## 5 Conclusions  
A 3D spatio-temporal kinematic analysis was conducted on a series of triaxial sand specimens sheared under different experimental conditions. This was achieved by introducing 3D kinematic operators under the cylindrical coordinates, which can appropriately account for the specimen geometry. Kinematic results showed 3D micro- to meso-scale localization effects during shearing of the triaxial specimens, leading to the following conclusions:  
1. The development of expansion bands and compaction bands occurred at different temporal stages. Expansion bands were seen initiating as early as in the hardening stage. With strain, softening began to occur and the expanding rate gradually declined, whereas the compaction band began to emerge in the adjacent areas of the expansion band.
2. When a shear band was fully formed, it could alter the radial deformation of the specimen according to the shape and orientation of the shear band(s). Furthermore, local axial strain become concentrated exclusively within the shear band.
3. The localization effects were less evident when confinement was low, or when tests were conducted on the loose specimen. On the other hand, high confining pressure could lead to multiple shear bands progressing simultaneously.
4. The kinematic characterization of the layered specimen showed a similar behavior to that of the loose specimen. Additionally, the failure mode was governed by the material response within the loose part of the specimen.  
It is worth noting that stochastic behavior was present in the sand localization effects, driven by the heterogeneous nature of the material. A statistical characterization of localization effects, which can account for the main deformation mode and uncertainty associated with the kinematics of the sand, is thus significant for further evaluation of sand's localization phenomena, and of interest for calibration and simulation, among other numerical studies on sand failure mechanisms. A follow up paper will introduce a spatio-temporal statistical analysis of the same kinematic operators discussed in this paper.  
Figure 29: Evolution of gradient along \(\hat{g}\) axis \(F_{33}\) of test 120704c: (**a**) axial strain 0.0%–1.0%; (**b**) axial strain 1.0%–30%; (**c**) axial strain 3.0%–5.0%; (**d**) axial strain 5.0%–7.0%; (**e**) axial strain 7.0%–9.0%.  
**Author Contributions:** Conceptualization, Y.Z. and Z.M.-C.; data curation, Y.Z. and Z.M.-C.; funding acquisition, Z.M.-C.; investigation, Y.Z. and Z.M.-C.; methodology, Y.Z.; project administration, Y.Z. and Z.M.-C.; resources, Z.M.-C.; supervision, Z.M.-C.; validation, Y.Z. and Z.M.-C.; visualization, Y.Z.; writing--original draft, Y.Z.; writing--review & editing, Z.M.-C. All authors have read and agreed to the published version of the manuscript.  
**Funding:** Work presented in this paper was sponsored by multiple agencies, including the National Science Foundation (NSF), Consejo Nacional de Ciencia y Tecnologia (CONACYT), and Texas A&M University (TAMU).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The experimental data supporting the findings of this paper are provided along with the article and made available at the Texas Data Repository ([[https://dataverse.tdl.org/dataverse/SGL-MDPI-Topic-StochasticGeomechanics-ForwardModeling](https://dataverse.tdl.org/dataverse/SGL-MDPI-Topic-StochasticGeomechanics-ForwardModeling)]([https://dataverse.tdl.org/dataverse/SGL-MDPI-Topic-StochasticGeomechanics-ForwardModeling](https://dataverse.tdl.org/dataverse/SGL-MDPI-Topic-StochasticGeomechanics-ForwardModeling))) (accessed on 1 August 2022). Additional queries may be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflict of interest.