## 3 Materials and Experimental Methods
### Experimental Methods  
In order to comprehensively characterize the yield surface, the following parameters need to be determined: \(\beta\), \(d\), \(P_{a}\), \(R\), \(P_{b}\) and \(\alpha\)[22]. To define the Drucker-Prager shear failure surface, the internal friction angle \(\beta\) and the cohesion of material \(d\) are required; to specify the cap surface, the cap shape parameter \(R\) and evolution \(P_{a}\), \(P_{b}\) are needed [24]. The various experiments shown in Table 2 were used to determine the DPC parameters. Lastly, for the DPC parameters determined from the experiments shown in Table 2, we have referenced a summary in our current study and provided detailed descriptions in our previous study [27]. Prior to the main compaction process, the samples were subjected to pre-compaction using a uniaxial press equipment. The die used for pre-compaction had a circular tube shape with an external diameter, an internal diameter, and a height of 20 mm,10 mm, and 50 mm, respectively. The punches, and the die utilized in the pre-compaction were made from 115 CrV3 steel, which has a Poisson ratio of 0.3 and Young's modulus of 210 GPa. Silicon spray was sprayed in the inner surface of die wall to decrease the friction of particles and the die surface [28]. Five samples of each composition were subjected to a series of Brazilian and uniaxial compression tests to determine the shear failure line parameters, the cohesion of material (\(d\)), and the friction angle (\(\beta\)). The specimens with height-to-diameter aspect ratios of 1.5:1 and 0.25 were used for the uniaxial and Brazilian compression tests. Axial loading for the tests was applied using the Hegewald & Peschke material testing system (Nossen, Sachsen, Germany). Figure 2a,b present the failure mode of the Brazilian and uniaxial compression tests, respectively. During the Brazilian compression test, the samples underwent compression diametrically. The resulting radial strength for Brazilian compression test \(\sigma_{T}\) of the samples can be determined using the following equation:  
\[\sigma_{T}=\frac{2P_{T}}{\Pi Dt} \tag{6}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample** & **Mg Average Particle Size** & **SiC Average Particle Size** & **SiC Particle Vol\%** \\ \hline
**M105\(\mu\)** (Mg+ Submicron size SiC) & 40 \(\mu\)m & \(\lessapprox\)1 \(\mu\)m & 10\% \\
**M105n** (Mg+ Nano size SiC) & 40 \(\mu\)m & 50 nm & 10\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The particle sizes of as-received Mg, SiC, and the particles volume percent (Vol%) of the Mg-SiC powder composites. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Testing Procedure** & **Description** & **Parameter, Unit** \\ \hline axial compression test & cohesion of material & \(d\), MPa \\ \hline Radial compression test & friction angle & \(\beta\), degree \\ \hline Instrumented die compaction test & Cap eccentricity & \(R\) \\ \hline CIP experiment & Pressure (Hardening law) & \(P_{b}\), Mpa \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The calibration of Modified Drucker-Prager Cap model. The failure load \(P_{T}\) and the dimensions of the cylindrical sample, including diameter \(D\) and thickness \(t\), are used to calculate tensile strength for Brazilian compression test \(\sigma_{T}\). The sample stress state experienced by diametrical loading can be mathematically represented by the following equations:  
\[\sigma_{C}=\frac{P_{C}}{A} \tag{7}\]  
where \(P_{C}\) and \(A\) represent the uniaxial breaking force and the sample cross-section area [26]. From this stress state, the cohesion of material \(d\) and friction angle \(\beta\) can be defined as follows:  
\[d=\frac{\sigma_{C}\ \sigma_{T}\ (\sqrt{13-2})}{\sigma_{C}\ -2\sigma_{T}} \tag{8}\]  
\[\beta=tan\!-\!1\!\left(\frac{3\ (\sigma_{C}\ -d)}{\sigma_{C}}\right) \tag{9}\]  
The cap shape parameters \(R,P_{a},\ \text{and}\ P_{b}\) were obtained using the die compaction test. The similar die used in the uniaxial and Brazilian compaction experiments was utilized for this test.