# Analysis of Microscopic Pore Characteristics and Macroscopic Energy Evolution of Rock Materials under Freeze-Thaw Cycle Conditions

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Progress and Methodology

#### 2.1. Raw Materials Selection and Sample Preparation

#### 2.2. Laboratory Test

#### 2.2.1. Freeze-Thaw Cycle Test

#### 2.2.2. NMR Test

_{2}relaxation time). In the process of sample detection, the magnetic field strength was 0.3 T, and the central principal frequency of NMR was 12.8 MHZ. The diameter of the probe coil was 60 mm and the sampling number was 4 times greater. The sampling interval was 6000 ms and the number of echoes was 7000 times. In addition, to saturate the rock sample, the specimens need to be vacuum saturated before the NMR test. The vacuum pressure inside the instrument cover was 0.1 MPa. The dry pumping time was set to 360 min and the wet pumping time was 240 min.

#### 2.2.3. Uniaxial Compressive Strength Test

#### 2.3. Pore Radius Decision

#### 2.4. Model Description

## 3. Results and Discussions

#### 3.1. The Effects of the Freeze-Thaw Cycle on Microscopic Pore Structure

_{2}distribution basically presents a three-peak feature character. The peak value of the micropore signal was significantly higher than that of mesopores and macropores. The internal microscopic pores are mainly micropores, followed by mesopores, and lastly macropores. As the number of freeze-thaw cycles increased, the T

_{2}distribution shifted significantly to the right, causing the microscopic pores inside the sample to change from a smaller radius to a larger radius. In addition, except for individual data points, with the increase in F-T cycles, the signal peak of micropores and mesopores also showed a significant increase. However, the change in the peak signal in macropores was not obvious.

^{2}= 0.9982), and the porosity increased exponentially with the increase in the number of F-T cycles.

#### 3.2. The Effects of the Freeze-Thaw Cycle on Macroscopic Properties

#### 3.2.1. The Effects of the Freeze-Thaw Cycle on Mechanical Properties

#### 3.2.2. The Effects of Freeze-Thaw Cycles on Energy Evolution

_{c}). As the force of the specimen gradually increased, the total energy, strain energy, and bond strain energy showed an increasing trend. At this stage, there were no cracks and no frictional energy. The total energy was basically converted into strain energy and cementation energy, which were stored in linear springs and parallel linear springs. The second stage is the slow growth of cracks (0.44σ

_{c}–0.75σ

_{c}). At this stage, the energy absorption rates for total energy, strain energy, and cementation energy increased, and the different energies showed an increasing trend. Cracks and frictional energy appeared inside the specimen, and there was a slow, increasing trend. It can be seen that the friction energy was synchronized with the crack propagation. When cracks appeared, the friction energy also appeared. Moreover, the friction energy showed an increasing trend with the increase in the number of cracks. When the strain between particles reached a certain degree, cracks occurred between different particles. The energy consumed by friction between particles when they are cracked is generated by both friction energies. Since the cracks inside the specimen were in the stage of germination and slow growth, the frictional energy was small. Therefore, the cementation energy and strain energy were much greater than the friction energy. The third stage is the crack acceleration growth stage (0.75σ

_{c}–σ

_{c}). In the stage of accelerated crack growth, the total energy, strain energy, and bond strain energy continued to show an increasing trend. The number of cracks increased, and the growth rate of friction energy increased gradually. More and more of the total energy is dissipated by the frictional energy that overcomes the sliding of the particles. The fourth stage is the rapid growth of cracks (σ

_{c}). In the post-peak phase, the crack spread rapidly, and the friction energy increased rapidly. The bearing capacity of the specimen was weakened, and the rate of total energy growth decreased. The strain energy and boundary energy stored by the linear spring and parallel bond spring were released rapidly due to the failure, and the variation trend changed from increasing to decreasing.

_{c}and 0.75σ

_{c}, the total energy, strain energy, bond strain energy, and friction energy have a good exponential relationship with the freeze-thaw cycles, the fitting coefficients of which are above 0.94. With the gradual increase in the number of freeze-thaw cycles, the total energy, strain energy, bond strain energy, and friction energy all showed an exponentially decreasing trend. At the point of peak strength (σ

_{c}), the number of freeze-thaw cycles was exponentially related to the total energy and strain energy, and the fitting coefficient was 0.81964~0.86064. However, the bond strain energy and friction energy conformed severally to a good exponential relationship with freeze-thaw cycles, for which the fitting coefficient was above 0.95. While the number of freeze-thaw cycles gradually increased, the total energy, strain energy, bond strain energy, and friction energy all showed the law of exponential decrease.

## 4. Conclusions

- (1)
- The microscopic structure is mainly composed of micropores, followed by mesopores, and lastly macropores. The micropores and mesopores showed an increasing trend with the increase in the number of freeze-thaw cycles, while the change in large porosity was not obvious. In addition, the porosity conformed to a good exponential relationship with the number of freeze-thaw cycles and increased exponentially with the increase in the cycles.
- (2)
- The influence of the F-T cycle on elastic modulus is the largest, followed by peak strength, and the influence of peak strain is the least. The peak strength and elastic modulus had a good exponential relationship with the number of freeze-thaw cycles. With the increase in the number of freeze-thaw cycles, the peak strength and elastic modulus showed an exponentially decreasing trend, while the peak strain showed an exponentially increasing trend. In addition, there was a good exponential relationship between the porosity and the uniaxial compressive strength. The uniaxial compressive strength decreases exponentially with the increase in porosity.
- (3)
- The failure mode of mechanical testing under different F-T cycle conditions was similar to that of numerical simulation. The failure is mainly tensile, accompanied by shear failure locally. The failure mode is mainly tensile failure, accompanied by shear failure. In energy evolution, strain energy and bond strain energy showed an increasing trend before the peak intensity and a decreasing trend after the peak intensity. The friction energy and crack synchronously showed an accelerated increasing trend before the peak strength and a rapid increasing trend after the peak. The total, strain, bond strain, and friction energy had an exponential relationship with the number of freeze-thaw cycles. With the increasing number of freeze-thaw cycles, different types of energy showed an exponentially decreasing trend.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NMR | Nuclear magnetic resonance |

F-T cycle | Freeze-thaw cycle |

Kratio | Normal-to-shear stiffness ratio |

Emod | Effective modulus |

Pb_Emod | Bond effective modulus |

Pb_coh | Cohesion |

Pb_ten | Tensile strength |

pb_fa | Friction angle |

## References

- Al-Omari, A.; Beck, K.; Brunetaud, X.; Török, Á.; Al-Mukhtar, M. Critical degree of saturation: A control factor of freeze-thaw damage of porous limestones at Castle of Chambord, France. Eng. Geol.
**2015**, 185, 71–80. [Google Scholar] [CrossRef] - Li, J.; Kaunda, R.; Zhou, K. Experimental investigations on the effects of ambient freeze-thaw cycling on dynamic properties and rock pore structure deterioration of sandstone. Cold Reg. Sci. Technol.
**2018**, 154, 133–141. [Google Scholar] [CrossRef] - Li, J.; Zhu, L.; Zhou, K.; Liu, H.W.; Cao, S.P. Damage characteristics of sandstone pore structure under freeze-thaw cycles. Rock Soil Mech.
**2020**, 9, 3524–3532. [Google Scholar] - Zhou, K.; Li, B.; Li, J.; Deng, H.; Bin, F. Microscopic damage and dynamic mechanical properties of rock under freeze-thaw environment. Trans. Nonferrous Met. Soc. China
**2015**, 25, 1254–1261. [Google Scholar] [CrossRef] - Li, J.; Zhou, K.; Liu, W.; Deng, H. NMR research on deterioration characteristics of microscopic structure of sandstones in freeze-thaw cycles. Trans. Nonferrous Met. Soc. China
**2016**, 26, 2997–3003. [Google Scholar] [CrossRef] - Gao, F.; Wang, Q.; Deng, H.; Zhang, J.; Tian, W.; Ke, B. Coupled effects of chemical environments and freeze–thaw cycles on damage characteristics of red sandstone. Bull. Eng. Geol. Environ.
**2017**, 76, 1481–1490. [Google Scholar] [CrossRef] - Liu, C.; Deng, H.; Zhao, H.; Zhang, J. Effects of freeze-thaw treatment on the dynamic tensile strength of granite using the Brazilian test. Cold Reg. Sci. Technol.
**2018**, 155, 327–332. [Google Scholar] [CrossRef] - Gao, F.; Xiong, X.; Xu, C.; Zhou, K. Mechanical property deterioration characteristics and a new constitutive model for rocks subjected to freeze-thaw weathering process. Int. J. Rock Mech. Min. Sci.
**2021**, 140, 104642. [Google Scholar] [CrossRef] - Gao, F.; Xiong, X.; Zhou, K.; Li, J.; Shi, W. Strength deterioration model of saturated sandstone under freeze-thaw cycles. Rock Soil Mech.
**2019**, 40, 926–932. [Google Scholar] - Yang, C.; Zhou, K.; Xiong, X.; Deng, H.; Pan, Z. Experimental investigation on rock mechanical properties and infrared radiation characteristics with freeze-thaw cycle treatment. Cold Reg. Sci. Technol.
**2021**, 183, 103232. [Google Scholar] [CrossRef] - Li, J.; Zhu, L.; Zhou, K.; Chen, H.; Gao, L.; Lin, Y.; Shen, Y. Non-linear creep damage model of sandstone under freeze-thaw cycle. J. Cent. South Univ.
**2021**, 28, 954–967. [Google Scholar] [CrossRef] - Xie, H.; Ju, Y.; Li, L. Criteria for strength and structural failure of rocks based on energy dissipation and energy release principles. Chin. J. Rock Mech. Eng.
**2005**, 17, 3003–3010. (In Chinese) [Google Scholar] - Xie, H.; Ju, Y.; Li, L.; Peng, R.D. Energy mechanism of deformation and failure of rock masses. Chin. J. Rock Mech. Eng.
**2008**, 9, 1729–1740. (In Chinese) [Google Scholar] - Deng, H.; Yu, S.; Deng, J.; Ke, B.; Bin, F. Experimental Investigation on Energy Mechanism of Freezing-Thawing Treated Sandstone under Uniaxial Static Compression. KSCE J. Civ. Eng.
**2019**, 23, 2074–2082. [Google Scholar] [CrossRef] - Feng, Q.; Jin, J.; Zhang, S.; Liu, W.; Yang, X.; Li, W. Study on a Damage Model and Uniaxial Compression Simulation Method of Frozen–Thawed Rock. Rock Mech. Rock Eng.
**2022**, 55, 187–211. [Google Scholar] [CrossRef] - Gao, F.; Cao, S.; Zhou, K.; Lin, Y.; Zhu, L. Damage characteristics and energy-dissipation mechanism of frozen-thawed sandstone subjected to loading. Cold Reg. Sci. Technol.
**2020**, 169, 102920. [Google Scholar] [CrossRef] - Lazar, M.; Apostu, I.; Faur, F.; Rotunjanu, I. Factors influencing the flooding process of former coal open-pits. Min. Miner. Depos.
**2021**, 15, 124–133. [Google Scholar] [CrossRef] - Jiang, C.; Guo, W.; Chen, H.; Zhu, Y.; Jin, C. Effect of filler type and content on mechanical properties and microstructure of sand concrete made with superfine waste sand. Constr. Build. Mater.
**2018**, 192, 442–449. [Google Scholar] [CrossRef] - Diambra, A.; Festugato, L.; Ibraim, E.; Peccin da Silva, A.; Consoli, N.C. Modelling tensile/compressive strength ratio of artificially cemented clean sand. Soils Found.
**2018**, 58, 199–211. [Google Scholar] [CrossRef] - Ueyendah, S.; Lezgy-Nazargah, M.; Eskandari-Naddaf, H.; Emamian, S.A. Predicting the mechanical properties of cement mortar using the support vector machine approach. Constr. Build. Mater.
**2021**, 291, 123396. [Google Scholar] [CrossRef] - Chuta, E.; Colin, J.; Jeong, J. The impact of the water-to-cement ratio on the surface morphology of cementitious materials. J. Build. Eng.
**2020**, 32, 101716. [Google Scholar] [CrossRef] - Deng, H.; Tian, G.; Yu, S.; Jiang, Z.; Zhong, Z.; Zhang, Y. Research on Strength Prediction Model of Sand-like Material Based on Nuclear Magnetic Resonance and Fractal Theory. Appl. Sci.
**2020**, 10, 6601. [Google Scholar] [CrossRef] - Tian, G.; Deng, H.; Xiao, Y. Correlation Analysis between Microscopic Pore Parameters and Macroscopic Mechanical Properties of Rock-like Materials from the Perspective of Water-Cement Ratio and Sand-Cement Ratio. Materials
**2022**, 15, 2632. [Google Scholar] [CrossRef] [PubMed] - Tian, G.; Deng, H.; Xiao, Y.; Yu, S. Experimental Study of Multi-Angle Effects of Micron-Silica Fume on Micro-Pore Structure and Macroscopic Mechanical Properties of Rock-like Material Based on NMR and SEM. Materials
**2022**, 15, 3388. [Google Scholar] [CrossRef] - SL/T 264-2020; Rock Test Regulations for Water Conservancy and Hydropower Engineering. People’s Republic of China Ministry of Water Resources: Beijing, China, 2020. (In Chinese)
- Cundall, P.; Strack, O. A discrete numerical model for granular assemblies. Geotechnique
**1979**, 29, 47–65. [Google Scholar] [CrossRef] - Zhao, Z.; Wang, X.; Wen, Z. Analysis of Rock Damage Characteristics Based on Particle Discrete Element Model. Geotech. Geol. Eng.
**2018**, 36, 897–904. [Google Scholar] [CrossRef] - Ning, J.; Liu, X.; Tan, Y.; Wang, J.; Tian, C. Relationship of box counting of fractured rock mass with Hoek–Brown parameters using particle flow simulation. Geomech. Eng.
**2015**, 9, 619–629. [Google Scholar] [CrossRef] - Chen, P.Y. Effects of microparameters on macroparameters of flat-jointed bonded-particle materials and suggestions on trial-and-error method. Geotech. Geol. Eng.
**2017**, 35, 663–677. [Google Scholar] [CrossRef] - Tan, X.; Chen, W.; Liu, H.; Wang, L.; Ma, W.; Chan, A.H.C. A unified model for frost heave pressure in the rock with a penny-shaped fracture during freezing. Cold Reg. Sci. Technol.
**2018**, 153, 1–9. [Google Scholar] [CrossRef] - Huang, S.; Ye, Y.; Cui, X.; Cheng, A.; Liu, G. Theoretical and experimental study of the frost heaving characteristics of the saturated sandstone under low temperature. Cold Reg. Sci. Technol.
**2020**, 174, 103036. [Google Scholar] [CrossRef] - Deprez, M.; Kock, T.; De Schutter, G.; Cnudde, V. A review on freeze-thaw action and weathering of rocks. Earth-Sci. Rev.
**2020**, 203, 103143. [Google Scholar] [CrossRef] - Huang, S.; Lu, Z.; Ye, Z.; Xin, Z. An elastoplastic model of frost deformation for the porous rock under freeze-thaw. Eng. Geol.
**2020**, 278, 105820. [Google Scholar] [CrossRef] - Huang, S.; Yu, S.; Ye, Y.; Ye, Z.; Cheng, A. Pore structure change and physico-mechanical properties deterioration of sandstone suffering freeze-thaw actions. Constr. Build. Mater.
**2022**, 330, 127200. [Google Scholar] [CrossRef]

**Figure 4.**The diagram of the parallel bond model. ${g}_{s}$ is the surface gap between particles and ${k}_{n}$ is the normal stiffness of a linear spring. ${k}_{s}$ is the shear stiffness of a linear spring, and ${\overline{k}}_{n}$ is the normal stiffness of the parallel bond. ${\overline{k}}_{s}$ and ${\overline{\sigma}}_{c}$ are the shear stiffness and the normal strength of the parallel bonds. $\left\{\overline{c}\right.,\left.\overline{\varphi}\right\}$ is the shear strength of the parallel bond and $\mu $ is the coefficient of friction.

**Figure 5.**The effect of freezing-thaw cycles on microscopic pore structure. (

**a**) Relaxation distribution; (

**b**) porosity.

**Figure 7.**The influence of the number of freeze-thaw cycles on physical and mechanical parameters. (

**a**) Stress-strain curve; (

**b**) peak strength and elastic modulus; and (

**c**) peak strain and porosity.

**Figure 9.**Sample destruction mode under different freeze-thaw cycles: (

**a**) 0 F-T cycle; (

**b**) 10 F-T cycle; (

**c**) 20 F-T cycle; and (

**d**) 30 F-T cycle.

**Figure 10.**The evolution of different types of energy in different F-T cycles: (

**a**) 0 cycle; (

**b**) 10 cycle; (

**c**) 20 cycle; and (

**d**) 30 cycle.

**Figure 11.**The intrinsic relationship between the number of freeze-thaw cycles and different types of energy: (

**a**) 0.44σ

_{c}; (

**b**) 0.75σ

_{c}; and (

**c**) σ

_{c}.

Material | Traits | Main Ingredients | |||
---|---|---|---|---|---|

3CaO·SiO_{2} | 2CaO·SiO_{2} | 3CaO·Al_{2}O_{3} | 4CaO·Al_{2}O_{3}·Fe_{2}O_{3} | ||

Portland cement | Taupe powder | 52.8% | 20.7% | 11.5% | 8.8% |

Material | Traits | Main Ingredients | Particle Size | Density (g/cm^{3}) |
---|---|---|---|---|

Quartz sand | Yellow and white particles | Quartz > 99% | 0.5–1.0 mm | 1.49 |

Naphthalene water reducer | Brown-yellow powder | β-Naphthal-enesulfonate sodium formaldehyde condensate | - | - |

Silica fume | White powder | SiO_{2} > 99% | 1 μm | 2.2–2.6 |

Material | Density (g/cm^{3}) | Porosity (%) | Uniaxial Compressive Strength (MPa) |
---|---|---|---|

Sandstone | 2.33 | 3.186 | 32.01 |

Sandstone-like material | 2.31 | 3.431 | 33.64 |

F-T Cycle | Porosity (%) | The Peak of Micropore (%) | The Peak of Mesopore (%) | The Peak of Macropore (%) |
---|---|---|---|---|

0 | 3.449 | 0.1197 | 0.00922 | 0.00188 |

10 | 3.553 | 0.11857 | 0.01483 | 0.00411 |

20 | 3.631 | 0.12249 | 0.01491 | 0.00366 |

30 | 3.709 | 0.12329 | 0.02102 | 0.00284 |

F-T Cycle | Peak Strength (MPa) | Elastic Modulus (GPa) | Peak Strain (%) |
---|---|---|---|

0 | 40.75 | 4.173 | 1.40612 |

10 | 39.73 | 4.021 | 1.49959 |

20 | 31.72 | 3.965 | 1.55147 |

30 | 17.87 | 1.263 | 2.25087 |

Change Ratio | 56.15% | 69.73% | 60.08% |

F-T Cycle | Density (kg/m ^{3}) | Radius (m) | Kratio | Porosity | Fric | Emod\Pb-Emod (GPa) | Pb_coh/Pb _ten (MPa) | pb_fa (°) |
---|---|---|---|---|---|---|---|---|

0 | 2000 | 0.002–0.005 | 1.5 | 0.03 | 0.5 | 5.66 | 37.7 | 20 |

10 | 5.53 | 42.6 | 50 | |||||

20 | 5.12 | 30.7 | 40 | |||||

30 | 1.16 | 16.3 | 20 |

**Table 7.**Indoor mechanics test results and numerical simulation results under different freeze-thaw cycles.

Mechanical Properties | Peak Stress (MPa) | Elastic Modulus (GPa) | ||||||
---|---|---|---|---|---|---|---|---|

F-T cycle | 0 | 10 | 20 | 30 | 0 | 10 | 20 | 30 |

Laboratory test | 40.75 | 39.73 | 31.72 | 17.87 | 4.173 | 4.021 | 3.965 | 1.263 |

Numerical simulation | 40.19 | 38.95 | 32.62 | 17.78 | 4.259 | 4.017 | 3.909 | 1.312 |

Differential value | 0.56 | 0.78 | 0.9 | 0.09 | 0.086 | 0.004 | 0.056 | 0.049 |

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**MDPI and ACS Style**

Xiao, Y.; Deng, H.; Tian, G.; Yu, S.
Analysis of Microscopic Pore Characteristics and Macroscopic Energy Evolution of Rock Materials under Freeze-Thaw Cycle Conditions. *Mathematics* **2023**, *11*, 710.
https://doi.org/10.3390/math11030710

**AMA Style**

Xiao Y, Deng H, Tian G, Yu S.
Analysis of Microscopic Pore Characteristics and Macroscopic Energy Evolution of Rock Materials under Freeze-Thaw Cycle Conditions. *Mathematics*. 2023; 11(3):710.
https://doi.org/10.3390/math11030710

**Chicago/Turabian Style**

Xiao, Yigai, Hongwei Deng, Guanglin Tian, and Songtao Yu.
2023. "Analysis of Microscopic Pore Characteristics and Macroscopic Energy Evolution of Rock Materials under Freeze-Thaw Cycle Conditions" *Mathematics* 11, no. 3: 710.
https://doi.org/10.3390/math11030710