In general, a shape drawing process consists of 2−3 passes of a multi-pass drawing process. Therefore, in this study, the pass schedule was designed based on the average reduction in area under the assumption of applying roll die drawing to the intermediate pass of the multi-pass shape drawing process. The number of passes in the multi-pass shape drawing process can be set by the average reduction in area equation expressed using Equation (1).

where r_avg is the average reduction in area, r_total is the total reduction in area, and n is the number of passes. In general, the multi-pass shape drawing process limits the reduction in area to less than 30% per pass. Figure 2 shows the cross-sectional shape and dimensions of the rectangular product used in this study. The initial shape diameter should be larger than the minimum circumscribed circle of the product's shape so that an under-cut is not generated during shape drawing. The minimum circumscribed circle diameter of the final product cross section is 38.3 mm, and by using Equation (1), the number of passes was set so that the average reduction in area is 30% or less. At the same time, the initial material diameter has an overall offset of 2 mm for each pass from the minimum circumscribed circle diameter of the product shape, considering the filling rate of each pass. Therefore, the total number of passes and initial material diameter were set to 2 and 42.3 mm, respectively, and the total reduction in area was 45.6%.

Figure 2: 
Shapes and dimensions of initial bar and final product

Most shape drawing products are fabricated through 2 to 3 passes of the multi-pass drawing process, so the shape design needs to have the appropriate intermediate pass dies to reduce the formation of load and local stress concentrations and improve the dimensional precision of the final product. In this study, we first referred to the electric field analysis method proposed by Lee et al. [10] for the design of intermediate die shapes.

The design method of intermediate die shape using electric field analysis (EFA) is as follows. First, different voltages are applied to two objects with the centers coinciding. This creates a finite equivalent potential line that does not overlap between the two objects. According to a previous study [11], the equivalent potential line showed a tendency which is similar to the minimum work path between two conductors. Based on this tendency, the shape with the area corresponding to the reduction in area of the pass schedule is extracted from the many equivalent potential lines, and it is used as the intermediate die shape for each pass. In the present study, a pass schedule based on equal reduction in area was applied. Two-dimensional electric field analysis was performed using ANSYS 2019 r2 to design intermediate die shapes. Figure 3 shows the equipotential lines formed by applying 1 V and 0 V to the initial shape and the final shape, respectively, and the intermediate die shape by equal reduction in area pass schedule.

Figure 3: 
Intermediate die shape by EFA

However, the intermediate shape design method using electric field analysis is generally applied to conventional shape drawing dies and cannot be directly applied to the roll groove design. In roll die drawing, there is a roll gap that results in a free-surface that does not come into contact with the die. Free-surface parts that do not undergo deformation by contact with the roll die will cause a strain inhomogeneity in the conventional drawing die pass, which will reduce the dimensional accuracy. Therefore, overall consideration of the free-surface part is required in the intermediate shape design. In addition, it is necessary to define the relevant parameters applicable to the process design method and the free-surface profile prediction model of this study for rectangular shapes with various aspect ratios. Therefore, the intermediate roll die shape for the multi-pass shape drawing process was determined by the following procedure as shown in Figure 4, referring to the electric field analysis results and the roll die groove design method proposed by Kim et al. [6].

Figure 4: 
Design of intermediate roll die shape

First, considering the dimensional precision in the second pass of drawing, the final product shape is overall offset with the shape factor so that the deformation is uniform throughout the product shape. At this time, the under-cut area between the enlarged final product and the initial shape is excluded from the intermediate shape. The shape factor (S.F) is defined using Equation (2) below and is calculated by the square root of the area ratio between the intermediate shape area and the drawn product shape according to the pass schedule [12]. However, the intermediate shape determined in the first step produces high strain deviations in roll die drawing. Therefore, in the second step, the intermediate shape is partially offset locally based on the intermediate shape from the electric field analysis to reduce such strain deviations. Here, the ratio of the local offset amount to the overall offset amount is defined as the local offset ratio (LR) as shown in Equation (3), and the ratio of the horizontal and vertical length of overall offset shape is defined as a process variable called the aspect ratio (AR), as shown in Equation (4). These values are used as the design variables for the subsequent experimental design method. Based on the designed intermediate shape information, that is, the process parameters, the free-surface profile of the intermediate shape is predicted, and the design suitability is determined by considering the presence of overfilled or underfilled defects in the next pass.

where A_i is the area of the intermediate shape for pass i in the schedule, A_f is the area of the final drawn product and r is the percent reduction in area.

where L_l is the length of the local offset and L_o is the length of the overall offset.

where L_h and L_v are the horizontal and vertical length of the overall offset shape. The process parameters explained above are shown in Figure 4.

The response surface method is a statistical analysis method for response surfaces that represent the change in responses when complex actions of several variables have an impact on response variables. The main purpose of using the response surface method is to estimate the functional relationship between the design variables and response variables from a given experimental dataset by linear or quadratic regression and to select a combination of design variables (or experimental factors) to obtain an optimal response. Box-Behnken design is a representative experimental design method used for response surface analysis. It has a smaller number of experimental points compared to the 3^k factorial design when the number of factors is k, and it is easier to make orthogonal blocks and to obtain a quadratic regression equation. A quadratic regression model with k design variables can be expressed in terms of linear, squared, and interaction terms, as shown in Equation (5).

Here, Y is the response value of W_v and W_s (see Figure 5), A is the local offset ratio, B is the aspect ratio, C is the reduction in area, D is the friction coefficient, and β is the regression coefficient.

Figure 5: 
Description of free-surface profile prediction

In this study, we propose a quadratic regression model to predict the free surfaces in the roll gap in four shape roll dies drawing process based on the Box-Behnken design. For this purpose, as key process parameters affected the material deformation behaviors, along with the local offset ratio and aspect ratio which are the geometric parameters defined in the intermediate shape design process, friction coefficient and reduction in area were also selected by referring to previous studies [4][13][14]. The response variables were set as the reduced width of contact area after the roll drawing in the contact area between the initial material and the vertical and side rolls, respectively. Based on the measured response variable, the predicted shape of the free surface part was determined as follows. First, it is assumed that the intermediate point (P_I) of the free surface part of the initial material moves in the direction of the center of the shape, and the distance of P_I is proportional to the square root of the reduction rate. Here, the square root of the reduction rate is defined as the reduction factor (RF), and is calculated as in Equation (6).

Thereafter, when the point of P_I moves in the horizontal and vertical directions independently by the length of the measured response variable multiplied by the reduction factor, the corresponding positions are defined as P_V and P_S, respectively. The point of P_I will be simultaneously affected by the reduction in the width and height directions. Thus, a triangle connecting the two points (P_V, P_S) and the free surface midpoint (P_I) is created. In order to apply the initial assumption, the final position (P_C) of the midpoint of the free surface shape after drawing can be determined as the point where the movement path of the intermediate point (P_I) toward the center direction intersects with the hypotenuse of the triangle. Finally, the arc shape connecting the boundary point of the contact area between the vertical roll and the side roll and the point (P_C) of the hypotenuse determined previously is illustrated. Figure 5 demonstrates the procedure explained above. The values of design variables at each level are given in Table 1.

Rigid-plastic finite element analysis was performed for each combination of design variables according to the Box-Behnken design using DEFORM-3D, a commercial forming analysis software. High purity stainless steel 316L was considered as the material in the analysis. The flow stress equation obtained from the uniaxial tensile test is shown in Equation (7).

A quarter model of the overall shape was applied considering the analysis time and the symmetry of the product shape. The length and diameter of the initial material were 120 mm and 42.3 mm, respectively, and a total of 105,369 tetrahedral elements were used. Considering the equipment specifications used in the field, the drawing speed was 10 mm/s, and the friction coefficient between the workpiece and the die was set to 0.12 based on previous studies [6][7]. The above finite element analysis model is shown in Figure 6.

Figure 6: 
3D finite element analysis model

Table 2 shows the results for a total of 27 finite element analyses. Based on the values of W_v and W_s measured from FE analysis, the coefficients of the regression model for W_v and W_s derived from the results are shown in Table 3 and Table 4, respectively.

In addition, the significance of the design variables for each response value was evaluated through analysis of variance (ANOVA). DF is the degree of freedom of design variables, SS is the sum of squares, MS is the mean square of SS divided by DF, F is the test statistics, and P is the significance probability.

Table 5 and Table 6 provide the results of the ANOVA calculations. It can be seen that the regression equation composed of the design variables selected in this study can have significant effects on the value of W_v because the P values of all regression terms relating to W_v were less than 0.05. In addition, in the case of W_s, the p-value of the square and the interaction term is slightly larger than 0.05, but since it is close to the significance level, it was included in the regression equation without pooling them into the error term.

To evaluate the accuracy of the regression equation, the coefficient of determination (R²), which represents the ratio of the variation explained by the regression equation to the total variation, was calculated using Equation (8).

Here, the total variance (SST) is divided into a regression sum of squares (SSR) and a residual sum of squares (SSE), which are respectively expressed as follows.

where y is the actual response, y¯ is the mean of the actual responses, and y^ is the response estimated by the regression model.

The coefficient of determination has a value between 0 and 1, and the closer it is to 1, the better the approximation. The coefficients of determination for W_v and W_s were 96.4% and 95.9%, respectively, which shows that the regression model explains well the relationship between design variables and response values.