decreases with the increase in *w*. At higher *f* values, *Fz*am decreases first and then increases when *w* increases. However, the range of *Fz*am changes is limited by only changing *w*. Therefore, the down-beating mode and low *f* are enough to significantly reduce *Fz*am.

As shown in Figure 12, the change trend of *Fy*max resulting from *f* and *w* varies for the different roll-beating modes, and the change range of *Fy*max in the up-beating mode is larger. For up-beating, with an increase in *w*, *Fy*max decreases first and then increases. Compared with *w*, *f* has a greater influence on *Fy*max. Increasing *f* leads to an obvious increase in *Fy*max. When *f* is small and *w* is near 900 r/min, *Fy*max is the smallest. In the down-beating mode, *Fy*max is more affected by *w* than *f*. *Fy*max and *w* are proportional to each other. In response to increasing *f*, *Fy*max increases rapidly and then decreases. As such, the down-beating mode, low *w*, and small *f* or *f* close to 780 mm/min are advantageous for reducing *Fy*max.

The whole change range of β is smaller in the up-beating mode. β decreases with the increase in *w*. In the low-*w* region, increasing *f* causes β to increase. In the high-*w* region, β decreases first and then increases when *f* grows, and the minimum β value is obtained around *f* = 450 mm/min. In the down-beating mode, β is weakly affected by *w* and decreases slightly with an increase in *w*. β is strongly influenced by *f*. As a result of increasing *f*, β decreases first and reaches the minimum; when *f* is near 240 mm/min, β increases and reaches the maximum near *f* = 840 mm/min. Figure 13 shows that in different roll-beating modes, setting *w* to 1280–1500 r/min and setting *f* to 30–590 mm/min allows β to be controlled at its lower level.

As shown in Figure 14, Sa increases when *w* increases. With increasing *f*, Sa increases rapidly first and then decreases gradually. Sa is the maximum at *w* = 1500 r/min and *f* = 350 mm/min. When *w* is within 450–870 r/min and *f* is within 600–960 mm/min, Sa is controlled under 0.1 μm.

In addition, for the single tooth forming process, a higher *f* value means higher forming efficiency.

### *4.4. Optimize*

From the above analysis, it can be seen that the influence of each process parameter on the forming force, forming quality, and forming efficiency indices has its own characteristics. There are some contradictions between the ranges of the process parameters, and these conflicts result in each index having its own optimal value at the same time; in order to determine the process parameters that allow each index to achieve comprehensive optimization, the linear weighted sum method was used to establish the comprehensive evaluation function *E* for *Fz*am, *Fy*max, β, Sa, and forming efficiency, as shown in Equation (2).

$$E = \sum\_{i=1}^{n} \mathbf{C}\_{i} \mathbf{e}\_{i\bar{j}} \tag{2}$$

In Equation (2), *i* denotes the ordinal number of the index (in this case, *i* = 1, 2, ..., 5); *j* is the number of the process parameter combination; *eij* represents the evaluation value of the *i*th index obtained under the *j*th process parameter combination; *C* is the weight coefficient, and the sum of the weight coefficients is 1. The function *E* allows corresponding weight coefficients to be assigned to each index according to the importance of each index, and it includes the linear combination of the multiple indices. It reduces the multi-objective optimization problem to the numerical optimization of the function *E*, which realizes the comprehensive optimization of the multiple objectives.

To solve the above, the first step is to get the evaluation value of each index so that they have the same order of magnitude. The Min-Max normalization function is used to convert the index data into the evaluation value. So, all the evaluation values of each objective range from 0 to 1. This Min-Max function takes two forms, as shown by Equations (3) and (4), in which *O* represents the data of each index. In this case, *Fz*am, *Fy*max, β, and Sa are normalized by Equation (3), and *f* is normalized by Equation (4).

$$c\_{i\bar{j}} = \frac{\max O\_{i\bar{j}} - O\_{i\bar{j}}}{\max O\_{i\bar{j}} - \min O\_{i\bar{j}}} \tag{3}$$

$$x\_{i\bar{j}} = \frac{O\_{i\bar{j}} - \text{min}O\_{i\bar{j}}}{\max O\_{i\bar{j}} - \text{min}O\_{i\bar{j}}} \tag{4}$$

In the range of process parameters, m = [−1, 1], w ∈ [475, 1500], f ∈ [30, 960], and the maximum and minimum values of each index are solved. The results are shown in Table 6.


**Table 6.** The max*Oij* and min*Oij* values of each index.

The entropy weight method was used to determine the weight coefficients, as it has strong objectivity and is widely applicable to practical engineering optimal problems [26]. This method uses the evaluation data of a sample to calculate the entropy weight coefficient of each index as the weight coefficient of each index. In order to calculate the entropy weight coefficient, it is first and foremost necessary to determine a large enough sample of evaluation values. Given the range of the process parameters, with the interval steps of *w* and *f* set to 5, there are 77,044 combinations of the different process parameters. Then, the evaluation values for each index under each group of process parameters are calculated by the regression model in Section 4.2, Equations (3)–(4), and Table 6. As such, we get a 5 × 77,044 evaluation matrix as the sample data. Then, the entropy weight coefficient of each index is obtained by Equations (5) and (6), in which *Hi* represents the entropy value of the *i*th index, and *ci* is the entropy weight coefficient for the *i*th index. During the calculation, it is important to note that when *f* is 30, calculating *H*<sup>5</sup> is meaningless. To avoid this, the value of 30 for *f* is replaced by 30.00001 in these calculations.

$$\begin{aligned} \sum\_{j=1}^{k} \frac{c\_{ij}}{\sum\_{j=1}^{k} c\_{ij}} \ln \frac{c\_{ij}}{\sum\_{j=1}^{k} c\_{ij}} \\ H\_i = -\frac{1}{\ln k} \frac{\sum\_{j=1}^{k} c\_{ij}}{\ln k} \end{aligned} \tag{5}$$

$$c\_i = \frac{1 - H\_i}{n - \sum\_{i=1}^{n} H\_i} \tag{6}$$

However, the weight coefficient obtained by this method has a lack of horizontal comparison among the objectives. Accordingly, to reflect the importance of the different objectives, an additional subjective weight coefficient λ is added to the entropy weight coefficient to get a composite weight coefficient, which is the *C* in Equation (2). The composite weight coefficient can be calculated by Equation (7). The subjective weight coefficient is given according to the importance of each objective. For the CRBF process, the first considered object is the forming quality, followed by the forming force and forming efficiency. Consequently, the subjective weight coefficients of β and Sa in the evaluation of tooth profile angle error and tooth wall roughness are large. Among them, considering that the tooth wall roughness is maintained at a certain level under the different process parameters, the subjective weight coefficient of Sa is smaller than that of β. For the indices used to evaluate the forming force, compared with *Fz*am, *Fy*max is more important since it directly reflects the torque required of the spindle and the load of the feed system during the forming process. A smaller *Fy*max implies lower energy consumption. Therefore, the subjective weight coefficient of *Fy*max is slightly higher than that of *Fz*am. The forming efficiency and *Fz*am are a set of contradictory indices for CRBF. High efficiency means a large feed speed. However, with a large feed speed, *Fz*am is larger, as shown in Figure 11. A larger *Fz*am means that the forming equipment is subjected to a greater load, which speeds up the wear of the equipment parts and increases the cost of the forming process. As such, it is appropriate to assign *Fz*am and *f* the same subjective weight coefficients. The results of the weight coefficient calculations are listed in Table 7.

$$\mathbf{C}\_{i} = \frac{\lambda\_{i}\mathbf{c}\_{i}}{\sum\_{i=1}^{n}\lambda\_{i}\mathbf{c}\_{i}}; \sum\_{i=1}^{n}\lambda\_{i} = 1\tag{7}$$


**Table 7.** The weight coefficients of objectives.

From Equations (2)–(4), the regression model of the indices, Table 6, and Table 7, the extended expression of the comprehensive evaluation function *E* is obtained, as shown in Equation (8).

$$E = \begin{cases} 0.5397 + 7.351 \times 10^{-5} w - 1.103 \times 10^{-3} f - 7.303 \times 10^{-8} w^2 + 5.934 \times 10^{-7} wf + 2.207 \times 10^{-6} f^2 \\ - 2.835 \times 10^{-10} w^2 f - 6.132 \times 10^{-11} wf^2 - 1.285 \times 10^{-9} f^3, & m = 1 \\\\ 0.6527 - 6.206 \times 10^{-5} w - 7.234 \times 10^{-9} f - 4.367 \times 10^{-8} w^2 + 2.757 \times 10^{-7} wf - 2.002 \times 10^{-7} f^2 \\ - 2.283 \times 10^{-10} w^2 f + 2.372 \times 10^{-10} wf^2 + 1.1175 \times 10^{-10} f^3, & m = -1 \end{cases} \tag{8}$$

For the process parameter ranges, *m* = [−1, 1], *w* ∈ [475, 1500], and *f* ∈ [30, 960], and the surface graph of the comprehensive evaluation function *E* is obtained and shown in Figure 15. The coordinates of each point on the surface represent a process parameter combination and the evaluation values resulting from this process parameter combination. A higher evaluation value corresponds to a more reasonable combination of process parameters in order to achieve the comprehensive process effect of a small forming force, high forming quality, and high forming efficiency. It can be seen that in the up-beating and down-beating modes, the evaluation value of the process parameter combination is higher in the ranges of *w* ∈ [475, 1200] and *f* ∈ [600, 960], respectively. In the case of *w* and *f*, with a minimum interval of 1, the vertex coordinates of the comprehensive evaluation function surface at *m* = −1 and *m* = 1 are (850, 882, 0.6354) and (801, 960, 0.6664), respectively. Subsequently, it is concluded that for CRBF, the optimum process parameters according to the comprehensive consideration of forming forces, forming quality, and forming efficiency are the following: Up-beating mode, a spindle rotation speed of 801 r/min, and a feed speed of 960 mm/min.

**Figure 15.** Surface graphs of comprehensive evaluation function: (**a**) Up-beating (*m* = 1); (**b**) down-beating (*m* = −1).

With the optimal process parameters, a confirmation experiment was repeated three times, and the average values of *Fz*am, *Fy*max, β, and Sa from the experiments are shown in Table 8. A comparison of predicted and experimental results reveals that the experimental results are close to the optimal solution obtained from the predicted model. The percentage errors between the prediction results and the confirmation experimental results are less than 7%.


**Table 8.** Optimal solution and confirmation experiment results.

Next, taking the experiments in Table 3 as control experiments and using the measured data of each index, the comprehensive evaluation value of each experiment was obtained by Equations (2)–(4). Comparing the comprehensive evaluation values from the confirmation experiment and the control experiments, it can be seen that the control experimental results are not higher than the confirmation experimental evaluation values, as shown in Figure 16. Therefore, it can be asserted that the regression models for *Fz*am, *Fy*max, β, and Sa are correct, and the method of process parameter selection for multi-objective optimization of the CRBF process in this paper is feasible.

**Figure 16.** Comprehensive evaluation value of control experiments and confirmation experiment.

### **5. Conclusions**

In this paper, the influences of several process parameters on the CRBF process were studied by forming an external tooth with ASTM 1045 material. The purpose was to derive a method for selecting process parameters that result in a balance between the forming forces, product quality, and forming efficiency. The following conclusions can be drawn:


**Author Contributions:** Conceptualization, L.L. and Y.L.; data curation, L.L. and T.T.; funding acquisition, L.L. and Y.L.; investigation, L.L.; methodology, M.Y.; project administration, Y.L.; writing—original draft, L.L.; writing—review & editing, M.Y.

**Funding:** This research was funded by National Natural Science Foundation of China (Grant No. 51475366, 51805433), Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2016JM5074), and Ph.D. Innovation fund projects of Xi'an University of Technology (Grant No. 310-252071601).

**Acknowledgments:** The authors are grateful to National Natural Science Foundation of China, Natural Science Basic Research Plan in Shaanxi Province of China, and Ph.D. Innovation fund projects of Xi'an University of Technology, which enabled the research to be carried out successfully.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Achievements of Nearly Zero Earing Defects on SPCC Cylindrical Drawn Cup Using Multi Draw Radius Die**

### **Rudeemas Jankree and Sutasn Thipprakmas \***

Department of Tool and Materials Engineering, King Mongkut's University of Technology Thonburi, 126 Prachautid Rd., Bangmod, Thungkru, Bangkok 10140, Thailand; rudeemas.jan@gmail.com

**\*** Correspondence: sutasn.thi@kmutt.ac.th

Received: 3 July 2020; Accepted: 27 August 2020; Published: 8 September 2020

**Abstract:** In recent years, the old-fashioned cylindrical cup shapes are still widely used, and there are many defects which could not be solved yet. In the present research, the classical earing defects, which are mainly caused by the material mechanical property of the anisotropic property of the material (*R*-value), are focused on. The multi draw radius (MDR) deep drawing die is applied and investigated to achieve nearly zero earing defects by encountering the *R*-value during the deep drawing process. Based on the experiments, in different directions in the sheet plane, the somewhat concurrent plastic deformation could be controlled, and the uniform elongated grain microstructure and uniform strain distributions on the cup wall could be achieved. Therefore, on the basis of these characteristics, the earing defects could be prevented, and the nearly zero earing defects could be achieved. However, to achieve the nearly zero earing defects, the suitable MDR die design relating to the *R*-value should be strictly considered. In the present research, to apply the MDR die for the medium carbon steel sheet grade SPCC cylindrical drawn cup, the following was recommended: the large draw radius positioned at 45◦ to the rolling direction and the small draw radius positioned along the plane and at 90◦ to the rolling direction. Therefore, in the present research, it was originally revealed that the nearly zero earing defects could be successfully performed on the process by using the MDR die application.

**Keywords:** cylindrical cup; deep drawing; draw radius; earing; microstructure

### **1. Introduction**

Sheet-metal products are increasingly fabricated to serve in various manufacturing industries especially the aerospace industry, electronics industry, and automobile industry. The complex shapes with high precision are also increasingly required in recent years. These products are commonly fabricated by sheet-metal forming processes such as bending, stamping, and deep drawing processes. Therefore, based on the experiments and finite element method (FEM) techniques, many studies have been performed and also reported by researchers and engineers to develop these sheet-metal forming processes and meet the mentioned requirements. Several researchers have focused on improving the quality of sheet-metal products as well as manufacturing the complex shapes with high precision through that associated with the experiments and FEM techniques [1–4]. In contrast, in terms of old-fashioned cylindrical cup shapes, the cylindrical cups are still widely used in various sheet-metal manufacturing industries. In general, almost all of them are conventionally fashioned by deep drawing process. On the basis of experimental and FEM works, the developments on this process have been continuously reported in many past studies [5–9]. For examples, Mahdavian and Tui Mei Yen [5] determined the effect of different punches' head profiles on the deep drawing of 5005H34 aluminium circular blanks to manufacture a cup-shape product. Nei et al. [6] investigated the deep drawing at various temperatures and studied the microstructure evolutions of three typical regions including

bottom, corner, and wall taken from the drawn cylindrical part with the largest limiting drawing ratio (LDR). Liu et al. [7] showed the increase in the formability and quality of the pure aluminium spherical bottom cylindrical parts (SBCP) by using magnetic medium-assisted sheet metal drawing process. Bassoil et al. [8] studied the effects of a draw bead working on an Al6014-T4 strip according to assigned industrial conditions by the handy draw bead simulator (DBS). Sezek et al. [9] educated the effects of the die radius on blankholder forces and drawing ratio. The results also showed that the major parameters affected cup wall thickness were blankholder force, die radius, and lubricant use. However, in terms of earing defects as shown in Figure 1, there are few studies that have carried out the investigation and prevention these defects [10–15]. Marton et al. [10] investigated the prediction of the earing defect on 0.3 and 3.0 mm thick cold rolled 1050 type aluminium sheets subjected to annealing heat treatments for different time intervals to promote the recrystallization processes and obtain different earing behaviors. It was shown that the proposed method was able to predict the type and magnitude of earing with satisfactory results for both 0.3 and 3.0 mm sheet thicknesses. Kishor and Kumar [11] studied the earing problem in deep drawing of flat bottom cylindrical cups using the FEM (LSDYNA). The optimization of the initial blank shape was proposed to meet the smallest earing defects. Walde and Riedel [12] investigated the earing defects on the magnesium alloy AZ31 that the crystallographic texture and plastic anisotropy were usually pronounced during the rolling process. Based on the FEM in conjunction with a viscoplastic self-consistent texture model, the results showed that not only on the initial texture of sheet metal but also the evolution of texture during drawing process caused the earing defects. Izadpanah et al. [13] studied the deformation behaviors of sheet metals using advanced anisotropic yield criteria by FEM simulation. The results revealed that the earing profile and thickness distribution obtained from experiments well corresponded with FEM simulations. Cazacu et al. [14] proposed the anisotropy plasticity CPB06 theory, and Singh et al. [15] applied and implemented it in an FE model to predict the nonuniform material flow characteristics, earing defects, and thickness distributions successfully. Some studies were performed to prevent earing defects by making the material property into isotropic material. Olaf Engler et al. [16] studied the microstructure evolutions resulting in earing profiles on the Al alloy (AA8011A) during the down-stream process. The results showed that the earing defects could be controlled. Tang et al. [17] studied the microstructure and texture, tensile mechanical properties in terms of strength and elongation, and the anisotropy of conventional unidirectional rolling (UR) and cross rolling (CR) sheets at room temperature. The results showed that the CR sheet produced a deeper drawn cup than that of the UR sheet due to its lower normal anisotropy *R* value and layer elongation compared with those of the UR sheet. However, these techniques cause the increases in additional rolling operations resulting in the increases in production cost and time consuming. Next, by using the FEM, the studies have been focused on the applications of models, while the proposal of new anisotropy models is not covered. In addition, Phanitwong and Thipprakmas [18], by using multidraw radius (MDR), showed the interesting results of material flow characteristics during the deep drawing process in which the nonaxisymmetric material flow characteristic due to the anisotropy property of the material could be encountered by using the MDR die, and the axisymmetric material flow characteristic could be formed. Therefore, in the present research, the new idea of MDR application for the earing prevention during the deep drawing process is proposed. Specifically, the different draw radius positioned in different directions in the sheet plane was designed to encounter the material mechanical property of the anisotropy property of the material, and then the earing could be prevented. The MDR die seemed to be difficult to design and fabricate due to the complicated three-dimensional shape of draw radius (die shoulder radius or hole edge corner radius). However, based on the computer-aided design (CAD) and computer-aided manufacturing (CAM) technologies nowadays, this MDR die could be designed and fabricated in general. In addition, by comparing with other techniques applied for the deep drawing process such as draw bead application, this MDR die application shows the less process parameters to be concerned. Therefore, as the benefits of this application, it is easier to set the process parameters, and it can be performed on the deep drawing operation with no any additional operations. This results in the

decreases in the production cost and time consumption. However, the conceptual design of MDR deep drawing die is strictly designed related to the R-value. As the results, they originally revealed that the MDR deep drawing die could be useful to reduce earing defects as well as to meet the nearly zero earing defects with no additional operations.

**Figure 1.** Earing defects and effective height of deep drawn parts.

### **2. Materials and Method**

In the present research, the medium carbon steel sheet grade SPCC (JIS) with the thickness of 0.975 mm was used as a workpiece material. The chemical compositions were listed in Table 1. The mechanical properties were also examined by the tensile testing technique, and they were listed in Table 2. As a major material property that affected the earing defects, the *R*-value along the plane, at 45 and 90◦ to the rolling direction were examined, and they were 2.1, 1.9, and 2.6, respectively. Table 2 also shows the other material properties of Young's modulus, Poisson's ratio, and ultimate tensile strength. The microstructures along the plane, at 45 and 90◦ to the rolling direction were examined as well. The sheet materials were also sectioned and further processed by subsequent mounting, polishing, and etching with 2% nital etchant. Optical microscopy was used to observe and capture microstructure images for microscopic examinations. The images of examined microscopic along the plane, at 45 and 90◦ to the rolling direction are, respectively, shown in Figure 2a–c. In addition, the grain sizes were also examined on both horizontal and vertical (thickness) directions as they were also reported in Figure 2. The cylindrical cup of 60.0 mm in diameter, 26.2 mm in height, and 5.0 mm in cup bottom radius, as shown in Figure 3, was used as a model of cylindrical deep drawn cup. To fabricate this cup, the initial blank size of 100.0 mm was calculated based on the deep drawing theory [19]. The initial blank was prepared by using a wire electrical discharge machine (Wire-EDM, Sodick Model AQ325L, Yokohama, Kanagawa, Japan). Figure 4a shows the punch and die designed based on the deep drawing theory [19]. As listed in Table 3, the punch diameter of 57.8 mm and die diameter of 60.0 mm were set. The die radius of 5.0 mm was designed. The deep drawing clearance of 1.1 mm was set. Figure 4b shows the press machine, which includes a universal sheet metal testing machine (JT TOHSI INC., Model SAS-350D, Minato-ku, Tokyo, Japan). After the deep drawing process, the obtained cylindrical cups were sectioned by a wire-EDM machine for material thickness examinations. The material thickness was measured using a digital micrometer (Insize, serie IS13108, Loganville, GA, USA). The cup height was measured using a vernier height gauge (Mitutoyo, Model CD-6" ASX, Kawasaki, Kanagawa, Japan), and the earing defects were calculated. Five samples from each deep drawing condition were used to inspect the obtained cylindrical cups. The amount of material thickness and earing defects were calculated based on these obtained cylindrical cups, and the average material thickness and earing defect values were reported.

**Table 1.** Chemical compositions of SPCC steel (JIS).



**Figure 2.** Microstructure examinations of SPCC steel (JIS): (**a**) plane xz (0◦ to rolling direction); (**b**) plane wz (at 45◦ to rolling direction); (**c**) plane yz (at 90◦ to rolling direction).

**Figure 3.** Model of cylindrical deep drawn part.

*Metals* **2020**, *10*, 1204

(**a**) Die set: (**aƺ1**) punch; (**aƺ2**) conventional die; (**aƺ3**) multi draw radius (MDR) die; (**aƺ4**) blank holder.

(**b**) Detail drawing of MDR die.

(**c**) Press machine

**Figure 4.** Die set for experiment and press machine: (**a**) die set; (**b**) detail drawing of MDR die; (**c**) press machine.


### **Table 3.** Experimental conditions.

### **3. Results and Discussion**

*3.1. Analysis of the Microstructure Evolutions, Strain Distributions, and Thickness Distributions on Deep Drawn Parts Using Conventional Dies*

As per the deep drawing theory [19], the cylindrical cup is initially formed after the blank is drawn over the draw radius. This characteristic resulted in the microstructure evolutions and generated strain distributions on the cup. Specifically, the crystal grain deformed adjusting to macroscopic deformation, and so the deformation of crystal grain would naturally correspond to macroscopic strain. On the basis of the fundamentals of the deep drawing mechanism, the material at the cup bottom zone was subjected to equi-biaxial tension and the stretching flange deformation was generated based on the equi-biaxial elongation. This deformation characteristic commonly results in the material thinning. The stretching flange characteristic also commonly formed at the cup bottom radius zone. The bending characteristic was formed at this zone, in addition. Therefore, in general, the material thinning was greatest, and the possibility of breakage was also the highest at the cup bottom radius zone. Next, to form a cup wall, the material at flange portion moves in the direction to the center of die, and then this material must shrink in circular direction. As these characteristics, the tension and compression stresses are commonly generated in radial and circular directions, respectively. Therefore, the material elongates and shrinks in radial and circular directions, respectively. Based on these characteristics, after the cup wall completely formed, the earing defects were simultaneously generated. Therefore, relating to the *R*-value, the analysis of the microstructure evolutions and strain distributions on the cup wall zone was focused on in the present research to clarify the earing defects as well as to use as fundamental information for a new die design to prevent earing defects. In comparison to the initial blank sheet, the microstructure evolutions along the plane at 45 and 90◦ to the rolling direction at the middle and near edge of the cup walls in the case of a draw radius of 5.0 and 13.0 mm are shown in Figure 5. The examined elongated grains in the horizontal and vertical directions were reported as well. Based on these examined grain sizes, the logarithmic strain in each direction was calculated based on Equation (1). Therefore, by the same token, the logarithmic strains in radial and thickness directions could be calculated as well. Next, by using the volume constancy law, the logarithmic strain in the circular direction could be calculated following Equation (2), and they were reported in the Figure 6.

$$\text{Logarithmic strain} = \ln\left(l/l\_0\right) \tag{1}$$

when *l*<sup>0</sup> is the initial grain size and *l* is the deformed grain size.

$$
\varepsilon\_{\partial} = - (\varepsilon\_{r} + \varepsilon\_{l}) \tag{2}
$$

when εθ is the logarithmic strain in the circular direction, ε*<sup>r</sup>* is the logarithmic strain in the radial direction, and ε*<sup>t</sup>* is the logarithmic strain in the thickness direction.

The same manner of strain distribution analysis in the cases of a draw radius of 5.0 and 13.0 mm could be observed. In terms of thickness strain distribution, it was negative in all directions in the sheet plane. The large negative thickness strain was generated in the direction along the plane and at 90◦ to the rolling direction, and the small negative one was generated in the direction at 45◦ to the rolling direction. In comparison to the middle cup wall zone, at the near edge of the cup wall, the positive thickness strain was generated instead of the negative one. Again, the large thickness strain was still generated in the direction at 45◦ to the rolling direction. Therefore, the material thinning was formed at the middle of cup wall, whereas the material thickening was formed at the near edge of the cup wall. To validate these results, the thickness examinations were carried out, and the thickness distributions are shown in Figure 7. There was a small change of material thickness on the cup bottom zone due to the stretching flange deformation characteristic. However, the maximum material thinning was generated on the cup bottom radius because the bending was added to the stretching flange deformation characteristics, and then the large plastic deformation was generated in that area. It was also observed that the changes in material thickness with respect to draw radii were somewhat at the same level. The effects of the rolling direction on the material thickness of the cup bottom zone and cup bottom radius zone were very small. With the cup wall zone, the results showed that the material thinning was decreased along the cup wall height. The effects of the rolling direction on material thickness were clearly illustrated, especially on the cup wall zone. These thickness distribution results agreed well with the thickness strain distribution results calculated on the basis of the deformed grain size. These results also corresponded well with the deep drawing theory and literature [18,19]. Next, in terms of radial strain and circular strain distributions, the radial strain was positive, but the circular strain was negative. The large radial strain was generated in the direction along the plane and at 90◦ to the rolling direction, and the small one was generated in the direction at 45◦ to the rolling direction. Again, the large circular strain was generated in the direction along the plane and at 90◦ to the rolling direction, and the small one was generated in the direction at 45◦ to the rolling direction. These results could be explained that, owing to the *R*-value as listed in Table 2, the plastic deformation would occur earlier in the direction along the plane and at 90◦ to the rolling direction due to the large R-value. Therefore, the material would flow into these directions from other portions, i.e., the portion at 45◦ to the rolling direction. Therefore, the nonconcurrent plastic deformation was generated and then caused the restriction of material flown into the die. In addition, the highly excessive elongating material flowed outward in the direction to edge of the blank sheet. Therefore, the earing defects were formed by the peak of the earing profile that was along the plane and at 90◦ to the rolling direction, whereas the bottom of the earing profile was at 45◦ to the rolling direction. Figure 8a,b show the earing defects in the case of a draw radius of 5.0 and 13.0 mm, respectively. The obtained earing defects were approximately of 1.7 mm in both cases of the draw radius of 5.0 and 13.0 mm. It was also observed that the peak of the earing profile formed along the plane and at 90◦ to the rolling direction, whereas the bottom of the earing profile formed at 45◦ to the rolling direction. They agreed well with radial and circular strain distributions calculated on the basis of the deformed grain size. These results also corresponded well with the deep drawing theory and literature [18,19]. As a result, based on the microstructure evolutions and strain distributions, the occurrence of earing defects could be clearly clarified. These data are valuable information for supporting MDR die design and development to achieve the nearly zero earing defects.

(**a**) Draw radius 5.0 mm.

(**b**) Draw radius 13.0 mm.

**Figure 5.** Comparison of microstructure evolutions on deep drawn parts with respect to draw radii: (**a**) draw radius 5.0 mm; (**b**) draw radius 13.0 mm.

**Figure 6.** Strain distributions on cup wall zone with respect to draw radii: (**a**) draw radius 5.0 mm; (**b**) draw radius 13.0 mm.

**Figure 7.** Thickness distributions on deep drawn parts with respect to draw radii: (**a**) draw radius 5.0 mm; (**b**) draw radius 13.0 mm.

(**a**) Draw radius 50 mm (**b**) Draw radius 130 mm

**Figure 8.** Earing defects with respect to draw radius obtained from the conventional die application: (**a**) draw radius 5.0 mm; (**b**) draw radius 13.0 mm.

### *3.2. Conceptual Design of Multi Draw Radius (MDR) Deep Drawing Die*

As per the author past research [18], the multi draw radius (MDR) deep drawing die is mainly proposed to increase the limiting drawing ratio of cylindrical deep drawn parts (LDR) by reducing the nonaxisymmetric material flow during deep drawing process. In the present research, this MDR die was proposed and applied to prevent the earing defects. Figure 9 shows the conceptual design of the MDR die. By using the conventional die, as shown in Figure 9a, the nonconcurrent plastic deformation characteristic was formed during the deep drawing process because of the effects of the *R*-value. As aforementioned, the plastic deformation would occur more early in the direction along the plane and at 90◦ to the rolling direction due to the large *R*-value. This resulted in the restriction of material flown into the die due to the highly excessive elongating material flow into these directions from other portions. Then, the nonaxisymmetric material flow characteristic on the flange portion during the deep drawing process was generated [18]. These plastic deformation and material flow characteristics caused the material in the direction along the plane and at 90◦ to the rolling direction were easier to stretch compared to that in direction at 45◦ to the rolling direction, and then, the earing defects were formed. To encounter these characteristics and prevent earing defects, the later plastic deformation generated in the direction at 45◦ to the rolling direction should be driven to generate earlier as it generated in the direction along the plane and at 90◦ to the rolling direction. Therefore, the plastic deformation in different directions in the sheet plane during the deep drawing process could be concurrently generated. As per the past research [18], by using the MDR die, the axisymmetric material flow characteristic could be achieved. Therefore, the MDR die might be able to solve the above-mentioned plastic deformation characteristic and earing defects. As suggested in the past research, the larger draw radius was positioned at 45◦ to the rolling direction and the smaller draw radius positioned along the plane and at 90◦ to the rolling direction. This resulted in the plastic deformation generating in the direction at 45◦ to the rolling direction, which could be made to occur earlier, and then the concurrent plastic deformation characteristic could be controlled, as shown in Figure 9b. The uniform material stretching in each direction in the sheet plane could be generated, and the uniform strain distribution in each direction in the sheet plane could be obtained. Concerning these plastic deformation mechanisms, by using the MDR die, the achievement of nearly zero earing defects by the process encountering the material mechanical property of the anisotropy property of the material could be met. However, the draw radius in each direction in the sheet plane should be positioned as follows: the larger draw radius positioned at 45◦ to the rolling direction and the smaller draw radius positioned along the plane and at 90◦ to the rolling direction. In the present research, the different MDR dies were investigated to achieve nearly zero earing defects, and the results are shown in Figure 10. Specifically, the small draw radius of 5.0 mm, as recommended for conventional deep drawing die, was positioned along the plane and at 90◦ to the rolling direction. The large draw radius of 6.0–14.0 mm was positioned at 45◦ to the rolling direction. The results showed the decreases, and again, the increases in the earing defects as the large draw radius increased. The smallest earing defects approximately of 0.1 mm could be generated with the large draw radius of 13.0 mm applied.

**Figure 9.** Illustration of plastic deformation characteristics with respect to die types: (**a**) conventional die application; (**b**) MDR die application.

**Figure 10.** Comparison of earing defects between conventional and MDR die applications.

### *3.3. Application of Multidraw Radius (MDR) Deep Drawing Die*

As aforementioned, the small draw radius of 5.0 mm, as recommended for conventional deep drawing die, was positioned along the plane and at 90◦ to the rolling direction, and the large draw radius of 13.0 mm positioned at 45◦ to the rolling direction was designed to achieve nearly zero earing defects. As this MDR die was applied, the nearly uniform elongated grain microstructure evolutions in each direction in the sheet plane could be achieved and the earing defects could be prevented. The examined microstructure evolutions are shown in Figure 11b, and the examined elongated grains in horizontal and vertical directions were reported as well. By comparing the conventional die use, these results revealed that by using the MDR die, the microstructure evolutions in each direction in the sheet plane during the deep drawing process, especially on the cup wall zone, could be encountered by the multi draw radius, and the nearly uniform elongated grain in each direction in the sheet plane on each cup wall height, i.e., the middle and near edge of cup walls, could be achieved. In terms of strain distribution, as shown in Figure 12b, by comparing with the conventional die use, the changes in radial, circular, and thickness strains in the direction at 45◦ to the rolling direction were clearly identified. Specifically, the radial strain was increased in the positive direction, but the circular and thickness strains were increased in the negative direction. In addition, the strain distributions also showed the interesting results that the nearly uniform strain distribution in each direction in the sheet plane on each cup wall height, i.e., the middle and near edge of cup walls, could be achieved. In terms of thickness distribution, as shown in Figure 13b, the thickness distributions showed the good agreement with the strain distribution results. As the thickness strain in the direction at 45◦ to the rolling direction increased in the negative direction, the thickness distribution showed the decreases in material thickness compared to that in the case of conventional die use. Moreover, the nearly uniform thickness distribution in each direction in the sheet plane on each cup wall height could be achieved. Based on these microstructure evolutions, strain distributions, and thickness distributions, in terms of

earing defects, as shown in Figure 14, the results revealed that the earing defects could be reduced, and the nearly zero earing defects could be achieved. The results showed that, by using the MDR die, the earing defects of approximately 0.1 mm were formed. To more clearly clarify the MDR die use, the MDR die of large die radius of 12.0 mm was also designed and investigated. Again, the results of microstructure evolutions, strain distributions, and thickness distributions are shown in Figure 11b, Figure 12b, and Figure 13b, respectively. The same characteristics of those MDRs with the large die radius of 13.0 mm use could be observed. However, it could be noted that the increases in radial, circular, and thickness strains were smaller than those in the case of MDR with a die radius of 13 mm use. Therefore, the earing defects were larger. These results confirmed that, by using the MDR die, the nearly zero earing defects could be achieved. However, the suitable MDR die design relating to the *R*-value of material should be strictly considered.

**(a)** MDR die 5-13-5: large radius 13.0 mm; small radius 5.0 mm.

**Figure 11.** *Cont*.

**(b)** MDR die 5-12-5: large radius 12.0 mm; small radius 5.0 mm.

**Figure 11.** Comparison of microstructure evolutions on deep drawn parts with respect to MDR dies: (**a**) MDR die 5-13-5; (**b**) MDR die 5-12-5.

**Figure 13.** Illustration of thickness distributions on deep drawn parts with respect to MDR dies: (**a**) MDR die 5-13-5; (**b**) MDR die 5-12-5.

**Figure 14.** Illustration of earing defects on deep drawn part by using the MDR die.

### **4. Conclusions**

In the present research, the earing defects as the major problem in the deep drawing process, especially for cylindrical drawn parts, were focused on. The MDR die application was proposed for reducing the earing defects as well as for achieving the nearly zero earing defects in the cylindrical deep drawing process. First, the effects of the draw radius on microstructure evolutions, strain distributions, thickness distributions, and earing defects were investigated. The results confirmed that, based on the material mechanical property of the R-value, the draw radius significantly caused the microstructure evolutions in different directions in the sheet plane. In addition, the changes in these microstructure evolutions significantly affected the strain and thickness distributions in different directions in the sheet plane, and then, the earing defects were formed. However, the changes in the draw radius rarely affected these microstructure evolutions, strain and thickness distributions, and earing defects. Next, the MDR die was applied and investigated to prevent earing defects. It was designed based on the principle of that, during the deep drawing process, the multi draw radius could encounter the effects of the *R*-value on microstructure evolutions and strain distributions in each direction in the sheet plane, especially on the cup wall zone. The plastic deformation generated in each direction in the sheet plane should be concurrently performed, and the axisymmetric material flow characteristic could be achieved. Therefore, the nearly uniform elongated grains in each direction in the sheet plane on each cup wall height could be achieved, and the nearly uniform strain and thickness distributions, especially on cup wall zone in each direction in the sheet plane, could be formed. Based on these

microstructure evolutions, strain distributions and thickness distributions, the earing defects could be reduced, and the nearly zero earing defects could be achieved by using the MDR die. However, to achieve the nearly zero earing defects, the suitable MDR die design relating to the *R*-value should be strictly considered. In the present research, it was suggested that the larger draw radius should be positioned at 45◦ to the rolling direction and the smaller draw radius positioned along the plane and at 90◦ to the rolling direction.

**Author Contributions:** Conceptualization, R.J. and S.T.; data curation, R.J. and S.T.; funding acquisition, R.J. and S.T.; investigation, R.J.; methodology, R.J. and S.T.; project administration, S.T.; supervision, S.T.; writing—original draft, R.J.; writing—review and editing, S.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the "Petchra Pra Jom Klao Master's Degree Research Scholarship" from King Mongkut's University of Technology Thonburi. The APC was also funded by King Mongkut's University of Technology Thonburi.

**Acknowledgments:** The authors would like to express their gratitude to Wiriyakorn Phanitwong, Department of Industrial Engineering, Rajamangala University of Technology Rattanakosin, and Arkarapon Sontamino, Department of Mechanical Engineering Technology, College of Industrial Technology, King Mongkut's University of Technology North Bangkok for their advice on experiments in this present research.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Multi Draw Radius Die Design for Increases in Limiting Drawing Ratio**

### **Wiriyakorn Phanitwong 1,\* and Sutasn Thipprakmas <sup>2</sup>**


Received: 22 May 2020; Accepted: 24 June 2020; Published: 30 June 2020

**Abstract:** As a major sheet metal process for fabricating cup or box shapes, the deep drawing process is commonly applied in various industrial fields, such as those involving the manufacture of household utensils, medical equipment, electronics, and automobile parts. The limiting drawing ratio (LDR) is the main barrier to increasing the formability and production rate as well as to decrease production cost and time. In the present research, the multi draw radius (MDR) die was proposed to increase LDR. The finite element method (FEM) was used as a tool to illustrate the principle of MDR based on material flow. The results revealed that MDR die could reduce the non-axisymmetric material flow on flange and the asymmetry of the flange during the deep drawing process. Based on this material flow characteristic, the cup wall stretching and fracture could be delayed. Furthermore, the cup wall thicknesses of the deep drawn parts obtained by MDR die application were more uniform in each direction along the plane, at 45◦, and at 90◦ to the rolling direction than those obtained by conventional die application. In the present research, a proper design for the MDR was suggested to achieve functionality of the MDR die as related to each direction along the plane, at 45◦, and at 90◦ to the rolling direction. The larger draw radius positioned for at 45◦ to the rolling direction and the smaller draw radius positioned for along the plane and at 90◦ to the rolling direction were recommended. Therefore, by using proper MDR die application, the drawing ratio could be increased to be 2.75, an increase in LDR of approximately 22.22%.

**Keywords:** deep drawing; limiting drawing ratio (LDR); draw radius; anisotropy; finite element method

### **1. Introduction**

In recent years, the available sheet metal components are able to serve almost all manufacturing industries, such as is the case for sheet metal components used in automobile and aerospace. The fabrication of such sheet metal components by means of sheet metal die is commonly classified according to the utilization of die bending, die deep-drawing, and die cutting processes [1]. Based on these processes, and through the associated experiments and finite element method (FEM) techniques, many studies have been carried out by researchers and engineers to overcome the major defects that occur on these sheet metal components [2–6] as well as to improve the formability for each sheet metal forming process [7–10]. In the present research, we focused on the deep drawing process. This process is extensively applied to fabricate various consumer products such as household utensils, medical equipment, electronics, and automobile parts. This process is also cost-effective because it is characterized by high production rates and gives finished parts of good quality without additional operations. Nowadays, for sheet metal components manufactured by deep drawing process, a more complex profile and higher formability are required [11–15]. The optimization of process parameters to enhance the formability of AA 5182 alloy in the deep drawing of square cups by hydroforming

was carried out [11]. A bead optimization algorithm was developed to increase the efficiency of the bead design while simultaneously considering manufacturability. The influences of the deep drawing depth, initial specimen geometry, and bead height on formability were subsequently investigated by means of sensitivity analysis [12]. Abe et al. proposed the technique of local work hardening with punch indentation to improve sheet metal formability [13]. The most important process parameter affecting thinning was the peak pressure, whereas the pressure path had the least effect on formability. The square deep draw steel/carbon fiber reinforced plastic (CFRP) hybrid composite material was investigated. The effects of fiber orientation on formability were also investigated [14]. In addition, the micro deep drawn parts are also focused on [16–19]. C. J. Wang. et al. carried out the research on the micro deep drawing process of a conical part with ultra-thin copper foil using a multi-layered DLC film-coated die [16]. The grain size effect on multi-stage micro deep drawing of a micro cup with domed bottom was investigated by W. T. Li et al. [17]. However, in terms of old-fashioned cylindrical cup shapes, deep drawing is also a conventional sheet metal forming process for wide application in industry at a very high production rate. The products of cylindrical cup shapes are also still widely used in various sheet metal manufacturing industries. Therefore, the developments on this process have been continuously reported in many previous studies on the basis of experimental and FEM works [20–29]. Some previous studies were carried out to prevent major defects, such as wrinkles, earing, and fracture defects [20–23]. In addition, as a common formability indicator of cylindrical cup forming, the limiting drawing ratio (LDR) has been also investigated in many previous studies [24–29]. Many techniques have been proposed to increase in LDR. A new technique for deep drawing of elliptic cups through a conical die without blankholder or draw beads was proposed to increase LDR [24]. The LDR of aluminum tailored friction stir welded blanks could be increased using a modified conical tractrix die technique. By using this technique, the improvements in LDR of approximately 27% and 14% were recorded, respectively, for the dissimilar grade and the dissimilar gauge aluminum tailor friction stir welded blanks [26]. Bandyopadhyay, K. et al. showed that the LDR of tailor welded blanks (TWBs) fabricated using two dissimilar material combinations of dual phase (DP) and interstitial free (IFHS) steels could be improved with restricted weld movement by shifting the initial weld line position [27]. However, in the previous studies, the increases in LDR were still limited. In addition, the way to increase LDR was also complex as additional operations were applied. In the present research, therefore, a new approach to increase in LDR is proposed and investigated. Based on the mechanical property of plastic strain ratio (R-value), the material flow on flange along the perpendicular differed, and the resulting cup wall stretching and fracture as well as the drawing ratio was limited. This new technique of multi draw radius, termed MDR, is proposed in the present research to encounter the material property of plastic strain ratio and generate the same manner of material flow on the flange along the perpendicular. Specifically, the MDR die could reduce the non-axisymmetric material flow on flange and the asymmetry of the flange during the deep drawing process. Furthermore, on the basis of this technique, the deep drawing process could be applied without additional operations, resulting in production cost and time being saved. However, for the proper design of MDR die, we recommend that the larger draw radius be positioned at 45◦ to the rolling direction and the smaller draw radius positioned along the plane and at 90◦ to the rolling direction. The application of the MDR was compared with the conventional die, and the results illustrated that the LDR could be increased.

### **2. Proposed Multi Draw Radius (MDR) Die and Its Principle**

As per the deep drawing theory [1], the LDR is the maximum ratio of initial blank diameter to punch diameter in which the cylindrical cup could be formed without any fractures. The LDR also depends upon the type of material used as well as relates to the mechanical property of material being of common concern in the deep drawing process. In terms of fracture, the fracture is commonly generated on the basis of two deep drawing characteristics [1]. Specifically, the first is that the wrinkle defects are generated on the flange due to the overly low blankholder pressure applied which causes

the obstacle of material flow into the die. This manner resulted in the generation of cup wall stretching and fractures. Next, the second is that the flange is tightly clamped by applying an overly high blankholder pressure. This manner resulted in the material not being able to be drawn into the die as well as subsequent cup wall stretching and generation of fractures. In addition, the material properties, especially plastic strain ratio (R-value), also affect fractures. As is well known, the R-value is the anisotropy property of the material and depends upon the direction along the plane, at 45◦, and at 90◦ to the rolling direction. Namely, the anisotropy property of the material causes the different material flow and formability in each direction along the plane, at 45◦, and at 90◦ to the rolling direction. As this characteristic causes non-axisymmetric material flow on the flange during the deep drawing process, cup wall stretching can easily be generated, as well as that flange wrinkles can be easily formed. Based on these reasons, to prevent the fracture and increase in deep drawing formability, the prevention of cup wall stretching and flange wrinkle should be strictly considered. As per previous studies [11,19,20], the working process parameters on deep drawing process, i.e., blankholder pressure, lubricant, and draw radius, were investigated. However, as aforementioned, the anisotropy property of the material in each direction along the plane, at 45◦, and at 90◦ to the rolling direction related to draw radius die design for reducing the non-axisymmetric material flow during deep drawing process has not yet been investigated. In the present research, the multi draw radius die (or so-called MDR die) is proposed to reduce the non-axisymmetric material flow during deep drawing process and prevent fracture as well as to increase in LDR. The schematic of MDR die was shown in Figure 1. The draw radius was designed related to the anisotropy property of the material in each direction along the plane, at 45◦, and at 90◦ to the rolling direction to reduce the non-axisymmetric material flow during the deep drawing process. As per deep drawing theory, the LDR should be treated as the baseline. It gives only estimated values and does not take into account many factors. Therefore, in the present research, to compare the formability based on the LDR between conventional die and MDR die applications, the other process parameters affecting LDR, excluding die types, were set with the same conditions for both cases of conventional and MDR die applications.

**Figure 1.** Schematic of deep drawing model and types of deep drawing die: (**a**) Deep drawing model; (**b**) Conventional die; (**c**) Multi draw-radius die.

Figure 2 shows the principle of MDR die based on the material flow characteristic. The comparison of schematic of material flow characteristic between conventional and MDR dies was illustrated. In the case of conventional die as shown in Figure 2a, owing to the effects of the anisotropy property of the material in each direction along the plane, at 45◦, and at 90◦ to the rolling direction, the non-axisymmetric material flow characteristic was formed. By contrast, to encounter these effects, the MDR die was proposed. The different draw radius positioned in each direction along the plane, at 45◦, and at 90◦ to the rolling direction was designed. Based on this MDR die, the non-axisymmetric material flow characteristic due to the effects of the anisotropy property of the material could be reduced by using a different draw radius in each direction along the plane, at 45◦, and at 90◦ to the rolling direction, as shown in Figure 2b. Therefore, the cup wall stretching and fracture prevention could be achieved, and the LDR could also be increased.

**Figure 2.** Principle of multi draw radius die based on the material flow characteristic: (**a**) Conventional die; (**b**) Multi draw-radius die.

### **3. The FEM Simulation and Experimental Procedures**

In the present research, the FEM simulation was used as a tool to clarify the deep drawing mechanism of MDR die based on the material flow. The nonlinear FEM commercial code HyperForm 14.0 with RADIOSS script (Altair Engineering Inc., Troy, MI, USA) as the solver was used for FEM simulation of the deep drawing process. The investigated model of MDR die application is shown in Figure 3a. In addition, to clearly understand the deep drawing mechanism of MDR die application, the deep drawing mechanism of conventional die application was also investigated as the model shown in Figure 3b. These 3-D deep drawing models were created by Cimatron 3 (3D Systems Inc., Givat Shmuel, Israel) and then imported as IGES file into HyperForm. The HyperMesh preprocessor was used to create the mesh. The initial blank was set as elastic–plastic and meshed into finite elements of "shell" type. The 4 node quadrilateral shape elements of approximately of 3500 elements were generated. The adaptive remeshing was also set. After remeshing, to lead rather smooth meshes, the combination of 4 node quadrilateral shape elements and triangular shape elements were generated. The tool including punch, die, and blankholder were meshed with the rigid mesh type to prevent their deformations during the deep drawing process. The blankholder pressure was set as gap type. The gap of material thickness was applied. In the present research, the forming limit diagram (FLD) was used to clarify the forming characteristics as well as to predict the fracture zone on deep drawn parts. The workpiece used in this present research was medium carbon steel grade SPCC (JIS standard) with the thickness of 0.5 mm. The material properties of flow curve equation and plastic strain ratio (R-value) were prepared as input parameters for FEM simulation. The workpiece material was described with an elastoplastic, power exponent, isotropic plasticity model of Hollomon power law hardening model, with the constitutive equation determined from the stress–strain curve using the tensile test data. The other necessary material properties, such as the Young's modulus, Poisson's ratio, and ultimate tensile strength are given in Table 1. As per the literature [30–32], for the static compression test, the friction coefficient was set to be from 0.1 to 0.3. In the present research, based on the deep drawing process with lubricant used, the contact surface model was defined by a Coulomb friction law, and friction coefficient (μ) of 0.10 was applied. It was applied for both cases of conventional and MDR deep drawing processes to ignore the effect of lubricant use. Next, the diameters of punch and die were 40.0 and 41.0 mm, respectively, in which the clearance of 0.5 mm was set. The tool

radius for conventional die was set following deep drawing theory [1]. Namely, a punch radius of 8.0 mm and die radius of 3.5 mm were set. The MDR draw radii of 3.5–5, 3.5–7, and 3.5–9 mm were investigated. On the basis of deep drawing theory [1], the LDR for this material was approximately of 2.25. Three levels of initial blank diameter of 90, 110, and 115 mm in which the drawing ratios of 2.25, 2.75, and 2.88 were investigated.

**Figure 3.** Deep drawing simulation models: (**a**) Conventional deep drawing model; (**b**) Multi draw-radius deep drawing model.


**Table 1.** FEM simulation and experimental conditions.

The laboratory experiments were performed to validate the FEM simulation results. Figure 4 shows the press machine, which includes a universal sheet metal testing machine (Model SAS-350D, JT Toshi Inc., Minato-ku, Japan) and the sets of conventional die and MDR die applications. The initial blanks were prepared using a wire electrical discharge machine (Wire-EDM) (Model AQ325L, Sodick Co., Ltd., Ishikawa, Japan). The obtained deep drawn parts were sectioned by wire electrical discharge machine for cup wall thickness examination. The cup wall thickness was measured. Five samples from each deep drawing condition were used to inspect the obtained deep drawing parts. The cup wall thickness was calculated based on these obtained deep drawing parts and the average cup wall thickness values were reported and compared with those analyzed by the FEM simulation.

**Figure 4.** Press machine and set of punch and die: (**a**) Press machine (Universal sheet metal testing machine); (**b**) Punch; (**c**) Blankholder; (**d**) Conventional die (radius 3.5 mm); (**e**) Multi draw-radius die (radius 3.5–7mm); (**f**) Multi draw-radius die (radius 3.5–9mm).

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

### *4.1. The Validation of FEM Simulation Use*

The FEM simulation was used, in the present research, as a tool for characterization of the deep drawing mechanism and prediction of the obtained deep drawn parts. Therefore, although the commercial finite element code HyperForm was used, the accuracy of the FEM simulation results should be again validated before starting the discussion section of FEM simulation results. As the validation of the FEM simulation results shows in Figure 5, by comparing with the laboratory experiments, the FEM simulation results showed the successful deep drawn parts and unsuccessful deep drawn parts which corresponded well with the experiments. In addition, the FEM simulation results also showed the earing defects and fracture which corresponded well with the experimental results. The unsuccessful deep drawn part in which the fracture generated as shown in Figure 5b, the FEM simulation result corresponded well with the experiments. On the basis of FLD, the fracture was generated on corner zone and a circumferential character was formed which agreed well with the fracture generated on deep drawn part obtained by experiment. The cup wall thickness was also examined. The comparisons of cup wall thickness distribution between FEM simulation and experimental results are illustrated in Figure 6. The FEM simulation results showed that the predicted cup wall thickness distributions corresponded well with the experiments, in which the errors in the analyzed cup wall thickness were approximately 3% compared with the experimental results. Finally, the deep drawing force was also recorded during experiments to validate the deep drawing force analyzed by FEM simulation. Figure 7 shows the comparison of deep drawing force obtained by FEM simulation and experiment. Again, the analyzed deep drawing force by FEM simulation corresponded well with the experiment, in which the error in the analyzed deep drawing force was approximately 1% compared with the experimental result.

**Figure 5.** Comparison of deep drawn parts obtained by FEM simulation and experimental results: (**a**) FEM Simulation result; (**b**) Experimental result.

**Figure 6.** Cup wall thickness distribution between FEM simulation and experimental results: (**a**) Position for measurement; (**b**) 0◦ to rolling direction; (**c**) 45◦ to rolling direction; (**d**) 90◦ to rolling direction.

**Figure 7.** Comparison of deep drawing force between FEM simulation and experimental results (DR: 2.25; Ø initial blank: 90 mm).

### *4.2. Comparison of Material Flow Analysis between Conventional Die and MDR Die Applications*

The principle of MDR die application, as aforementioned, was clearly characterized based on the material flow obtained by FEM simulation. As shown in Figure 8, the comparison of material flow between conventional and MDR die applications was illustrated. For the deep drawing stroke of approximately 17 mm, the material flow showed that the same manner in both cases of conventional and MDR die applications could be observed as shown in Figure 8a. These results corresponded well with deep drawing theory [1]. For the deep drawing stroke of approximately 25 mm, in the case of conventional die application as shown in Figure 8b-1, the effects of the anisotropy property of the material on material flow were clearly illustrated. The non-axisymmetric material flow on flange was clearly illustrated as depicted by dashed lines. These results corresponded well with deep drawing theory [1]. By contrast, in the case of MDR die application as shown in Figure 8b-2, the effects of the anisotropy property on material flow were compensated by multi draw radius especially on the large draw radius zone. However, owing to that there were large radius zones formed on MDR die, it was observed that the material flow velocity in the case of MDR die application was larger than that in the case of conventional die application especially on the large radius zone of MDR die. Moreover, the reduction of the non-axisymmetric material flow characteristic on flange could be obtained and clearly illustrated as depicted by dashed lines. Next, the deep drawing stroke was increased to approximately 37 mm, as shown in Figure 8c, and the effects of the anisotropy property of the material on material flow were clearly evidenced in the case of conventional die application as shown in Figure 8c-1. It was clearly observed that the non-axisymmetric material flow on flange was clearly illustrated as depicted by dashed lines. The flange shape was not a circular but had become square. By contrast, in the case of MDR die application as shown in Figure 8c-2, the effects of the anisotropy property of the material on material flow were continuously compensated by multi draw radius. The reduction of the non-axisymmetric material flow characteristic on flange and the asymmetry of the flange were clearly illustrated. The flange was in a more circular shape. Finally, in the case of conventional die application, the effects of the anisotropy property of the material on material flow were stronger as the deep drawing stroke increased as shown in Figure 8d-1. Vice versa, in the case of MDR die application, the effects of the anisotropy property of the material on material flow were continuously compensated by multi draw radius. The reduction of the non-axisymmetric material flow characteristic on flange were clearly illustrated as well as that the asymmetry of the flange could be continuously reduced and the flange was more circular in shape as shown in Figure 8d-2. These results revealed that by using MDR die application, the effects of the anisotropy property of the material on material flow could be compensated during the deep drawing process. The non-axisymmetric material flow characteristic on the flange and the asymmetry of the flange could be reduced during the deep drawing process. This resulted in preventing cup wall stretching and fracture as well as that the LDR could be increased.

**Figure 8.** Comparison of the deep drawing mechanism between conventional and multi draw radius (MDR) die applications: (**a**) Die stroke 17 mm; (**b**) Die stroke 25 mm; (**c**) Die stroke 37 mm; (**d**) Die stroke 45 mm.

### *4.3. MDR Die Design Related to the Anisotropy Property of the Material*

As mentioned in the previous section, the results illustrated that the effects of the anisotropy property of the material on material flow were related to the draw radius. Therefore, the MDR die should be strictly designed by relating to the anisotropy property of the material in each direction along the plane, at 45◦, and at 90◦ to the rolling direction. Figure 9b,c show the deep drawn parts obtained by MDR die application related to the anisotropy property of the material. Figure 9b shows the MDR die application by designing the larger draw radius positioned for at 45◦ to the rolling direction and the small draw radius positioned for along the plane and at 90◦ to the rolling (named MDR type I). By contrast, Figure 9c shows the MDR die application by designing the larger draw radius positioned for along the plane and at 90◦ to the rolling direction and the small draw radius positioned for at 45◦ to the rolling (named MDR type II). The results showed that, in the case of LDR 2.25 (initial blank diameter of 90 mm), the deep drawn parts could be formed by conventional die application as shown in Figure 9a-1. This result corresponded well with the deep drawing theory and literature that by using conventional die application, the deep drawn part could be formed with LDR [1]. The results also showed that the deep drawn parts could be formed by using MDR die application in both cases of MDR die designs as shown in Figure 9b-1,c-1. However, it was observed that in terms of earing defect, the MDR type II showed a larger earing defect than that of MDR type I as well as than that of conventional die application. In addition, it was also observed that the earing defect obtained by using MDR type I was smaller than that of conventional die application. As these results show, to deep draw with LDR, the deep drawn parts could be achieved by using MDR die application. In addition, the quality of deep drawn parts obtained by MDR type I in terms of earing defects was better than that obtained by conventional die. The results illustrate that the MDR die design should be strictly considered as related to the anisotropy property of the material in each direction along the plane, at 45◦, and at 90◦ to the rolling direction. Next, to deep draw over LDR, the result showed that by using conventional die application, the deep drawn part could not be achieved, and a fracture was generated as shown in Figure 9a-2. Based on the FLD, the FEM simulation result clearly showed that the fracture characteristic is formed as a circumferential character. This result agreed well with deep drawing theory [1]. By contrast, in using the MDR die application, the results illustrated that the deep drawn part could be achieved by MDR die type I application as shown in Figure 9b-2. However, using MDR die type II, the deep drawn part could not be achieved. The six ears were also characterized on the basis of FLD, and a fracture was also generated on the top of the deep drawn part as shown in Figure 9c-2. Owing to the anisotropy property of the material in each direction along the plane, at 45◦, and at 90◦ to the rolling direction related to the formability [1], therefore, the design of the larger draw radius positioned at 45◦ to the rolling direction and the smaller draw radius positioned along the plane and at 90◦ to the rolling direction was suggested. As shown in Figure 10, the MDR die type II resulted in that the larger non-axisymmetric material flow on flange was formed compared with those in the cases of conventional and MDR die type I applications. As these material flow analyses show, as aforementioned, the non-axisymmetric material flow characteristic on flange and the asymmetry of the flange could be increased and cup wall stretching and fracture were then easier to generate.

**Figure 9.** MDR die design related to the anisotropy property of the material: (**a**) Conventional die (Radius 3.5 mm); (**b**) MDR type I; (**c**) MDR type II.

### *4.4. Examination of Drawing Ratio with Respect to Multi Draw Radius Dies*

Figure 11 shows the obtained deep drawn parts with respect to various MDR dies and drawing ratios. The drawing ratios of 2.75 and 2.88 which were larger than LDR were investigated, as respectively shown in Figure 11a,b. On the basis of deep drawing theory [1], the draw radius of 3.5 mm was recommended and then it was set as a small draw radius in MDR die. Next, the large draw radius values of 5, 7, and 9 mm were set as the large draw radius. With the drawing ratio of 2.75 as shown in Figure 11a, the results showed that the deep drawn parts could not be achieved when the large radius of 5 mm was set, as shown in Figure 11a-1. This result could be explained by the draw radius of 5 mm, set as the large radius, was too small to reduce the non-axisymmetric material flow characteristic during the deep drawing process. Conversely, as the large radius was increased, the greater reduction of non-axisymmetric material flow characteristic could be achieved, and then the deep drawn parts could be achieved as for the large radius values set as 7 and 9 mm, as shown in Figure 11a-2,a-3, respectively. The increases in large radius resulted in that, as aforementioned, the non-axisymmetric material flow characteristic on the flange was reduced, and the asymmetry of flange could also be reduced. This resulted in the more circular flange shape. However, it was also observed that owing to the large radius of 9 mm, the non-axisymmetric material flow characteristic on the flange could be reduced, getting a more circular flange shape than that for the large radius of 7 mm during the deep drawing process. The earing defect in the case of large radius 9 mm was smaller than that obtained in the case of the large radius of 7 mm. Next, with the drawing ratio of 2.88 as shown in Figure 11b, the results showed that the deep drawn parts could not be achieved. Namely, owing to the overly large drawing ratio (overly large initial blank diameter) applied, the non-axisymmetric material flow on the flange could not be effectively reduced during a whole deep drawing process. These results revealed that in the present research, the LDR could be increased by approximately 22.22% using MDR die application. In addition to the increases in LDR, the quality of deep drawn parts in terms of cup wall thickness and earing defects were also increased. Specifically, the earing defect could be reduced approximately 40% compared with the use of conventional die as shown in Figures 9 and 11. Next, the more uniform cup wall thickness in each direction along the plane, at 45◦, and at 90◦ to the rolling direction could be obtained by comparing with the use of conventional die as shown in Figure 12. However, the MDR die should be strictly design related to the anisotropy property of the material in each direction along the plane, at 45◦, and at 90◦ to the rolling direction which was suggested in the previous section.

### *4.5. Confirmation of MDR Die Application*

To validate the accuracy of the MDR die application obtained by FEM simulation, the FEM simulation results were compared with those obtained by experimental results, as shown in Figure 13. The FEM simulation results showed that the predicted deep drawn parts corresponded well with the experiments as shown in Figure 13a-1,b-1 in the cases of MDR draw radius of 3.5–7 and 3.5–9 mm, respectively. In terms of cup wall thickness, the FEM simulation results showed that the predicted cup wall thickness corresponded well with the experiments as shown in Figure 13a-2,b-2 in the cases of MDR draw radius of 3.5–7 and 3.5–9 mm, in which the errors in the analyzed cup wall thickness were approximately 3% compared with the experimental results.

**Figure 11.** Multi draw radius die application with respect to various DRs: (**a**) DR: 2.75 (Ø Initial blank 110 mm); (**b**) DR: 2.88 (Ø Initial blank 115 mm).

**Figure 12.** Comparison of cup wall thickness between conventional and MDR die applications obtained by FEM simulation (DR: 2.25; initial blank diameter: 90 mm). (**a**) Conventional die (Radius 3.5 mm); (**b**) Multi draw radius die type I (Radius 3.5–7 mm).

**Figure 13.** *Cont.*

**Figure 13.** Comparison of theoretical and experimental results of multi draw radius die. (DR: 2.75; Ø initial blank: 110 mm): (**a**) Draw radius 3.5-7 mm; (**b**) Draw radius 3.5-9 mm.

### **5. Conclusions**

To increase the drawing ratio and overcome the LDR, the MDR die application was proposed in the present research. First, the conceptual design of MDR die was proposed, and its principle was also clearly elucidated in the present research by FEM simulation based on the material flow. The absolute validation of FEM simulation use was also performed. By using MDR die application, it was revealed that the non-axisymmetric material flow characteristic on the flange as well as the asymmetry of flange shape could be reduced. Specifically, during deep drawing process, the MDR could compensate the effects of the anisotropy property of the material on material flow characteristics in each direction along the plane, at 45◦, and at 90◦ to the rolling direction as well as that the non-axisymmetric material flow characteristic on flange could be effectively reduced and, by reducing the asymmetry of flange, a more circular flange shape could be obtained. Therefore, wall cup stretching could be reduced as well as the delay of fracture. Based on this principle, the LDR could be increased. However, in the present research, the proper MDR die design related to the anisotropy property of the material in each direction along the plane, at 45◦, and at 90◦ to the rolling direction was suggested. Specifically, the larger draw radius positioned for at 45◦ to the rolling direction and the smaller draw radius positioned for along the plane and at 90◦ to the rolling direction were recommended. Again, the experiments were also carried out to validate the FEM simulation results in the case of MDR die application. The FEM simulation

results showed that the predicted deep drawn parts corresponded well with the experimental results. In addition, the FEM simulation results showed that the predicted cup wall thickness corresponded well with the experiments, in which the errors in the analyzed cup wall thickness were approximately 3% compared with the experimental results. The results of the present research reveal that the LDR could be increased approximately 22.22% using MDR die application. In addition to the increases in LDR, in terms of quality of deep drawn part, the deep drawn parts obtained by MDR die application showed a smaller earing defect compared with those obtained by conventional die. The decreases in earing defect of approximately 40% could be achieved as well as the more uniform cup wall thickness in each direction along the plane, at 45◦, and at 90◦ to the rolling direction could be obtained. However, the MDR die should be strictly design related to the anisotropy property of the material in each direction along the plane, at 45◦, and at 90◦ to the rolling direction as suggested in the present research.

**Author Contributions:** Conceptualization, W.P. and S.T.; data curation, W.P.; funding acquisition, W.P.; investigation, W.P.; methodology, W.P. and S.T.; project administration, W.P.; supervision, S.T.; writing—original draft, W.P.; writing—review & editing, S.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by a grant from The Thailand Research Fund (TRF) and Rajamangala University of Technology Rattanakosin under Grant No. MRG6280205.

**Acknowledgments:** The authors would especially like to thank Rudeemas Jankree and Jaksawat Sriborwornmongkol, for their help in this research. The authors would like to express our gratitude to Pravitr Paramaputi, Srisahawattanakij Co. Ltd., for his support in the Cimatron v.3 program.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Prediction and Control Technology of Stainless Steel Quarter Buckle in Hot Rolling**

### **Hui Li 1, Chihuan Yao 1, Jian Shao 1,\*, Anrui He 1, Zhou Zhou <sup>2</sup> and Weigang Li <sup>3</sup>**


Received: 12 July 2020; Accepted: 4 August 2020; Published: 6 August 2020

**Abstract:** To obtain the good flatness of hot-rolled stainless-steel strips, high-precision shape control has always been the focus of research. In hot rolling, the quadratic wave (centre buckle and edge wave) can usually be controlled effectively. However, the quarter buckle of the strip is still a challenge to solve. In this paper, a prediction model of roll deflection and material flow is established to study the change rule and control technology of the quarter buckle. The effects of the process parameters on the quarter buckle are analysed quantitatively. The process parameters affect the quarter buckle and the quadratic wave simultaneously. This coupling makes the control of the quarter buckle difficult. The distribution of lateral temperature and the quartic crown of the strip have less effect on the quadratic wave but have a great effect on the quarter buckle. Finally, a new technology for work roll contour is developed to improve the quarter buckle. Through industrial application, the model and the new contour are proved to be effective.

**Keywords:** quarter buckle; roll stack deflection; strip material flow; roll contour optimisation; hot-rolled stainless steel

### **1. Introduction**

Hot-rolled strips, especially of stainless steel, are commonly used directly in product processing. The strip should have a good flatness to meet the requirement of users. He et al. [1] divided strip flatness defects into three types: the linear wave, the quadratic wave, and the high-order wave, according to the patterns of these waves in the width direction of the strip. The linear wave is also called the single-edge wave, which is adjusted by roll tilting. The quadratic wave refers to the centre buckle and the edge wave, which can be controlled by the roll bending system. The high-order wave cannot be characterised by linear or quadratic equations. In the production of hot-rolled stainless steel, high-order wave primarily include the flatness defect of the bilateral symmetrical quarter buckle. Figure 1 shows the quarter buckle of the strip after rolling detected by the flatness meter in a 1450 mm hot-rolled stainless steel line. It not only causes the strip edge to easily crack during hot rolling, or even pile-up accidents, but also affects the surface quality of the products in subsequent processes such as pickling and annealing. It is a challenge for researchers to overcome this problem.

**Figure 1.** Quarter buckle of the strip after rolling: (**a**) flatness measured by on-line meter; (**b**) diagram of the quarter buckle.

The essence of the flatness problem has been widely recognised. Zhang et al. [2] described the generation mechanism of flatness when studying the shape control of a temper mill. In the process of the plastic deformation of the strip, when the distribution of the longitudinal extension difference along the strip width is uneven, residual internal stress is generated, including positive tensile stress and negative compressive stress. When the difference of the internal stresses exceeds the ultimate load of the strip buckling, a visible wave will appear. This can also explain the generation of the quarter buckle of hot-rolled stainless steel.

Some researchers believe that the roll wear and the grinding error of the roll contour cause the quarter buckle. Du et al. [3] heated the local position of the roll to compensate for the uneven roll wear to eliminate the quarter buckle. Li et al. [4] believed the uneven wear of the back-up roll and the low wear of the work roll to have little effect on the strip flatness, but that the large grinding error of the roll contour caused the quarter buckle. In production, we find that within the working period of the work roll, the quarter buckle appears at all stages. Meanwhile, with the use of high-precision roll grinders and the improvement of roll grinding technology, the grinding accuracy of the roll contour can usually be well controlled. He et al. [5] proposed intelligent roll precision grinding and management technology. Hence, the effects of these on the quarter buckle are ignored in this paper. Under some process parameters, the longitudinal strain distribution of the quarter-buckle after strip rolling is related to the high-order deflection of the loaded roll gap profile. By calculating the deformations of the roll stack and the strip, the distribution of the strip longitudinal strain can be obtained to help study how the changes of process parameters affect the quarter buckle.

Nowadays, with the development of computer performance, the finite element method (FEM) has become popular. Jiang et al. [6] used rigid-plastic FEM to study the friction in thin strip rolling. The study was based on the strip deformation, and the effect of the roll stack was simplified. Chandra et al. [7] analysed the temper rolling process using rigid-plastic FEM. The calculation of roll stack deformation is optimised. Wang et al. [8] studied the effect of roll contour configuration on the flatness of hot-rolled thin strip using two-dimensional varying thickness FEM. The elastic deformation of the roll stack is calculated by this method, which is a simplification of the three-dimensional FEM. The optimisation of the roll contour reduced the concentration of contact stress between the back-up roll and the work roll and improved the flatness control of the bending force. Kong et al. [9] built an elastic-plastic FEM model for the calculation of roll thermal expansion, which improved the calculation accuracy. Kim et al. [10] established a three-dimensional FEM model of thermal–mechanical coupling for the research of lubrication rolling. Hao et al. [11] built an FEM model of aluminium alloy rolling by using the ANSYS software and calculated the deformations of the roll stack and the strip. The calculation accuracy of the FEM has been recognised by many researchers. However, it has the shortcomings of longer calculation and difficulty of attaining convergence. The modelling process with multiple

constraints and iterative calculations is complex and difficult to achieve for the commercial finite element software.

The influence function method has high accuracy and short calculation time, which is convenient for on-line applications. Jiang et al. [12] analysed the contact of roll edge during cold rolling of the thin strip using the influence function method. Hao et al. [13] used the influence function method to calculate the deformation of the roll stack for aluminium alloys. Wang et al. [14] described the distributions of the contact pressure between the rolls and the rolling force using a high-order polynomial and achieved a fast calculation of the influence function method. In the influence function method, the stress and displacement fields are obtained from the influence function superposition, so it is not suitable for calculating the plastic deformation of the strip. Wang et al. [15] solved the lateral metal flow of the strip in the hot rolling process using the variational method. Lian et al. [16] believed that the accuracy of the variational method depended too much on assumptions. The three-dimensional difference method was adopted to calculate the strip deformation. Zhang et al. [17] built a rigid-plastic FEM model to calculate the strip deformation, which was combined with the influence function method to get the lateral distribution of the strip thickness. Shao et al. [18] analysed the limitations of the finite difference method, and used the finite volume method to solve the plastic deformation of the strip. Analysis of the quarter buckle with different process parameters requires a large number of simulations. Considering the time cost of the FEM model, in this paper, the rapid influence function method and the finite volume method are used to calculate the distribution of the strip's longitudinal strain after rolling.

The key to solving the problem of the quarter buckle is to make up the high-order loaded roll gap profile and improve the distribution of the strip's longitudinal strain after rolling. In cold rolling, Ma et al. [19] added the roll bending system of the intermediate and work rolls for the researched universal crown mill. In this way, the high-order deflection of the loaded roll gap profile can be adjusted. Guo et al. [20] studied the relationship between the spray cooling of the work roll and the strip cross section. Wang et al. [21] explored the local heat transfer characteristic of the spray cooling of the work roll. Although spray cooling is a solution for local higher-order waves, the effect of changing the local crown of the work roll by heat transfer is slow. Li et al. [22] designed the Baosteel universal roll contour for the intermediate rolls, which was used to adjust the quarter buckle in cooperation with roll shifting. Li et al. [23] upgraded the original roll contour of the continually variable crown to a new roll contour of the quintic curve, which can improve the quarter buckle. Seilinger et al. [24] reduced the quarter buckle by adjusting the sinusoidal component of the roll contour of Smart Crown. In addition, Hara et al. [25] showed the quarter buckle could be corrected by using a partial concave profile to the first intermediate rolls for the Sendzimir mill. Kubo et al. [26] improved the quarter buckle for the Sendzimir mill by using the flexible shaft backing assemblies and the concave roll contour. In the hot rolling, there have been limited reports about the applications of the continually variable crown plus and the Smart Crown. When the quarter buckle is adjusted, the quadratic crown of the loaded roll gap profile is also changed. Hence, it is difficult for them to control the quarter buckle individually, which makes their application more difficult. Therefore, it is of great significance to develop a new control technology for the quarter buckle in the hot rolling.

In this paper, to study the change rule and control technology of the quarter buckle of hot-rolled stainless steel, a prediction model of roll deflection and material flow (RDMF) is established. The rapid influence function method and the finite volume method are used to achieve the iterative computations between the elastic deformation of the roll stack and the three-dimensional plastic deformation of the strip. The model is used to quantitatively analyse the effect of the shape process parameters on the quarter buckle, such as the bending force, the rolling force, the lateral temperature distribution of the strip, the quartic crown of the strip before rolling and so forth. Finally, we develop a new work roll contour to improve the quarter buckle, which is called the middle variable crown (MVC).

### **2. Methods and Materials**

The RDMF model that is built to research the change rule of the quarter buckle incorporates the roll stack deflection and the strip material flow. The former is used to solve the loaded roll gap profile, while the latter can obtain the metal flow and stress distribution. The iterative calculations are performed between them. The calculation process of the RDMF model is shown in Figure 2.

**Figure 2.** Flow chart of the RDMF model.

### *2.1. Roll Bending*

Because of the symmetry of the rolling system, the calculation process involves the upper rolls. According to the load characteristic of the rolls, both the work roll and the back-up roll can be abstracted into a simply supported beam, respectively. The simplified mechanical model of the upper rolls is shown in Figure 3.

**Figure 3.** Simplified mechanical model of the upper rolls.

In Figure 3, DS represents the driving side of the mill and OS represents the operating side. *xcd*, *xbd* are the positions of the rolling force and bending force of DS. *xco*, *xbo* are those of OS. The equilibrium relation in the *y* direction is as follows:

$$\begin{cases} F\_{cd} + F\_{co} + \int\_{\mathbf{x}\_{cd}}^{\mathbf{x}\_{cp}} -q(\mathbf{x})d\mathbf{x} = 0\\ \int\_{\mathbf{x}\_{cd}}^{\mathbf{x}\_{co}} \int\_{\mathbf{x}\_{cd}}^{\mathbf{x}\_{co}} -q(\mathbf{x})d\mathbf{x}d\mathbf{x} - F\_{co}L\_{\mathbf{F}c} + M\_{cd} + M\_{c0} = 0 \end{cases} \tag{1}$$

where *q*(*x*) is the distribution of the contact pressure between the rolls, *Fcd* is the rolling force of DS and *Fco* is that of OS, *LFc* is the distance between the rolling forces, *Mcd* is the bending moment of DS for back-up roll and *Mco* is that of OS.

For the work roll:

$$\begin{cases} F\_{bd} + F\_{bv} + \int\_{x\_{bd}}^{x\_{bv}} [p(\mathbf{x}) - q(\mathbf{x})] d\mathbf{x} = 0\\ \int\_{x\_{bd}}^{x\_{bv}} \int\_{x\_{bd}}^{x\_{bv}} [p(\mathbf{x}) - q(\mathbf{x})] d\mathbf{x} d\mathbf{x} - F\_{bv}(L\_{Fb} \pm S\_w) + M\_{bd} + M\_{bo} = 0 \end{cases} \tag{2}$$

where *p*(*x*) is the distribution of rolling force, *Fbd* is the bending force of DS and *Fbo* is that of OS, *LFb* is the distance between the bending forces, *Sw* is the shifting value of the work roll in the axial direction. We define the shifting in the direction of OS to be positive. *Mbd* is the bending moment of DS for work roll and *Mbo* is that of OS.

The roll barrel is discretely divided into *N* slab elements with the same spacing Δ*x*, as shown in Figure 4. The coordinate *xi* of the roll barrel is:

$$\mathbf{x}\_{i} = \mathbf{i} \times \Delta \mathbf{x} \quad \mathbf{i} = \mathbf{1}, \ \mathbf{2}, \ \dots, \ \mathbf{N} \tag{3}$$

**Figure 4.** Discretisation of the back-up roll and loads.

According to the bending theory of the beam, the total bending deformation of the roll is expressed by the following equation:

$$y\_i = y\_M^i + y\_Q^i \tag{4}$$

where *yi* is the total bending deformation at element *i*, *y<sup>i</sup> <sup>M</sup>* is the deformation at element *i* caused by the bending moment, *y<sup>i</sup> <sup>Q</sup>* is the deformation at element *<sup>i</sup>* caused by the shearing force. *yi <sup>M</sup>* and *<sup>y</sup><sup>i</sup> <sup>Q</sup>* can be expressed as:

$$\begin{cases} \frac{d^4 y\_M^i}{dx^4} = \frac{\varphi\_i}{EI} \\\frac{d^2 y\_Q^i}{dx^2} = 2k\_{sf}(1+\mu)\frac{\varphi\_i}{GS} \end{cases} \tag{5}$$

where ϕ*<sup>i</sup>* is the concentrated force at element *i*, *E* is Young's modulus, *I* is the polar moment of inertia, *ksf* is the shear factor of the cylindrical beam, μ is Poisson's ratio, *G* is the shear elastic modulus, *S* is the cross-section area of the roll.

According to the boundary conditions of the simplified mechanical model of the rolls and the discretisation of the rolls, the total bending deformation at each element can be derived as:

$$y\_{i+1} = y\_i + \frac{M\_1 + M\_2 + \dots + M\_i}{EI} (\Delta x)^2 + \frac{M\_{i+1}}{2EI} (\Delta x)^2 + \frac{k\_{sf}q\_i}{GS} (\Delta x) \tag{6}$$

### *2.2. Roll Flattening*

Based on the Hertz theory, the contact width of an element in the contact zone between the back-up roll and the work roll is assumed to be *b*:

$$b = \sqrt{\frac{8}{\pi}q \left(\frac{1-\mu^2}{E}\right) \frac{R\_1 R\_2}{R\_1 + R\_2}}\tag{7}$$

where *q* is the specific contact pressure between the rolls at the element, *R*<sup>1</sup> is the radius of the work roll and *R*<sup>2</sup> is that of the back-up roll.

As shown in Figure 5, the proximity of the axes of the back-up roll and the work roll caused by the contact flattening is expressed as *w*:

$$w = \frac{2q\left(1 - \mu^2\right)}{\pi E} \left(\frac{2}{3} + \ln \frac{2R\_1}{b} + \ln \frac{2R\_2}{b}\right) \tag{8}$$

$$\left<\frac{}{R\_2}\right>\tag{9}$$

**Figure 5.** Schematic diagram of contact flattening between the back-up roll and the work roll.

Combined with Equations (7) and (8), this can be obtained as:

$$w = \frac{2q\left(1 - \mu^2\right)}{\pi E} \left( \ln \frac{\pi^3 \sqrt{q^2} E}{4(1 - \mu^2)} + \ln(2R\_1 + 2R\_2) - \ln q \right) \tag{9}$$

Because of the coupling relationship between the flattening *w* and the specific contact pressure *q*, we fit the functions by the quartic polynomial, respectively. The two expressions are repeatedly iterated and corrected during the calculation of the elastic deformation of the rolls.

$$w = a\_0 + a\_1 q + a\_2 q^2 + a\_3 q^3 + a\_4 q^4 \tag{10}$$

$$q = b\_0 + b\_1 w + b\_2 w^2 + b\_3 w^3 + b\_4 w^4 \tag{11}$$

where *ai*, *bi* are the corresponding polynomial coefficients *i* = 0, 1, 2, 3, 4.

The contact flattening equation of the infinite cylinder is not suitable for the calculation of the flattening between the work roll and the strip. Therefore, the influence function method is used in this paper. The work roll is equidistantly discretised in the direction of the roll barrel according to the above discretisation method. The strip is divided into n slab elements, as shown in Figure 6. The discretised coordinates are expressed as:

$$\begin{cases} n = \frac{\overline{B}}{\Delta x} \\ x\_i = \dot{i} \cdot \Delta x - \frac{\Delta x}{2} \\ x\_j = \dot{j} \cdot \Delta x - \frac{\Delta x}{2} \end{cases} \tag{12}$$

where *B* is half the strip width, *xi* is the coordinate of the roll barrel at element *i*, *xj* is the coordinate of the strip at element *j*.

**Figure 6.** Schematic diagram of contact flattening between the work roll and the strip.

At the coordinate *xi* of the roll barrel, the flattening *ws*(*xi*) between the work roll and the strip is:

$$w\_s(\mathbf{x}\_i) = \frac{\left(1 - \mu^2\right)}{\pi E} \sum\_{j}^{n} \left[a\_F\right]\_{ij} p(\mathbf{x}\_j) \tag{13}$$

where *p*(*xj*) is the specific rolling force at the coordinate *xj* of the strip, [*aF*]*ij* is the influence coefficient matrix of the elastic flattening of the work roll caused by the specific rolling force.

### *2.3. Strip Material Flow*

The strip material flow model is used to calculate the metal flow and stress distribution so as to provide the lateral distribution of the rolling force for the calculation of the roll stack deflection. Figure 7 shows the stress state of the segment of the strip in the deformation zone. The coordinate axes *x*, *y* and *z* correspond to the strip length, thickness and width direction, respectively. The origin o is located at the cross-section centre of the strip passing through the axis of the work roll.

**Figure 7.** Stress state of the segment of the strip in the deformation zone.

Basic equations include equilibrium differential equation, constitutive equation, the condition of constant volume and the condition of plastic yield. Ignoring the effect of body force, the equilibrium differential equation is:

$$\begin{cases} \frac{\partial \boldsymbol{\upbeta}\_{\rm x}}{\partial \boldsymbol{\upbeta}} + \frac{\partial \boldsymbol{\upbeta}\_{\rm xy}}{\partial \boldsymbol{\upbeta}} + \frac{\partial \boldsymbol{\upbeta}\_{\rm x}}{\partial \boldsymbol{\upbeta}} = \boldsymbol{0} \\\ \frac{\partial \boldsymbol{\upbeta}\_{\rm xy}}{\partial \boldsymbol{\upbeta}} + \frac{\partial \boldsymbol{\upbeta}\_{\rm y}}{\partial \boldsymbol{\upbeta}} + \frac{\partial \boldsymbol{\upbeta}\_{\rm y}}{\partial \boldsymbol{\upbeta}} = \boldsymbol{0} \\\ \frac{\partial \boldsymbol{\upbeta}\_{\rm x}}{\partial \boldsymbol{\upbeta}} + \frac{\partial \boldsymbol{\upbeta}\_{\rm y}}{\partial \boldsymbol{\upbeta}} + \frac{\partial \boldsymbol{\upbeta}\_{\rm z}}{\partial \boldsymbol{\upbeta}} = \boldsymbol{0} \end{cases} \tag{14}$$

where σˆ*<sup>i</sup>* is the normal stress, τˆ*ij* is the shear stress, *i*, *j* = *x*, *y*, *z*.

For the segment of the contact surface between the strip and the work roll, the equation of its upper surface is:

$$
\hat{g} = \frac{1}{2}\hat{h}(\hat{x}, \hat{z})\tag{15}
$$

where ˆ *h* is the strip thickness at the point ( *x*ˆ, *z*ˆ) in the contact surface between the strip and the work roll.

The equilibrium differential equation of the segment of the contact surface is expressed as:

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ σˆ *x* 2 ∂ˆ *h* <sup>∂</sup>*x*<sup>ˆ</sup> <sup>−</sup> <sup>τ</sup>ˆ*xy* <sup>+</sup> <sup>τ</sup>ˆ*xz* 2 ∂ˆ *h* <sup>∂</sup>*z*<sup>ˆ</sup> <sup>+</sup> *<sup>p</sup>*<sup>ˆ</sup> 2 ∂ˆ *h* <sup>∂</sup>*x*<sup>ˆ</sup> + *q*ˆ*<sup>x</sup>* → **n** → **i** = 0 <sup>σ</sup><sup>ˆ</sup> *<sup>y</sup>* <sup>−</sup> <sup>τ</sup>ˆ*xy* 2 ∂ˆ *h* <sup>∂</sup>*x*<sup>ˆ</sup> <sup>−</sup> <sup>τ</sup><sup>ˆ</sup> *yz* 2 ∂ˆ *h* <sup>∂</sup>*z*<sup>ˆ</sup> <sup>+</sup> *<sup>p</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>q</sup>*ˆ*<sup>x</sup>* 2 ∂ˆ *h* ∂*x*ˆ → **n** → **i** − *q*ˆ*z* 2 ∂ˆ *h* ∂*z*ˆ → **n** → **m** = 0 σˆ *z* 2 ∂ˆ *h* <sup>∂</sup>*z*<sup>ˆ</sup> <sup>−</sup> <sup>τ</sup><sup>ˆ</sup> *yz* <sup>+</sup> <sup>τ</sup>ˆ*xz* 2 ∂ˆ *h* <sup>∂</sup>*x*<sup>ˆ</sup> <sup>+</sup> *<sup>p</sup>*<sup>ˆ</sup> 2 ∂ˆ *h* <sup>∂</sup>*z*<sup>ˆ</sup> + *q*ˆ*<sup>z</sup>* → **n** → **i** = 0 (16)

where <sup>→</sup> **<sup>n</sup>** is the outward normal vector of the surface, <sup>→</sup> **i** is the tangent vector of the friction stress in the *x* direction and <sup>→</sup> **m** is that in the *z* direction.

The plastic constitutive equation of the strip describes the relationship between plastic stress and strain. The Levy–Mises incremental constitutive theory is used in this paper:

$$\begin{cases} \begin{aligned} \hat{\lambda}\hat{s}\_{xx} &= \frac{\partial \vartheta}{\partial \dot{x}} & \hat{\lambda}\hat{s}\_{xy} &= \frac{1}{2} \Big( \frac{\partial \vartheta}{\partial \dot{\vartheta}} + \frac{\partial \vartheta}{\partial \dot{x}} \Big) \\ \hat{\lambda}\hat{s}\_{yy} &= \frac{\partial \vartheta}{\partial \dot{\vartheta}} & \hat{\lambda}\hat{s}\_{yz} &= \frac{1}{2} \Big( \frac{\partial \vartheta}{\partial \dot{z}} + \frac{\partial \vartheta}{\partial \dot{\vartheta}} \Big) \\ \hat{\lambda}\hat{s}\_{zz} &= \frac{\partial \vartheta}{\partial \dot{z}} & \hat{\lambda}\hat{s}\_{xz} &= \frac{1}{2} \Big( \frac{\partial \vartheta}{\partial \dot{x}} + \frac{\partial \vartheta}{\partial \dot{z}} \Big) \end{aligned} \tag{17}$$

where *s*ˆ*ij* is the deviator stress, *i*, *j* = *x*, *y*, *z*; λ is the plastic multiplier factor, which changes with the loading process; *u*ˆ, *v*ˆ and *w*ˆ represent the strain rates in the *x*, *y* and *z* directions, respectively.

The equation of constant volume is expressed as:

$$\frac{\partial \hat{\mathfrak{U}}}{\partial \hat{\mathfrak{X}}} + \frac{\partial \hat{\mathfrak{Y}}}{\partial \hat{\mathfrak{Y}}} + \frac{\partial \hat{\mathfrak{U}}}{\partial \hat{\mathfrak{z}}} = 0 \tag{18}$$

Mises yield criterion is applied in the plastic deformation of the strip:

$$\left(\text{s}\_{\text{xx}}^2 + \text{s}\_{yy}^2 + \text{s}\_{zz}^2 + 2\text{ s}\_{xy}^2 + 2\text{ s}\_{xz}^2 + 2\text{ s}\_{yz}^2 = 2\text{\hat{k}}\_s^2 = \frac{2}{3}\text{\hat{o}}\_F^2\tag{19}$$

where ˆ *ks* is the shear strength, σˆ *<sup>F</sup>* is the yield strength.

The ratio of the centreline thickness of the strip ˆ *hr* to the length of contact arc ˆ *lc* is defined as the bite ratio δ, as follows:

$$
\delta = \frac{\hat{h}\_r}{\hat{l}\_c} \tag{20}
$$

According to the asymptotic analysis method, the variables in the material flow model are normalised:

For the dimension variables:

$$\begin{cases} \begin{aligned} \hat{x} &= x\hat{l}\_c \\ \hat{y} &= y\hat{l}\_r \\ \hat{z} &= z\hat{l}\_c \end{aligned} \end{cases} \tag{21}$$

For the stress variables:

$$\begin{cases} \dot{p} = p\hat{k}\_s\\ \hat{s}\_{ij} = s\_{ij}\delta\hat{k}\_s\\ \dot{\sigma}\_i = (s\_{ii}\delta - p)\hat{k}\_s\\ \dot{\tau}\_{ij} = \tau\_{ij}\delta\hat{k}\_s\\ \dot{q}\_i = q\_i\delta\hat{k}\_s \end{cases} \tag{22}$$

*Metals* **2020**, *10*, 1060

For the rate variables:

$$\begin{cases} \begin{aligned} \hat{u} &= \mu \hat{v}\_0 \\ \hat{v} &= v \pounds\_0 \\ \hat{w} &= w \pounds\_0 \end{aligned} \tag{23}$$

For the plastic multiplier factor:

$$
\hat{\lambda} = \lambda \frac{\hat{x}\_0}{\delta \,\hat{k}\_s \hat{l}\_c} \tag{24}
$$

The equilibrium differential equation after normalisation is:

$$\begin{cases} -\frac{\partial p}{\partial x} + \frac{\partial \tau\_{xy}}{\partial y} + \delta \frac{\partial \epsilon\_{xx}}{\partial x} + \delta \frac{\partial \tau\_{xz}}{\partial z} = 0\\ -\frac{\partial p}{\partial y} + \delta \frac{\partial \epsilon\_{yy}}{\partial y} + \delta^2 \frac{\partial \tau\_{xy}}{\partial x} + + \delta^2 \frac{\partial \tau\_{yz}}{\partial z} = 0\\ -\frac{\partial p}{\partial z} + \frac{\partial \tau\_{yz}}{\partial y} + \delta \frac{\partial \epsilon\_{xz}}{\partial z} + \delta \frac{\partial \tau\_{xz}}{\partial x} = 0 \end{cases} \tag{25}$$

The strip thickness is much smaller than the radius of the work roll, so the bite ratio δ is much smaller than one. The first-order asymptotic expansion of the equilibrium equation is deduced, neglecting the small items with δ:

$$\begin{cases} \frac{\partial \mathbf{r}\_{xy}^{(0)}}{\partial y} = \frac{\partial p^{(0)} }{\partial x} \\\\ \frac{\partial p^{(0)}}{\partial y} = 0 \\\\ \frac{\partial \mathbf{r}\_{yx}^{(0)}}{\partial y} = \frac{\partial p^{(0)} }{\partial z} \end{cases} \tag{26}$$

Combined with all the basic equations and constraints, the governing equation can be obtained:

$$f\_x h^2 \left(\frac{\partial^2 p}{\partial \mathbf{x}^2} + \frac{\partial^2 p}{\partial z^2}\right) + 2hf\_x \left(\frac{\partial ph}{\partial \mathbf{x}^2} + \frac{\partial ph}{\partial z^2}\right) = \frac{(f\_x + f\_z)v\_0}{h^2} \left(\frac{\partial h}{\partial \mathbf{x}} + f\_{\overline{z}} \frac{\partial h}{\partial \mathbf{z}}\right) \tag{27}$$

In this paper, the finite volume method is used to solve the partial differential Equation (27). Rectangular element meshing is applied to discretise the deformation zone, as shown in Figure 8.

**Figure 8.** Dispersion of the deformation zone of the strip.

In Figure 8, taking the node *A* of the shadow region as an example, the points *c*, *d*, *e* and *f* are the midpoints of the line *AC*, *AD*, *AE* and *AF*, respectively. The nodes *C*, *D*, *E* and *F* are the adjacent nodes of the node *A* in the *x* and *z* directions. Based on the finite volume method, Equation (27) is solved in the control volume. According to the Gauss formula, the following equation can be acquired:

$$\begin{aligned} f\_f & \text{f}\_f \text{h}^3 \left[ \text{S}\_f \left( \frac{\partial p}{\partial \mathbf{x}} \right)\_f - \text{S}\_d \left( \frac{\partial p}{\partial \mathbf{x}} \right)\_d + \text{S}\_\varepsilon \left( \frac{\partial p}{\partial z} \right)\_\varepsilon - \text{S}\_\varepsilon \left( \frac{\partial p}{\partial z} \right)\_\varepsilon \right] + 2\text{h}^2 f\_\mathbf{x} \left[ \text{S}\_f \left( \frac{\partial \text{ph}}{\partial \mathbf{x}} \right)\_f \right. \\ & \left. - \text{S}\_d \left( \frac{\partial p\mathbf{h}}{\partial \mathbf{x}} \right)\_d + \text{S}\_\varepsilon \left( \frac{\partial \text{ph}}{\partial z} \right)\_\varepsilon - \text{S}\_\varepsilon \left( \frac{\partial \text{ph}}{\partial z} \right)\_\varepsilon \right] = \left( f\_\mathbf{x} + f\_\mathbf{z} \right) \text{vo} \left( \frac{\mathbf{S}\_d + \mathbf{S}\_f}{2} + f\_\mathbf{z} \frac{\mathbf{S}\_c + \mathbf{S}\_\varepsilon}{2} \right) \end{aligned} \tag{28}$$

where *Sf* is the area of the control volume at the point *f* in the *x* direction and *Sd* is that at the point d in the *x* direction, *Sc* is the area of the control volume at the point *c* in the *z* direction and *Se* is that at the point *e* in the *z* direction.

Each partial derivative in Equation (28) is obtained by the linear interpolation of stress values at adjacent nodes:

$$\begin{cases} \left(\frac{\partial p}{\partial x}\right)\_f = \frac{P\_F - P\_A}{dx} \quad \left(\frac{\partial p\mathbf{h}}{\partial x}\right)\_f = \frac{P\_F \mathbf{h}\_F - P\_A \mathbf{h}\_A}{dx} \\\\ \left(\frac{\partial p}{\partial z}\right)\_c = \frac{P\_A - P\_C}{d\_{CA}} \quad \left(\frac{\partial p\mathbf{h}}{\partial z}\right)\_c = \frac{P\_A \mathbf{h}\_A - P\_C \mathbf{h}\_C}{d\_{CA}} \\\\ \left(\frac{\partial p}{\partial x}\right)\_d = \frac{P\_A - P\_D}{dx} \quad \left(\frac{\partial p\mathbf{h}}{\partial x}\right)\_d = \frac{P\_A \mathbf{h}\_A - P\_D \mathbf{h}\_D}{dx} \\\\ \left(\frac{\partial p}{\partial z}\right)\_c = \frac{P\_E - P\_A}{d\_{EA}} \quad \left(\frac{\partial p\mathbf{h}}{\partial z}\right)\_c = \frac{P\_E \mathbf{h}\_E - P\_A \mathbf{h}\_A}{d\_{EA}} \end{cases} \tag{29}$$

where *PA*, *PC*, *PD*, *PE* and *PF* are the rolling forces at nodes *A*, *C*, *D*, *E* and *F*, respectively; *dCA* is the length of the line *CA* and *dEA* is that of the line *EA*.

The large sparse linear equation is built as Equation (30). We used the preprocessing method of the incomplete LU decomposition and the bi-conjugate gradient stabilised method to achieve a fast calculation of Equation (30):

$$\mathbf{Q}\_{n \times n} \mathbf{P}\_{n \times 1} = \mathbf{R}\_{n \times n} \tag{30}$$

where **Q***n*×*<sup>n</sup>* is the coefficient matrix, **R***n*×*<sup>n</sup>* is the constant matrix.

### *2.4. Model Verification*

To verify the calculation accuracy of the RDMF model, the last stand of the 1450 mm tandem hot mills in Southwest Stainless Steel Co., Ltd. is taken as the simulation object. The actual parameters of the strip are taken to the RDMF model for numerical simulation and the calculated lateral thickness difference of the strip is compared with the measured one. In addition, considering that the accuracy of FEM is recognised by many researchers, an implicit model of roll-strip coupling is built by using the large-scale commercial finite element software ANSYS [27]. The calculation result of the FEM model is also used to verify the calculation accuracy of the RDMF model. The parameters of the geometrical dimension and the material are listed in Table 1. The FEM model is shown in Figure 9. The element type of the roll stack is SOLID45 and that of the strip is SOLID185. The total number of model elements is 74444, and the number of model nodes is 79185.

**Figure 9.** Finite element model of roll-strip coupling.


**Table 1.** Parameters of the geometrical dimension and the material.

As shown in Figure 10, the distributions of the lateral thickness difference of the strip after rolling are compared. The lateral thickness difference is defined as the difference between the cross section of the strip and the thickness of the strip centre. Taking the midpoint of the strip width as an origin, the normalised coordinate values of the strip width are used as the abscissa axis. It can be found that the calculation results of the RDMF model and the FEM model are basically consistent with the measured distribution of the lateral thickness difference. Without the effect of the edge-drop region, the relative error of the RDMF model is 10.61%, and that of the FEM model is 6.69%. The calculation error of the strip edge may be caused by the friction condition and deformation state in the edge region, but it has a limited effect on the calculation of the strip cross section. Therefore, the calculation accuracy of the RDMF model and the FEM model are able to meet the requirements of actual production. However, the RDMF model has a better solving speed. In the same configuration of calculation (CPU: Intel Core2 Quad Q6600, RAM: 2GB), the RDMF model takes 0.005 s to calculate, while it takes 4.55 h for the FEM model. Because of the complex grade and various specification of the strip in the hot-rolled line, the simulation workload is huge. In the case of ensuring the calculation accuracy and reducing calculation time, the RDMF model proposed in this paper is more practical for the numerical calculation in the hot rolling process.

**Figure 10.** Comparison of the lateral thickness difference.

### **3. Results and Discussion**

Using the proposed RDMF model, a lot of simulated calculations are carried out with different working conditions. Based on the calculation results of the RDMF model, the effects of process parameters on the quarter buckle are analysed. According to the actual production of the last stand of the tandem hot mill, the basic working condition is determined as the control group, as listed in Table 2.


**Table 2.** Parameters of the basic working condition.

### *3.1. E*ff*ect of Bending Force*

The bending system of the work roll is an on-line adjusting method for controlling the flatness. To study the effect of the change of bending force on the quarter buckle, the distributions of the longitudinal strain differences of the strip are calculated during the increase of the bending force from –16 kN to 384 kN based on the control group, as shown in Figure 11.

**Figure 11.** Longitudinal strain difference with different bending forces.

To parameterise the distribution characteristic of the longitudinal strain difference, the curve is fitted to a quartic polynomial, as shown below:

$$
\Delta \varepsilon(\mathbf{x}) = c\_0 + c\_2 \mathbf{x}^2 + c\_4 \mathbf{x}^4 \quad \mathbf{x} \in [-1, 1] \tag{31}
$$

where *x* is the normalised coordinate of the strip width; *ci* is the polynomial coefficient, *i* = 0, 2, 4.

According to the Chebyshev polynomial expansion, Equation (31) can also be expressed as:

$$
\Delta \varepsilon(\mathbf{x}) = d\_0 + d\_2 T\_2(\mathbf{x}) + d\_4 T\_4(\mathbf{x}) \tag{32}
$$

where *di* is the Chebyshev polynomial coefficient, *i* = 0, 2, 4; *T2*(*x*) is the Chebyshev quadratic component; *T*4(*x*) is the Chebyshev quartic component.

According to Equations (31) and (32), the Chebyshev polynomial coefficients can be expressed as:

$$\begin{cases} d\_0 = c\_0 + \frac{1}{2}c\_2 + \frac{3}{8}c\_4 \\\ d\_2 = \frac{1}{2}(c\_2 + c\_4) \\\ d\_4 = \frac{1}{8}c\_4 \end{cases} \tag{33}$$

In the distribution of the longitudinal strain difference, the convexity of the quadratic component is defined as Δε<sup>2</sup> and the convexity of the quartic component is Δε4:

$$\begin{cases} \Delta \varepsilon\_2 = d\_2 T\_2(0) - \frac{1}{2} [d\_2 T\_2(1) + d\_2 T\_2(-1)] = -(c\_2 + c\_4) \\\ \Delta \varepsilon\_4 = \frac{1}{2} [d\_4 T\_4 \left(\frac{\sqrt{2}}{2}\right) + d\_4 T\_4 \left(-\frac{\sqrt{2}}{2}\right)] - d\_4 T\_4(0) = -\frac{c\_4}{4} \end{cases} \tag{34}$$

The values of Δε<sup>2</sup> and Δε<sup>4</sup> with different bending forces are shown in Figure 12. As the bending force increases, both of them increase simultaneously. When the value of Δε<sup>2</sup> is close to zero and the value of Δε<sup>4</sup> is large, there is the mode of a quarter buckle. When the bending forces are 284 and 384 kN, the distributions of longitudinal strain difference are in the mode of a centre buckle. The values of Δε<sup>2</sup> are 108.2 <sup>×</sup> <sup>10</sup>−<sup>5</sup> and 216.7 <sup>×</sup> <sup>10</sup><sup>−</sup>5, the values of <sup>Δ</sup>ε<sup>4</sup> are also large. When the bending forces are <sup>−</sup>16 and 84 kN, the distributions of longitudinal strain difference are in the mode of an edge wave. In addition, the values of <sup>Δ</sup>ε<sup>2</sup> are <sup>−</sup>219.9 <sup>×</sup> <sup>10</sup>−<sup>5</sup> and <sup>−</sup>110.5 <sup>×</sup> <sup>10</sup><sup>−</sup>5, while the values of <sup>Δ</sup>ε<sup>4</sup> are much smaller than that of the centre buckle mode. When the bending force is changed from −16 kN to 384 kN, the quarter buckle mode occurs in the change from the edge wave mode to the centre buckle mode. This is similar to the situation in production: when a quarter buckle appears, increasing the bending force will turn it into a centre buckle, while reducing the bending force will turn it into an edge wave.

**Figure 12.** The values of Δε<sup>2</sup> and Δε<sup>4</sup> with different bending forces.

It is difficult to improve the quarter buckle solely by the bending force. When studying the effect of other parameters on the quarter buckle, it is necessary to decouple the quadratic wave and the quarter buckle, namely, by adjusting the bending force to eliminate the quadratic wave, and then studying the change of the Δε4.

### *3.2. E*ff*ect of Strip Lateral Temperature Di*ff*erence*

In hot rolling, the temperature of the strip edge is lower than the middle. Because the thermal conductivity of stainless steel is poor, the strip lateral temperature difference (Δ*T*) is usually greater than that of plain carbon steel. The uneven temperature distribution of the strip makes the distribution of the rolling force uneven, which is related to the generation of the quarter buckle. The lateral temperature distribution of the strip in the last stand, detected by a thermal imager, is shown in Figure 13. The temperature difference between the middle and the edge of the strip is 50 ◦C. By keeping the

measured value of the temperature in the middle of the strip constant and decreasing the temperature value at the edge of the strip, the distribution curves with lateral temperature differences of 75 ◦C and 100 ◦C were obtained as control groups. In Figure 14, due to the different distribution of the lateral temperature, the deformation resistance of the strip in the width direction is also different.

**Figure 13.** Lateral temperature distribution of the strip in the last stand measured by a thermal imager.

**Figure 14.** Distribution of the deformation resistance with difference values of Δ*T*.

In the simulation of the rolling process, the value of Δε<sup>2</sup> is close to zero by changing the bending force. As the Δ*T* of the strip increases, the bending force required to improve the quadratic wave decreases. The bending force and the Δε<sup>4</sup> are shown in Figure 15. With an increase in Δ*T*, the value of <sup>Δ</sup>ε<sup>4</sup> increases linearly. When <sup>Δ</sup>*T* increased by 10 ◦C, and the value of <sup>Δ</sup>ε<sup>4</sup> increased by 3.1 <sup>×</sup> 10−5. However, the change of bending force is very small with different temperature differences.

**Figure 15.** Bending force and Δε<sup>4</sup> of the strip with different values of Δ*T*.

### *3.3. E*ff*ect of Rolling Force*

Research into the effect of rolling force on the quarter buckle can provide a basis for the optimisation of load distribution. With different distributions of lateral temperature, the effect of rolling force on the quarter buckle is considered. The Δ*T* of the strip is selected to be 50 ◦C, 75 ◦C and 100 ◦C, respectively. Taking 1000 kN as the interval, two rolling force control groups are selected above and below the rolling force of 10,720 kN in Table 2. The calculation results are shown in Figures 16 and 17.

**Figure 16.** The Δε<sup>4</sup> of the strip with different rolling forces.

**Figure 17.** Bending force and the Δε<sup>4</sup> of the strip with different rolling forces (Δ*T* = 50 ◦C).

The results show that at the same Δ*T* the value of Δε<sup>4</sup> decreases linearly with the rolling force increasing, but the bending force required to eliminate the quadratic wave increases gradually. The effect of rolling force on the reduction of the quarter buckle is different with different values of Δ*T*. When the Δ*T* is relatively small, as the rolling force increases, the bending force required to improve the quadratic wave is increased, but the effect of reducing the quarter buckle is more obvious as a whole.

### *3.4. E*ff*ect of Strip Quartic Crown before Rolling*

The key to the shape control of hot-rolled strips is the change of the strip crown distribution before and after rolling. To study the change rule of the quarter buckle of the strip, the effect of the strip crown before rolling cannot be ignored. The strip crown before rolling can be divided into the quadratic crown and the quartic crown. The former is related to the quadratic wave of the strip, and the latter is related to the high-order wave [23]. In this paper, the effect of the strip quartic crown before rolling on the quarter buckle is studied. The quartic crown is set to 0, 2, 4, 6 and 8 μm. Other parameters remain unchanged. The calculation results are shown in Figures 18 and 19.

**Figure 18.** The value of Δε<sup>4</sup> of the strip with different quartic crowns before rolling.

**Figure 19.** Bending force and Δε<sup>4</sup> of the strip with different quartic crowns before rolling (Δ*T* = 50 ◦C).

At the same Δ*T*, the value of Δε<sup>4</sup> gradually increases with the increase of the quartic crown. With different values of Δ*T*, as the quartic crown increases, the values of Δε<sup>4</sup> increase at almost the same rate. This indicates that with different values of Δ*T*, the effect of the quartic crown on the quarter buckle of the strip is only different in the basic value. The magnitude of the quartic crown can reflect the severity of the quarter buckle to a certain extent. According to the comparison of the calculation results, it is found that when the quartic crown changes, the bending force does not change significantly. It has a great impact on the quarter buckle.

### *3.5. E*ff*ect of Back-Up Roll Chamfer Length*

The chamfer of the back-up roll helps to reduce the harmful contact zone between the rolls, and affects the deflection of the roll stack. The chamfer length is an important parameter for the back-up roll contour, which affects the range of the contact zone between the rolls. The chamfer length of the back-up roll is selected to be 50, 100, 150, 200 and 250 mm. The effect on the quarter buckle are studied through simulation.

It can be seen in Figure 20 that at the same Δ*T*, the Δε<sup>4</sup> decreases in a parabola with the increase of the chamfer length, which reduces the possibility of a quarter buckle. At the same time, increasing the chamfer length will reduce the deflection of the roll stack, thereby reducing the strip crown. To compensate for this effect and avoid the quadratic wave, the bending force is significantly reduced, as shown in Figure 21. Therefore, the reduction of Δε<sup>4</sup> in the calculation results from the combined effects of increasing the chamfer length and reducing the bending force.

**Figure 20.** The value of Δε<sup>4</sup> of the strip with different chamfer lengths of back-up roll.

**Figure 21.** Bending force and Δε<sup>4</sup> of the strip with different chamfer lengths of back-up roll (Δ*T* = 50 ◦C).

### **4. Design of MVC and Industrial Application**

Aiming at the quarter buckle of hot-rolled stainless steel, a new technology of work roll contour is developed in this paper. The MVC designed by the superposition of the quadratic curve and the sextic curve not only causes the work roll to have the ability to control the quadratic wave, but also improves the quarter buckle. The coefficient of the quadratic curve is designated by the magnitude of the quadratic wave. The coefficients of the sextic curve are determined by the position and magnitude of the quarter buckle. Then, they will be superimposed in different regions. Assuming the strip width is 2*B* and the roll barrel is 2*L*, the curve equation of MVC is:

$$y(\mathbf{x}) = \begin{cases} \ \mathbf{e}\_2 \mathbf{x}^2 + \left( f \mathbf{x}^2 + f\_4 \mathbf{x}^4 + f\_6 \mathbf{x}^6 \right) & \mathbf{x} \in \left[ -B, \ B \right] \\\ \mathbf{e}\_2 \mathbf{x}^2 & \mathbf{x} \in \left[ -L, B \right) \cup \left( B, L \right] \end{cases} \tag{35}$$

where *e*<sup>2</sup> is the coefficient of the quadratic curve; *fi* is the coefficient of the sextic curve, *i* = 2, 4, 6.

Figure 22a is a diagram of the compensation for the quarter buckle by the roll contour of MVC. To determine the coefficients of the sextic curve, Equation (36) needs to be satisfied. The contour of the sextic curve is shown in Figure 22b. The key is to determine the values of the extreme point *x*<sup>0</sup> and the compensation *y*0. The position *x*<sup>0</sup> is generally determined by the position of the quarter buckle, and the compensation *y*<sup>0</sup> is optimised based on the RDMF model.

$$\begin{cases} f\_2 \mathbf{x}\_0^2 + f\_4 \mathbf{x}\_0^6 + f\_6 \mathbf{x}\_0^6 = y\_0 \\ 2f\_2 + 4f\_4 \mathbf{x}\_0^2 + 6f\_5 \mathbf{x}\_0^4 = 0 \\ f\_2 B^2 + f\_4 B^4 + f\_6 B^6 = 0 \end{cases} \tag{36}$$

**Figure 22.** Compensation for the quarter buckle by the MVC: (**a**) schematic diagram; (**b**) contour of the sextic curve.

Taking the quarter buckle problem (Figure 1) of the 1450 mm hot rolling line in Southwest Stainless Steel as an example, we use the MVC technology to optimise the original contour of the work roll (quadratic parabola, −200 μm). The roll contours before and after the optimisation are depicted in Figure 23.

**Figure 23.** Roll contours before and after optimisation.

According to the actual production, the changes of the Δε<sup>4</sup> with different values of the quartic crown before and after the optimisation are listed in Table 3. When the original contour is used, the Δε<sup>4</sup> increases from 34.9 to 86.2. However, when the optimised contour is used, as the quartic crown of the strip increases, the bending force for adjusting the quadratic wave is little changed, and the Δε<sup>4</sup> can be kept relatively small. To improve the quarter buckle problem, the calculation results show that the optimised contour needs the compensation value of 2.71 to 6.74 μm.

**Table 3.** Changes of the Δε<sup>4</sup> under different values of the quartic crown before and after the optimisation (Δ*T* = 100 ◦C).


In Figure 24a, the cross section of the strip before and after the optimisation is compared. Using the optimised contour, the cross section of the strip is compensated in the region of the quarter buckle, thereby reducing the relative longitudinal strain of the strip in the region. Correspondingly, in Figure 24b, the relative rolling force of the strip in the region is reduced, and the overall distribution of the rolling force is more uniform, which is beneficial for achieving a more uniform reduction.

**Figure 24.** Comparison of calculation results before and after the optimisation (Δ*T* = 100 ◦C, quartic crown is 8 μm): (**a**) lateral thickness difference of the strip; (**b**) specific rolling force.

By comparing the strip flatness detected by the flatness meter before and after the optimisation, the effectiveness of the MVC technology is verified. Figure 25a shows the quarter buckle before solving the problem. With the optimised contour, the quarter buckle of the strip is effectively improved, as shown in Figure 25b. In addition, the research in this paper was successfully applied to other hot rolling lines, as listed in Table 4.

**Figure 25.** Comparison of the strip flatness detected by the on-line meter: (**a**) before optimisation; (**b**) after optimisation.


**Table 4.** Other hot rolling lines with a successful application.

### **5. Conclusions**

To solve the problem of the quarter buckle in the production of hot-rolled strips, the change rule and control technology of the quarter buckle are studied in this paper. The conclusions are as follows:

(1) A model of roll deflection and material flow for predicting the quarter buckle is established. Compared with the finite element method and the measured data, the accuracy of the RDMF model is verified, and is able to meet the requirements of actual production. However, the RDMF model is much faster in terms of calculation speed than the finite element model, which is suitable for online application and performing a large number of offline calculations.

(2) Quantitative analyses of the effects of the shape process parameters on the quarter buckle are carried out using the RDMF model. The coupling relationship between the quarter buckle and the quadratic wave increases the difficulty of the quarter buckle control. By decoupling analysis, it is found that the lateral temperature difference of the strip and the quartic crown of the strip before rolling have a great impact on the quarter buckle, but their effects on the quadratic wave are small. In addition, increasing the rolling force and the chamfer length of the back-up roll could reduce the quarter buckle, but this would also obviously change the bending force and affect the control of the quadratic wave.

(3) A new control technology for the work roll contour, MVC, is developed for the quarter buckle. The roll contour of MVC improves the strip shape by compensating the loaded roll gap profile at the position of the quarter buckle. It not only has the ability to control the quadratic wave, but also effectively improves the quarter buckle. The roll contour of MVC proved to be effective in industrial applications.

**Author Contributions:** H.L. and C.Y. carried out the simulation work; A.H. and Z.Z. conceived and designed the industrial verification of the model; J.S. and A.H. developed the work roll contour; H.L. and C.Y. analysed the data. J.S., W.L. and Z.Z. performed the industrial applications. H.L. wrote the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Fund of China, grant number 51674028; Innovative Method Project of Ministry of Science and Technology of China, grant number 2016IM010300; and Guangxi Special Funding Program for Innovation-Driven Development, grant number GKAA17202008.

**Acknowledgments:** This work was supported by the National Natural Science Fund of China (No.51674028); the Innovative Method Project of Ministry of Science and Technology of China (2016IM010300); and the Guangxi Special Funding Program for Innovation-Driven Development (GKAA17202008).

**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.

### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Finite Element Analysis on Ultrasonic Drawing Process of Fine Titanium Wire**

### **Shen Liu, Xiaobiao Shan, Hengqiang Cao and Tao Xie \***

School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China; joseliu2013@outlook.com (S.L.); shanxiaobiao@hit.edu.cn (X.S.); yzhinv@outlook.com (H.C.)

**\*** Correspondence: xietao@hit.edu.cn; Tel.: +86-451-8641-7891; Fax: +86-451-8641-6119

Received: 31 March 2020; Accepted: 23 April 2020; Published: 28 April 2020

**Abstract:** Ultrasonic drawing is a new technology to reduce the cross-section of a metallic tube, wire or rod by pulling through vibrating dies. The addition of ultrasound is beneficial for reducing the drawing force and enhancing the surface finish of the drawn wire, but the underlying mechanism has not been fully understood. In this paper, an axisymmetric finite element model of the single-pass ultrasonic drawing was established in commercial FEM software based on actual wire length. The multi-linear kinematic hardening (MKINH) model was used to define the elastic and plastic characteristics of titanium. Influences of ultrasonic vibration on the drawing process were investigated in terms of four factors: location of the die, ultrasonic amplitude, drawing velocity, and friction coefficient within the wire-die contact zone. Mises stresses, as well as contact and friction stress, in conventional and ultrasonic drawing conditions, were compared. The results show that larger ultrasonic amplitude and lower drawing velocity contribute to greater drawing force reduction, which agrees with former research. However, their effectiveness is further influenced by the location of the die. When ultrasonic amplitude and drawing speed remain unchanged, the drawing force is minimized when the die locates at the half-wavelength position, while maximized at the quarter-wavelength position.

**Keywords:** ultrasonic drawing; titanium wire; drawing force; finite element analysis; Mises stress; contact stress; work hardening; numerical simulation

### **1. Introduction**

Titanium wires are widely used in various industries, including aerospace, automobile, biomedicine, petrochemistry, fishery, due to their exceptional characteristics, such as high strength, lightweight, good corrosion resistance, excellent biocompatibility, etc. [1–4]. However, unlike many other metallic wires, the manufacturing of titanium wire is usually conducted at elevated temperature, as their cold workability is degraded by high yield stress to tensile strength (Y/T) ratio and the strain hardening phenomenon [5]. The conventional drawing process involves complicated heat preparation, lubrication, and surface treatment, which is neither eco-friendly nor energy-saving. What is more, the wire products often have low dimensional accuracy, poor surface finish, and high breakage ratio [6]. Ultrasonic drawing is a metalworking process to reduce the diameter of a wire, tube, or rod by pulling them through oscillating dies with converging cross-section shapes [7]. Compared with the conventional drawing process, the new technology has the following advantages: reduced drawing force, enhanced surface finish, lower breakage ratio, prolonged tool life, greater area reduction per pass, etc. Therefore, ultrasonic drawing is a promising technology to increase productivity, lower costs, and improve the quality of the products in the industrial manufacturing of metallic wires [8]. The benefits associated with ultrasound are not limited to titanium and the drawing process, but to varied machining processes, such as welding, forging, rolling, and cutting of a wide range of metallic and non-metallic materials like aluminum, steel, and carbon fiber [9–11].

The idea of introducing ultrasonic vibrations into the drawing process was first reported by Blaha et al. in 1955 [12]. They discovered the acoustic softening phenomenon, which is similar to thermal softening effects, with an abrupt 40% reduction in drawing force when ultrasound was superimposed. The observations were explained as the result of activation and increased mobility of dislocations at the lattice defects. In 1968, Winsper et al. reviewed various experimental results on ultrasonic-assisted metal forming processes and pointed out that all these phenomena could be explained by the superposition of an alternating load on a static load, which is later known as stress superposition hypothesis [13]. The above theories triggered numerous studies on the influence of ultrasound in metal forming processes, which remains active to the present day. In 2003, Hayashi et al. compared the influence of longitudinal and radial ultrasonic vibrations on the aluminum wire drawing process using the finite element method (FEM) [14]. It was found that radial vibration yielded superior results than longitudinal vibration with respect to drawing force reduction, surface quality improvement, and critical drawing speed. In 1999, Susan et al. conducted the ultrasonic drawing experiment of steel ball-bearing steel wire. They attributed the drawing force reduction to the surface effect of ultrasound utilizing the friction reversion mechanism [15]. The friction coefficient and the drawing force were calculated according to the relations between three factors: drawing velocity, ultrasonic frequency, and amplitude of the die. In 2004, the proposed calculation method was further revised to calculate the drawing force in the ultrasonic-assisted tube drawing process [16]. They also pointed out that ultrasonic vibration contributes to weakening the strain hardening effect and helps to increase the plasticity of the specimen. In 2009, Qi et al. experimentally examined the effects of longitudinal ultrasonic vibrations in the industrial production of brass wire [17]. Results proved the advantages of ultrasonic drawing over the conventional drawing with a 17% increment in drawing speed, 7% decrease in drawing force, prolonged tension regulation period from 0.5 s to about 1.5 s, and improved surface finish of the drawn wire. In 2012, Shan et al. proposed a new mathematic model to describe the ultrasonic drawing process based on the nonlocal friction theorem and obtained the stress distribution, contact pressure, and drawing force [18]. The ultrasonic anti-friction effect of stainless-steel wire, copper wire, and titanium wires at room temperature were analyzed. In 2016, Yang et al. compared the influences of longitudinal-torsional composite vibration and longitudinal vibration on the titanium wire drawing process through numerical and experimental methods [19]. The results indicated that longitudinal vibration is more beneficial for reducing the friction force and improving the surface finish of the drawn wire than composite vibration. In 2018, Liu et al. performed an experimental study on two-pass titanium wire drawing with two oscillating dies and achieved a drawing force reduction of over 50% [20]. Results showed that increased vibration amplitude leads to greater drawing force decrement, whereas increased drawing velocity brings about the reverse effect. Higher drawing speed with moderate ultrasonic amplitude is more preferable in removing surface defects.

Although the feasibility and advantages of ultrasonic wire drawing have been verified by previous researchers, the underlying mechanisms were explained in different ways. The correctness of presuppositions in quantitative calculations remains to be verified. The recent works mainly focused on experimental research and theoretical studies, whereas little attention was paid to the finite element analysis of the ultrasonic-assisted drawing process. In this paper, the ultrasonic drawing process was described as contact and friction problems between the elastic-plastic traveling string and the rigid vibrating die. The axisymmetric finite element (FE) model was established in commercial FEM software Abaqus, considering the actual length of the wire and the strain hardening characteristics of the material. Influences of ultrasound on the drawing force were discussed and compared in terms of four factors: vibration amplitude, drawing speed, location of the die, and the friction coefficient. Mises stress distribution inside the wire, as well as contact and friction stress within the contact region, was investigated.

### **2. Finite Element (FE) Model Establishment and Simulation Procedure**

The titanium wire drawing process could be explained as the frictional contact problem between a certain length of deformable wire and the internal surfaces of the rigid oscillating die. Considering the inherent symmetry of the geometry and boundary condition, an axisymmetric finite element analysis model was established in commercial software Abaqus (Dassault Systèmes Simulia Corp., Providence, RI, USA) to minimize computational costs, as shown in Figure 1. In this particular problem, the diameter of the raw titanium wire, Ø 0.4 mm, is drawn into Ø 0.36 mm, with a cross-section area reduction of 19%. The generation of heat due to plastic dissipation inside the wire and frictional heat generation within the contact region are not considered, as this paper emphasizes on the independent effects caused by the addition of ultrasound in the general drawing process of long thin metallic wires. The drawn wire is wound up around the drum reel driven by an electrical motor. The length of unwound wire, or the distance between the die and the reel, denoted as viable *L* in the figure, is usually 100~200 mm and should be larger than the radius of the drum, 40 mm, in practice. The flexibility and elastic deformation of slightness titanium wire would exert a remarkable influence on the ultrasonic drawing process, therefore, the uncoiled drawn wire should be modeled with the full length. In contrast, the length of the raw wire, denoted as *l* in the figure, has little influence on simulation results and therefore is assigned as a constant of 10 mm. In the discretization process, titanium wire is defined as a deformable part with an overall meshing size of 0.02 mm, whereas the die is modeled as a rigid body with an element size of 0.03 mm. The two parts are all modeled using the 4-node reduced axisymmetric continuum quadrilateral solid element. Penalty and kinematic formulations are employed in the definition of contact interactions. The contact type between the die and the wire is specified as surface-to-surface contact, and the contact behavior is assumed to abide by Coulomb friction law, with an initial coefficient of 0.1. The simulation is also performed with Arbitrary Lagrangian-Eulerian (ALE) adaptive meshing and enhanced hourglass control.

**Figure 1.** Simulation settings and FE model of single-die ultrasonic wire drawing.

In the meshed model, all the nodes situated on the central axis are constrained at the radial (X) and circumferential (Z) direction. Velocity constraints along the wire drawing direction (−Y) are prescribed to the bottom nodes of the wire near the reel. A reference point NREF is assigned to control the movement of the rigid die. The whole simulation process, which lasts for 5 ms, can be divided into three steps. The duration for the first and the third step is 2 ms, which are separately divided into 100 substeps. In these two steps, the amplitude of the die is assigned as 0 to simulate the conventional drawing process. The second step lasts for 1 ms, which equals to 20 oscillation cycles for, and is divided into 400 substeps, during which ultrasonic vibrations of 20 kHz is applied to the die to stimulate the ultrasonic drawing process. In the study, the frequency of the die remains the same, whereas its amplitude changes from 1 to 10 μm. The variation range of the drawing speed and the distance from the die to the reel are 100~1200 mm/s and 15.8~126.4 mm, respectively. The friction coefficient varies from 0.1 to 0.5. The wire drawing forces could be captured by extracting the reaction force along the Y direction at the NREF point.

In the simulation, the material of wire is selected as TA2 commercial pure titanium, the density, elastic modulus, and Poisson ratio of which are 4500 kg/m3, 115 GPa, and 0.37, respectively. The wire drawing die mainly consists of two sections: a stainless casing and the diamond nib [21]. In the FE model, only the nib section, which directly contacts the wire, is considered. The diamond mandrel, with 3520 kg/m3, 1100 GPa, 0.07, respectively, in density, elastic modulus, and Poisson ratio, could be divided into three areas: the reduction area with the semi-cone angle of 7◦, the bearing area, and the exit area. Plastic deformation mainly occurs in the reduction area; however, the bearing area is essential for maintaining the dimensional accuracy of the wire products. The stress-strain curve of TA2, as well as the geometry and dimension of the diamond nib, is shown in Figure 2. The plastic deformation of the titanium is assumed to follow the Mises yield criterion, and a multi-linear kinematic hardening (MKINH) model was employed to describe the plastic behavior of the material [22]. Considering the changing stress state at the wire-die interface caused by the oscillation of the die, the simulation process involves not only material nonlinearity, but also nonlinear geometry and boundary conditions; therefore, the transient analysis is conduct in the Abaqus/Explicit module to improve the efficiency and convergence of the calculation.

**Figure 2.** Material property and geometry definition of the die specified in the simulation program: (**a**) Stress-strain curve of TA2 commercial pure titanium in FE simulation; (**b**) Structure and dimensions about the diamond nib of the wire drawing die.

### **3. Results and Discussion**

Based on the above finite element analysis (FEA) model, influences of ultrasonic vibration on titanium wire drawing process are investigated in terms of four factors: the location of the die, the vibration amplitude of the die, the wire drawing velocity, and the friction coefficient. The relationships between each factor and the averaged drawing force are analyzed using the control variates technique. The internal equivalent stress distribution of titanium wire, as well as the contact

and friction stress distribution at the die-wire interface, is also displayed to understand the underlying mechanism of the ultrasonic drawing process.

### *3.1. Influence of Die Location on Drawing Force*

In previous research, Hayashi et al. built an FEA model and discussed the influence of vibration amplitude and direction on drawing force and Mises stress distribution inside the wire when drawn with the assistance of axial and radial ultrasonic vibrations [14]. However, the established model was confined to a very narrow region of 1~2 mm in length adjacent to the die. The flexibility and stretching of the elastic slender wire are neglected. In industrial production, the wire drawing velocity is provided by the reel drum, which rotates at a constant speed. For the conventional drawing process, the velocity of the wire adjacent to the reel drum is equal to that near the die. However, for the ultrasonic wire drawing process, when the die oscillates periodically, they are not equal, as the uncoiled drawn wire would be tightened and relaxed intermittently along with the die. The influence of this neglected factor on the drawing process is determined by the length of uncoiled drawn wire, i.e., the distance between the reel drum and the wire.

To investigate the influence of reel-die distance *L* on the ultrasonic drawing process, the other factors maintain constant, with ultrasonic amplitude, drawing speed, and friction factor specified as 10 μm, 300 mm/s, and 0.1, respectively. As the drawn wire is forced to vibrate and its natural frequency is determined by the length, simulations are conducted when the wire length amounts to λ/8, λ/4, 3λ/8, and λ/2, respectively, where λ denotes the wavelength of ultrasonic vibration when prorogating in the titanium wire. The wavelength can be calculated as 252.8 mm, based on Young's modulus, density of the TA2, and the oscillation frequency of the die, 20 kHz. The variations of drawing force with time under different uncoiled wire lengths are illustrated in Figure 3.

**Figure 3.** Time-variation of the drawing force for different distances between the die and the reel: (**a**) *L* = 31.6 mm; (**b**) *L* = 63.2 mm; (**c**) *L* = 94.8 mm; (**d**) *L* = 126.4 mm.

It can be seen that the whole drawing process lasts for 5 ms. Ultrasonic vibrations were imposed during the period of 2~3 ms and then were removed. At the beginning of the first 2 ms, the drawing force ramped up from 0 N to around 20 N within 1 ms, which represents the elastic deformation phase of the wire drawing process. Afterward, the growth of the drawing force gradually slowed down, which means the start of plastic deformation. The drawing force finally stabilized at around 22.5 N, and wire products were steadily pulled through the die. Comparing Figure 3a,d, we can easily find that the distance between the die and the wire does not influence the conventional drawing force, however, it would affect the duration of the elastic deformation period. When ultrasonic vibrations were applied, the drawing force began to fluctuate in a certain range, with the maximum values slightly larger than or equal to the normal drawing force, whereas the minimum values are significantly lower than that, which results in a decrease in averaged drawing forces. The fluctuation range of ultrasonic drawing force, or the reduction ratio of averaged ultrasonic drawing forces compared to the conventional drawing value, is heavily influenced by the location of the die. Specifically, at the same ultrasonic amplitude and drawing velocity, the drawing force fluctuates most violently when the reel locates 126.4 mm (λ/2) from the die, while the fluctuation is the minimum when the reel-die distance equals 63.2 mm (λ/4). Besides, there is a difference between the four subfigures with respect to the envelope curve shapes under the ultrasonic drawing condition. When the die is distributed λ/2 from the reel, the envelope of the ultrasonic drawing force is shaped like a trapezoid, instead of rectangular when located at other positions. For these positions, the amplitudes of drawing force fluctuation reach the maximum as soon as ultrasounds are imposed, and the drawing forces immediately restored to around 22.5 N when vibrations are removed. At the half-wavelength position, however, the minimum drawing force drops steadily in the first five oscillating cycles before reaching a steady state. When ultrasound is removed, the drawing force remains lower than 22.5 N in about 0.7 ms, instead of going back to the original value right away. Finally, even in the half-wavelength condition and even without the consideration of the pliability of the metallic wire, the ultrasonic drawing forces remains above 0, which means no separations occur between the wire and die at their contact interface, and the validity of the presuppositions for reversed friction mechanism should be further verified. Unlike a bar, a long thin wire could not bear the pressure, which is another reason why the drawing force stays above 0 and the separation could not happen.

Figure 4 shows the variation of averaged drawing force with the distance between the die and the reel. It can be seen that, as the reel-die distance increases from 15.8 mm (λ/16) to 126.4 mm, the averaged drawing force first goes up and then drops, peaking at the quarter-wavelength position, however, remains below the conventional drawing force value of 22.5 N. At the half-wavelength position, the averaged drawing force reaches the minimum value of 10.28 N, with a reduction of more than 50% compared with conventional drawing force. The influence of the reel-die distance on the ultrasonic drawing force can be attributed to the stretching vibration of the uncoiled drawn wire. This section of titanium wire resonates under the drive of vibrating die when its length approximates half-wavelength, as the frequency of the die is approaching the first-order longitudinal resonant frequency of the wire.

**Figure 4.** FEM calculation results of average ultrasonic drawing forces at different distances between the die and the reel drum.

### *3.2. Influence of Ultrasonic Amplitude on Drawing Force*

It can be seen from the above analysis, with the same ultrasonic amplitude and drawing speed, maximum drawing force reduction can be achieved when the distance between the reel and the die approximates half-wavelength of ultrasound traveling in the titanium wire. Therefore, the influences of other factors, including ultrasonic amplitude *A*, wire drawing speed *V*, and friction coefficient μ, on the ultrasonic drawing force will be further considered under this condition. From previous studies, it can be known that the effects of ultrasonic amplitude and drawing speed on ultrasonic drawing force are correlated [14–20]. Drawing force reduction can only be realized when the prerequisite *V* < 2πf*A* is satisfied.

Figure 5 illustrates the changing of wire drawing force along with time at the drawing velocity of 300 mm/s and with the friction coefficient of 0.1. It can be found that the upper limit of the drawing force remains around 22.5 N, which will not be affected by the intensity of ultrasonic vibrations. In Figure 5a, where ultrasound amplitude is 2 μm, there is almost no change in drawing force compared with the conventional drawing value, because the drawing speed, 300 mm/s, exceeds the threshold value 2πf*A*, which is determined by the oscillation amplitude and frequency of the die. In other subfigures, where pre-condition is satisfied, an obvious decrease in drawing force can be observed. The envelope lines of ultrasonic drawing force present the shape of the triangle or trapezoid instead of rectangular, as the die locates half-wavelength off the reel drum, which is consistent with previous analysis. The influence of ultrasound will also last for a short time after the die stops vibrating, instead of disappearing instantly. However, the fluctuation range of the ultrasonic drawing force will be widened with the increment of ultrasonic amplitude. When ultrasound is turned on, the minimum values of the drawing force go down consistently until arriving at the stable phase. The time consumption for this period is shortened with larger ultrasonic amplitude, and the decrement of minimum drawing force for each oscillation period will be enlarged correspondingly.

**Figure 5.** Influence of ultrasonic amplitude on time-variation of the drawing force: (**a**) *A* = 2 μm; (**b**) *A* = 4 μm; (**c**) *A* = 6 μm; (**d**) *A* = 8 μm.

Figure 6 shows the numerical calculation results of averaged ultrasonic drawing force under different ultrasonic amplitudes at drawing speed of 300 mm/s and 600 mm/s, respectively. Overall, the two variation curves show the same trend, that is, the averaged ultrasonic drawing force goes down with the increment of ultrasonic amplitude. However, the drawing force declines faster at a relatively lower drawing speed of 300 mm/s. A flat section could be found at the initial segment of the two curves, where there is almost no reduction in drawing force compared with 22.5 N. Therefore, at a certain drawing velocity, to achieve drawing force reduction, the intensity of ultrasonic vibration has to be large enough. The flat region expands at a relatively higher drawing velocity because the threshold drawing speed (V = 2πf*A*) increases with ultrasonic intensity.

**Figure 6.** FEM calculation results of average ultrasonic drawing force with different ultrasonic amplitudes at drawing speed of 300 mm/s and 600 mm/s.

Based on the above simulation results, the following hypothesis could be put forward. With the addition of ultrasonic vibration, the conventional continuous drawing process is turned into an intermittent ultrasonic drawing process. A vibration cycle of the die can be divided into two phases: the plastic deformation phase, and the stretching vibration phase. For the former, titanium wire is drawn through the die and the drawing force equals the conventional drawing force. For the latter, however, the wire vibrates along with the die. The drawing force is determined by the stretching force of the wire and should be lower than the conventional drawing force. The time allocation between the two phases is determined by both the amplitude of the die and the wire drawing velocity. With the increment of drawing speed, the proportion of the plastic deformation stage will rise correspondingly, therefore, averaged ultrasonic drawing force will be closer to normal drawing force 22.5 N. On the contrary, when ultrasonic amplitude increases, the time duration of the deformation stage is decreased, and the stretching force vibration is aggravated simultaneously; therefore, the overall drawing force goes down.

### *3.3. Influence of Drawing Speed on Drawing Force*

Figure 7 displays the influence of wire drawing velocity on the time variation of drawing forces when the distance between the die and reel drum equals half-wavelength and the ultrasonic amplitude remains 10 μm. By comparison, it can be found that the steady-state value of conventional drawing force is not affected by drawing speed, which fluctuates slightly around 22.5 N, because strain rate dependence of the flow stress is not included in the material model. However, the time duration of the elastic deformation stage is prolonged when drawing speed increases, with 1.8 ms for 200 mm/s to 0.5 ms for 800 mm/s. For the ultrasonic drawing stage, the fluctuation range of drawing forces is narrowed with the increment of drawing velocity. And the time consumption for ultrasonic drawing forces to reach the steady value increases correspondingly from 5 oscillation cycles for 200 mm/s to 13 oscillation cycles for 800 mm/s. After the die stops vibrating, the fluctuation of the drawing force continues for a while, but the time duration is shortened with the increment of drawing velocity. According to the hypothesis proposed in Section 3.2, with the increment of drawing speed, the plastic

deformation phase takes up a greater proportion in a vibration period, therefore the overall drawing force is increased. Meanwhile, the stretching vibration time is reduced, more cycles are undergone before the minimum drawing force gets stabilized. In addition, as the drawing speed approaches the threshold value 2πf*A*, the decrement of minimum drawing force in each cycle drops, making the total drawing force reduction decreased.

**Figure 7.** Influence of drawing speed on time-variation of the drawing force with reel-die distance of 126.4 mm: (**a**) *V* = 200 mm/s; (**b**) *V* = 400 mm/s; (**c**) *V* = 600 mm/s; (**d**) *V* = 800 mm/s.

Figure 8 illustrates the time variation of drawing forces at different drawing speed when the die locates 7.9 mm (λ/32) to the reel drum and the ultrasonic amplitude remains 10 μm. In general, the drawing forces present a similar upward trend as the drawing speed rises. However, the envelope lines of the drawing forces are shaped as rectangular instead of the trapezoid. An abrupt decline in drawing force could be observed in these figures when ultrasonic vibration is imposed. The drawing force will restore to the initial value of 22.5 N immediately when ultrasonic vibrations are removed. Besides, compared with the half-wavelength conditions, the fluctuation range of the corresponding drawing force appears to be more sensitive to drawing speed, which is narrowed down sharply, especially at higher drawing speeds. This phenomenon might be caused by the weakened stretching vibration of uncoiled drawn wire, as the oscillating frequency of the die is far below the first-order longitudinal frequency of the wire.

Figure 9 compares the influence of drawing speed on averaged ultrasonic drawing force when the reel-die distance equals 126.4 mm and 7.9 mm, respectively. Although the two curves present a similar upward trend with the increment of drawing speed, the drawing force at the half-wavelength position is lower than the λ/32 position. The gap between the two curves is gradually narrowed as the drawing speed rises. When approaching the critical drawing speed 1256 mm/s, which is calculated at the amplitude of 10 μm, and frequency of 20 kHz, the two curves all converge to the conventional drawing force of 22.5 N. The difference between the two curves might be attributed to the stretching vibration of the wire. At the 7.9 mm condition, drawing force improvement caused by increased drawing velocity could not be compensated by the strengthened oscillation of the wire, as occurs in the half-wavelength condition.

**Figure 8.** Influence of drawing speed on time-variation of the drawing force with reel-die distance of 7.9 mm: (**a**) *V* = 200 mm/s; (**b**) *V* = 400 mm/s; (**c**) *V* = 600 mm/s; (**d**) *V* = 800 mm/s.

**Figure 9.** FEM calculation results of average ultrasonic drawing force with different drawing speeds with reel-die distance of 126.4 mm and 7.9 mm.

### *3.4. Influence of Friction Coe*ffi*cient on Drawing Force*

To further investigate the influence of friction coefficient on the variation of drawing force, ultrasonic amplitude and drawing speed are specified as 10 μm and 300 mm/s, and remain unchanged. Figure 10 shows the time variation of drawing forces when the reel-die distance equals 15.8 mm (λ/16) with friction coefficient assigned as 0.3 and 0.5, respectively. Compared with simulation results in

Figures 3 and 4, in the two subfigures of Figure 10, conventional drawing forces climb up to 36.13 N and 45 N from 22.5 N, respectively, and the averaged ultrasonic drawing forces correspondingly ascend to 32.07 N and 41 N from 18.68 N. Therefore, it is suggested that the friction coefficient influences the drawing force no matter whether ultrasonic vibrations are imposed. In addition, the increments of conventional drawing force, when friction coefficient rises, are very close to that of averaged ultrasonic drawing forces. In other words, ultrasonic vibration does not affect the wire-die friction coefficient. However, with excessive friction, plastic deformation would occur to the drawn wire between the die and the reel, resulting in wire breakage, as plotted in Figure 10b.

**Figure 10.** Time variation of drawing force with different friction coefficient when the distance between the reel and the die is 15.8 mm: (**a**) μ = 0.3; (**b**) μ = 0.5.

Figure 11 shows the time variation of drawing forces with friction coefficient of 0.3 and 0.5, respectively, when the die locates 126.4 mm from the reel. Conventional drawing forces are 36.1 N and 42.8 N, which are consistent with the results in Figure 10. Averaged ultrasonic drawing forces increase to 17.15 N and 20.83 N from 10.28 N. Similarly, ultrasonic vibration exerts influence on both the conventional and the ultrasonic drawing forces. However, on this occasion, the increments of conventional drawing force caused by increased friction are obviously larger than that of averaged ultrasonic forces. Therefore, at the half-wavelength condition, ultrasonic vibration helps to decrease the equivalent friction with the contacting region. Compared with Figure 3d, severe distortion could be found in the two subfigures. In the conventional drawing phase, time-consuming to reach the steady-state increases to 1.6 ms and 1.8 ms from 1 ms, respectively. This might be caused by the plastic deformation of the uncoiled drawn wire. In the ultrasonic drawing phase, many negative values appear in the drawing force curve. As the flexibility of the titanium is not fully considered in the FE model, the negative section of the curves will be chopped off in the data post-processing procedure.

**Figure 11.** Time variation of drawing force with different friction coefficient when the distance between the reel and the die is 126.4 mm: (**a**) μ = 0.3; (**b**) μ = 0.5.

Figure 12 illustrates the influences of friction coefficient on conventional and ultrasonic drawing forces. In both subfigures, the drawing forces ascend with an increased friction coefficient. For conventional drawing, the simulation results are in good agreement with a friction coefficient below 0.4 and are consistent with theoretical results calculated according to Avitzur's theory, when the friction coefficient is lower than 0.2 [23]. For ultrasonic drawing, the drawing force curve has the same shape with the conventional drawing force curve when reel-die distance equals λ/16. However, at the half-wavelength condition, the increment of ultrasonic drawing force is much smaller than that of conventional drawing force, which means a decreased equivalent friction.

**Figure 12.** Influence of friction coefficient on drawing forces when the die locates at different positions: (**a**) Influence of friction on conventional drawing force; (**b**) Influence of friction coefficient on averaged ultrasonic drawing forces.

### *3.5. Influence of Ultrasonic Vibration on Stress Distribution*

Figure 13 depicts the Mises equivalent stress distribution of titanium wire under conventional drawing conditions with drawing velocity of 300 mm/s and friction coefficient of 0.1. In the contour, the equivalent stress remains 0~105.9 MPa in a large portion of the feeding area, which locates to the left of the die, when the influence of the back-pull force is neglected. At the entrance region, the Mises stress surged to around 423 MPa. The maximum stress 423.6~529.5 MPa occurs at the reduction area and bearing area, where the die is in direct contact with the wire. However, at the bearing area and the right adjacent area to it, the maximum stress only concentrates on the outer layer of the wire. These are mainly residual stresses caused by the inhomogeneous deformation between the surface and core section of the metallic wire, which declines sharply along the radial direction (−X) of the wire. At the reduction area, however, the equivalent stress maintains 423.6~529.5 MPa throughout the whole internal region of the wire. This is where plastic deformation mainly occurs. For the wires far from the right side of the die, the drawing stress is kept between 211.8 MPa to 311.7 MPa.

**Figure 13.** Mises equivalent stress distribution of the wire at the contact region under conventional drawing conditions.

To investigate the influence of ultrasonic vibration on the stress distribution of the wire, the simulation is carried out with an ultrasonic amplitude of 10 μm and a wire-die distance of 15.8 mm. The equivalent contours around the contact region for different movement states of the die are demonstrated in Figure 14. It can be seen that, when the die moves, along the reversed wire drawing direction, from the equilibrium position to the leftmost position, the overall stress distribution is similar to that of conventional drawing, except for slight difference at the core segment of the drawn wire, with a maximum stress of 525.9 MPa. In contrast, when the die moves, along the drawing direction, from the central position to the right extreme position, more significant changes could be observed with the maximum equivalent stress dropping to 515 MPa and shrinkage in its area. In addition, the equivalent stress rises to 206 MPa at the feeding area, 308.8 MPa at the core of the drawn wire, and reduced to below 411.8 MPa at the outer surface layer. The more uniform distribution of the equivalent stress is beneficial for eliminating the defects and residual stress caused by inhomogeneous deformation between the core and surface layer of the titanium wire.

**Figure 14.** Mises equivalent stress distribution of the wire at the contact region under ultrasonic drawing conditions when the die locates 15.8 mm from the reel and in various motion states: (**a**) Equilibrium position and moves to left; (**b**) Left extreme position; (**c**) Equilibrium position and moves to the right; (**d**) Right extreme position.

Figure 15 illustrates the contact and friction stress distribution at the wire-die interface under conventional drawing conditions. From Figure 15a, it can be seen that the maximum value of contact stress occurs at the two sides of the reduction area, adjacent to the entrance and the bearing areas. This is where the surface plastic deformation mainly happens, as the contact stress range, 513.1~641.4 MPa, is apparently above the yield stress of titanium, which is calculated as around 460 MPa based on theoretical and simulation results. In the middle of the reduction area, contact stress falls into the range of 384.8~513.1 MPa, both elastic and plastic deformations coexist. In the entrance and bearing regions, which locate at the two sides of the reduction region, the contact stress plummets to below 128.3 MPa, which means no plastic deformation happens. In Figure 15b, we can find that the friction stress has the same distribution with the contact stress, but is 10% of the latter in value, as the Coulomb friction model is adopted in the setup of the FEA model with a coefficient of 0.1. Therefore, no additional simulation result about the friction stress will be separately presented. In general, the contact and friction stress distribution is compatible with the equivalent stress distribution.

Figure 16 shows the contact stress contours for the ultrasonic drawing process with the die in different motion states. Compared with traditional drawing, there is almost no change in the distribution of contact stress, except for a slight change in the values. When the die locates at the equilibrium position and moves to the left, the maximum contact stress goes up to 660.4 MPa, which is slightly above 641.4 MPa, as shown in Figure 16a. In other positions, the contact stresses are slightly lower than conventional drawing values. In the whole ultrasonic drawing process, the contact stress at the reduction region is apparently above 0. In other words, no separations would occur at the

wire-die interface, as presupposed in the reverse friction mechanism theory. Although with shortened reel-die distance, the minimum contact force could be further decreased, to the point of achieving an intermittent separation between the two parts, which has no practical meaning.

**Figure 15.** Contact stress and friction stress distribution at the contact interface under conventional drawing conditions: (**a**) Contact stress distribution; (**b**) Friction stress distribution.

**Figure 16.** Contact stress distribution at contact interface under ultrasonic drawing conditions when the die locates 15.8 mm from the reel and in various motion states: (**a**) Equilibrium position and moves to left; (**b**) Left extreme position; (**c**) Equilibrium position and moves to the right; (**d**) Right extreme position.

### **4. Conclusions**

An axisymmetric FE model for a single-pass titanium wire drawing process was built in the commercial software Abaqus based on the practical length of wire. Then, the influence of ultrasonic vibrations on the wire drawing process was investigated with respect to four factors: the location of the die, ultrasonic amplitude, drawing velocity, friction coefficient. At last, the stresses distribution contours at the contact region with and without ultrasonic vibrations were compared. The main conclusions of this study can arrive as follows:


them is determined by the ultrasonic amplitude and wire drawing speed. The drawing force reduction increases with the increment of ultrasonic amplitude and decrease in drawing speed.


**Author Contributions:** S.L. wrote the paper and completed the numerical simulation; T.X. and X.S. provided the funding and supervised the project; H.C. revised the manuscript and helped in the simulation part. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China, grant number 51575130.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article*
