# A Simulation of the Densification Process of NdFeB Bulks by a Modified Drucker-Prager Cap Model
## 2 Materials and Methods  
As already demonstrated by studies, the density of samples subjected to SPS at approximately 800 \({}^{\circ}\)C attained 99% of the theoretical value, with the augmentation of sintering temperatures rising within the 650-900 \({}^{\circ}\)C range. Grain growth was obvious when the T-sps reached about 900 \({}^{\circ}\)C [16]. This study aimed to explore the densification process across two distinct sintering temperature ranges. NdFeB bulks were sintered at 780 \({}^{\circ}\)C, 850 \({}^{\circ}\)C, and 940 \({}^{\circ}\)C via the spark plasma sintering technique. The density of NdFeB powder (2000 meshes) is 7.4 g/cm\({}^{-3}\). NdFeB powder was placed into a graphite mold for SPS sintering. The SPS parameters were set as follows: a vacuum atmosphere of 10\({}^{-3}\) Pa; a pressure of 60 MPa; sintering temperatures of 780 \({}^{\circ}\)C, 850 \({}^{\circ}\)C, and 940 \({}^{\circ}\)C; a heating rate of 30 \({}^{\circ}\)C/min; and a holding duration of 10 min. The equipment used for the preparation and testing of NdFeB is displayed in Figure 1 and listed as follows: mold with 10 mm diameter; Zeiss Ultra 55 Field Emission Scanning Electron Microscope from Carl Zeiss AG, Oberkochen, Germany; YLJ-SPS-T20 spark plasma sintering furnace; X-ray residual stress tester from Lianying Precision Machinery Technology Co., Ltd., Beijing, China. The X-ray residual stress tester setting program is as follows: The sin\({}^{2}\)\(\Downarrow\) method was adopted to measure the residual stress on the sample's surface. First and foremost, a stress-free standard sample is employed to calibrate the diffraction angle's zero point. Aside from this, replace the target with a Cr target and set the X-ray tube parameters to 20 kV and 4 mA. The detector is moved to collect diffraction peaks at different \(\Downarrow\) angles, and the \(\Downarrow\) angle range is usually set from 0\({}^{\circ}\) to 45\({}^{\circ}\) and needs to be evenly distributed (such as 0\({}^{\circ}\), 12.24\({}^{\circ}\), and 17.45\({}^{\circ}\)). At the end, the software (such as the XrdWin 2.0 system) is utilized to fit the diffraction peak shifts, and the residual stress is calculated. The SEM instrument parameters are as follows: The acceleration voltage and working distance were set to 5 kV and 4 mm. The InLens high-resolution detector from Carl Zeiss AG, Oberkochen, Germany was selected, and the contrast/brightness was adjusted to take images at different magnifi  
Figure 1: (**a**) NdFeB bulks sintered at 780 \({}^{\circ}\)C, 850 \({}^{\circ}\)C, and 940 \({}^{\circ}\)C; (**b**) mold (10 mm diameter); (**c**) scanning electron microscope (SEM); (**d**) spark plasma sintering furnace (SPS); (**e**) X-ray residual stress tester. cations (50\(\times\), 500\(\times\), and 1000\(\times\)) in multiple regions for subsequent selection analysis. The spark plasma sintering processing procedure is as follows: Prior to sintering, the powder undergoes fine grinding and is coated with nano-sized BN on the mold's inner wall to avert adhesion, guaranteeing homogeneous filling. During sintering, the temperature is elevated to the target level under a vacuum, followed by pressure maintenance and temperature stabilization. Ultimately, the material is cooled under an inert atmosphere to prevent oxidation. The modified DPC model has found widespread application in powder compaction simulations, concurrently serving as a density-dependent constitutive model employed for densification research. Figure 2[17] illustrates the yield surface of the modified DPC model, which is composed of two parts. One part is the linear shear failure surface that exists within itself, while the other is the cap surface. The shear failure surface is described by Equation (1):  
\[F_{\mathrm{s}}=t-d-p_{a}\tan\beta \tag{1}\]  
where \(t\) symbolizes the deviatoric stress, \(p_{a}\) defines the hydrostatic pressure, and \(\beta\) and \(d\) represent the angle of friction of the material and its cohesion.