# A Simulation of the Densification Process of NdFeB Bulks by a Modified Drucker-Prager Cap Model
## 3 Results  
### Analysis of Sintering Temperature Fields  
In sequentially coupled thermal stress analysis, the influence of the stress field on the temperature field is negligible, and the determination of the stress field mainly depends on the calculation results of the temperature field. When the temperature field is analyzed in ABAQUS, the control equation of the temperature field is established in accordance with the law of energy conservation and the law of Fourier heat conduction [20]. \[\rho_{\text{i}}C_{i}\frac{\partial T}{\partial t}=k_{i}(\frac{\partial^{2}T}{ \partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}})+Q \tag{11}\]  
Thermal stress arises when an object undergoes temperature variations while being constrained, as it is unable to expand freely with temperature changes. To start with, the generalized [PERSON]'s law describing the association between stress and strain is defined in Equation (12) [21]:  
\[ \begin{array}{l}\epsilon_{x}^{e}=\frac{1}{E_{x}}(\sigma_{x}-(\mu_{yx}\sigma_{ y}+\mu_{zx}\sigma_{z}))\\ \epsilon_{y}^{e}=\frac{1}{E_{y}}(\sigma_{y}-(\mu_{xy}\sigma_{x}+\mu_{zy}\sigma _{z}))\\ \epsilon_{z}^{e}=\frac{1}{E_{z}}(\sigma_{z}-(\mu_{xz}\sigma_{x}+\mu_{yz}\sigma _{y}))\\ \gamma_{xy}^{e}=\frac{1}{G_{xy}}\tau_{xy},\gamma_{yz}^{e}=\frac{1}{G_{xz}} \tau_{xz},\gamma_{zx}^{e}=\frac{1}{G_{yz}}\tau_{yz}\end{array}  \tag{13}\]  
where superscript \(e\) denotes the elastic strain component; \(E_{x}\), \(E_{y}\), and \(E_{z}\) are the [PERSON]'s modulus in the \(x\), \(y\), and \(z\) directions, respectively; \(\mu_{xy}\), \(\mu_{yx}\), \(\mu_{xz}\), \(\mu_{zx}\), \(\mu_{yz}\), and \(\mu_{zy}\) are the [PERSON]'s ratios in the \(xy\), \(yx\), \(xz\), \(xz\), \(yz\), and \(zy\) directions, respectively; \(\tau_{xy}\), \(\tau_{xz}\), and \(\tau_{yz}\) are the shear stresses in the \(xy\), \(xz\), and \(yz\) directions, respectively; \(G\) is the shear modulus. Objects will undergo expansion or contraction during heating and cooling processes. The thermal strain experienced by a unit within a temperature field can be expressed in Equation (14) [22]:  
\[ \begin{array}{l}\epsilon_{x}^{T}=\alpha_{x}T\\ \epsilon_{y}^{T}=\alpha_{y}T\\ \epsilon_{z}^{T}=\alpha_{z}T\end{array}  \tag{14}\]  
where \(T\) represents the thermal expansion strain components: \(\alpha_{x}\), \(\alpha_{y}\), and \(\alpha_{z}\) are the thermal expansion coefficients along the \(x\), \(y\), and \(z\) directions. Subsequently, the total strain can be articulated using Equation (15). \[\varepsilon=\varepsilon^{e}+\varepsilon^{T} \tag{15}\]  
Consequently, the integration of Equation (12) with Equation (14) yields Equation (16):  
\[ \begin{array}{l}\varepsilon_{x}^{e}=\frac{1}{E_{x}}(\sigma_{x}-(\mu_{yx} \sigma_{y}+\mu_{zx}\sigma_{z}))+\alpha_{\text{x}}T\\ \varepsilon_{y}^{e}=\frac{1}{E_{y}}(\sigma_{y}-(\mu_{xy}\sigma_{x}+\mu_{zy} \sigma_{z}))+\alpha_{y}T\\ \varepsilon_{z}^{e}=\frac{1}{E_{z}^{e}}(\sigma_{z}-(\mu_{xz}\sigma_{x}+\mu_{ yz}\sigma_{y}))+\alpha_{z}T\end{array}  \tag{16}\]  
\[ \begin{array}{l}\gamma_{xy}^{e}=\frac{1}{G_{xy}}\tau_{xy},\gamma_{yz}^{e}= \frac{1}{G_{xz}}\tau_{xz},\gamma_{zx}^{e}=\frac{1}{G_{yz}}\tau_{yz},\phi= \sigma_{\text{x}}+\sigma_{y}+\sigma_{\text{z}}\end{array}  \tag{17}\]where \(E\) is the [PERSON]'s modulus of the material, \(\mu\) refers to the [PERSON]'s ratio, \(G\) denotes the shear modulus, and \(\psi\) defines the volume stress. The generalized Hooke's law expressed by strain and temperature differences (Equation (18)) can be derived from Equations (16) and (17). \[ \begin{array}{l}\sigma_{x}=2G\varepsilon_{x}+\frac{\mu}{1+\mu}\phi-2G\alpha t \\ \sigma_{y}=2G\varepsilon_{y}+\frac{\mu}{1+\mu}\phi-2G\alpha t\\ \sigma_{z}=2G\varepsilon_{z}+\frac{\mu}{1+\mu}\phi-2G\alpha t\\ \tau_{xy}=G_{xy}\gamma_{xy},\tau_{yz}=G_{yz}\gamma_{yz},\tau_{zx}=G_{zx}\gamma _{zx}\end{array}  \tag{18}\]  
By incorporating the relationship of volumetric strain e from elastic mechanics, the trio of equations in Equation (18) can be superimposed to derive Equation (20).