## 2 Experimental Method  
### Triaxial Compression Test  
Dry sand, classified as SP, was used to constitute the testing specimens for triaxial compression tests. The median particle size was 0.50 mm, and the coefficients of uniformity and curvature were 2.34 and 1.11, respectively. In this research, a subset of the database as populated in [PERSON] et al. [1] was used, in which the experimental conditions were nominally similar, to study the dominant mode and uncertainty of localization effects in sand. Table 1 presents the specimen characteristics of 17 participating tests and the basic statistics of each specimen parameter. Thirteen testing specimens were vibratory compacted in three layers, and four specimens were prepared through dry pluviation with a carefully controlled drop height to reach a similar initial density. In the bottom row of Table 1, the standard deviation quantities show that the magnitude of variation of the specimen parameters is relatively small, suggesting that all the specimens were nominally similar in the global sense and thus suitable for representing the data population under the current experimental conditions. The experimental process was practically similar to a conventional triaxial test. However, the Plexiglas cell used for confinement was removed, and the specimen was consolidated at a 40 kPa confining pressure through the use of a vacuum pump. The axial loading was applied to the specimen with a strain-controlled manner (\(\hat{e}=0.2\%/\min\)) along the di \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Test Name**} & \multirow{2}{*}{**Aspect Ratio**} & **Initial Density (kg/m\({}^{3}\))** & **Relative Density (\%)** & **Friction Angle (Deg)** & **Peak (\(\sigma^{*}\)1/\(\sigma^{*}\)3)** & **Sample Preparation Method** \\ \hline  
& 2.24 & 1721.73 & 94.73 & 49.30 & 7.27 & Dry pluviation \\ \multicolumn{6}{c}{**First-order statistics of experimental data ensemble**} \\ Mean & 2.22 & 1712.83 & 91.72 & 48.48 & 6.68 & - \\ Standard deviation & 0.03 & 7.20 & 2.45 & 0.62 & 0.61 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of sample characteristics. rection from the bottom up. Two digital cameras were placed in front of the testing sample, namely Q-Imaging PMI-4201, capturing synchronous images of the deforming specimen at every 0.05% of axial strain. Figure 1 shows the stress-strain and volumetric-axial strain responses of all 17 tests, from 0.0 to 9.8% of axial strain. The volumetric strains in this study were estimated through the reconstruction of specimen geometry by taking advantage of the DIC-sampled boundary evolution. Specifically, we assume the specimen's volume is a sum of a series of \"stacked disks\". The height of each disk was determined as 1 mm, and the diameter was calculated from the average radius derived from the DIC analysis at the corresponding height of the specimen. Equation 1 shows the method to integrate all volumetric disks to estimate the bulk volume of the specimen. The precision and accuracy of this indirect approach to measure the specimen's volumetric strain are thus dependent on the quality and characteristics of the sampled DIC datasets. The calibrations show that this technique was in good agreement with the conventional volumetric strain measurement as demonstrated by several previous studies [1; 20; 21]:  
\[\mathrm{V=}\int_{0}^{\mathrm{H}}\int_{0}^{2\pi}\int_{0}^{\mathrm{R}}\mathrm{r} \;\mathrm{d}\theta\;\mathrm{d}z=\int_{0}^{\mathrm{H}}\int_{0}^{2\pi}\frac{1}{2 }\mathrm{R}^{2}\;\mathrm{d}\theta\;\mathrm{d}z=\frac{1}{2}\mathrm{R}^{2}2\pi \int_{0}^{\mathrm{H}}\mathrm{d}z=\pi\int_{0}^{\mathrm{H}}\mathrm{R}^{2}\mathrm{ d}z \tag{1}\]  
The dashed intervals indicate the strain windows for the kinematic analysis that is detailed in the proceeding sections. The axial stress-strain curves anticipate the development of strain bifurcation effects among all tests in the pre-peak regime, while the variability remains stable after the peak stress.