  Citation: Zhu, Y.; Medina-Cetina, Z.; Pineda-Contreras, A.R. Spatio-Temporal Statistical Characterization of Boundary Kinematic Phenomena of Triaxial Sand Specimens. Materials 2022, 15, 2189. https://doi.org/10.3390/ ma15062189 Academic Editor: Giovanni Garcea Received: 1 February 2022 Accepted: 10 March 2022 Published: 16 March 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). materials Article Spatio-Temporal Statistical Characterization of Boundary Kinematic Phenomena of Triaxial Sand Specimens Yichuan Zhu 1 , Zenon Medina-Cetina 2, * and Alma Rosa Pineda-Contreras 3 1 Civil & Environmental Engineering Department, Temple University, Philadelphia, PA 19122, USA; yichuan.zhu@temple.edu 2 Zachry Department of Civil & Environmental Engineering, Texas A&M University, College Station, TX 77843, USA 3 Laboratorio de Geoinformática, Instituto de Ingeniería, Universidad Nacional Autónoma de México UNAM, Torre de Ingeniería2 ◦ Piso, Circuito Escolar C.U., Coyoacan, Mexico City 04510, Mexico; apinedac@iingen.unam.mx * Correspondence: zenon@tamu.edu Abstract: This paper follows up on a reference paper that inspired MDPI’s Topic “Stochastic Ge- omechanics: From Experimentation to Forward Modeling”, where global and local deformation effects on sand specimens are fully described from high resolution boundary displacement fields, and supported by its experimental database, which is open to the scientific community for further study. This paper introduces the use of spatio-temporal statistics from a subset of such an experimental database to characterize the specimens’ spatio-temporal displacement fields, populated by repeating a set of triaxial compression tests on drained, dry, vacuum-consolidated sand specimens, tested under similar experimentally controlled conditions. A three-dimensional digital image correlation (3D-DIC) technique was used to measure the specimens’ boundary displacement fields throughout the course of shearing under axial compression. Spatio-temporal first- and second-order statistics were computed for different data dimensionality conditions (0D, 0D-T, 1D-T, 3D-T) to identify and characterize the dominant failure mechanisms across different testing specimens. This allowed us to quantify localization phenomena’s spatio-temporal uncertainty. Results show that the uncertainty captured along the deformation process across different dimensionality conditions can be directly associated with different failure mechanisms, including localization patterns, such as the onset and evolution of shear, compression, and expansion bands. These spatio-temporal observations show the dependencies between locally distinctive displacement regions over a specimen’s surface, and across different times during a specimen’s shearing process. Results of this work provide boundary spatio-temporal statistics of experimental evidence in sands, which sets the basis for the development of research on the numerical simulation of sand’s constitutive behavior. Moreover, it allows to add a new understanding on the effect of uncertainty on the mechanistic interpretation of sands’ kinematic phenomena. Keywords: statistical analysis; 3D-DIC; spatio-temporal process; localization effects; triaxial compres- sion test 1. Introduction Soils in their natural environment have an inherent variability associated with their geologic origin observed through their specific physical and mechanical properties, and their stratigraphic spatial distribution. These are associated with a wide range of material properties. Soils’ variability in particular represents a unique random (space) and stochastic (time, or space and time) geoscientific and geoengineering challenge, starting with the quantification of such variability into a metric of uncertainty. For instance, in typical labora- tory triaxial compression tests of soil specimens, various mechanistic failure modes can be observed among soils with similar physical characteristics, even if the experimental Materials 2022, 15, 2189. https://doi.org/10.3390/ma15062189 https://www.mdpi.com/journal/materials