# Statistical Characterization of Boundary Kinematics Observed on a Series of Triaxial Sand Specimens

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Method

#### 2.1. Triaxial Compression Test

#### 2.2. 3D-Digital Image Correlation (3D-DIC)

## 3. 3D Kinematics of the Boundary Displacement Field

## 4. Statistical Characterization of Boundary Kinematics on Triaxial Sand Specimens

#### 4.1. Experimental Design

#### 4.2. Statistical Characterization of the Evolution of the ${F}_{11}$ Field

#### 4.3. Statistical Characterization of the Evolution of the ${F}_{33}$ Field

#### 4.4. Statistical Characterization of the Evolution of the $div\hspace{0.17em}U$ Field

#### 4.5. Statistical Characterization of the Evolution of the ${(curl\hspace{0.17em}U)}_{\rho}$ Field

## 5. Conclusions

- (1)
- The onsets of expansion and compaction bands follow a chronological order and dominate the main volumetric behavior of the specimen at different loading stages, with a watershed point around an axial strain of ${\epsilon}_{a}=5.0\%$ that corresponds to the early softening stage;
- (2)
- The inter-particle rotation and axial compression are two main kinematic phenomena that appeared from the persistent occurrence of shear band developments. The former is more evident when shear bands develop further within the specimen’s central expansion region, and the latter is as a result of interactions between the shear band and the compaction bands. These kinematic properties can be further related to the formation and buckling of force chains, which warrants a future study to investigate such phenomena according to sands’ particulate behaviors;
- (3)
- The orientation of a shear band can be influenced by the development of expansion and compaction bands. In addition, the local axial strain can be localized inside a persistent shear band once it is fully formed;
- (4)
- The uncertainty analyses show that more variability is associated with the development of compaction and shear bands, compared to that of expansion regions. Also, the intensity of the kinematic phenomena and the location of these may contribute to the increased randomness captured closer to the upper and lower boundaries of the specimen.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The triaxial stress–strain and volumetric–axial strain curves of 17 tests. Dashed intervals indicate the temporal increments of kinematic analysis.

**Figure 2.**The experimental setup: (

**a**) triaxial compression system; (

**b**) the 3D-DIC system; (

**c**) a schematic illustration of a common section captured by two camera systems.

**Figure 3.**Incremental displacement fields of test 092903b between an axial strain of 3.0% and 9.0% (post-peak of the stress–strain curve). The first row from left to right are displacements along horizontal (u), vertical (v), and out-of-plane (w) directions. The second row describes the same displacement field, but decomposed into radial (r), tangential (t), and axial (v) directions in cylindrical coordinates.

**Figure 4.**Schematic illustration of the 3D basic unit for kinematic analysis in cylindrical coordinates.

**Figure 5.**An example of a divergence field of test 092903b at axial strains between ${\epsilon}_{a}=0.0\%$ and ${\epsilon}_{a}=9.0\%$: (

**a**) specimen image at an axial strain of ${\epsilon}_{a}=9.0\%$; (

**b**) stress– strain curve of the test and strain moment of ${\epsilon}_{a}=9.0\%$; (

**c**) divergence field calculated between ${\epsilon}_{a}=0.0\%$ and ${\epsilon}_{a}=9.0\%$; (

**d**) schematic illustration of the positive divergence field in cylindrical coordinates.

**Figure 6.**Experimental design: four kinematic properties—gradient along the $\widehat{\rho}$ axis (${F}_{11}$), gradient along the $\widehat{y}$ axis (${F}_{33}$), divergence ($div\hspace{0.17em}U$), and curl along the $\widehat{\rho}$ axis (${(curl\hspace{0.17em}U)}_{\rho}$)—were calculated on all 3D-DIC-sampled displacement fields, and used for subsequent statistical characterization.

**Figure 7.**The evolution of the mean fields of gradient along the $\widehat{\rho}$ axis ${F}_{11}$: (

**a**–

**e**) mean fields of ${F}_{11}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with varying range colormaps; (

**f**–

**j**) mean fields of ${F}_{11}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with consistent range colormaps.

**Figure 8.**The evolution of the standard deviation fields of gradient along the $\widehat{\rho}$ axis ${F}_{11}$: (

**a**–

**e**) the standard deviation fields of ${F}_{11}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with varying range colormaps; (

**f**–

**j**) the standard deviation fields of ${F}_{11}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with consistent range colormaps.

**Figure 9.**The evolution of the mean fields of gradient along the $\widehat{y}$ axis ${F}_{33}$: (

**a**–

**e**) the mean fields of ${F}_{33}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with varying range colormaps; (

**f**–

**j**) the mean fields of ${F}_{33}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with consistent range colormaps.

**Figure 10.**The evolution of the standard deviation fields of the gradient along the $\widehat{y}$ axis ${F}_{33}$: (

**a**–

**e**) the mean fields of ${F}_{33}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with varying range colormaps; (

**f**–

**j**) the mean fields of ${F}_{33}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with consistent range colormaps.

**Figure 11.**The evolution of the mean fields of divergence $div\hspace{0.17em}U$: (

**a**–

**e**) the mean fields of $div\hspace{0.17em}U$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with varying range colormaps; (

**f**–

**j**) the mean fields of $div\hspace{0.17em}U$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with consistent range colormaps.

**Figure 12.**The evolution of the standard deviation fields of divergence $div\hspace{0.17em}U$: (

**a**–

**e**) the standard deviation fields of $div\hspace{0.17em}U$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with varying range colormaps; (

**f**–

**j**) the standard deviation fields of $div\hspace{0.17em}U$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with consistent range colormaps.

**Figure 13.**The evolution of the mean fields of ${(curl\hspace{0.17em}U)}_{\rho}$: (

**a**–

**e**) the mean fields of ${(curl\hspace{0.17em}U)}_{\rho}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with varying range colormaps; (

**f**–

**j**) the mean fields of ${(curl\hspace{0.17em}U)}_{\rho}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with consistent range colormaps.

**Figure 14.**The evolution of the standard deviation fields of ${(curl\hspace{0.17em}U)}_{\rho}$: (

**a**–

**e**) the standard deviation fields of ${(curl\hspace{0.17em}U)}_{\rho}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with varying range colormaps; (

**f**–

**j**) the standard deviation fields of ${(curl\hspace{0.17em}U)}_{\rho}$ at axial strains of 0.0–1.0%, 1.0–3.0%, 3.0–5.0%, 5.0–7.0%, and 7.0–9.0%, respectively, with consistent range colormaps.

Test Name | Aspect Ratio | Initial Density (kg/m ^{3}) | Relative Density (%) | Friction Angle (Deg) | Peak $({{\mathit{\sigma}}^{\prime}}_{1}/{{\mathit{\sigma}}^{\prime}}_{3})$ | Sample Preparation Method |
---|---|---|---|---|---|---|

092903b | 2.18 | 1710.95 | 91.09 | 49.51 | 7.35 | Vibratory compaction |

093003b | 2.19 | 1696.00 | 85.96 | 47.98 | 6.78 | Vibratory compaction |

100103a | 2.21 | 1702.22 | 88.10 | 48.66 | 7.03 | Vibratory compaction |

100103b | 2.19 | 1717.13 | 93.18 | 47.96 | 6.77 | Vibratory compaction |

100103d | 2.18 | 1702.41 | 88.17 | 47.37 | 6.57 | Vibratory compaction |

100203a | 2.20 | 1715.32 | 92.57 | 48.90 | 7.12 | Vibratory compaction |

100203b | 2.17 | 1711.91 | 91.41 | 47.96 | 6.77 | Vibratory compaction |

100303b | 2.22 | 1718.70 | 93.71 | 48.56 | 6.98 | Vibratory compaction |

120604c | 2.25 | 1717.48 | 93.30 | 48.89 | 7.11 | Vibratory compaction |

120904b | 2.25 | 1720.40 | 94.28 | 48.76 | 5.86 | Vibratory compaction |

120904c | 2.25 | 1713.13 | 91.83 | 48.77 | 5.86 | Vibratory compaction |

120904d | 2.24 | 1707.89 | 90.04 | 47.68 | 5.44 | Vibratory compaction |

120904e | 2.25 | 1718.70 | 93.71 | 47.79 | 5.51 | Vibratory compaction |

101204a | 2.24 | 1708.03 | 90.09 | 48.03 | 6.89 | Dry pluviation |

120604a | 2.23 | 1721.06 | 94.50 | 49.46 | 7.33 | Dry pluviation |

120604b | 2.25 | 1715.13 | 92.50 | 48.54 | 6.98 | Dry pluviation |

121304a | 2.24 | 1721.73 | 94.73 | 49.30 | 7.27 | Dry pluviation |

First-order statistics of experimental data ensemble | ||||||

Mean | 2.22 | 1712.83 | 91.72 | 48.48 | 6.68 | - |

Standard deviation | 0.03 | 7.20 | 2.45 | 0.62 | 0.61 | - |

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**MDPI and ACS Style**

Zhu, Y.; Medina-Cetina, Z.
Statistical Characterization of Boundary Kinematics Observed on a Series of Triaxial Sand Specimens. *Appl. Sci.* **2022**, *12*, 11413.
https://doi.org/10.3390/app122211413

**AMA Style**

Zhu Y, Medina-Cetina Z.
Statistical Characterization of Boundary Kinematics Observed on a Series of Triaxial Sand Specimens. *Applied Sciences*. 2022; 12(22):11413.
https://doi.org/10.3390/app122211413

**Chicago/Turabian Style**

Zhu, Yichuan, and Zenon Medina-Cetina.
2022. "Statistical Characterization of Boundary Kinematics Observed on a Series of Triaxial Sand Specimens" *Applied Sciences* 12, no. 22: 11413.
https://doi.org/10.3390/app122211413