# Spatio-Temporal Statistical Characterization of Boundary Kinematic Phenomena of Triaxial Sand Specimens

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## Abstract

**:**

## 1. Introduction

## 2. Soil Experiments

#### 2.1. Triaxial Compression Test

#### 2.2. 3D-DIC

## 3. Statistical Characterization of Spatio-Temporal Boundary Displacement Fields

#### 3.1. “0D-0T” Data Ensemble

#### 3.2. “0D-T” Data Ensemble

#### 3.3. “1D-T” Data Ensemble

#### 3.3.1. “1D-T” Vertical Displacement Field

#### 3.3.2. “1D-T” Radial Displacement Field

#### 3.4. “3D-T” Data Ensemble

#### 3.4.1. First-Order Statistics

#### 3.4.2. Second-Order Statistics

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Triaxial stress–strain curves of the 17-test ensemble (

**a**) and axial strain–volumetric strain curves of the 17-test ensemble (

**b**).

**Figure 2.**Example of 3D displacement field. Test 092903b at 7% of axial strain. (

**a**) Displacement field projections of Cartesian components in horizontal, vertical, and out-of-plane directions (left to right). (

**b**) Displacement field projections of cylindrical components in radial, tangential, and axial directions.

**Figure 3.**Empirical cumulative density function of each constitutive parameter, including Gaussian and lognormal model fits (as descriptive reference only): Young’s modulus (

**a**), Poisson’s ratio (

**b**), Friction angle (

**c**), and dilation angle (

**d**).

**Figure 4.**First-order statistics of axial stress–strain curves (

**a**) and axial strain–volumetric strain curves (

**b**).

**Figure 5.**Correlation coefficients computed from axial stress–strain data ensemble (

**a**) and axial strain–volumetric strain data ensemble (

**b**).

**Figure 6.**(

**a**) Averaged vertical data ensembles calculated from 17 tests at four loading stages—0.8%, 3.2%, 7.0%, and 9.6% of axial strain—for 1D-T data ensembles. The vertical displacement at each specimen height is estimated through averaged vertical displacements that are captured by 3D-DIC. (

**b**) Mean profiles of data ensembles of averaged vertical displacements shown in (

**a**). (

**c**) Standard-deviation profiles of data ensembles of averaged vertical displacements shown in (

**a**).

**Figure 7.**Cases illustrating computing correlation coefficients toward the 1D-T vertical displacement field. (

**a**) Calculation of auto-correlations of data ensemble at loading stage of 7.0% of axial strain. Red triangles represent the computing case that has positive spatial lag, and blue squares represent the computing case that has negative spatial lag. The resulting correlation coefficients are plotted against spatial lags along the vertical profile of the specimen. (

**b**) Calculation of cross-correlations of data ensembles at loading stages of 7.0 and 9.6% of axial strain. The procedure is similar to that of Figure 6a, except that time lag, ${\delta}_{t}$, is non-zero and needs to be interpreted from two loading stages.

**Figure 8.**Smooth hypersurface representing the spatio-temporal empirical correlation structure for the 1D-T data ensemble of averaged vertical displacements.

**Figure 9.**(

**a**) 1D-T data ensembles of averaged radial displacements calculated from 17 tests at four loading stages—0.8%, 3.2%, 7.0%, and 9.6% of axial strain. The radial displacement at each specimen height is estimated through averaged radial displacements that are captured by 3D-DIC. (

**b**) Mean of data ensembles shown in (

**a**). (

**c**) Standard deviation of data ensembles shown in (

**a**).

**Figure 11.**A 3D-T data ensemble (clouds of points) under the Cartesian coordinate system at four loading stages—0.8%, 3.2%, 7.0%, and 9.6% of axial strain. Coordinates are normalized according to the specimen’s diameter. (

**a**) Horizontal (u) displacement data ensembles. (

**b**) Vertical (v) displacement data ensembles. (

**c**) Out-of-plane (w) displacement data ensembles.

**Figure 12.**A 3D-T data ensemble (clouds of points) under the cylindrical coordinate system at four loading stages—0.8%, 3.2%, 7.0%, and 9.6% of axial strain. Coordinates are normalized according to the specimen’s diameter. (

**a**) Radial (r) displacement data ensembles. (

**b**) Tangential (t) displacement data ensembles. (

**c**) Axial (v) displacement data ensembles.

**Figure 13.**Mean and standard deviation distributions of 3D-T data ensemble under the Cartesian coordinate system, where each column defines a specific loading stage, and each row shows either the mean or standard deviation of a displacement’s data ensemble.

**Figure 14.**Mean and standard deviation distributions of 3D-T data ensemble under the cylindrical coordinate system, where each column defines a specific loading stage, and each row shows either the mean or standard deviation of a displacement’s data ensemble.

**Figure 15.**Computation of spatio-temporal correlation coefficients for 3D-T data ensembles. (

**a**) Spatial coordinates of first displacement vector, $u\left(P1\right)$. (

**b**) Spatial lags between $u\left(P1\right)$ and $u\left(P2\right)$. (

**c**) Spatial coordinates of second displacement vector, $u\left(P2\right)$. (

**d**) Resultant correlation coefficient defined by spatio-temporal lags.

**Figure 16.**Spatio-temporal empirical correlation structures of 3D-T data ensembles under Cartesian coordinates. (

**a**) Smooth representation of correlation structures for u, v, and w displacement fields (left to right). (

**b**) Spatial correlation maps for u, v, and w displacement fields when ${\delta}_{t}=0\mathrm{min}$ (i.e., floor of (

**a**)).

**Figure 17.**Spatio-temporal empirical correlation structures of 3D-T data ensembles under cylindrical coordinates. (

**a**) Smooth representation of correlation structures for r, t, and v displacement fields (left to right). (

**b**) Spatial correlation maps for r, t, and v displacement fields when ${\delta}_{t}=0\mathrm{min}$ (i.e., floor of (

**a**)).

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

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 | - |

Statistics | Young’s Modulus (MPa) | Poisson’s Ratio | Friction Angle (Deg) | Dilation Angle (Deg) |
---|---|---|---|---|

Mean | 25.70 | 0.25 | 43.89 | 21.23 |

Standard deviation | 5.70 | 0.16 | 1.19 | 2.60 |

Minimum | 20.67 | 0.07 | 41.74 | 12.94 |

Maximum | 40.68 | 0.49 | 47.14 | 24.55 |

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**MDPI and ACS Style**

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

**AMA Style**

Zhu Y, Medina-Cetina Z, Pineda-Contreras AR.
Spatio-Temporal Statistical Characterization of Boundary Kinematic Phenomena of Triaxial Sand Specimens. *Materials*. 2022; 15(6):2189.
https://doi.org/10.3390/ma15062189

**Chicago/Turabian Style**

Zhu, Yichuan, Zenon Medina-Cetina, and Alma Rosa Pineda-Contreras.
2022. "Spatio-Temporal Statistical Characterization of Boundary Kinematic Phenomena of Triaxial Sand Specimens" *Materials* 15, no. 6: 2189.
https://doi.org/10.3390/ma15062189