# A Simulation of the Densification Process of NdFeB Bulks by a Modified Drucker-Prager Cap Model
## 3 Results
### Analysis of Sintering Temperature Fields
\[\varepsilon=\varepsilon_{x}+\varepsilon_{y}+\varepsilon_{z}=\frac{1-2\mu}{1+ \mu}\cdot\frac{\phi}{2G}+3\alpha t=\frac{1-2\mu}{E}\phi+3\alpha t \tag{20}\]  
So,  
\[\phi=\frac{E}{1-2\mu}(E\varepsilon-3\alpha t) \tag{21}\]  
In this step, Equation (22) can be derived by substituting Equation (19) into Equation (17) and leveraging the relationship of the Lammer constant:  
\[ \begin{array}{l}\sigma_{x}=2G\varepsilon_{x}+\lambda\varepsilon-\beta t\\ \sigma_{y}=2G\varepsilon_{y}+\lambda\varepsilon-\beta t\\ \sigma_{z}=2G\varepsilon_{z}+\lambda\varepsilon-\beta t\end{array}  \tag{22}\]  
where \(\beta\) denotes the thermal stress, and \(\lambda\) refers to the Lame constant. The equations are as follows:  
\[\beta=\frac{\alpha E}{1-2\mu}=\alpha(3\lambda+2G) \tag{23}\]  
\[\lambda=\frac{E\mu}{(1+\mu)(1-2\mu)} \tag{24}\]  
Eventually, the equation to express stress according to strain and temperature differences is as follows  
\[ \begin{array}{l}\tau_{xy}=G\gamma_{xy}=2G\varepsilon_{xy}\\ \tau_{yz}=G\gamma_{yz}=2G\varepsilon_{yz}\\ \tau_{zx}=G\gamma_{zx}=2G\varepsilon_{zx}\end{array}  \tag{25}\]  
where \(\varepsilon_{xy}\), \(\varepsilon_{yz}\), and \(\varepsilon_{zx}\) are the shear strains of elastic theory. The stress and strain caused by external forces can be calculated through elastic-mechanical concepts, whereas those resulting from temperature differentials are determined using the thermoelastic principle. Eventually, these two parts are stacked together. Nonetheless, it is noteworthy that the calculation precision of this method may be affected to a certain degree when the structure undergoes significant deformation or involves extensive plastic deformation scenarios. During the sintering process, temperature plays an important role in the densification of NdFeB powder [23]. As a consequence, the temperature field's cloud chart of the finite element model is analyzed first. Figure 5 suggests the temperature field at three time points in the sintering process. Figure 5a reveals the preliminary stage of rising temperatures. Figure 5b demonstrates the holding stage, and Figure 5c illustrates the final stage of falling temperatures. As Figure 5 reveals, the powder has a large temperature gradient throughout the heating and cooling processes. The temperature difference between the inside and the outside is about 56 \({}^{\circ}\)  
(Figure 5a) to 146 \({}^{\circ}\)C (Figure 5c), which can be attributed to the rapid heating and cooling of the outer wall. Prolonged temperature and pressure holding allow the temperature inside to reach equilibrium. During the temperature rise stage, the temperature of the outer layer is much higher than the internal temperature, attributable to the heat transfer from the outer wall of the rigid body. In such a case, the densification rate of the powder's outside compactation is much faster than the inside. Meanwhile, as the densification rate rises, the volume of the powder's compaction contracts while its density increases [24]. Therefore, both the temperature gradient and disparity in relative density progressively intensify as the powder's compaction absorbs more heat. Until the temperature and pressure holding stage, the internal and external temperature of the powder compact homogenizes gradually, with the densification rate slowing down.