# Numerical Simulation of Frost Heave Deformation of Concrete-Lined Canal Considering Thermal-Hydro-Mechanical Coupling Effect

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## Abstract

**:**

## 1. Introduction

## 2. Multi-Physics Coupling Equations of Soil

#### 2.1. Seepage Field

#### 2.2. Temperature Field

#### 2.3. Dynamic Equilibrium Relationship of Ice-Water Phase Transition

#### 2.4. Stress Field

## 3. Multi-Physics Coupling Equations of Concrete

#### 3.1. Stress Field

#### 3.2. Seepage Field

#### 3.3. Temperature Field

## 4. Computational Model and Results Analysis

#### 4.1. Computational Model

#### 4.1.1. Geometry Modeling and Mesh Generation

#### 4.1.2. Initial and Boundary Conditions

#### 4.1.3. Calculation Parameters

^{−6}, ${a}_{2}=$ 9.366 × 10

^{−4}, ${a}_{3}=$ 9.580986 × 10, ${a}_{4}=$ 2.27832.

#### 4.2. Calculation Results and Analysis

#### 4.2.1. Evolution of Temperature Distribution

#### 4.2.2. Evolution of ice Content and Frost Depth

#### 4.2.3. Evolution of Frost heave Deformation

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kong, L.Z.; Quan, J.; Yang, Q.; Song, P.B.; Zhu, J. Automatic control of the middle route project for south-to-north water transfer based on linear model predictive control algorithm. Water
**2019**, 11, 1873. [Google Scholar] [CrossRef][Green Version] - Grigg, N. Large-scale water development in the United States: TVA and the California State Water Project. Int. J. Water Resour. Dev.
**2021**, 39, 70–88. [Google Scholar] [CrossRef] - Li, S.Y.; Lai, Y.M.; Zhang, M.Y.; Pei, W.S.; Zhang, C.; Yu, F. Centrifuge and numerical modeling of the frost heave mechanism of a cold-region canal. Acta Geotech.
**2019**, 14, 1113–1128. [Google Scholar] [CrossRef] - Li, S.Y.; Zhang, M.Y.; Tian, Y.B.; Pei, W.S.; Zhong, H. Experimental and numerical investigations on frost damage mechanism of a canal in cold regions. Cold Reg. Sci. Technol.
**2015**, 116, 1–11. [Google Scholar] [CrossRef] - Taber, S. The mechanics of frost heaving. J. Geol.
**1930**, 38, 303–317. [Google Scholar] [CrossRef] - Everett, D.H. The thermodynamics of frost damage to porous solids. Trans. Faraday Soc.
**1961**, 57, 1541–1551. [Google Scholar] [CrossRef] - Miller, R.D. Freezing and heaving of saturated and unsaturated soil. Highw. Res. Rec.
**1972**, 393, 1–11. [Google Scholar] - O’Neill, K.; Miller, R.D. Exploration of a rigid ice model of frost heave. Water Resour. Res.
**1985**, 21, 281–296. [Google Scholar] [CrossRef] - Harlan, R.L. Analysis of coupled heat-fluid transport in partially frozen soil. Water Resour. Res.
**1973**, 9, 1314–1323. [Google Scholar] [CrossRef][Green Version] - Mo, T.F.; Lou, Z.K. Numerical simulation of frost heave of concrete lining trapezoidal channel under an open system. Water
**2020**, 12, 335. [Google Scholar] [CrossRef][Green Version] - Liu, Y.; Wang, Z.Z.; Wang, Y.; Liu, Q.H.; Guo, R.; Xiao, M. Frost heave model of canal considering influence of moisture migration and phase transformation on temperature field. Trans. Chin. Soc. Agric. Eng.
**2016**, 32, 83–88. [Google Scholar] [CrossRef] - Wang, Z.Z.; Liu, S.J.; Wang, Y.; Liu, Q.H.; Ge, J.R. Size effect on frost heave damage for lining trapezoidal canal with arc-bottom in cold regions. J. Hydraul. Eng.
**2018**, 49, 803–813. [Google Scholar] [CrossRef] - Shang, H.S.; Song, Y.P. Experimental study of strength and deformation of plain concrete under biaxial compression after freezing and thawing cycles. Cem. Concr. Res.
**2006**, 36, 1857–1864. [Google Scholar] [CrossRef] - Liu, D.Y.; Tu, Y.M.; Sas, G.; Elfgren, L. Freeze-thaw damage evaluation and model creation for concrete exposed to freeze-thaw cycles at early-age. Constr. Build. Mater.
**2021**, 312, 125352. [Google Scholar] [CrossRef] - Qiu, W.L.; Peng, R.X.; Jiang, M. Meso equivalent calculation model for frost evaluation of concrete. Constr. Build. Mater.
**2021**, 272, 121867. [Google Scholar] [CrossRef] - Tahiri, I.; Dangla, P.; Vandamme, M.; Vu, Q. Numerical investigation of salt-frost damage of pervious concrete at the scale of a few aggregates. Cem. Concr. Res.
**2022**, 162, 106971. [Google Scholar] [CrossRef] - Coussy, O.; Monteiro, P. Poroelastic model for concrete exposed to freezing temperatures. Cem. Concr. Res.
**2008**, 38, 40–48. [Google Scholar] [CrossRef] - Wang, Z.D.; Zeng, Q.; Wang, L.; Li, K.F.; Xu, S.L.; Yao, Y. Characterizing frost damages of concrete with flatbed scanner. Constr. Build. Mater.
**2016**, 102, 872–883. [Google Scholar] [CrossRef] - Gong, F.Y.; Maekawa, K. Proposal of poro-mechanical coupling among ASR, corrosion and frost action for damage assessment of structural concrete with water. Eng. Struct.
**2019**, 188, 418–429. [Google Scholar] [CrossRef] - Peng, R.X.; Qiu, W.L.; Jiang, M. Frost resistance performance assessment of concrete structures under multi-factor coupling in cold offshore environment. Build. Environ.
**2022**, 226, 109733. [Google Scholar] [CrossRef] - Powers, T.C. A working hypothesis for further studies of frost resistance of concrete. J. Am. Concr. Inst.
**1945**, 16, 245–272. [Google Scholar] [CrossRef] - Coussy, O.; Dormieux, L.; Detournay, E. From mixture theory to Biot’s approach for porous media. Int. J. Solids Struct.
**1998**, 35, 4619–4635. [Google Scholar] [CrossRef] - Coussy, O. Poromechanics of Freezing Materials. J. Mech. Phys. Solids
**2005**, 53, 1689–1718. [Google Scholar] [CrossRef] - Zuber, B.; Marchand, J. Modeling the deterioration of hydrated cement systems exposed to frost action: Part 1: Description of the mathematical model. Cem. Concr. Res.
**2000**, 30, 1929–1939. [Google Scholar] [CrossRef] - Zuber, B.; Marchand, J. Predicting the volume instability of hydrated cement systems upon freezing using poro-mechanics and local phase equilibria. Mater. Struct.
**2004**, 37, 257–270. [Google Scholar] [CrossRef] - Duan, A. Numerical simulation of the freezing process of concrete. J. Mater. Civ. Eng.
**2013**, 25, 1317–1325. [Google Scholar] [CrossRef] - Pan, C. Research of Damage Mechanism of Cement Concrete Pavement under Freezing Environment Based on FEM Numerical Simulation. Master’s Thesis, Harbin Institute of Technology, Harbin, China, 2010. [Google Scholar]
- Taylor, G.S.; Luthin, J.N. A model for coupled heat and moisture transfer during soil freezing. Can. Geotech. J.
**1978**, 15, 548–555. [Google Scholar] [CrossRef] - Jame, Y.W.; Norum, D.I. Heat and mass transfer in a freezing unsaturated porous medium. Water Resour. Res.
**1980**, 16, 811–819. [Google Scholar] [CrossRef] - Bai, Q.B.; Li, X.; Tian, Y.H.; Fang, J.H. Equations and numerical simulation for coupled water and heat transfer in frozen soil. Chin. J. Geotech. Eng.
**2015**, 37, 131–136. [Google Scholar] [CrossRef] - Tai, B.W.; Liu, J.K.; Li, X.; Yue, Z.; Shen, Y.P. Numerical model of frost heaving and anti-frost heave measures of high speed railway subgrade in cold region. China Railw. Sci.
**2017**, 38, 1–9. [Google Scholar] [CrossRef] - Liu, X.C. The Study of Frost Heaving Model on Water-Heat-Deformation Coupled Fields of Canal Slope in Seasonally Frozen Area. Master’s Thesis, Northeast Agricultural University, Harbin, China, 2016. [Google Scholar]
- Li, K.F.; Zeng, Q. Influence of freezing rate on cryo-damage of cementitious material. J. Zhejiang Univ. Sci. A
**2009**, 10, 17–21. [Google Scholar] [CrossRef]

Composition | $\mathit{\rho}$ (kg/m ^{3})
| $\mathit{\lambda}$ (W/m·k) | $\mathit{C}$ (J/kg·K) | $\mathit{\alpha}$ (GPa) | $\mathit{K}$ (GPa) | $\mathit{E}$ (GPa) | $\mathit{D}$ (m ^{2})
| $\mathit{b}$ | $\mathit{n}$ |
---|---|---|---|---|---|---|---|---|---|

Concrete | 2400 | — | — | — | 11.1 | 35 | 8.3 × 10^{−12} | 0.657 | 0.155 |

Skeleton | — | 1.80 | 950 | 2.0 × 10^{−5} | 32.4 | — | — | — | — |

Water | 1000 | 0.54 | 4200 | −9.2 × 10^{−5} | 2.0 | — | — | — | — |

Ice | 900 | 2.22 | 2100 | 1.2 × 10^{−4} | 8.0 | — | — | — | — |

Composition | $\mathit{\rho}$ (kg/m ^{3})
| $\mathit{C}$ (J/kg·°C) | $\mathit{\lambda}$ (W/m·k) | T_{f}(°C) | $\mathit{E}$ (GPa) | $\mathit{A}$ |
---|---|---|---|---|---|---|

Soil | 1940 | 1680 | 1.22 | −0.15 | 0.30 | 0.56 |

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**MDPI and ACS Style**

Teng, R.; Gu, X.; Xia, X.; Zhang, Q.
Numerical Simulation of Frost Heave Deformation of Concrete-Lined Canal Considering Thermal-Hydro-Mechanical Coupling Effect. *Water* **2023**, *15*, 1412.
https://doi.org/10.3390/w15071412

**AMA Style**

Teng R, Gu X, Xia X, Zhang Q.
Numerical Simulation of Frost Heave Deformation of Concrete-Lined Canal Considering Thermal-Hydro-Mechanical Coupling Effect. *Water*. 2023; 15(7):1412.
https://doi.org/10.3390/w15071412

**Chicago/Turabian Style**

Teng, Renjie, Xin Gu, Xiaozhou Xia, and Qing Zhang.
2023. "Numerical Simulation of Frost Heave Deformation of Concrete-Lined Canal Considering Thermal-Hydro-Mechanical Coupling Effect" *Water* 15, no. 7: 1412.
https://doi.org/10.3390/w15071412