These efforts are focused on the wrinkling problems associa- ted with the forming  operations  of  simple  shapes  only, such as a circular cup. In the early 1990s, the successful application of the 3D dynamic/explicit finite-element method to the sheet- metal forming process made it possible to analyse the wrinkling problem involved in stamping complex shapes. In the present study, the 3D finite-element method was employed to analyse the effects of the process parameters on the metal flow causing wrinkles at the draw wall in the stamping of a tapered square cup,  and  of a  stepped  rectangular part.

A tapered square cup, as shown in Fig. 1(a), has an inclined draw wall on each side of the cup, similar to that existing in a conical cup. During  the  stamping  process,  the  sheet metal on  the  draw  wall  is  relatively  unsupported,  and  is  therefore

Fig. 1. Sketches of (a) a tapered square cup and  (b) a stepped rectangular cup.

prone to wrinkling. In the present study, the effect of various process parameters on the wrinkling was investigated.  In the case of a stepped rectangular part, as shown in  Fig.  1(b), another type of wrinkling is observed. In order to estimate the effectiveness of the analysis, an actual production part with stepped geometry was examined in the present  study.  The cause of the wrinkling was determined using finite-element analysis, and an optimum die design was proposed to eliminate the wrinkles. The die design obtained from finite-element analy- sis was validated by observations on an actual production    part.

2. Finite-Element Model

The tooling geometry, including the punch, die and blank- holder, were designed using the CAD program PRO/ ENGINEER. Both the 3-node and 4-node shell elements were adopted to generate the mesh systems for the above tooling using the same CAD program. For the finite-element simul- ation, the tooling is considered to be rigid, and the correspond- ing  meshes  are used  only to define  the tooling  geometry   and

Fig. 2. Finite-element mesh.

are not for stress analysis. The same CAD program using 4- node shell elements was employed to construct  the mesh system for the sheet blank. Figure  2 shows the  mesh  system for the complete set of tooling and the sheet-blank used in the stamping of a tapered square cup. Owing to the symmetric conditions, only a quarter of  the  square  cup  is  analysed.  In the simulation, the sheet blank is put on the blank-holder  and the die is moved down to clamp the sheet blank against the blank-holder. The punch is then moved up to draw the sheet metal into the die   cavity.

In order to perform an accurate finite-element analysis, the actual stress–strain relationship of  the sheet  metal  is required as part of the input data.  In  the  present  study,  sheet  metal with deep-drawing quality is used in the simulations. A tensile test has been conducted for the specimens cut along planes coinciding with the rolling direction (0) and at angles of 45

and 90to the rolling direction. The average flow stress o, calculated from the equation o= (o0 + 2o45 + o90)/4, for each measured  true  strain,  as  shown  in  Fig.  3,  is  used  for      the

simulations for the stampings of the tapered  square  cup and also  for the  stepped  rectangular cup.

All the simulations performed in the present  study were    run

on an SGI Indigo 2 workstation using the finite-element pro- gram PAMFSTAMP. To complete the set of input data  required