## 2.1 Necessity and concept of dividing blank holder into multi-ring

During the forming process, BHF is applied to the flange area of the metal plate to increase the radial tensile stress of the metal plate to increase the forming height and to reduce the circumferential compressive stress as much as possible to suppress wrinkles [16].

Theoretically, due to restricted material flow and reduced outer diameter, when the BHF is absent, the outer thickness of the flange would increase far greater than the inner flange. But in fact, using the conventional single blank holder, most of the BHF is forced in a narrow range of flange edge since the blank holder is a whole block. This will cause wrinkles to appear far away from the flange edge in the critical state of metal instability when BHF applied on the flange edge is sufficient to suppress wrinkling. This distribution of BHF increases the risk of drawing defects. Figure 1 and Fig. 2 show the simulated drawn cup and experimental results that the wrinkles appear somewhere inside because of the lack of effective holding.

For this reason, the multi-ring blank holder technique was designed here, and it was expected that appropriate BHF can be effectively applied on every radial flange position. The schematic diagram of the new blank holder can be understood from Fig. 3, where the blank holder is divided into several segments along the radial direction. Based on this configuration, the unreasonable distribution of BHF caused by the uneven metal thicknesses along the flange radial regions can be overcome. It is possible to enhance the forming quality using the new blank holder technique.

## 2.2 Theoretical basis for the feasibility of using multi-ring blank holder

Generally, the wrinkle height on the flange can be expressed by the following formula:

(1)

where , , *ρ* is the particle radial coordinate of the flange, *r*0 is the opening radius of die, *y*m is the maximum value of wrinkle height (MWH), *R*w is the radius of metal plate, *φ* is polar angle and *φ*0 is the angle of one wrinkle.

It can be seen from Eq. (1) that wrinkle height is monotonically increasing from the die opening to the plate edge along the radial direction, and the maximum value *y*m is at the outer edge of the plate (*ρ* = *R*w). However, as mentioned above, BHF applied using the conventional method will be acted in a very small range of flange edge since the large thickness here. That is, the blank holder force per unit area (FUA) on the periphery region of the flange is much larger than the inner position, which is unreasonable and the forming effect will not good. If FUA on the flange periphery is sufficient to suppress wrinkles, then the MWH may appear some internal points due to lack of sufficient FUA. Under this critical state, the mathematical expression of wrinkle height can be improved as:

(2)

where , , *R*c is the radial coordinate of MHW point, and *λ* is the ratio between wrinkle height on flange edge and *y*m.

In this case, a simplified wrinkle curve along the radial direction can be seen in Fig. 4, which consists of two lines with slopes *k*1 and *k*2. And according to the point where BHF is applied, the total BHF can be divided into two parts: *F*1 applied on flange edge, and *F*2 applied on critical point *R*c.

According to the radial wrinkle curve shown in Fig. 4, it can be seen that the wrinkle cannot be suppressed using the single blank holder. As shown in Fig. 5, in order to overcome the deficiency, a deep drawing setup using the multi-ring blank holder shown in Fig. 3 was designed.

Based on this structure of the new blank holder, the FUA applied on different regions can be independent and different. If the value of *F*2 applied on critical point *R*c is suitable, the wrinkles at point *R*c shown in Fig. 4 can be suppressed. At this time, the radial wrinkle curve can be improved as shown in Fig. 6, in which the curve has the same MHW between point *R*c and point *R*w.

Correspondingly, the mathematical wrinkling curve using the new blank holder technique is expressed as Eq. (3).

(3)

From the above analysis, it can be predicted that since all deformation areas can be effectively suppressed by independent and different FUA, the effect of the new technique will be better.