## 4 Results and Discussions
### Extrusion Loads  
In the CFBE process of AZ31B Mg alloys, the important process parameters affecting the extrusion loads are forming temperature, punch velocity, and the forward-backward extrusion ratio. First, the effects of temperature on the extrusion loads were examined. Figure 6 shows the variation of punch load along a punch stroke that is obtained from the CFBE experiments under the punch speed of 2 mm/s, FER = 2.25, and BER = 7.75. As expected, the punch load increases rapidly at the beginning stage of the CFBE process and decreases slowly after its maximum value. This maximum punch load is often called the extrusion load. As shown in Figure 6, the extrusion load at the forming temperature  
Figure 4: Distribution of (**a**) effective stress; (**b**) effective strain rate; (**c**) effective strain; and (**d**) temperature predicted by finite element analyses for the CFBE processes of AZ31B Mg alloy.  
Figure 5: Distribution of predicted damage and experiments, described in Reference [13] for BER of (**a**) 2.16; (**b**) 4.02; (**c**) 7.75 under the punch speed of 2 mm/s and temperature of 200 \({}^{\circ}\)C.  
of 200 \({}^{\circ}\)C is lower than that at 180 \({}^{\circ}\)C. The increase of forming temperature induces a lower flow stress, and it correspondingly reduces the punch load required for a CFBE process.  
Although the effects of punch speed on the extrusion loads are not shown in this paper, the increase of punch velocity from 2 mm/s to 20 mm/s reduces the extrusion load slightly. It can be explained as follows: The plastic work dissipation rate increases as the deformation rate increases from the high punch speed. Such high plastic work dissipation induces the increase of work piece temperature, and then reduces the flow stress. There is a little increase in flow stress due to the strain-rate dependency of AZ31B Mg alloy, but this increase in flow stress is less than the decrease in flow stress due to the temperature increase by plastic work dissipation.  
Figure 7 shows the variation of punch load along a punch stroke, obtained from the CFBE experiments under the punch speed of 20 mm/s, BER = 7.75, and forming temperature of 200 \({}^{\circ}\)C. Three FER values were considered in this case, and they were 2.25, 4.0, and 9.0. Again, the punch load increases rapidly at the beginning stage and decreases slowly after passing the maximum point of extrusion load. It seems natural that a high FER causes a large punch load. As a FER becomes larger, more material flows in the backward direction.  
Figure 8 shows the comparisons between the extrusion loads by finite element analyses and by experiments for the various FER's and BER's under the forming temperature of 200 \({}^{\circ}\)C and punch speed of 20 mm/s. As the extrusion ratio becomes smaller, the deviation between the predictions and experimental measurements becomes slightly larger. However, the predicted extrusion loads matched those in the experiments well. Similar comparisons were observed for the punch speed at  
Figure 6: Effects of temperature on the punch loads. (Punch speed = 2 mm/s, FER = 2.25, BER = 7.75).  
Figure 7: Variation of punch loads along the punch stroke. (Temperature = 200 \({}^{\circ}\)C, BER = 4.02, and punch speed = 20 mm/s).  
2 mm/s, although it is not given here. It may be concluded that the finite element analysis coupled with damage evolution presented in this work is valid for modeling a CFBE process of AZ31B Mg alloys at warm temperatures.