## 1 Introduction  
Two reference papers precede this work: one introducing the supporting experimental database used to produce the analyses on localization effects discussed in this paper [1], and its companion paper, which presents the statistical analyses of the experimental database [2]. This paper introduces a study of strain localization in sand specimens, which is assumed to be associated with non-homogeneous deformation occurring in materials when they are subjected to compressive or tensile stress [3]. The accumulation of strain localization is frequently presented as a thin zone of intense deformation [4], such as shear, expansion, or compaction band, which encompass the main material responses once it is fully formed [5]. Since the onset and evolution of localization effects underlie soil mechanical failure processes, its study plays a key role in advancing the understanding of geomechanics, from its phenomenological behavior (global stress-strain responses) to fundamental physics (particulate interactions), to provide the best interpretation of the micro-deformation mechanisms associated with macro-deformation mechanisms observed in geo-structure behavior (e.g., foundations, dams, excavations, etc.). The analysis of localization based on laboratory experimentation requires access to full displacement fields and to standard global (axial) measurements. The latter typically yield at best the averaged material response (global/axial stress-strain behavior), disregarding the occurrence of any localization effects. On the other hand, modeling of full displacement fields, including the use of the discrete element method (DEM) [6; 7], has played an important role, owing to its ability to account for particle-particle interactions and granular morphology, thus enabling the strain localization to be revealed on the grain scale [8; 9; 10; 11; 12]. For example, [PERSON] and [PERSON] [12] used DEM to reveal the formation and collapse of column-like structures, parallel to the principal stress direction during the hardening and softening phases of a sample compressed under the plane strain condition. Further, the ability of DEM to directly measure particle contact maps and kinematics allows for inspection of nonaffine deformation features, including microbands [13], vortex structures [8; 14], and translational/rotational behaviors of soil clusters [15; 16], among other strain localization phenomena. Some experimental efforts to measure full-field displacement fields, and to study grain-scale kinematics in particular, include the study by [PERSON] et al. [17], who combined X-ray imaging and digital image correlation to study the three-dimensional translation and rotation of sand grains in a sand specimen. Based on this work, [PERSON] et al. [18] further developed the 1D-Track technique, which labeled particle volume as the feature to identify the trajectory of each particle during the triaxial compression tests. [PERSON] et al. [19; 20] focused on more sand particle properties, including volume, surface area, and dimension, which are incorporated into the particle-tracking method. Results showed that the particle kinematics and micro shear bands developed prior to the formation of persistent shear bands. More recent works have considered the local deformation gradient to examine the local shear component of displacement fields by using micro-strain [21] and translation gradient [22; 23]. The digital image correlation (DIC) method has emerged as a promising method to elucidate full-field displacement fields that can provide a spatio-temporal continual description of sand deformation. This is based on pixel analysis of images taken during a specimen compression test, from which local displacements are evaluated by comparison of overlapping subsets of images captured at different deformation stages of the test. The effectiveness of DIC in the study of geomechanics has been demonstrated by many previous studies [17; 24; 25].