# Freeze–Thaw Cycle Effects on the Energy Dissipation and Strength Characteristics of Alkali Metakaolin-Modified Cement Soil under Impact Loading

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## Abstract

**:**

## 1. Introduction

## 2. Test

#### 2.1. Testing Material

^{2}/g specific surface area and a 1 μm average particle size with a fineness of 1250 meshes and an activity index of more than 110 were adopted. The characteristic parameters are listed in Table 3.

#### 2.2. Specimen Preparation

#### 2.3. Test Scheme

#### 2.4. Testing Equipment

_{i}(t) + ε

_{r}(t) and ε

_{t}(t) curves were consistent. The number of intersection points of the two curves was greater than three, and the two curves basically coincided. Therefore, under impact load, the cement soil sample satisfied dynamic stress equilibrium condition.

## 3. Test Results and Analysis

#### 3.1. Mass Loss Rate

#### 3.2. Strength

#### 3.3. Shape Features

_{r}/M]-lnr curve and performing linear fitting, the value of b was obtained to be equal to the slope of the curve. It was seen from Figure 10 that the correlation coefficient (R

^{2}) of linear fitting was higher than 0.99, which revealed the self-similarity of cement soil under impact load.

#### 3.4. Energy Dissipation Characteristics

_{i}), reflected energy (W

_{r}) and transmitted energy (W

_{t}) in SHPB tests were as follows [26]:

_{i}is incident stress, σ

_{r}is reflected stress, σ

_{t}is transmitted stress (all in MPa), ε

_{i}(t) is incident strain, ε

_{r}(t) is reflected strain, ε

_{t}(t) is transmitted strain, E

_{0}is elastic modulus, A

_{0}is cross-sectional area and C

_{0}is the longitudinal wave velocity of elastic bars.

_{t}(t) = ε

_{i}(t) + ε

_{r}(t). Combining Equations (3)–(5), Equation (6) was simplified as:

_{r}is reflected stress, σ

_{t}is transmitted stress, E

_{0}is elastic modulus, A

_{0}is cross-sectional area and C

_{0}is compression bar longitudinal wave velocity.

## 4. The Modified Dynamic Constitutive Model

#### 4.1. Derivation Process

^{−4}~10

^{3}s

^{−1}). The model consisted of a transient response element, which was unrelated to strain rate, one is the transient response element, which is unrelated to the strain rate, and the other is the transient response, which is related to the strain rate and is composed of two Maxwell elements, as shown in Figure 17. The mathematical expression of the model was expressed as Equation (8) [24]:

_{0}, E

_{1}, E

_{2}, α and β are elastic constants, and θ

_{1}and θ

_{2}are relaxation times of low-frequency and high-frequency Maxwell elements, respectively.

^{−1}), the time relaxation of a low-frequency Maxwell element with a characteristic time of 0.1 s was not sufficient, which could be simplified as a Hooke element with an E

_{1}elastic constant, as shown in Equation (10) and Figure 14b:

_{0}and E

_{1}) existed. To further simplify the equation, an equivalent Hooke element (E

_{a}) was used instead, as shown in Equation (11) and Figure 14c:

#### 4.2. Damage Variable

_{12}), as expressed in Equation (15):

_{1}and D

_{2}are damage variables due to freeze–thaw cycle and strain rate, respectively, and D

_{12}is damage variable considering both strain rate and freeze–thaw cycle. The mathematical expressions of D

_{1}and D

_{2}are stated in Equations (16) and (17), respectively:

_{n}and σ

_{0}are peak stress after n and 0 freeze–thaw cycles, respectively, (MPa):

_{n}and σ

_{0}are the peak stress after n and 0 freeze–thaw cycles, respectively, (MPa); ${\dot{\epsilon}}_{0}={10}^{-4}{\mathrm{s}}^{-1}$ is the reference strain rate and A, α and β are material constants; θ

_{2}is the relaxation time of high-frequency Maxwell elements; and E

_{2}and E

_{a}are the elastic constants of high-frequency Maxwell element and the equivalent Hooke element, respectively.

#### 4.3. Model Verification

_{test}and σ

_{pre}are the test and predicted values, respectively, (MPa).

## 5. Conclusions

**Limitations:**In the current research, the effects of impact pressure and the freeze–thaw cycle number on the dynamic mechanical characteristics of cement soils were investigated, but the weakening mechanism of the freeze–thaw cycles on sample strength was not studied. Scanning electron microscope and nuclear magnetic resonance tests need to be performed to further evaluate the influence of the freeze–thaw cycles on cracks and pores inside the samples.

**Application:**In cold regions, cement soil undergoes freeze–thaw cycles. In the safety design of cement soil reinforcement projects, the soil bears not only the static load generated by external earth pressure and the self-weight of overlying structure, but also dynamic loads, such as instantaneous impact (simple airport runway base reinforcement) as well as mechanical and blasting vibration impacts (adjacent building demolition construction, earthquake, etc.).

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 13.**Crushing particle size under different test conditions. (

**a**) Freeze–thaw cycle: 0, 0.10 MPa; (

**b**) Freeze–thaw cycle: 10, 0.10 MPa; (

**c**) Freeze–thaw cycle: 1, 0.08 MPa; (

**d**) Freeze–thaw cycle: 1, 0.12 MPa.

**Figure 17.**Comparison of theoretical and test results. (

**a**) impact pressure: 0.08 MPa; (

**b**) impact pressure: 0.10 MPa; (

**c**) impact pressure: 0.12 MPa.

**Figure 19.**Relative error of cement soils under different test conditions: (

**a**) 0.08 MPa; (

**b**) 0.10 MPa; (

**c**) 0.12 MPa.

Composition | SiO_{2} | Al_{2}O_{3} | CaO | Fe_{2}O_{3} | TiO_{2} | MgO | K_{2}O | Na_{2}O | Others |
---|---|---|---|---|---|---|---|---|---|

Percentage/% | 72.3 | 14.3 | 0.8 | 5.4 | 1.3 | 2.1 | 3.1 | 0.5 | 0.2 |

Composition | SiO_{2} | Al_{2}O_{3} | CaO | Fe_{2}O_{3} | SO_{3} | MgO | Na_{2}O | K_{2}O |
---|---|---|---|---|---|---|---|---|

Percentage/% | 19.6 | 6.5 | 66.3 | 3.5 | 2.5 | 0.7 | 0.6 | 0.3 |

Composition | SiO_{2} | Al_{2}O_{3} | Fe_{2}O_{3} | Na_{2}O | K_{2}O | Others |
---|---|---|---|---|---|---|

Percentage/% | 54 | 43 | 1 | 0.3 | 0.2 | 1.5 |

Impact Pressure/MPa | Strain Rate | Freeze–Thaw Times | Strength/MPa | Fractal Dimension |
---|---|---|---|---|

0.08 | 182 | 0 | 4.79 | 1.60 |

1 | 3.83 | 1.75 | ||

3 | 3.52 | 1.82 | ||

6 | 3.32 | 1.86 | ||

10 | 3.26 | 1.88 | ||

0.10 | 219 | 0 | 6.90 | 1.71 |

1 | 6.08 | 1.82 | ||

3 | 5.69 | 1.92 | ||

6 | 5.51 | 1.96 | ||

10 | 5.29 | 1.93 | ||

0.12 | 252 | 0 | 9.55 | 1.92 |

1 | 8.06 | 1.94 | ||

3 | 7.79 | 2.01 | ||

6 | 7.44 | 2.03 | ||

10 | 7.23 | 2.04 |

Freeze–Thaw Cycle | W_{i}/J | W_{r}/J | W_{t}/J | W_{s}/J | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.08 MPa | 0.10 MPa | 0.12 MPa | 0.08 MPa | 0.10 MPa | 0.12 MPa | 0.08 MPa | 0.10 MPa | 0.12 MPa | 0.08 MPa | 0.10 MPa | 0.12 MPa | |

0 | 11.06 | 17.11 | 23.92 | 8.94 | 14.10 | 19.99 | 0.68 | 0.76 | 0.82 | 1.44 | 2.25 | 3.11 |

1 | 11.21 | 17.02 | 23.86 | 9.47 | 14.38 | 20.49 | 0.60 | 0.73 | 0.77 | 1.14 | 1.91 | 2.60 |

3 | 10.96 | 16.98 | 24.02 | 9.56 | 14.63 | 20.87 | 0.52 | 0.69 | 0.74 | 0.88 | 1.66 | 2.41 |

6 | 11.11 | 17.14 | 23.79 | 9.90 | 14.97 | 20.88 | 0.48 | 0.67 | 0.72 | 0.73 | 1.50 | 2.19 |

10 | 11.12 | 17.05 | 23.88 | 10.01 | 15.80 | 21.13 | 0.46 | 0.63 | 0.69 | 0.65 | 1.34 | 2.06 |

Impact Pressure/MPa | Freeze–Thaw Cycles |
$$\dot{\mathit{\epsilon}}$$
| A | α | β | E_{a} | E_{2} | θ_{2} |
---|---|---|---|---|---|---|---|---|

0.08 | 0 | 182 | 2.97 | −1.54 | −6.40 | −4.60 × 10^{6} | 4.60 × 10^{6} | 3 |

1 | 182 | 0.15 | 0.59 | 2.09 | 4.63 × 10^{6} | −4.63 × 10^{6} | 3 | |

3 | 182 | 1.38 | 0.95 | 4.16 | 3.57 × 10^{6} | −3.57 × 10^{6} | 3 | |

6 | 182 | 0.49 | 0.47 | 1.99 | 4.35 × 10^{6} | −4.35 × 10^{6} | 3 | |

10 | 182 | 0.84 | 0.46 | 1.80 | 1.38 × 10^{6} | −1.38 × 10^{6} | 3 | |

0.10 | 0 | 219 | 0.04 | −0.26 | −0.21 | −1.23 × 10^{4} | 1.23 × 10^{4} | 3 |

1 | 219 | 322.92 | 1.25 | 8.39 | 4.23 × 10^{5} | −4.23 × 10^{5} | 3 | |

3 | 219 | 0.16 | −0.54 | −1.41 | 1.61 × 10^{6} | −1.61 × 10^{6} | 3 | |

6 | 219 | −9.24 | −0.36 | −1.22 | 9.81 × 10^{6} | 9.81 × 10^{6} | 3 | |

10 | 219 | 44.75 | −1.19 | −2.16 | −1.75 × 10^{6} | 1.75 × 10^{6} | 3 | |

0.12 | 0 | 252 | 12.24 | −0.99 | −0.85 | −3.62 × 10^{3} | 3.62 × 10^{3} | 3 |

1 | 252 | 0.03 | 1.06 | 3.88 | 2.94 × 10^{6} | −2.94 × 10^{6} | 3 | |

3 | 252 | 0.20 | 0.41 | 1.75 | 3.54 × 10^{6} | −3.54 × 10^{6} | 3 | |

6 | 252 | 135.49 | 1.87 | 11.29 | −1.30 × 10^{6} | 1.30 × 10^{6} | 3 | |

10 | 252 | 0.13 | 1.12 | 4.26 | 9.71 × 10^{6} | −9.71 × 10^{6} | 3 |

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## Share and Cite

**MDPI and ACS Style**

Huang, K.; Wang, H.; Huang, K.
Freeze–Thaw Cycle Effects on the Energy Dissipation and Strength Characteristics of Alkali Metakaolin-Modified Cement Soil under Impact Loading. *Water* **2023**, *15*, 730.
https://doi.org/10.3390/w15040730

**AMA Style**

Huang K, Wang H, Huang K.
Freeze–Thaw Cycle Effects on the Energy Dissipation and Strength Characteristics of Alkali Metakaolin-Modified Cement Soil under Impact Loading. *Water*. 2023; 15(4):730.
https://doi.org/10.3390/w15040730

**Chicago/Turabian Style**

Huang, Kun, Heng Wang, and Kai Huang.
2023. "Freeze–Thaw Cycle Effects on the Energy Dissipation and Strength Characteristics of Alkali Metakaolin-Modified Cement Soil under Impact Loading" *Water* 15, no. 4: 730.
https://doi.org/10.3390/w15040730