Our proposed method for the design of the intermediate die profile mainly involves two phases; the construction of a preform using the virtual die method and the application of a local offset to the preform on the basis of DOCF method. The following steps were involved:

• Step 1. The shape of the final product was enlarged using a scale factor to induce uniform deformation of the material. Then, the center of the enlarged final and the initial shapes were aligned, and the overlapped region was determined as the preform.
• Step 2. The shape of the preform was modified by applying a local offset to its flat parts to minimize the difference of the radial flow velocity and the plastic strain between the corners and at the flat parts of the workpiece.
• Step 3. Finally, the transformed shape was magnified using the scale factor to have an equal cross-section reduction ratio.

In general, the intermediate die shape should be similar to the final shape to make the local reduction ratio uniform and result in a decrease of the drawing force during the final drawing and an increase of the dimensional accuracy of the final product. In this study, the preform, whose shape was as similar to that of the final product as possible, was constructed as a basis of the intermediate die shape via the virtual die method, as shown in Figure 3 (a).

Figure 3: 
Design method of intermediate roll-die profile

First, the enlarged shape of the final product was considered using a scale factor. The scale factor was a function of the areas of the initial material, and the final product calculated based on the schedule-pass of equal reduction area, as shown in Equation (1).

where, Ai is the area of i-th pass, and γ is the reduction ratio of the pass. Then, the center of the enlarged final shape by S.F. was aligned with that of the initial shape. Finally, the overlapped region was determined as the preform for the intermediate die shape of our process.

It was necessary to control the material flow to prevent unfilled regions at the corners of the rectangular cross-section of the final product, in particular, it was crucial to control the flow along the radial direction of the workpiece (i.e., the flow from the edge to the center) during the final drawing process. When the workpiece was in contact with the die, the direction of the material flow within the workpiece was determined by the deformation distance between the initial and the final points of the workpiece-die. Since the workpiece was a continuum, the flow velocity increased with increasing deformation distance. As a result, the likelihood of unfilled defects in the region where the smallest deformation distance occurs increased. In this study, unfilled defect was most likely to occur at the corner of the rectangular cross-section of the final product.

To improve the corner filling of the product, the preform needs to be modified by introducing a local offset so that more material can be involved to fill the corners during the 1st and 2nd drawing passes. Based on this idea, Park et al. [5] applied the DOCF method to design the shape of the intermediate die for a shape drawing process by expanding the corners of the workpiece.

Since the corners of the workpiece remain a free surface during the tandem roll-die drawing process, it is impossible to apply a local offset expanding the corner like the original DOCF method. Therefore, a modified method applying a local offset to the flat parts of the preform instead of the corners was introduced, leading to higher plastic deformation values at the corners during the 2nd converging die drawing. This design method was called as the local offset method (LOM).

By using the LOM, the flat parts of the preform were transformed into concavely curved parts, as shown in Figure 3 (b). The amount of local offset was 50% of the width between the flat parts of the preform and the final shape. Since the area of the transformed shape varies according to the amount of the local offset applied using the LOM, the reduction ratio varies from that of the initial preform. Therefore, the transformed shape was finally magnified using the scale factor to achieve the desired area reduction ratio.

Although the LOM was expected to reduce the difference in the radial-direction of metal flow between the flat parts and the corners of the workpiece during the converging die drawing, it can increase the free surface during the roll-die drawing process affecting shape accuracy by discontinuous deformation. Therefore, another intermediate shape was designed to minimize the free surface area and to increase uniform deformation during the roll-die drawing process. As shown in Figure 3 (c), the flat parts of the preform become convex-shaped once the local offset was set in the opposite direction to that of LOM method. This process was called the reverse local offset method (RLOM).

Figure 5 shows the distribution of the radial flow velocities on the cross-section of the workpiece during the 2nd converging die drawing process. Figure 6 shows the measured radial flow velocities along the outer surface. The unfilled corners after the 2nd converging die drawing were closely related to the metal flow along the radial direction, which is the direction from the center to the outer surface [5].

Figure 5: 
Distribution of radial flow velocity during 2^nd converging drawing processes designed via (a) local offset method (LOM) and (b) reverse local offset method (RLOM)

Figure 6: 
Radial flow velocity along the surface during 2nd converging drawing process

If the radial flow velocity at a specific position of the outer surface is higher than in other positions, a material sink is likely to be formed at that point. In the process designed via the LOM, the difference between the minimum and maximum radial flow velocities was 0.52 mm/s, which indicated as expected that the radial velocity distribution observed by the LOM was more uniform than the one observed for the RLOM (0.64 mm/s). However, the radial flow velocity of the corners of the workpiece in the process designed via the LOM was approximately 45% higher (~ -0.85 mm/s) than that of the RLOM (~ -1.24 mm/s).

These results can be attributed to the difference of free surface area between the two processes during the 1st roll-die drawing. Since the region near the free surface is relatively soft due to little work-hardening during the roll-die drawing, the region has a relatively high radial flow velocity, and easily extends by the axial drawing force and the back tension. Therefore, the process designed via the LOM, where larger free surface area forms during the roll-die drawing, leads to higher radial flow velocity along the corner.

Before the two designed processes were compared in terms of corner filling, the behavior of plastic deformation was analyzed. Figure 7 shows the effective strain distributions during the processes designed via the two methods. The process designed via the RLOM showed a more uniform strain distribution than the LOM after the 1st roll-die drawing. The process designed via the LOM caused significant deformation at the flat parts. The effective strain distributions after 2nd converging die drawing showed a similar tendency compared to those after the 1st roll-die drawing. The process designed via the LOM led to higher plastic deformation at the flat parts than the one designed via the RLOM.

Figure 7: 
Effective strain distributions for the processes designed via (a) local offset method (LOM) and (b) reverse local offset method (RLOM

The standard deviations of plastic strain (SDP) were calculated using Equation (3) for the quantitative evaluation of the strain uniformity after the 2nd converging die drawing [4].

ε_i,S is the effective strain at node i in the FE analysis, ε_avg,S is the average effective strain of section S, and i and N are the node number and the total number of nodes in section S, respectively. In this study, the cross-section of the FE model was divided into nine sections (S= 1~9) and the average effective strain was calculated by extracting the strains of 50 nodes (i= 1~50, N= 50) after the final pass (Figure 8). As a result of the SDP analysis for the two processes, it was confirmed that the process designed via the RLOM resulted in a more homogeneous effective strain distribution within the final product, and the corner region (S3) for the process designed via the LOM had the highest SDP.

Figure 8: 
Sections for standard deviations of plastic strain calculation

Figure 9: 
Standard deviation of plastic strain at each section in final product

Figure 10 shows the changes of the cross-sectional shape of the workpiece during the two new tandem drawing process designed via the LOM and RLOM. The concave offset of the LOM increased the size of the free surface during the roll-die drawing because the transformed shape was magnified after applying the offset (Figure 3 (b)). On the other hand, the convex offset of the RLOM led to a smaller size of the free surface because the shape was reduced (Figure 3 (c)). Therefore, the free surface area of for the process designed via the LOM was larger than the one designed via the RLOM during the roll-die drawing.

Figure 10: 
Cross-sectional shapes for the processes designed via (a) local offset method (LOM) and (b) reverse local offset method (RLOM)

After the final converging die drawing, the size of the unfilled region at the corner, i.e., the diagonal distance between the drawn shape, and the die shape at the corner, was 1.40 mm for the process designed via the LOM and 1.05mm for the process designed via the RLOM.

These results clearly show that the size of the free surface during roll-die drawing, which was determined by the design method (LOM/RLOM), plays an important role in the corner filling during converging die drawing in the new tandem drawing process. Smaller free surface area during roll-die drawing for the RLOM led to smaller unfilled areas. Larger free surface area during the roll-die drawing led to relatively easy plastic deformation, caused by less work-hardening and higher radial flow velocity toward the center at the corner during the converging die drawing. Consequently, a larger unfilled area in the final product was observed. Additionally, the uniformity of the plastic strain, as well as of the radial velocity, is thought to influence the characteristics of the corner filling.

As an indicator of the reliability of the proposed processes, the damage factor was calculated on the basis of the normalized Cockroft–Latham model. As shown in Figure 11, the bar through the process designed via the LOM showed higher damage factor at the corner than the one designed via the RLOM, after both the intermediate roll-die drawing and the final converging die drawing. This is because the intermediate die shape by LOM gave rise to a wider free surface near the corner than the one designed via the RLOM. Since the free surface has no contact with the die, tensile stress dominantly exists near the free surface region during the process. Tensile stress plays a key role in the propagation of internal cracks of the material, and therefore the damage factor. Consequently, it is thought that the accumulated damage near the corner region within the free surface during the roll die drawing results in higher damage factor at the corner after the final drawing.

Figure 11: 
Damage factor distributions for the processes designed via (a) local offset method (LOM) and (b) reverse local offset method (RLOM)