## 3 Materials and Experimental Methods
### Experimental Methods
Figure 2c depicts the die compaction set-up, while Figure 2d illustrates the die compaction test schematic, which includes four strain gauges on the outer wall of the die for measuring hoop strains. The die was carefully filled with the powders to ensure consistent packing. The load and unload procedures were conducted at a consistent speed of 2 mm/min, which was controlled by the movement of the upper punch, while the bottom punch stayed unmoved. Within the die compaction test, the height of the compact underwent continuous changes in response to the applied pressure. To determine the radial stress, the measured hoop strains by the strain gauges pasted on the die wall were analyzed. Finite element analysis was conducted to calculate the hoop strains on the die outer surface under various uniform interior pressures at different heights. Moreover, a histogram was generated according to the calculated finite element outcomes for radial stress, hoop strain, and compact height on the outer surface of the die [23]. The hoop strains resulting from the die compaction test at different compact heights are depicted in Figure 3.  
Figure 2: (**a**) the failure mode of the Brazilian compression test mode of failure, (**b**) the failure mode of the uniaxial compression test, (**c**) the die compaction set-up, and (**d**) the die compaction schematic with four strain gauges. By utilizing the experimental data of a particular height of compaction and the corresponding hoop strain, the radial stress was determined by resourcing the histogram provided from the simulation results. Finally, the cap eccentricity (\(R\)) was determined by utilizing the maximal amount of radial stress, as calculated using Equation (10). \[R=\frac{1}{3}\sqrt{6}\sqrt{\frac{\left(1+\alpha-\frac{\alpha}{\cos(\beta)} \right)^{2}(p-p_{a})}{q}} \tag{10}\]  
where \(\alpha\) was assumed to be 0.01 for both samples, and the evolution parameter \(P_{a}\) was defined from Equation (3) in Section 2[22], and the hydrostatic pressure (\(p\)) and deviatoric stress (\(q\)) were determined utilizing Equation (11). \[P=\frac{1}{3}(\sigma_{z}+\sigma_{r}) \tag{11}\]  
\[q=|\sigma_{z}+\sigma_{r}| \tag{12}\]  
where \(\sigma_{z}\) and \(\sigma_{r}\) represent the maximal magnitudes of the axial and radial stresses during loading, respectively.\(P_{b}\) as a function of the volumetric plastic strain is needed to determine the cap hardening/softening rule [26]. A series of CIP experiments under pressures ranging from 100 to 700 MPa and with 10 min holding time in a latex cover was performed for every sample. All samples were hand-pressed prior to the CIP experiment for 5 min under 89 MPa pressure in a glove box under an atmosphere with high purity argon. To mitigate the friction between the particles and the die wall, silicone spray was applied. The volumetric plastic strain is derived from the relative density as:  
\[\varepsilon_{v}^{pl}=\ln\left(\frac{\rho}{\rho_{0}}\right) \tag{13}\]  
where \(\rho\) represents the sample's relative density after cold isostatic press, and \(\rho_{0}\) is the sample's relative density after hand press. The Archimedes method in ethanol was applied to determine the relative density of samples. Figure 4c depicts the plotted curves of volumetric plastic strain versus pressure, which were utilized to derive the equation for the hardening law of the samples. Table 3 represents the accomplished tests results and the used DPC model parameters in this work. \[\sigma_{C}=\frac{P_{C}}{A} \tag{7}\]0 The measured hoop strain \(\varepsilon_{\theta}\) for the upper and lower strain gauges versus various compacts height under an interior pressure of 700 MPa for M10 Sn and M10S\(\mu\).