# A Simulation of the Densification Process of NdFeB Bulks by a Modified Drucker-Prager Cap Model
## 3 Results
### Scanning Electron Microscope Experiment Results  
Figure 11 illustrates the grain size measurement at three sintering temperatures and the histograms of the grain size distribution obtained through software (Nano Measure 2.0). The calculated grain sizes are 7.82 \(\upmu\)m, 9.95 \(\upmu\)m, and 12.48 \(\upmu\)m. As illustrated in Figure 11, the grain size undergoes a significant increase as the sintering temperature rises.  
Figure 10: Stress cloud charts after sintering temperatures of **(a,b)** 850 \({}^{\circ}\)C and **(c,d)** 940 \({}^{\circ}\)C.  
Figure 9: Temperature cloud charts after sintering temperatures of **(a)** 780 \({}^{\circ}\)C, **(b)** 850 \({}^{\circ}\)C, and **(c)** 940 \({}^{\circ}\)C.  
Figure 12 illustrates the grain microstructure of NdFeB bulks at three different sintering temperatures. The dashed line refers to the microstructure corresponding to the central area of the cross-section's temperature field. When the distribution of the neodymium-rich phase and oxide is not uniform, the aggregation of grains easily occurs [29]. The aggregation of grains depends on the movement of the grain boundary, while the movement of the grain boundary depends on the diffusion of atoms. In such a case, a high sintering temperature will facilitate the aggregation of grains [30]. This corresponds to the stress transformation trend mentioned above. As suggested in Figure 12, at a sintering temperature of 780 \({}^{\circ}\)C (Figure 12a), the sample exhibits numerous small grains and pores with denser internal structures. In contrast, at 850 \({}^{\circ}\)C (Figure 12b), the pore area diminishes, and the grains grow considerably larger compared to the prior sample. At a sintering temperature of 940 \({}^{\circ}\)C, the grain size of the sample increases again, and densification displays remarkable elevation (Figure 12c).  
Figure 13 further elucidates that temperature elevation facilitates the densification process of NdFeB, with the bulk cross-section grains displaying distinct stratification at three sintering temperatures. The dashed line refers to the microstructure corresponding to the stratification area of the cross-section's temperature field. The grain's size increases from the top to the bottom, and the grain distribution homogenizes as the temperature rises. Aside from this, densification is further improved, which highly fits with the cross-section simulation in cloud charts and verifies the accuracy of the simulation results.  
Figure 11: Microstructure and crystal size distribution calculation diagram of NdFeB bulks at sintering temperatures of (**a**) 780 \({}^{\circ}\)C, (**b**) 850 \({}^{\circ}\)C, and (**c**) 940 \({}^{\circ}\)C.  
Figure 12: Microstructure and temperature cloud charts of NdFeB bulks at sintering temperatures of (**a**) 780 \({}^{\circ}\)C, (**b**) 850 \({}^{\circ}\)C, and (**c**) 940 \({}^{\circ}\)C.  
Figure 14 demonstrates the microstructure at the corners of the cross-section in the simulated cloud charts of the stress field. The dashed line refers to the microstructure corresponding to the side of the cross-section's stress field. The material will crack when the maximum primary stress exceeds the tensile strength of the material. Moreover, the expansion direction of the fracture can be inferred by observing the direction of the maximum principal stress. As illustrated in Figure 14, it can be seen that the fracture occurs in the high-tensile-stress concentration area, where the value exceeds the tensile strength of NdFeB (80 MPa). This verifies the feasibility and practicability of the simulation prediction. Under such circumstances, FEA can predict the regions prone to fracture occurrences and optimize the process parameters to minimize trial-and-error efforts.