## 3 3D Kinematic Operators under Cylindrical Coordinates
\begin{split} grad\ U=\
abla\otimes U=\begin{bmatrix}F_{11}&F_{12}&F_{13} \\ F_{21}&F_{22}&F_{23}\\ F_{31}&F_{32}&F_{33}\end{bmatrix}=\begin{bmatrix}\frac{\partial U_{L}}{ \partial\rho}&\frac{1}{2}\Big{(}\frac{\partial U_{L}}{\partial\rho}-U_{\hat{ \rho}}\Big{)}&\frac{\partial U_{L}}{\partial\psi}\\ \frac{\partial U_{L}}{\partial\rho}&\frac{1}{2}\Big{(}\frac{\partial U_{L}}{ \partial\rho}+U_{\hat{\rho}}\Big{)}&\frac{\partial U_{L}}{\partial\psi}\\ \frac{\partial U_{L}}{\partial\rho}&\frac{1}{\rho}\frac{\partial U_{L}}{ \partial\psi}&\frac{\partial U_{L}}{\partial\psi}\end{bmatrix}\\ div\ U=\
abla\cdot U=\frac{1}{\rho}\frac{\partial}{\partial\rho}\big{(}U_{ \hat{\rho}}\cdot\rho\big{)}+\frac{1}{\rho}\frac{\partial U_{\hat{\rho}}}{ \partial\phi}+\frac{\partial U_{L}}{\partial y}\\ curl\ U=\
abla\times UI=(curl\ U_{\hat{\rho}}+(curl\ U)_{\hat{\rho}}+(curl \ U)_{y}\\ =\Big{(}\frac{1}{\rho}\frac{\partial U_{y}}{\partial\phi}-\frac{ \partial U_{\hat{\rho}}}{\partial y}\Big{)}\hat{\rho}+\Big{(}\frac{\partial U _{L}}{\partial y}-\frac{\partial U_{y}}{\partial\rho}\Big{)}\hat{\phi}\\ +\Big{(}\frac{1}{\rho}\frac{\partial}{\partial\rho}\big{(}U_{\hat{\rho}}\cdot \rho\big{)}-\frac{1}{\rho}\frac{\partial U_{\hat{\rho}}}{\partial\phi}\Big{)} \hat{y}\end{split}  \tag{3}\]  
Figure 4 provides an example of nine components of the _gradient_ field (Equation (1)) based on the state of a specimen's boundary displacements (_092903b_), from undeformed stage to critical state (axial strain 0.0% to 9.6%). The x-y coordinates are normalized according to specimen diameter to enable comparison of the results with other tests. Expansal behavior presented in radial and tangential directions, as shown by positive values in \(F_{11}\) and \(F_{22}\). Along the vertical direction, however, compression was observed inside the shearing zone, especially located at the top-left portion, as shown in \(F_{33}\). The overall volumetric behavior in terms of these two competing phenomena can be found in the plot of the divergence field, as shown in Figure 5. A general volumetric dilation was seen in the middle of the specimen; in particular, a local concentrating zone was found to coincide with areas of shear band. Nevertheless, inside the compression band, which was characterized as negative _divergence_, intense volumetric contraction was found matching with areas of large axial compression, as shown in \(F_{33}\). This revealed that the shear band can introduce highly volumetric expansion or compaction, depending on its overlap with the expansion band or compression band. This observation complements those from previous studies about volumetric change influenced by shear band, which focused on the large voids (volumetric dilation) generated by intense rolling of soil particles under the plane strain conditions [34; 37]. Figure 6 presents the curl fields with respect to three orthogonal axes. Sub-plots (d), (e), and (f) illustrate the positive rotational directions for each axis. In Figure 6a, a shear band is explicitly depicted by highlighted clockwise rotational areas. This corroborates, again, intense rolling of soil particles occurring inside the shear band [34; 37]. For rotation with respect to \(\hat{\phi}\) axis (as shown in Figure 6b), two bands are seen rotating in opposite directions and separated by a neutral zone in the middle, suggesting the \"barrel\" shape of expansion band. As for curl along \(\hat{y}\) axis, an approximate zero rotation is seen across the whole domain, except for some deviations found along the shear banding areas, suggesting that the variations in axisymmetric deformation of the specimen were introduced by the development of the shear band.