non-homogeneous deforming features in a deforming body, in contrast to conventional global-scale measurements such as axial stress versus strain and volumetric strain versus axial strain descriptions that can produce, at best, the volume-averaged deformational characteristics. That is, it is assumed that the specimen has a homogeneous structure. Within the geotechnical engineering community, the development of X-ray tomography and digital image correlation (DIC) methods have provided effective means to advance the elucidation of strain localizations in granular materials and sands, in particular. For instance, [PERSON] [5] used a 150 kV X-ray apparatus to check the large dilatancy within the shear band in a retaining wall model. Starting from the 1980s, [PERSON] et al. [6; 7; 8] used X-ray tomography to study various localization effects in sands, including the thickness, orientation, and volumetric dilation of the shear band. The more recent advent of the microfocus X-ray tomography system has enhanced the resolution of specimen imaging, which allows for microstructure and particulate behavior to be illustrated during the shearing of geo-material. For instance, [PERSON] et al. [9] used microfocus X-ray tomography to examine column-like structures in a shear band. [PERSON] et al. [10; 11] used 3D synchrotron microcomputed tomography to investigate particle translation and rotation, as well as the microband development that precedes the occurrence of persistent shear bands. Apart from X-ray tomography methods, DIC represents another emerging sensing technique that can sample high-resolution displacement fields along the exterior surface of the testing specimens. Compared to X-ray tomography, internal sampling is usually not allowed for DIC methods. Nevertheless, the high-frequency imaging process permits the almost continuous spatio-temporal description of the deforming surface that is critical in the study of localization effects considering that the onset or evolution of strain localization and bifurcation can take place over a very short time window [8], which could be smeared out if the sampling step is large. The DIC methods used in strain localization research have also offered insights on the triggering of the persistent shear band formation, vortex structures, and the build-up and collapse of force chains, among others that can be found in [12; 13; 14; 15; 16; 17]. Despite these studies considering the discrete nature of sands when investigating the localization effects, the spatial derivative of the displacement field, which associates to the relative movements of soil particles and the calculation of strain tensors, has not been directly characterized. To address this challenge, the seminal work of [PERSON] and [PERSON] [18] proposed seven finite strain measures that can derive the deformation _gradient_\(\overline{F}_{n+1}\) based on the 3D discrete element method (DEM) simulations. Building on this concept, [PERSON] et al. [19] calculated the Eulerian strain tensor based on the experimental data of 3D X-ray-computed tomography. The methodology provided in such analyses bridged the findings of strain localization at the micro- and meso-scales, which are complementary to each other, thus both particulate and relative deformations at the local areas were identified. In a more recent work, [PERSON] and [PERSON] [3] proposed the illustration of the kinematic phenomena in the cylindrical coordinate system, in which the orthogonal axes coincide with the principal stress directions of the triaxial compression setting on a cylindrical specimen. Their findings provided a new research perspective by presenting the interactions between different localization effects consisting of dilation, contraction, and rotation, during the specimen's failure process.