Fig. 3. The stress–strain relationship for the  sheet   metal.

for the simulations, the punch speed is set to 10 m s—1 and a coefficient of Coulomb friction  equal  to 0.1  is  assumed.

3. Wrinkling in  a Tapered  Square Cup

A sketch indicating some relevant dimensions of the tapered square cup is shown in Fig. 1(a). As seen in  Fig.  1(a), the length of each side of the square punch head (2Wp), the die cavity opening (2Wd), and the drawing height (H) are con- sidered as the crucial dimensions  that  affect  the  wrinkling. Half of the difference between the dimensions of the die cavity opening and the punch head is termed the die gap (G) in the

present study, i.e. G = Wd — Wp. The extent of the relatively unsupported  sheet  metal  at  the  draw  wall  is  presumably due

to the die gap, and the wrinkles are supposed to be suppressed by increasing the blank-holder force. The  effects of  both  the die gap and the blank-holder force in relation to the occurrence of wrinkling in the stamping of a tapered square cup are investigated in the following   sections.

3.1 Effect  of  Die Gap

In order to examine the  effect  of  die  gap  on  the wrinkling, the stamping of a tapered square cup with three different die gaps of 20 mm, 30 mm, and 50 mm was simulated. In each simulation, the die cavity opening is fixed at 200 mm, and the cup is drawn to the same height of 100 mm. The sheet metal

used in all three simulations is a 380 mm  × 380 mm square sheet with thickness of 0.7 mm, the stress–strain curve for the

material is shown in Fig.    3.

The simulation results show that wrinkling occurred in all three tapered square cups, and the simulated  shape  of  the drawn cup for a die gap of  50 mm  is shown  in Fig.  4. It  is seen in Fig. 4 that the wrinkling  is  distributed  on  the draw wall and is particularly obvious at the corner between adjacent walls. It is suggested that the wrinkling is due to the large unsupported area at the draw wall during the stamping process, also,  the  side  length  of  the  punch  head  and  the  die   cavity

Fig. 4. Wrinkling  in  a tapered  square cup  (G  = 50 mm).

opening are different owing to the die gap. The sheet metal stretched between the punch head and the die cavity shoulder becomes unstable owing to the presence of compressive trans- verse stresses. The unconstrained stretching of the sheet metal under compression seems to be the main cause for the wrink- ling at the draw wall. In order to compare the results for the

three different die gaps, the ratio þ of the two principal strains is introduced, þ being cmin/cmax, where cmax and  cmin  are the major and the minor principal strains,  respectively.  Hosford and Caddell [5] have shown that if the absolute value of þ is greater than a critical value, wrinkling  is  supposed  to occur, and the larger the absolute value of þ, the greater is the possibility  of wrinkling.

The þ values along the cross-section M–N at the same drawing height for the three  simulated  shapes  with different die gaps, as marked in Fig. 4, are plotted in Fig. 5. It is noted

from Fig. 5 that severe wrinkles are located close to the corner and fewer wrinkles occur in the middle of the draw wall for all three different die gaps. It is also noted  that the bigger  the die gap, the larger is the absolute value of þ. Consequently, increasing the die gap will increase the possibility of   wrinkling