# Study on Mechanical Properties of Deep Expansive Soil and Coupling Damage Model of Freeze–Thaw Action and Loading

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experiments Design

#### 2.1. Sample Preparation

#### 2.2. Experimental System

#### 2.3. Experimental Program

## 3. Experiments Results and Analysis

## 4. Damage Constitutive Model of Deep Expansive Clay under Freeze–Thaw Cycles

#### 4.1. Establishment of Damage Variables

_{ijkl}is the tensor component of material elastic stiffness, σ

_{ij}is the stress tensor component, σ* is the effective stress tensor component, ε

_{ij}is the strain tensor component, and D is the damage variable.

- When the sample is subjected to the loading, macroscopically, it can be regarded as an isotropic microelement, while at the microscopic level it contains some of the basic information that causes damage, and at this point, it can be regarded as a nonhomogeneous microscopic material.
- The clay microelement can be regarded as linearly elastic before damage is produced under loading. At this time, its strength follows the Hooke law, and the nonlinearity of stress–strain results from the generation of material damage. The initial tangential modulus can also be replaced by the elastic modulus of the undamaged material.
- The damage caused by microelements is considered to be strength damage in clay materials.

_{f}, to the total number of cells, N. The damage equation is expressed as follows:

_{0}are parameters of the Weibull distribution. They are determined by the mechanical properties of the clay.

#### 4.2. Determination of the Clay Microelement Strength

_{1}is the axial strain of soil, μ is the Poisson ratio of soil, and σ

_{3}is the surrounding pressure of soil.

_{0}is the elastic modulus of the soil without freeze–thaw cycles, and E

_{n}is the elastic modulus of the soil with freeze–thaw cycles of n times.

_{m}is the damage variable under the combined loading and freeze–thaw action; D is the damage variable under loading; D

_{n}is the damage variable under freeze–thaw actions alone.

#### 4.3. Determination of the Parameters of the Damage Constitutive Equation

_{n}and μ are the elastic modulus and the Poisson ratio, respectively, under freeze–thaw cycles of n times.

## 5. Discussion

- This paper mainly studies some properties of deep expansive clay, but it does not take into account the great differences in particle size, mineral composition, and structure of expansive clay at different depths. We can further explore the influence of these factors on the mechanical properties of soil.
- Whether the artificial soil can represent the deep environment and whether it is different from the deep in situ soil samples needs further discussion.
- Due to the limited amount of undisturbed soil and the test period and test equipment and other factors, the test sample used in the article is slightly insufficient; it can be supplemented by some relevant tests to make the conclusions of the article more reliable.

## 6. Conclusions

- In conventional triaxial test, the compressive strength of deep expansive clay gradually decreased with an increase in the water content. At this time, the stress–strain curve of clay under high confining pressure tended toward strain hardening, while low confining pressure shows strain softening.
- In the triaxial shear test under freeze–thaw cycles, the growth rate of stress with strain gradually decreased as the number of freeze–thaw cycles increased. Moreover, the ultimate peak stress also decreased as a result of the freeze–thaw cycles. Under different freeze–thaw cycles, the stress–strain curves of the triaxial tests all showed strain hardening; as the number of freeze–thaw cycles increased, the cohesion tended to decrease, while the internal friction angle tended to increase.
- Based on Lenaitre’s strain equivalence hypothesis and the Druck–Prager damage criterion, the parameters of the damage constitutive equation are calculated and determined, and the calculated data are substituted into the final damage constitutive equation for verification. The fitting degree between the calculated strength value and the theoretical strength value is as high as 99%.
- The damage constitutive equation can reasonably predict the damage evolution of soil under the combined action of loading and freeze–thaw cycles. This study can serve as an available reference for well wall construction and disaster prediction in deep coal mining.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The stress–strain curve under different moisture contents: (

**a**) ω =14%, (

**b**) ω = 17%, (

**c**) ω = 21%, and (

**d**) ω = 24%.

**Figure 3.**Stress–strain curves under different numbers of freeze–thaw cycles: (

**a**) 0 time, (

**b**) 3 times, (

**c**) 6 times, (

**d**) 9 times, and (

**e**) 12 times.

**Figure 5.**The Mohr stress circle under different numbers of freeze–thaw cycles: (

**a**) 0 time, (

**b**) 3 times, (

**c**) 6 times, (

**d**) 9 times, and (

**e**) 12 times.

**Figure 6.**The change diagram of mechanical index with the number of freeze–thaw cycles. (

**a**) The cohesion, (

**b**) The internal friction angle.

**Figure 7.**The comparison of the damage theoretical curve and test curve of deep expansive clay under different confining pressures. (

**a**) 100 kPa, (

**b**) 200 kPa, and (

**c**) 300 kPa.

Natural Moisture Content (%) | Natural Density (g·cm ^{−3}) | Dry Density (g·cm^{−3}) | Specific Gravity | Liquid Limit (%) | Plastic Limit (%) | Free Expansion Rate (%) |
---|---|---|---|---|---|---|

19.31 | 1.965 | 1.647 | 2.467 | 39.33 | 18.77 | 58.54% |

Particle Size (mm) | Particle Size Ratio (%) |
---|---|

<0.075 | 12.23 |

0.075~0.25 | 37.97 |

0.25~0.5 | 30.19 |

0.5~1.0 | 13.64 |

1.0~2.0 | 5.7 |

The Content of Moisture ω (%) | s | t | R^{2} |
---|---|---|---|

14 | 0.2254 | 132.145 | 0.9845 |

17 | 0.1647 | 112.014 | 0.9756 |

21 | 0.1225 | 97.014 | 0.9874 |

24 | 0.1014 | 80.257 | 0.9565 |

**Table 4.**The mechanical parameters of the damage model under the combined action of freeze–thaw cycles and loading.

The Number of Freeze–Thaw Cycles | ${\mathit{\sigma}}_{3}=100\mathbf{k}\mathbf{P}\mathbf{a}$ | ${\mathit{\sigma}}_{3}=200\mathbf{k}\mathbf{P}\mathbf{a}$ | ${\mathit{\sigma}}_{3}=300\mathbf{k}\mathbf{P}\mathbf{a}$ | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{E}}_{\mathit{n}}(\mathbf{M}\mathbf{P}\mathbf{a})$ | $\mathit{m}$ | ${\mathit{F}}_{0}$ | ${\mathit{E}}_{\mathit{n}}(\mathbf{M}\mathbf{P}\mathbf{a})$ | $\mathit{m}$ | ${\mathit{F}}_{0}$ | ${\mathit{E}}_{\mathit{n}}(\mathbf{M}\mathbf{P}\mathbf{a})$ | $\mathit{m}$ | ${\mathit{F}}_{0}$ | |

0 | 647 | 0.384 | 0.457 | 674 | 0.341 | 0.445 | 660 | 0.316 | 0.471 |

3 | 584 | 0.324 | 0.387 | 485 | 0.387 | 0.381 | 543 | 0.384 | 0.249 |

6 | 502 | 0.318 | 0.754 | 534 | 0.345 | 0.241 | 526 | 0.365 | 0.426 |

9 | 424 | 0.336 | 0.714 | 496 | 0.319 | 0.674 | 507 | 0.314 | 0.874 |

12 | 388 | 0.329 | 0.646 | 403 | 0.327 | 0.429 | 469 | 0.352 | 0.773 |

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**MDPI and ACS Style**

Zhu, Z.; Lin, B.; Chen, S.
Study on Mechanical Properties of Deep Expansive Soil and Coupling Damage Model of Freeze–Thaw Action and Loading. *Appl. Sci.* **2023**, *13*, 11099.
https://doi.org/10.3390/app131911099

**AMA Style**

Zhu Z, Lin B, Chen S.
Study on Mechanical Properties of Deep Expansive Soil and Coupling Damage Model of Freeze–Thaw Action and Loading. *Applied Sciences*. 2023; 13(19):11099.
https://doi.org/10.3390/app131911099

**Chicago/Turabian Style**

Zhu, Zhuliang, Bin Lin, and Shiwei Chen.
2023. "Study on Mechanical Properties of Deep Expansive Soil and Coupling Damage Model of Freeze–Thaw Action and Loading" *Applied Sciences* 13, no. 19: 11099.
https://doi.org/10.3390/app131911099