Figure 1 schematically illustrates the cross-sectional division method. The first step involves dividing the cross-sections of the initial and final products into finite elements at a small angle (θ). Each divided element can be treated as an axisymmetric drawing process, where the inlet and exit radii are denoted as R_(i,i) and R_(o,i), respectively, and αi is the half-die angle. Subsequently, the drawing force for an element (F_d,i) was computed with a theoretical model for axisymmetric drawing. Finally, the total shape drawing force (F_d,total) was determined by summing the drawing forces from each divided element as follows.

Figure 1: 
Schematic of cross-section division method

where n is the number of total elements.

According to Wistreich [12], the analytical solution for the axisymmetric drawing force (F_d) can be expressed as

where Φ is the redundant work factor; A₂ is the cross-sectional area of the die outlet; Y_m is the mean yield stress; μ is the friction coefficient; α is the half die angle. ε_h is the homogeneous strain defined by

where A₁ represents the cross-sectional area of the die inlet. For a material with a strain-hardening behavior following a power law, the mean yield stress is given by

where K is the strength coefficient of the material, and m is the strain hardening exponent.

In Equation (2), Y_mε_h represents the drawing force for homogeneous deformation. The term μcotα reflects the additional force required to overcome the friction between the material and the die. The redundant work factor Φ relates to the internal distortion of the material, which does not contribute to the dimensional change but induces a redundant strain other than that required for homogeneous deformation [13]. Φ can be determined experimentally by measuring the drawing force and dividing it by the term A₂Y_m(1+μcotα)ε_h which is denoted as F_I hereafter. The drawing force F_I can be readily calculated, provided the friction coefficient and the stress–strain relationship are known. Generally, it is believed that Φ depends on a single geometric parameter of the die’s deformation zone, namely Parameter Δ, which can be given by Equation (5) [13]:

where r is the reduction ratio between the initial shape and the final one.

Because the cross-section division method assumes that shape drawing comprises multiple axisymmetric drawing processes with different reduction ratios and half-die angles, it is essential to establish a general relationship between Φ and Δ. This allows the redundant work factor associated with a specific Δ to be estimated easily across a wide range of scenarios. To derive an analytical expression for the redundant work factor Φ, Atkins and Caddell [14] conducted extensive axisymmetric drawing tests on stainless steel and aluminum and found that a linear equation could be formulated between Φ and Δ. However, drawing tests for all conditions in question can be impractical and time-consuming. Therefore, here, FE analysis was conducted on axisymmetric drawing involving diverse combinations of reduction ratios and half-die angles to investigate the relationship between the redundant work factor and the parameter Δ.

Rigid-plastic analysis for axisymmetric drawing was conducted via the commercial forming analysis software DEFORM 2D. Table 1 presents the 20 values for the parameter Δ that were chosen for FE analysis, considering the typical ranges of half-die angles and reduction ratios utilized in shaped drawing [1]. Figure 2 illustrates the FE analysis model. The initial workpiece was 30-mm long and 20-mm in diameter. A total of 6,030 elements were employed, with a minimum element size of approximately 0.22 mm. Regarding the equipment specifications typically utilized in the industrial field, a drawing speed of 10 mm/s was utilized, and the friction coefficient μ between the material and die was set to 0.0577, as per previous research [10].

Figure 2: 
FE analysis model for axisymmetric drawing

The material under investigation was a highly pure stainless steel (STS 316L) manufactured via a two-stage vacuum melting process that included vacuum induction melting and vacuum arc remelting. Mechanical properties were determined via uniaxial tensile tests conducted in accordance with the KS B 0802 standard. The tensile test specimens had a gauge length of 25 mm and a diameter of 5 mm. The mechanical properties obtained from the tests are presented in Table 2, and Figure 3 illustrates the flow-stress curve employed for the FE analysis.

Figure 3: 
Flow-stress curve of STS 316L

The drawing forces evaluated by FE analysis for various Δ values are presented in Table 3, along with the theoretically calculated drawing forces (referred to as F_I) that do not consider the contribution of redundant deformation. The redundant work factor Φ was subsequently determined by comparing the FE analysis results with the corresponding drawing force, F_I. The values of Φ as a function of Δ are presented in Figure 4. It is noteworthy that, similar to the results obtained from the literature [14], all Φ values for the experimental material exhibit a dependence on Δ, which leads to the following linear regression equation.

Figure 4: 
Redundant work factor Φ as function of geometric parameter Δ

With the established relationship between Φ and Δ for the material described in Equation 6, Wistreich’s equation can now be applied to predict the forming load in drawing processes that create products with arbitrary cross-sectional shapes.

The drawing force prediction method proposed here was verified via an FE analysis of various shape-drawing processes. Figure 5 presents the cross-sectional profiles of the final products with rectangular and hexagonal shapes manufactured from the round initial bars. To prevent undercutting, the diameter of the initial material was set to be larger than the minimum circumference of the product. For the rectangular profile, a multistage drawing process was implemented, which included an intermediate heat treatment owing to the substantial reduction ratio of 45.6%. Therefore, to consider the effect of heat treatment, the effective strain accumulated in the material after 1^st pass drawing was set to zero when a 2^nd pass forming analysis was performed, as depicted in Figure 6. The intermediate die shape of the multistage drawing process (i.e., the final shape of the 1^st pass) was determined via electric field analysis [15]. In addition, a pass schedule was designed based on the average reduction ratio (r_avg), defined as follows.

Figure 5: 
Shapes and dimensions of initial bars and final products

Figure 6: 
FE analysis model for multistage rectangular shape drawing

where r_total is the total reduction ratio, and p is the number of passes.

Considering the symmetry of all final products, a quarter-section was analyzed with DEFORM 3D software, as demonstrated in Figure 6. The analysis utilized approximately 140,000 and 95,000 3D tetrahedral elements for multistage rectangular shape drawing and hexagonal shape drawing, respectively. The minimum element size was 0.14 mm for the former and 0.84 mm for the latter. The material properties and analysis conditions, such as drawing speed and friction coefficient, were consistent with those utilized in the FE analysis of the axisymmetric drawing process, as detailed in Section 2.2.

Figure 7 compares the drawing forces predicted by the proposed method and those obtained via FE analysis. The cross-section division method was employed for the evaluation utilizing the proposed method, involving the division of cross-sectional profiles into 30 elements with an angle (θ) of 3°. Following this, the redundant work factor for each divided element was computed with Equation (6) based on the reduction ratio and halfdie angle (i.e., parameter Δ). Finally, Wistreich’s equation was applied to predict the drawing force for each divided element, and the overall drawing force was determined by combining the drawing forces of all the elements.

Figure 7: 
Comparison between drawing forces predicted by Wistreich’s equation and FE analysis

The values of the drawing forces and associated errors of the proposed method are compared with the results obtained from the FE analysis in Table 4. For comparison, the drawing forces estimated by the method proposed by Lee et al. [9] utilizing Geleji’s equation was also documented. In the multistage drawing, the prediction made by the proposed method is in good agreement with the FE analysis results. In contrast, existing methods tend to underestimate the drawing force. The proposed method substantially decreased the error from 12.4% to 0.2% in the 1^st pass and from 17.4% to 2.1% in the 2^nd pass when compared with the previous method based on the Geleji equation. Similarly, in the shape drawing of a hexagonal profile with a reduction ratio of 24.6%, Wistreich’s equation led to a significant reduction in the error from 15.4% to 0.5% compared to Geleji’s equation.

Atkins and Caddell [14] highlighted that utilizing the mean yield stress leads to errors in the computation of the drawing force. Moreover, the deviation became more pronounced as the values of the strain and strain-hardening exponent increased, given a specific friction coefficient and half-die angle. This is likely why the existing method utilizing Geleji’s equation, which relies on the mean yield stress, inherently contains errors when estimating the drawing force for materials with high strain-hardening exponents, such as the STS 316L utilized here. Although Wistreich’s equation also employs the mean yield stress, it can offset this error by incorporating the redundant work factor Φ derived from FE analysis utilizing the actual flow stress, allowing it to provide a more accurate prediction.

In contrast, for the hexagonal drawing process with a reduction ratio of 33.4%, notable errors in the prediction results were observed in both models, amounting to 11.0% and 9.5%, respectively. These results appear to be associated with the limitation of the cross-sectional division method, particularly in shape-drawing processes with significant reduction ratios. This approach fundamentally assumes that the material in each element flows primarily in the radial direction, similar to an axisymmetric drawing process. However, because the shape-drawing process entails a more complex material flow, errors inevitably increase as the reduction ratio increases. Therefore, further extensive studies are required to develop a method that effectively applies the cross-sectional division method to shape drawing processes with high reduction ratios. In addition, while this study assumed that the shape-drawing process is performed at room temperature, it is worth noting that a temperature increase, especially at higher drawing speeds, can significantly alter the drawing force [16]. Considering that the drawing speed is a critical parameter for productivity, the impact of the drawing speed on the drawing force should be considered, which will be the focus of future research.