## 2 Experimental Method  
### Triaxial Compression Test  
Specimens constituted by dry sand, classified as SP, were tested under three-dimensional stress conditions [1]. The coefficients of uniformity and curvature were 2.34 and 1.11, respectively, providing the specimens an adequate range when viewed in grayscale; thus, the varying color patterns manifested on the specimen membrane can be distinguished through imaging correlation analysis. A subset from the experimental database of five tests was selected to for kinematic analysis. Basic physical properties and sample preparation methods of these tests are summarized in Table 1. All specimens from this subset were constituted approximately 160 mm in height and 70 mm in diameter. Three specimens had a relative density of more than 90.00% (092903b, 121304d, 121304c), one loose specimen had a relative density of 46.39% (121304b), and one two-layer specimen was constituted as bottom half \"dense\" (relative density of 98.97%) and top half \"loose\" (relative density of 30.54%) (120704c) [1]. The distinct density distribution produced by the two-layer specimen was intended for investigation of the effect of induced heterogeneity on the constitutive behavior of specimens (Table 1). A conventional triaxial apparatus performed the compression tests, except that the Plexiglass cell was removed to better capture the specimen's boundary displacement fields; thus, all specimens were vacuum consolidated. Three tests were consolidated to 40 kPa isotropic confining pressure (092903b, 121304c, and 120704c), and two tests were consolidated at confining pressures of 20 kPa (121304d) and 60 kPa (121304c), respectively. Axial loading compression was applied at a controlled axial strain rate of 0.2%/min until the specimen was fully sheared. Global axial stress-strain and strain-volumetric strain responses are shown in Figure 1. Note that due to the removal of the confining cell, the direct measurement of volumetric strain was not possible using the triaxial device; an \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Test Name**} & **Height** & **Diameter** & **Relative** & **Friction Angle** & **Peak** & **Sample** & **Precision** & **Notes** \\  & **(mm)** & **(mm)** & **Density** & **(\%)** & **(deg)** & **(\(\sigma^{*}\)/\(\sigma^{*}\))** & **Method** & \\ \hline
092903b & 155.50 & 71.33 & 91.09 & 49.51 & 7.35 &
\begin{tabular}{c} Vibratory \\ compaction \\ \end{tabular} & 40 kPa confinement \\
121304c & 159.50 & 71.38 & 99.71 & 50.95 & 7.95 & Dry pluxation & 20 kPa confinement \\
121304c & 160.00 & 70.48 & 93.72 & 48.59 & 7.00 & Dry pluxation & 60 kPa confinement \\
121304b & 158.17 & 70.86 & 46.39 & 40.88 & 4.79 & Dry pluxation & Loose specimen (40 kPa) \\  & 157.67 & 70.88 & 68.90 & 43.71 & 5.43 & \begin{tabular}{c} Two-layer specimen (40 kPa) \\ Lower part of two-layer \\ specimen: dense sand \\ Upper part of two-layer \\ specimen: loose sand \\ \end{tabular} & 79.50 & 71.27 & 98.87 & - & - & \
\begin{tabular}{c} Dry pluxation \\ Upper part of two-layer \\ specimen: loose sand \\ \end{tabular} & 78.17 & 70.68 & 30.54 & - & - &
\begin{tabular}{c} Upper part of two-layer \\ specimen: loose sand \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of sample characteristics. alternative methodology was therefore implemented to estimate the sample volume at every deformation stage through the reconstruction of the sample geometry using the stereodigital images and the DIC to produce the material displacement field, and from there to estimate the volumetric strain. A detailed description of the methodology developed to assess the volumetric strain can be found in [25; 31]. Figure 1a shows peak stress occurred for dense specimens (tests 092903b, 121304d and 121304c), but not for loose and two-layer specimens (121304b and 120704c respectively). Comparison of constitutive behaviors of three dense specimens showed the peak stress increase with higher confinement.