© T. Muraoka et al., Published by EDP Sciences 2026

1 Introduction

Cylindrical deep drawing is a basic process for forming cups from sheet materials. The formed cups are used in various applications, including motor cases, battery cases, and beverage cans [1–3]. A schematic of the forming method is shown in Figure 1. The cup is formed by pushing the punch into the die while the blank holder holds the sheet. Here, since the sheet material has anisotropy due to rolling, the edges of the cup are uneven, referred to as “ears” [4]. In actual products, the trimming process is used to prepare the end edges, resulting in material loss and an increase in the number of processes. These problems have become an issue of reduced productivity. Studies on the earing of deep drawing were performed to predict ear shape using finite element analysis (FEA). Low-carbon aluminum-killed steel and pure titanium [5], pure titanium CPB06 [6,7], and DC06 steel [8] were investigated for ear shape using numerical analysis applied to various anisotropic yield functions, and compared with experimental results. The focus of earing reduction is to suppress material anisotropy, and material manufacturers are taking the lead in development. Therefore, it is difficult for the drawing manufacturer who performs the forming process to solve the earing. As a method for suppressing earing, drawing using a deformed blank has been proposed [9]. However, in deep drawing—which typically involves multiple process steps—this approach can contribute only a single step. A technique involving laminating materials with different anisotropies and then drawing them to suppress earing has also been proposed [10]. However, this method makes it difficult to suppress the earing using a single material. Variable blank holder force has also been proposed to suppress earing [11], and in recent years, optimization using neural networks has been reported [12]. However, a special device is required for the variable blank holder. As a method to suppress ear height without the need for material development, Phanitwong et al. proposed a die with a die radius that varies in the circumferential direction, which has been shown to equalize the material flow [13,14]. However, the details of the die radius shape and the relationship between the forming conditions and the formed shape have not been reported yet.

In contrast, the authors proposed “Multiple R die (MR)” in which the die radius shape is defined by a trigonometric function, and confirmed that the ear height can be suppressed [15]. The die shape of MR is shown in Figure 2. In the conventional “Single R die (SR)”, where the die radius shape is uniform in the circumferential direction, the material inflow differs in the circumferential direction due to material anisotropy. The mechanism for suppressing ear height by MR is shown in Figure 3. Figure 3 shows the nodal displacement of the material flange section during forming. In SR drawing shown in Figure 3a, material displacement within the die is radially non-uniform due to material anisotropy. On the other hand, in MR drawing of isotropic material shown in Figure 3b, the area in contact with a small die shoulder R is actively displaced into the die cavity. In areas in contact with large die shoulder R dimensions, the bending stress generated in the surrounding material is low, causing the flow into the die cavity to be delayed and resulting in the earing. Therefore, as shown in Figure 3c, aligning the direction in which the earing naturally forms with the material contacting the area with a large die shoulder R is thought to suppress ear height [15]. The principle of ear shape suppression in MR and the effects of die radius dimensions and die radius shape have been clarified [15,16]. In this paper, the effects of blank initial thickness, blank diameter, and material anisotropy on MR in cylindrical drawing were investigated by experiment and FEA, and the effect of these items on the ear shape was clarified.

Fig. 1 Cylindrical deep drawing.

Fig. 2 Comparison of die shape, (a) Single R die (SR), (b) Multiple R die (MR).

Fig. 3 Mechanism for suppressing ear height by MR, (a) SR anisotropy, (b) MR isotropy, (c) MR anisotropy.

2 Forming method

2.1 Die radius shape in multiple R die

The die radius shape of MR can have various possible shapes. In this study, the die radius shape was defined by the trigonometric function in equation (1).

$$R_{\text{D}}\left({\theta }_{\text{D}}\right)=\frac{R_{\text{Dmax}}\text{+}{R}_{\text{Dmin}}}{2}-\frac{R_{\text{Dmax}}{R}_{\text{Dmin}}}{2}\text{cos}(f\cdot {\theta }_{\text{D}}).$$(1)

R_Dmin and R_Dmax are the minimum and maximum values of the die radius dimensions, f is the shape factor that determines the approximation of the die radius shape, and θ_D is the angular position of the die. Previous studies have confirmed that f = 4 is the most ear height suppressing for materials with four ears [15,16]. Therefore, in this study, forming was also performed at f = 4.

The shape of MR draw depends on the positional relationship between the anisotropy and the die radius shape. Therefore, the angle between the rolling direction and the die reference direction θ_D = 0° was defined as α, shown in Figure 4, and the effect of α on the ear height was investigated.

Fig. 4 Material and die position angle α.

2.2 Material testing

The anisotropy of the materials was investigated for a survey of forming behavior. Tensile test conditions and specimen shape are shown in Table 1 and Figure 5. The Lankford value was measured by tensile test as an indicator of anisotropy [17]. The specimens were used for austenitic stainless steel SUS304 (Japan Industrial Standard, JIS), ferritic stainless steel SUS430, heat-treated aluminum alloy A6061-T6, and tough pitch copper C1100. Only SUS430 was used, with a thickness of 0.25 mm, while the others had a thickness of 0.5 mm. SUS430 was tested at a different sheet thickness than other materials to investigate the effects of both different material properties and sheet thickness. For tensile testing, a universal testing machine “AG-100 kN Xplus” and video-based non-contact extensometer “TRViewX800D” (SHIMADZU CORPORATION) were used. The distances between gauge length and width of the parallel sections of the specimens were measured before testing, and tensile tests were conducted until the nominal strain was 15% for SUS304, SUS430, and C1100, and 5% for A6061. After the test, the distances between gauge length and width were measured again, and Lankford values were calculated. Specimens were tested at angular positions from rolling direction θ_B of 0, 45, and 90°.

The nominal stress-nominal strain curve from the tensile test when measuring anisotropy is shown in Figure 6. No local elongation was observed at any material or angular position from rolling direction θ_B. The test results for Lankford value are shown in Table 2. SUS304 and A6061 had the highest Lankford values at θ_B = 45°, and SUS430 and C1100 had the highest Lankford values at θ_B = 90°. As for Δr [18], which expresses the in-plane anisotropy shown in equation (2), SUS304 and A6061 had negative values, while SUS430 and C1100 had positive values.

$$\mathrm{\Delta }r=\frac{r_0\text{+}{r}_{90}}{2}\text{-}{r}_{45}.$$(2)

Fig. 5 Specimen sheet shape of tensile test.

Fig. 6 Nominal stress − nominal strain diagram, (a) SUS304, (b) SUS430, (c) A6061, (d) C1100.

2.3 Experimental condition

The experimental conditions, experimental equipment, and measurement methods are shown in Table 3 and Figure 7. Each dimension of the die is shown in Figure 1. The combinations of initial sheet thickness and blank material are the same as in Table 1. This paper investigates the effects of the initial blank thickness t₀ and initial blank diameter D_o on earing formation in SUS304, and the other materials were compared finally. Clearance C was set to 0.8 mm because, based on preliminary evaluations using SR, wrinkles did not occur at C = 0.8 mm when the blank folder force was adjusted for t₀ = 0.3 and 0.5 mm. Furthermore, since wrinkles became significantly apparent at t₀ = 0.25 of SUS430, C was set to 0.4 mm. In experiments with SR, R_Dmin was set to 3 mm because the die radius fractured when it was less than 3 mm. Based on previous studies, R_Dmax was standardized to 5 mm in this study, taking into account the effects of material properties and blank diameter. Blank holder force was adjusted for each blank thickness, diameter, and material to prevent difficulties for comparison due to wrinkling. A universal testing machine “SERVOPULSER” (SHIMADZU CORPORATION) was used for deep drawing, as shown in Figure 7a. The measurement of the ear shape after forming was performed using a measuring device that combines a digital indicator “ID-H0530” (Mitutoyo Corporation) and a rotary table, as shown in Figure 7b. The experimental cup shape was organized as the cup height H_C at an angular position from the rolling direction θ_B, as shown in Figure 7c.

Fig. 7 Experimental equipment and measurement methods, (a) Universal testing machine for cylindrical deep drawing, (b) Ear measurement device, (c) Measurement section.

2.4 Analysis condition

The analysis conditions and analysis model are shown in Table 4 and Figure 8. In this study, the dynamic explicit method of the finite element analysis code “Ansys LS-Dyna” was applied to the analysis. The blanks were modeled using solid elements, and all tools were modeled using shell elements. The number of blank elements was set to 50–120 divisions in the radial direction and 3 divisions in the thickness direction, depending on the initial sheet thickness and blank diameter. Work hardening was calculated using equation (3) of Swift's law [19], with parameters derived from the stress-strain relationship at θ_B = 0°.

$${\mathrm{\sigma }}_{\text{T}}=K{\left({\epsilon }_{\text{p}}+{\epsilon }_0\right)}^{\text{n}}.$$(3)

Here, σ_T is true stress, ε_p is plastic strain, n is the strain hardening exponent, K is the strength coefficient, and ε₀ is the offset strain. Hill's quadratic yield function [20] was used for anisotropic materials. The input material data are shown in Table 5. The material data of the specimen was used for SUS304, and the results of Lankford values from Table 2. The coefficient of friction was determined by referring to a previous study [21]. The formed shape was organized by the formed cup height H_C, as shown in Figure 7c, similar to the experimental results.

A preliminary study using SR method was conducted to evaluate the validity of FEA and the experiment. The ear shape and punch load − punch stroke diagram by SR are shown in Figure 9. Both the ear shaping and the punch load showed roughly agreement between experiments and FEA. Therefore, assuming the number of elements and the coefficient of friction in this analysis are valid, they were applied to other analysis conditions.

Fig. 8 Analysis model.

Fig. 9 Comparison of Experiment and FEA, (a) Ear shape, (b) Punch load − punch stroke diagram.

3 Formed results and discussion

3.1 Effect of initial thickness on formed shape

The effect of initial sheet thickness on the forming of MR was investigated. Experimental results for the formed cup height in initial thickness t₀ = 0.3, 0.5 mm, and maximum ear height ΔH_e are summarized in Figures 10 and 11. ΔH_e was calculated by the following equation (4).

$$\mathrm{\Delta }{H}_{\text{e}}=H_{\text{Cmax}}-H_{\text{Cmin}}.$$(4)

H_Cmax and H_Cmin are the maximum and minimum values of the formed cup height, respectively. As shown in the previous experimental results [15], SUS304 produced ears at θ_B = 45° in SR, and the ear height increased at α = 0° in MR and decreased at α = 45°. If the maximum die radius dimension R_Dmax is placed at the point where the highest Lankford value, the ear height can be suppressed [15]. The trend of ear shape change with the initial thickness t₀ change did not change in MR forming. As the initial sheet thickness t₀ was increased, the ear height became large under all forming conditions. Furthermore, in Figure 10, no symmetry in ear height was observed in any of the results. This may be due to misalignment between the material and the die center, with factors such as the blank roundness, set position, and the dimensional accuracy of the die's inner diameter being cited.

FEA was performed to examine the effect of initial sheet thickness. The analysis results of the formed cup height are shown in Figure 12. Figure 12d shows the analysis with isotropic materials in MR forming. By using isotropic materials, the effect of the die radius shape can only be considered. For isotropic materials, the angular position from the rolling direction θ_B is synonymous with the die reference angle position θ_D. Cup height and ear height are higher for initial sheet thickness t₀ = 0.5 mm than for t₀ = 0.3 mm. Comparing the effect of die radius shape only in Figure 12d, without considering the effect of material anisotropy, the thicker the sheet, the larger the cup edge deformation and the larger material inflow due to die radius shape. Furthermore, the experimental results and the analysis results do not quantitatively match. This may be due to the possibility that factors such as springback, elastic deformation of the die, or anisotropic yield functions have not been fully reproduced in FEA. The nodal displacement of the cup during the forming process is shown in Figure 13. Figure 13 shows a vector of xy-nodal displacements of the cup top surface at stroke S = 9 mm. It can be seen that nodal displacement and material inflow are large for initial sheet thickness t₀ = 0.5 mm. Therefore, the thicker the sheet, the greater the material inflow and consequently the higher the ear height.

Fig. 10 Effect of initial thickness on formed cup shape in experimental results, (a) SR, (b) MR α = 0°, (c) MR α = 45°.

Fig. 11 Comparison of maximum ear height in experimental results.

Fig. 12 Relationship between initial sheet thickness and formed cup shape in the analysis results, (a) SR, (b) MR α = 0°, (c) MR α = 45°, (d) MR Isotropy.

Fig. 13 Nodal displacement of vectors during drawing deformation in analysis of isotropic materials, (a) t0 = 0.3 mm, (b) t0 = 0.5 mm.

3.2 Effect of blank diameter on formed shape

The effect of blank diameter on the forming of MR was investigated. Experimental results for the formed cup height in blank diameter D_B = 50, 60 mm, and maximum ear height ΔH_e are summarized in Figures 14 and 15. To compare changes in ear shape only, Figure 14 shows the ear height distribution, the difference between cup height H_C and minimum cup height H_Cmin. As the blank diameter D_B increases, the ear height increases under all forming conditions. The large blank diameter D_B resulted in a slightly larger change in the ear height of MR compared to SR. However, while the initial sheet thicknesses of t₀ = 0.3 and 0.5 mm correspond to a sheet thickness of 0.2 mm and a volume of approximately 393 mm³, the blanks have diameters of D_o = 50 and 60 mm, resulting in a volume of approximately 432 mm³. Therefore, the blank diameter exhibits a greater volume change. This difference is thought to have influenced the degree of change in ear height. The experimental results all lack symmetry, but like the effect of initial sheet thickness, misalignment between the blank and the die center is considered a cause.

FEA was also used to investigate the effect of the blank diameter. The analysis results of the cup height after forming are shown in Figure 16. It can be seen that the larger the blank diameter D_B, the larger the ear height. Figure 16d shows the change in material inflow due to the die radius shape, because the analysis is based on isotropic material. Therefore, when the same die radius shape is used for forming, the material inflow is larger for the large blank diameter D_B, and the change in the ear shape inflow by MR is also considered to have been large. The quantitative discrepancy between experimental and analytical results can be attributed to the fact that FEA does not fully reproduce springback, elastic deformation of the die, or anisotropic yield functions, similar to the effect of initial sheet thickness.

Fig. 14 Effect of blank diameter on formed cup shape in experimental results, (a) SR, (b) MR α = 0°, (c) MR α = 45°.

Fig. 15 Comparison of maximum ear height in experimental results.

Fig. 16 Relationship between initial blank diameter and formed cup shape in the analysis results, (a) SR, (b) MR α = 0°, (c) MR α = 45°, (d) MR Isotropy.

3.3 Effect of blank material on formed shape

The effect of material anisotropy on the forming of MR was investigated. Experimental results for the formed cup height in SUS430, A6061, and C1100, maximum ear height ΔH_e, and photographs of formed products are shown in Figures 17–19. Figure 19 also shows the formed results of SUS304 for reference, and the rolling direction is the left-right direction in the photographs. The blank holder force P_B, and punch diameter D_P are changed to match each material's drawability and sheet thickness. For SUS430 and C1100, contrary to SUS304, ear height decreased at α = 0° and increased at α = 45° in MR. As with SUS304, A6061 showed an increase in ear height at α = 0° and a decrease in ear height at α = 45° in MR. As shown in Table 2, the in-plane anisotropy Δr was negative for SUS304 and A6061, and positive for SUS430 and C1100. Therefore, the ear height can be suppressed at α = 45° when the in-plane anisotropy Δr is negative, and at α = 0° when Δr is positive.

The relationship between the die radius shape of MR and the in-plane anisotropy was investigated by FEA. The cup shape and cup appearance images in the analysis results are shown in Figures 20 and 21. Similar to the experimental results, SUS304 and A6061 showed a tendency for ear height to decrease at α = 45° and increase at α = 0°. The quantitative discrepancy between experimental and analytical results can be attributed to the fact that FEA does not fully reproduce springback, elastic deformation of the die, or anisotropic yield functions, similar to the other effects. A graph summarizing the ear heights from Figures 17–19 are shown in Figure 22. The vertical axis of the graph represents the maximum ear height ratio, calculated by dividing MR ear height ΔH_e by SR ear height ΔH_es. When ΔH_e/ΔH_es is less than 1, ear height decreases; when it is greater than 1, the ear height increases. The ear height decreases at α = 45° and increases at α = 0° when Δr is negative. On the other hand, no correlation was observed between Δr and the degree of earing suppression. Previous studies have demonstrated clear trends in Δr and ear height in MR drawing by employing virtual anisotropic materials in FEA [16]. On the other hand, in this experiment and FEA, factors other than Δr—such as sheet thickness, blank diameter, and blank holder force—differed, which likely prevented a clear relationship from being shown.

According to the in-plane anisotropy equation (2), when Δr is negative, the r value is high in θ_B = 45°. For a normal SR, the flange circumferential compression inflow increases in the direction of the high r value and results in the earing [22]. Therefore, for example, as shown in the experimental results of SUS304 in Figure 9a and A6061 in Figure 16b, the earing is formed at θ_B = 45° in SR. As shown in the analysis results for isotropic materials in Figures 11 and 12, for example, in MR, material flows into the die at the minimum die radius R_Dmin angular position, faster than at other angular positions. If the minimum die radius position is placed at the point where the anisotropic material forms the earing, the ear height can be suppressed, and if the in-plane anisotropy Δr and the ear shape are correlated, the ear shape can be controlled.

Fig. 17 Effect of material formed cup shape in experimental results, (a) SUS430, (b) A6061, (c) C1100.

Fig. 18 Comparison of maximum ear height in experimental results.

Fig. 19 Appearance of formed cups in experiment.

Fig. 20 Relationship between material and formed cup shape in the analysis results, (a) SUS430, (b) A6061, (c) C1100.

Fig. 21 Appearance of formed cups in FEA.

Fig. 22 Comparison of maximum ear height and material anisotropy.

4 Conclusion

In this study, the forming conditions of the work material were mainly investigated in forming by Multiple R die (MR) in cylindrical deep drawing. The results are shown below.

The reduction rate in ear height due to MR forming did not vary significantly with the initial sheet thickness. Additionally, thicker sheets result in a greater inflow of material, relatively increasing the ear height.

A larger blank diameter results in greater ear height. The reduction rate in ear height due to MR is slightly greater with a larger blank diameter.

The ear shape changes due to in-plane anisotropy Δr. The forming conditions for suppressing ear height via MR also change. When Δr is negative, the forming condition requires setting α = 0°, and when Δr is positive, it requires setting α = 45°.

Acknowlegments

This research was supported by a grant from the Die and Mould Technology Promotion Foundation for “Research and development for dies and related fields”. The authors would like to express our gratitude for this support.

Funding

This research was supported by a grant from the Die and Mould Technology Promotion Foundation for “Research and development for dies and related fields”.

Conflicts of interest

The authors declare no conflicts of interest.

Data availability statement

The data supporting this paper are included within the paper.

Author contribution statement

Muraoka: Research coordination, conducting experiments, and writing papers. Okude: Conducting finite element analysis. Nakamura: Investigation of the anisotropy of work material. Katagiri: Investigation of shape measurement of experimental results

References

All Tables

All Figures

	Fig. 1 Cylindrical deep drawing.
In the text

	Fig. 2 Comparison of die shape, (a) Single R die (SR), (b) Multiple R die (MR).
In the text

	Fig. 3 Mechanism for suppressing ear height by MR, (a) SR anisotropy, (b) MR isotropy, (c) MR anisotropy.
In the text

	Fig. 4 Material and die position angle α.
In the text

	Fig. 5 Specimen sheet shape of tensile test.
In the text

	Fig. 6 Nominal stress − nominal strain diagram, (a) SUS304, (b) SUS430, (c) A6061, (d) C1100.
In the text

	Fig. 7 Experimental equipment and measurement methods, (a) Universal testing machine for cylindrical deep drawing, (b) Ear measurement device, (c) Measurement section.
In the text

	Fig. 8 Analysis model.
In the text

	Fig. 9 Comparison of Experiment and FEA, (a) Ear shape, (b) Punch load − punch stroke diagram.
In the text

	Fig. 10 Effect of initial thickness on formed cup shape in experimental results, (a) SR, (b) MR α = 0°, (c) MR α = 45°.
In the text

	Fig. 11 Comparison of maximum ear height in experimental results.
In the text

	Fig. 12 Relationship between initial sheet thickness and formed cup shape in the analysis results, (a) SR, (b) MR α = 0°, (c) MR α = 45°, (d) MR Isotropy.
In the text

	Fig. 13 Nodal displacement of vectors during drawing deformation in analysis of isotropic materials, (a) t0 = 0.3 mm, (b) t0 = 0.5 mm.
In the text

	Fig. 14 Effect of blank diameter on formed cup shape in experimental results, (a) SR, (b) MR α = 0°, (c) MR α = 45°.
In the text

	Fig. 15 Comparison of maximum ear height in experimental results.
In the text

	Fig. 16 Relationship between initial blank diameter and formed cup shape in the analysis results, (a) SR, (b) MR α = 0°, (c) MR α = 45°, (d) MR Isotropy.
In the text

	Fig. 17 Effect of material formed cup shape in experimental results, (a) SUS430, (b) A6061, (c) C1100.
In the text

	Fig. 18 Comparison of maximum ear height in experimental results.
In the text

	Fig. 19 Appearance of formed cups in experiment.
In the text

	Fig. 20 Relationship between material and formed cup shape in the analysis results, (a) SUS430, (b) A6061, (c) C1100.
In the text

	Fig. 21 Appearance of formed cups in FEA.
In the text

	Fig. 22 Comparison of maximum ear height and material anisotropy.
In the text