# Simulation of Triaxial Tests for Unsaturated Soils under a Tension–Shear State by the Discrete Element Method

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Selection of the Grain-Scale Contact Model

_{n}is the spring normal stiffness and X

_{n}is the normal relative displacement. The tensile force exists when elements are connected to each other. When X

_{n}exceeds the fracture displacement X

_{b}, the connection is broken. Therefore, the maximum normal force between elements F

_{nmax}is

_{s}is also considered in the linear elastic contact model. When two elements contact and slide against each other, the sliding friction force opposite to the sliding direction is generated. The two elements are connected by a breakable spring in the tangent direction, and the tangential spring force F

_{s}is defined by

_{s}is the shear stiffness and X

_{s}is the tangential relative displacement. For a complete element connection, the maximum shear force F

_{smax}is determined by the Coulomb criterion:

_{s}

_{0}is the shear resistance in the tangential direction between elements, ${\mu}_{p}$ is the friction coefficient between elements, and F

_{n}is the normal force (compressive force is negative). When an external force exceeds the maximum shear force F

_{smax}, the tangent connection between elements is broken, and the shear resistance F

_{s}

_{0}disappears. In this case, the tangential force F

_{s}is less than or equal to the maximum shear force F

_{smax}, which is defined by

_{smax}. When the two elements are separated from each other (${X}_{n}>0$), the normal force and the tangential force between elements are zero.

## 3. Back Analysis of Grain-Scale Parameters

#### 3.1. Determining Grain-Scale Parameters by Complex Uniaxial Tensile Test Simulation

#### 3.2. Impact Analysis of Material Parameters on the Discrete Element Simulation

#### 3.2.1. Creating a MatDEM Model for the Complex Uniaxial Tensile Test

^{2}) are controlled. Considering that particle grading and void ratio affect the mechanical behavior of soil, the discrete element particle samples with specified particle grading and void ratio were generated by the Monte Carlo method.

#### 3.2.2. Loading Mode of the Numerical Model and the Calculation Rule of Tensile Strength

^{2}). The peak tensile strength is the maximum tensile stress. The two wedges move in the opposite direction until a continuous failure surface is formed. The average value of contact stress along the tensile direction of two wedges at the failure state is taken as the tensile strength of the specimen. The tensile failure surface is shown in Figure 3.

#### 3.2.3. Influences of Material Parameters on the Complex Uniaxial Tensile Strength

- Impact analysis of the tensile strength on the complex uniaxial tensile strength

_{z}) acting on the element, the relative displacement X

_{n}

_{1}of the top particles increases. When X

_{n}

_{1}> X

_{b}, the connection between particles breaks. If X

_{n}

_{1}= X

_{b}, the element tensile strength (T

_{u}) is obtained:

_{u}is related to the spring normal stiffness ${K}_{n}$, tangential stiffness ${K}_{s}$, and the critical fracture displacement X

_{b}. The values of the material parameters are shown in Table 1.

_{u}, it was found that the critical fracture displacements X

_{b}obtained by automatic material training are different, and they also have a great influence on the tensile failure displacement of the complex uniaxial tensile simulation. As a result, the complex uniaxial tensile strength of the specimen increases with the increasing of tensile strength T

_{u}. The simulation results are shown in Table 2 and Figure 4.

_{z}. When the relative displacement of the bottom particles exceeds the limit deformation (X

_{n}

_{2}> X

_{b}), the horizontal connection breaks. C

_{u}is the stress value in the vertical direction when the connection is broken horizontally, which is obtained by

_{u}, it is found that C

_{u}affects the simulation results, and affects the spring normal and tangential stiffnesses.

- Impact analysis of Young’s modulus on the complex uniaxial tensile strength

^{7}, 2 × 10

^{7}, 3 × 10

^{7}, 4 × 10

^{7}, 5 × 10

^{7}, and 6 × 10

^{7}Pa). Figure 5 shows the tensile stress–displacement curve with different Young’s moduli. It is shown that with the increasing of the Young’s modulus, the tensile displacement decreases.

- Impact analysis of Poisson’s ratio on the complex uniaxial tensile strength

#### 3.3. Relationship between Water Content and the Complex Uniaxial Tensile Strength of Unsaturated Clay

^{3}, respectively, and the corresponding initial void ratios were 0.820, 0.706, and 0.606, respectively.

_{t}significantly depends on the water content w. With the increasing of w, σ

_{t}increases rapidly, and reaches the maximum value at the critical water content w

_{c}, corresponding to the peak value of uniaxial tensile strength. When the water content exceeds a certain value, the change of σ

_{t}is very small with the further increasing of water content.

- (1)
- Dry side (0 < w < 11.5%)$${\sigma}_{t}={A}_{d}+{B}_{d}\cdot w+{C}_{d}\cdot {w}^{2}$$
_{d}, B_{d}, and C_{d}are coefficients related to the initial void ratio under dry side. The values of A_{d}, B_{d}, and C_{d}for specimens with different initial void ratios are shown in Table 4. - (2)
- Wet side (11.5% < w < 35%)$${\sigma}_{t}={A}_{w}+{B}_{w}\cdot w+{C}_{w}\cdot {w}^{2}$$
_{w}, B_{w}, and C_{w}are coefficients under wet side. The values of A_{w}, B_{w}, and C_{w}for specimens with different initial void ratios are shown in Table 5.

#### 3.4. Relation between the Tensile Strength T_{u} and Complex Uniaxial Tensile Strength σ_{t}

_{u}on the complex tensile strength σ

_{t}is significant. Therefore, a series of numerical tests with different tensile strengths T

_{u}was carried out. The numerical simulation of complex uniaxial tensile strength can be divided into three steps: (1) creating samples with the same particle size distribution as the laboratory experiment; (2) cutting the model, giving the material parameters, and balancing the model; and (3) applying the strain-controlled load by the user-defined function, and outputting the calculation results.

^{−4}m, the particle diameter dispersion coefficient (rate) was between 0 and 0.8, the minimum particle radius was 1.25 × 10

^{−4}m, the maximum particle radius was 7.31 × 10

^{−4}m, and the total number of particles was 65,139. The information of discrete element samples with specific initial void ratios is shown in Table 6.

_{u}are listed in Table 1.

_{u}are shown in Table 7.

_{u}and complex uniaxial tensile strength σ

_{t}is obtained by

_{u}is 0 to 20 kPa, and the correction determination coefficient is 0.98722. Figure 8 is the simulation result with a tensile strength T

_{u}of 4 kPa.

#### 3.5. Relationship between MatDEM Material Parameters and the Water Content w of Unsaturated Clay

_{u}and the complex uniaxial tensile strength with a given initial void ratio is obtained by numerical simulation. In the following, the relationship between the tensile strength T

_{u}and water content w with a given initial void ratio is obtained through the intermediate value of complex uniaxial tensile strength.

_{u}and the complex uniaxial tensile strength σ

_{t}is

- (1)
- For the dry side (0 < w < 11.5%),$${\sigma}_{t}=-3.14821+3.88897\cdot w-0.08224\cdot {w}^{2}$$$${T}_{u}=\frac{1.7272-\sqrt{0.0066\cdot {w}^{2}-0.3125\cdot w+3.9980}}{0.0402}$$
- (2)
- For the wet side (11.5% < w < 35%),$${\sigma}_{t}=55.8452-2.40208\cdot w+0.03514\cdot {w}^{2}$$$${T}_{u}=\frac{1.7272-\sqrt{-0.0028\cdot {w}^{2}+0.1930\cdot w-0.7423})}{0.0402}$$

_{u}is obtained with different water contents. In order to verify the applicability of the fitting formula, the complex uniaxial tensile numerical simulation was carried out for unsaturated compacted clay with an initial void ratio of 0.706 and 0.606, respectively. According to the fitting results, numerical simulations for samples with different water contents were carried out, and the simulation results of the relationship between tensile strength and water content for the sample with an initial void ratio of 0.706 are shown in Figure 9. Figure 10 is the simulation result of the sample with an initial void ratio of 0.816 and water content of 20%.

_{u}and the water content w is suitable for simulating the tensile strength of unsaturated clay. According to this relationship, the discrete element numerical simulation of the triaxial tension–shear test for unsaturated soil will be carried out as the follows.

## 4. Simulation of Triaxial Tests for Unsaturated Soils under a Tension–Shear State

_{3}< 0. Finally, the specimen undergoes tensile failure. According to the results of numerical simulation, the variation of tensile–shear strength for unsaturated soil with confining pressures and water contents is revealed, and the triaxial tensile failure modes of unsaturated soil are analyzed.

#### 4.1. Simulation Steps

_{u}, materials are trained automatically, and the trained materials are given to the model. (3) The stress is applied by the upper and lower pressure plates with the water confining pressure σ

_{1}. (4) The upper and lower pressure plates move to the opposite direction with a constant speed until the sample is damaged or a continuous failure surface is formed.

#### 4.2. Simulation Results

#### 4.2.1. The Relationship between Deviatoric Stress and Axial Displacement

_{1}, and the axial stress is the small principal stress σ

_{3}. The deviatoric stress is defined by σ

_{1}–σ

_{3}. Figure 13 shows the discrete element numerical simulation results of samples with an initial void ratio of 0.872 with different confining pressures and water contents. For a given water content, the initial slopes of the deviatoric stress–axial displacement curve with different confining pressures are basically the same; however, the peak and residual strengths with different confining pressures are different. When the confining pressure is small (0 < σ

_{1}< 200 kPa), the peak strength appears early, and the peak value is larger than the confining pressure σ

_{1}, which means that the axial stress σ

_{3}reaches a tensile state. The deviatoric stress drops fast after the peak strength. Finally, the sample undergoes tensile failure, and the axial stress goes to 0. When the confining pressure is large (200 < σ

_{1}< 500 kPa), the peak deviatoric stress increases with the increasing of confining pressure. The peak deviatoric stresses are basically less than or equal to the confining pressure σ

_{1}, which indicates that the axial stress σ

_{3}is greater than 0. It should be noted that the water content affects the peak deviatoric stress and the hardening/softening characteristic. The strength increases with the decreasing of water content and the increasing of confining pressure. Moreover, the dilatancy phenomena is obvious for the samples with a low confining pressure and water content.

#### 4.2.2. Displacement Field

_{1}= 100 kPa), the axial stress σ

_{3}gradually changes from a compressive state to a tensile state. The fracture surface for tensile failure is basically horizontal.

_{1}= 200, 300 kPa), the failure modes of shear elongation and tensile fracture occur simultaneously. At the initial stage of loading, the four sides of the specimen remain straight. Then, a local inclined shear plane on the surrounding side of specimen appears when a threshold of tensile axial displacement is achieved. However, the shear plane does not develop to the interior of the specimen. Finally, the specimen fractures with the continuous increasing of the tensile axial displacement. According to the displacement field, the middle part of the fracture surface is basically horizontal, and local shear zones generate around the specimen.

_{1}= 400 kPa), the axial stress σ

_{3}of specimen is always in the compressive state under the tensile loading path. With the increasing of tensile displacement, pure shear failure occurs; however, axial tensile stress does not appear. According to the displacement field, a shear band generates inside the specimen.

#### 4.2.3. Heat and Energy Field

## 5. Conclusions

- The water content affects the peak deviatoric stress, dilatancy behavior, and failure mode.
- The strength increases with the decreasing of water content and the increasing of confining pressure.
- The dilatancy phenomena is obvious for the specimens with a low confining pressure range and water content.
- The specimens undergo pure tensile failure under a small confining pressure condition, shear elongation and tensile failure under a middle confining pressure condition, and shear failure under a large confining pressure condition.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Force–displacement relationships of contact model: (

**a**) normal direction and (

**b**) tangential direction.

**Figure 2.**Schematic diagram of simulation model: (

**a**) numerical wedges and (

**b**) complex uniaxial tensile numerical model.

**Figure 3.**The simulation results of complex uniaxial tensile test at failure state: (

**a**) stress field and (

**b**) interparticle contact force chain.

**Figure 9.**Relationship between tensile strength and water content for the sample with initial void ratio of 0.706.

**Figure 10.**Simulated result of sample with initial void ratio of 0.816 and water content of 20%: (

**a**) stress field and (

**b**) displacement field.

**Figure 11.**Relationship between tensile strength and water content for the sample with initial void ratio of 0.606.

**Figure 12.**Simulated result of “X-displacement” for sample with initial void ratio of 0.762 and water content of 15%.

Average Particle Radius r _{ave}/m | Particle Diameter Dispersion Coefficient Rate | Specific Gravity G _{s} | Young’s Modulus E/MPa | Poisson’s Ratio ν | Compressive Strength C _{u}/kPa | Internal Friction Coefficient of Material μ _{i} |
---|---|---|---|---|---|---|

0.002 | 0.6 | 2.73 | 20 | 0.3 | 20 | 0.4 |

Variable | Value | |||||||
---|---|---|---|---|---|---|---|---|

Tensile strength T_{u}/kPa | 0.1 | 0.2 | 0.5 | 1 | 2 | 4 | 6 | 8 |

Tensile failure displacement /mm | 0.230 | 0.306 | 0.349 | 0.470 | 0.449 | 0.570 | 0.669 | 0.749 |

Complex uniaxial tensile strength /kPa | 8.081 | 13.872 | 18.182 | 26.263 | 33.401 | 48.754 | 62.896 | 77.979 |

Soil Properties | Specific Gravity | Consistency Limit | USCS Classification | Compaction Characteristics | Particle Size Analysis | |||||
---|---|---|---|---|---|---|---|---|---|---|

Liquid Limit (%) | Plastic Limit (%) | Plasticity Index (%) | Optimum Water Content (%) | Maximum Dry Density (g/cm^{3}) | Sand (%) | Silt (%) | Clay (%) | |||

Value | 2.73 | 37 | 20 | 17 | CL | 16.5 | 1.7 | 2 | 76 | 22 |

Initial Void Ratio e | A_{d} | B_{d} | C_{d} | Determination Factor |
---|---|---|---|---|

0.820 | −3.14821 | 3.88897 | −0.08224 | 0.99546 |

0.706 | −12.37397 | 8.52281 | −0.22218 | 0.98924 |

0.606 | −29.40861 | 18.88517 | −0.81092 | 0.99389 |

Initial Void Ratio e | A_{w} | B_{w} | C_{w} | Determination Factor |
---|---|---|---|---|

0.820 | 55.8452 | −2.40208 | 0.03514 | 0.9753 |

0.706 | 132.56575 | −8.20754 | 0.15516 | 0.9791 |

0.606 | 184.43806 | −11.7487 | 0.24232 | 0.99729 |

Initial Dry Density g/cm^{3} | Particle Size Analysis % | Initial Void Ratio e | Total Number of Particles | ||
---|---|---|---|---|---|

1.5 | Laboratory test | Sand/Silt/Clay | 2/76/22 | 0.820 | 65,139 |

Numerical test | Sand/Silt/Clay | 1.3/74/24.7 | 0.872 | ||

1.6 | Laboratory test | Sand/Silt/Clay | 2/76/22 | 0.706 | 65,338 |

Numerical test | Sand/Silt/Clay | 1.8/78/20.2 | 0.816 | ||

1.7 | Laboratory test | Sand/Silt/Clay | 2/76/22 | 0.606 | 66,327 |

Numerical test | Sand/Silt/Clay | 2.2/76/21.8 | 0.762 |

Initial Void Ratio for Laboratory Test e _{0} | Initial Void Ratio for Numerical Test e _{0} | Total Number of Particles | Tensile Strength T_{u}/kPa | Complex Uniaxial Tensile Strength /kPa | Complex Uniaxial Tensile Failure Displacement /mm |
---|---|---|---|---|---|

0.820 | 0.872 | 65,139 | 0.5 | 10.12 | 0.12 |

1 | 12.23 | 0.15 | |||

2 | 13.45 | 0.25 | |||

3 | 14.11 | 0.34 | |||

4 | 15.45 | 0.38 | |||

5 | 16.89 | 0.40 | |||

6 | 17.98 | 0.42 | |||

7 | 19.56 | 0.45 | |||

8 | 22.46 | 0.48 | |||

9 | 24.56 | 0.51 | |||

10 | 25.79 | 0.58 | |||

11 | 26.89 | 0.67 | |||

12 | 27.36 | 0.75 | |||

13 | 28.45 | 0.80 | |||

14 | 29.35 | 0.86 | |||

15 | 30.15 | 0.89 | |||

16 | 32.12 | 0.95 |

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**MDPI and ACS Style**

Cai, G.; Li, J.; Liu, S.; Li, J.; Han, B.; He, X.; Zhao, C.
Simulation of Triaxial Tests for Unsaturated Soils under a Tension–Shear State by the Discrete Element Method. *Sustainability* **2022**, *14*, 9122.
https://doi.org/10.3390/su14159122

**AMA Style**

Cai G, Li J, Liu S, Li J, Han B, He X, Zhao C.
Simulation of Triaxial Tests for Unsaturated Soils under a Tension–Shear State by the Discrete Element Method. *Sustainability*. 2022; 14(15):9122.
https://doi.org/10.3390/su14159122

**Chicago/Turabian Style**

Cai, Guoqing, Jian Li, Shaopeng Liu, Jiguang Li, Bowen Han, Xuzhen He, and Chenggang Zhao.
2022. "Simulation of Triaxial Tests for Unsaturated Soils under a Tension–Shear State by the Discrete Element Method" *Sustainability* 14, no. 15: 9122.
https://doi.org/10.3390/su14159122