# A Simulation of the Densification Process of NdFeB Bulks by a Modified Drucker-Prager Cap Model
## 3 Results
### Analysis of Stress Field After Powder Sintering  
This work integrates the temperature field with the stress field, as the temperature field alone fails to identify the region where cracks may occur. This integration enables a clearer observation of NdFeB's stress variations in the stress field chart and facilitates the prediction of potential crack areas. The accuracy of the model can be validated by comparing the numerical outcomes with experimental findings. The equivalent residual stress on the upper surface of the NdFeB bulk was measured by an X-ray residual stress tester. As depicted in Figure 6, three measurement locations were chosen to determine values on the upper surface of the NdFeB bulk, which are point A, point B, and point C. Each point was tested three times, and then the results were averaged, as revealed in Table 3.  
As suggested in Figure 7, the test values and simulated values exhibit the same trend but significant discrepancies. This can be attributed to the natural aging of the material and the change in the internal structure of the material during the aging process, which releases the residual stresses on the surface. As already validated by studies, the aging temperature and aging time are directly proportional to the release of residual stress [25]. The highest temperature point on the upper surface (point A) released more residual stress after cooling, which confirms this finding. The model calculates the equivalent residual stress on the upper surface at the end of the cooling process. Nonetheless, over time, the  
Figure 5: Temperature cloud charts at 780 \({}^{\circ}\)C: (**a**) 10 th incremental step; (**b**) 108 th incremental step; (**c**) 623 rd incremental step. release of residual stress causes discrepancies between test and simulation results. To explore the aging-induced decay of residual stress, distinct experimental groups featuring varied natural aging durations can be devised. Specific measurement techniques, like X-ray diffraction, can then be employed for quantitative residual stress analysis. The in-depth exploration of disparities between models and experiments can be conducted through experimental endeavors, thereby enhancing the comprehensiveness of the research. Moreover, the source of residual stress is non-uniform deformation. In the case of powder compaction, residual stresses are mainly caused by mold friction and non-uniform densification [13]. As illustrated in Figures 7 and 8, the residual stress distribution of NdFeB varies greatly after sintering. In combination with the analysis of the temperature field, the green body undergoes non-uniform deformation during the hot pressing of the powder, ultimately shifting the residual stress gradients from the center to the edge. It can be concluded that the green body is affected by the sintering temperature, which results in uneven residual stress. In the powder cylinder, high residual stress at the edges may adversely impact the green body's densification. In the ABAQUS visualization module, the X, Y, and Z axes are defined as 1, 2, and 3 axes, respectively. For the solid unit, \"S22\" and \"S33\" represent the two components of the stress tensor. In this work, the cross-section is the main observation, so we selected \"S22\" (transverse) and \"S33\" (longitudinal) to observe the two stress components. Figure 8 demonstrates the distribution of residual stress in the cross-section of the NdFeB bulk after 780 \({}^{\circ}\)C sintering. It can be seen that the stress distribution in the middle region of the cross-section is extremely uneven and primarily exhibits compressive stress, with a maximum value of 169 Mpa. This phenomenon is attributed to the unidirectional pressure applied through the mold during hot-pressing sintering, thereby giving rise to local plastic deformations at the particle contact points [26].