# Modeling of Sand Triaxial Specimens under Compression: Introducing an Elasto-Plastic Finite Element Model to Capture the Impact of Specimens’ Heterogeneity

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## Abstract

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## 1. Introduction

## 2. Application of 3D-DIC on Triaxial Tests

#### 2.1. Triaxial Test

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

## 3. Finite Element Modeling of Triaxial Tests

#### 3.1. Experimental Behavior of Dense, Loose, and Half-Dense Half-Loose Specimens

#### 3.2. Finite Element Model

#### 3.3. Proposed Elasto-Plastic Constitutive Model and Parameters

#### 3.4. Analysis Cases of Homogeneous and Heterogeneous Specimens

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Photo images of deforming specimen 120904c at (

**a**) 0.2% (

**b**) 3.6% (

**c**) 7% (

**d**) 12% of axial strain levels [1].

**Figure 3.**Photo images and cumulative displacement fields in mm at 0.2% of axial strain, representing the elastic state.

**Figure 4.**Photo images and cumulative displacement fields in mm at 12.0% of axial strain, representing the critical state.

**Figure 5.**Averaged displacements of the dense sand specimen (120409c): (

**a**) vertical displacement (

**b**) Radial displacement.

**Figure 6.**Experimental behavior curves (

**a**) axial stress vs. axial strain (

**b**) volumetric strain vs. axial strain for the dense, loose, and half-dense half-loose specimens.

**Figure 7.**Schematic diagram of the triaxial compression test, and of its corresponding finite element model.

**Figure 8.**Vertical profiles of each of the three specimens analyzed (notice that the mean radius figures are the same observations but plotted at different scales).

**Figure 9.**Global stress-strain behavior curves: (

**a**) axial stress vs. axial strain, (

**b**) volumetric strain vs. axial strain.

**Figure 10.**Illustrative hardening and softening model based on the implementation of the ABAQUS Mohr–Coulomb model.

**Figure 11.**Finite element models to reproduce different sand specimen configurations: (

**a**) dense specimen (dense), (

**b**) loose specimen (loose), (

**c**) half-dense half-loose specimen modeled as a homogeneous material (layered_hom), (

**d**) half-dense half-loose specimen modeled as a heterogeneous material (layered_het) (

**e**) half-loose half-dense specimen modeled as a heterogenous material considering a transition zone between the loose and the dense specimen’s segments (layered_het_transition).

**Figure 16.**Contours of model cases at the 12% of axial strain representing the critical state of models (

**a**) dense (

**b**) loose and (

**c**) layered_het_transition for Stress (

**1**), plastic strain (

**2**), and total displacement (

**3**).

**Figure 17.**Displacement vector fields of layered models (

**a**) layered_hom (

**b**) layered_het (

**c**) layered_het_transition.

**Table 1.**Summary of experimental tests used in this study [1].

Case | Test Name | Height (mm) | Diameter (mm) | Initial Density (kg/m ^{3}) | Relative Density (%) | Confinement (kPa) | Sample Preparation |
---|---|---|---|---|---|---|---|

Dense | 120904c | 159.67 | 71.11 | 1713.13 | 91.83 | 40 | Vibratory compaction |

Loose | 121304b | 158.17 | 70.86 | 1588.84 | 46.39 | 40 | Dry pluviation |

Half-densehalf-loose layered | 120704c | 157.67 | 70.88 | 1648.06 (avg.) | 68.90 (avg.) | 40 | Vibratory compaction (two layers) |

Upper | 78.17 | 70.68 | 1549.61 | 30.54 | 40 | ||

Lower | 79.50 | 71.27 | 1764.17 | 98.87 | 40 |

Case | Configuration | Unit Weight (kN/m ^{3}) | Young’s Modulus (kPa) | Poisson’s Ratio | Friction Angle (deg) | Dilation Angle (deg) |
---|---|---|---|---|---|---|

Dense | Dense | 20 | 21,559 | 0.44 | 43.09 | 22.78 |

Loose | Loose | 20 | 15,818 | 0.25 | 32.86 | 14.48 |

Layered_hom | Medium | 20 | 18,164 | 0.20 | 32.12 | 11.97 |

Layered_het | Upper loose | 20 | 15,818 | 0.25 | 32.86 | 14.48 |

Lower dense | 20 | 21,559 | 0.44 | 43.09 | 22.78 | |

Layered_het_transition | Upper loose | 20 | 15,818 | 0.25 | 32.86 | 14.48 |

Transition zone | 20 | 20,361 | 0.37 | 36.86 | 18.19 | |

Lower dense | 20 | 21,559 | 0.44 | 43.09 | 22.78 | |

Porous stone | - | 20 | 1,000,000 | 0.20 | - | - |

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**MDPI and ACS Style**

Song, A.; Pineda-Contreras, A.R.; Medina-Cetina, Z. Modeling of Sand Triaxial Specimens under Compression: Introducing an Elasto-Plastic Finite Element Model to Capture the Impact of Specimens’ Heterogeneity. *Minerals* **2023**, *13*, 498.
https://doi.org/10.3390/min13040498

**AMA Style**

Song A, Pineda-Contreras AR, Medina-Cetina Z. Modeling of Sand Triaxial Specimens under Compression: Introducing an Elasto-Plastic Finite Element Model to Capture the Impact of Specimens’ Heterogeneity. *Minerals*. 2023; 13(4):498.
https://doi.org/10.3390/min13040498

**Chicago/Turabian Style**

Song, Ahran, Alma Rosa Pineda-Contreras, and Zenon Medina-Cetina. 2023. "Modeling of Sand Triaxial Specimens under Compression: Introducing an Elasto-Plastic Finite Element Model to Capture the Impact of Specimens’ Heterogeneity" *Minerals* 13, no. 4: 498.
https://doi.org/10.3390/min13040498