## 4 Results and Discussions
### Forming Limits and Deformation Behavior  
Effects of forming temperature and punch speed on forming limits for FEB = 9 and BER = 4.02 are shown in Figure 3. Figure 3a,b show extruded specimens at the forming temperatures of 180 \({}^{\circ}\)C and 200 \({}^{\circ}\)C, respectively. The specimens (a-1) and (a-2) in Figure 3a were obtained under the punch speed of 2 mm/s and 20 mm/s, respectively. Both specimens showed fractures in the forward extruded parts. On the other hand, the specimens (b-1) in Figure 3b showed minor fractures in the forward extruded part, as in Figure 3a, whilst specimen (b-2) under the punch speed of 20 mm/s does not have any defects. Thus, the elevated forming temperature of 200\({}^{\circ}\) C could result in less severe process conditions for a successful operation. As for the effects of punch speed, more plastic work dissipation under the punch speed of 20 mm/s would increase formability of the AZ31B Mg alloy. This is because the temperature of the work piece could increase due to plastic work dissipation from the high plastic deformation with increased punch speed.  
Deformation behaviors were examined for the CFBE process of AZ31B alloy using thermomechanical finite element analysis coupled with damage evolution. In this study, typical deformation behaviors were given specifically for the CFBE process under the forming temperature of 200 \({}^{\circ}\)C, punch speed of 2 mm/s, and FER = 2.25. Figure 4 shows the distributions of effective stress (Figure 4a), effective strain rate (Figure 4b), effective strain (Figure 4c), and temperature (Figure 4d), for three different BER's of 2.16, 4.02, and 7.75.  
As shown in Figure 4a, effective stresses become bigger as the BER gets larger. It can be explained as follows: The metal flow into the forward direction becomes large, since the exit area in the backward direction becomes small due to a large BER. Therefore, the effective stress near the punch corner becomes large. Distributions of effective strain rate and strain in Figure 4b,c show shear bands from the punch corner to the die corner. In general, ductile damage develops fast when the mean stress is large. Moreover, high shearing deformation amplifies such damage evolution. Thus, such shear bands could be the source of damage evolution and eventual fracture. As shown in Figure 4d, temperature in an extruded work piece becomes higher as BER gets larger, since a high BER induces more plastic work dissipation than the others as expected.  
The distribution of accumulated damage predicted by finite element analysis for the above process conditions is given in Figure 5. Additionally, the photo of the extruded products in experiments under the same process conditions is given in Figure 5, which was reported in the literature [13]. Finite element simulation predicts high values of accumulated damage in the shear band zone, and also near the surface of the forwardly extruded part. This is because more material in the shear band zone flows in the forward direction as a punch advances, especially when FER is low, such as at 2.25. The extruded work pieces for BER of 2.16 and 4.02, are shown in the photo in Figure 5b,c, where it shows some defects at the surface of the forwardly extruded part. When the finite element predictions on the accumulated damage were compared to the experimental observations under the same process conditions, the current finite element predictions matched very well with the experiments qualitatively. To accurately compare them, a critical damage or a criterion is required, which can identify the initiation of fracture in an extruded work piece. Such further work on critical damage is left for the future.  
Figure 3: Extruded specimens at the forming temperature of (**a**) 180 \({}^{\circ}\)C and (**b**) 200 \({}^{\circ}\)C under the punch speed of (1) 2 mm/s, (2) 20 mm/s for FER = 9.0 and BER = 4.02.