## 2 Experimental Method
### 3D-Digital Image Correlation (3D-DIC)  
The 3D-DIC is a non-intrusive experimental method to sense the displacement field on a deforming surface. The basic unit of DIC analysis is a subgroup of image pixels, called a subset, which is comprised by a cluster of sand colors manifested through the latex specimen membrane. Due to sand grains being naturally characterized with color  
Figure 1: The triaxial stress–strain and volumetric–axial strain curves of 17 tests. Dashed intervals indicate the temporal increments of kinematic analysis.  
variations, each subset is in stereo images possessing a unique mathematical entity that can be identified through pattern recognition. In the present research, two digital cameras were set up in front of the specimen with different orientation angles while focusing on the common area of the specimen surface. The 3D specimen boundary can be reconstructed through synchronous images captured by these two cameras, based on the principle which is similar to the way that human eyes perceive position and shape of an object. Displacement is derived by mapping the overlapping subsets between two digital images, and consequently the deforming vectors are defined by translational, rotational, and shearing quantities of subsets in 3D space. Cubic spline interpolation was used to track the trajectory of subsets over several incremental images referenced to the initial material coordinates. The displacement measurements covered a sector about 85 degrees around the specimen circumference, and yielded approximately 40,000 displacement vectors in each sampling with a resolution of 0.4 mm from center-to-center of the subsets. The 3D-DIC-coupled triaxial compression system is described in Figure 2.  
Figure 3 plots a typical 3D-DIC result of the displacement field for test 092903b between a global axial strain of 3.0% and 9.0%. The contours that indicate displacement quantities (in millimeters) along each axis are superimposed on the deformed body of axial strain at 9.0% (Eulerian description). The first row of Figure 3 shows displacements along horizontal (\(u\)), vertical (\(v\)), and out-of-plane (\(w\)) directions from left to right, respectively. The horizontal displacement field (\(u\) field) indicates the specimen was expanded in the middle and that can be associated with the development of the expansion band. The vertical displacement field shows that significant motion was concentrated at the bottom of the specimen, which is caused by the bottom-up loading method. The out-of-plane displacement field (\(w\) field) explicitly depicts the areas of expansion, in contrast to the top and bottom of the specimen, which presents approximately zero out-of-plane displacements. The second row of Figure 3 is the same displacement field, but decomposed in the cylindrical coordinate system, showing displacement fields along the radial (\(r\)), tangential (\(t\)), and axial (\(v\)) directions. Notice that the radial displacement (\(r\)) correlates with the developed expanding region at the present stage of loading. A shear banding area, suggested by intense clockwise rotation in the tangential displacement field (\(t\) field), is observed along the diagonal direction of the specimen's surface. The vertical displacement field (\(v\)) is the same vector component \(v\) as defined in Cartesian coordinates.  
Figure 2: The experimental setup: (**a**) triaxial compression system; (**b**) the 3D-DIC system; (**c**) a schematic illustration of a common section captured by two camera systems.