## 4 Statistical Characterization of Boundary Kinematics on Triaxial Sand Specimens
### Statistical Characterization of the Evolution of the div \(U\) Field  
The _divergence_ field _div U_ indicates the local volumetric dilation or contraction behaviors. Figure 11 presents the evolution of the mean field of _div U_ in space and time, which indicates a chronological order of expansion-compaction bands' behavior. Since this represents the sum of all components on the principal directions, predominant rates of change may dominate the highest and lowest divergence estimates. The local dilation, which is mainly presented in the middle of specimen, was initiated immediately after the test started (\(\varepsilon_{a}=0.0\)-\(1.0\%\)). The bulging of the specimen led to the volumetric expansion gradually localized in the middle of the specimen where a uniform expansion region formed, which reached its peak value around the start of the softening stage (\(\varepsilon_{a}=3.0\)-\(5.0\%\)). In the next strain level (\(\varepsilon_{a}=5.0\)-\(7.0\%\)), the expanding behavior declined, and two compaction bands formed at the top and bottom of the specimen, illustrating a change of the dominant volumetric behavior. Figure 12 presents the evolution of the standard deviation field of _div U_. The uncertainty is seen localized during the late softening stage (\(\varepsilon_{a}=5.0\)-\(7.0\%\)), and is concentrated in the compaction band areas, which may be an indication of the varying intensity and/or location of occurrence of the compression bands. Overall, the statistical characterization of the _div U_ field suggests a chronological order regarding the appearance of expansion and compaction bands, with the latter possessing more uncertainty as shown in its spatio-temporal descriptions. Figure 10: The evolution of the standard deviation fields of the _gradient_ along the \(g\) axis \(F_{33}\): (**a–e**) the mean fields of \(F_{33}\) at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with varying range colormaps; (**f–j**) the mean fields of \(F_{33}\) at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with consistent range colormaps. Figure 9: The evolution of the mean fields of _gradient_ along the \(y\) axis \(F_{33}\): (**a–e**) the mean fields of \(F_{33}\) at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with varying range colormaps; (**f–j**) the mean fields of \(F_{33}\) at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with consistent range colormaps. Statistical Characterization of the Evolution of the (\(\mathit{curl}\)\(U\))\({}_{\rho}\) Field  
The \(\mathit{curl}\) field along the \(\hat{\rho}\) axis (\(\left(\mathit{curl}\;U\right)_{\rho}\)) represents the rotational deformation gradient along the \(\hat{\rho}\) axis, which can be used to indicate the development of a shear band. The mean plots of the \(\left(\mathit{curl}\;U\right)_{\rho}\) fields (Figure 13) suggest that the strain bifurcation initiated from the hardening stage (\(\varepsilon_{a}=1.0\)-\(3.0\%\)), which can be associated with the development of micro shear bands [11]. In total, three persistent shear bands were fully formed at the loading stage of \(\varepsilon_{a}=5.0\)-\(7.0\%\). The inclination angle measures show that the shear band has a steeper orientation when it appears to the middle compared to the upper and lower parts of the specimen. The authors consider this is due to the development of compaction bands that decreased the vertical movement of the soil particles in their corresponding regions, which resulted in a more horizontal orientation in the upper and lower parts of the specimen. Figure 14 shows the standard deviation of the \(\left(\mathit{curl}\;U\right)_{\rho}\) fields, in which the uncertainty is seen highlighted within the shear bands once they were fully established (\(\varepsilon_{a}=5.0\)-\(7.0\%\)), suggesting varying strain localizations may be due to different a rotation intensity and/or location when compared with other regions within the specimen.