# The Dynamic Compressive Properties and Energy Dissipation Law of Sandstone Subjected to Freeze–Thaw Damage

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material Preparation and Testing Method

#### 2.1. Sandstone Specimens

#### 2.2. Testing Schemes of Freeze–Thaw Cycling

#### 2.3. Static Loading and Impact Loading Testing Schemes

^{−5}s

^{−1}.

^{−1}, 92.7 s

^{−1}, 117.3 s

^{−1}and 152.0 s

^{−1}. There were 12 working conditions, and 3 parallel specimens were assigned in each working condition. To ensure the accuracy of the testing data, waveform pulse-shaping technology was adopted. The rubber sheets of 35 mm in diameter and 2 mm in thickness were pasted on the impacting surface of an incident rod. Figure 4 displays the typical waveforms of specimens. There is a good correspondence of transmitted wave and incident wave plus reflected wave, satisfying the dynamic stress equilibrium state of specimens in the loading process.

## 3. Results

#### 3.1. Strength and Fracturing Characteristics of Freeze–Thawed Sandstone

#### 3.1.1. Strength Characteristics of Freeze–Thawed Sandstone

^{−1}, 92.7 s

^{−1}, 117.3 s

^{−1}and 152.0 s

^{−1}, the peak strengths of the sandstones are decreased by 18.26%, 19.42%, 14.81% and 10.50%, respectively, which are smaller than the degrees of decrease for the specimens under static loading. This finding indicates that the increase in the number of freeze–thaw cycles leads to a reduction in the peak strength of the sandstone; however, the peak strength of the freeze–thawed sandstone increases with increasing strain rate.

^{−1}, the increase in the DIF slows down with an increasing number of freeze–thaw cycles; when strain rate is higher than 105.96 s

^{−1}, the increase in the DIF is accelerated with an increase in the number of freeze–thaw cycles. This result shows that after reaching a certain strain rate, the strength of the sandstone subjected to multiple freeze–thaw cycles responds to strain rate more sensitively, which is similar to the experimental results in paper [37]. The variation in the DIF is influenced by both the strain rate and the number of freeze–thaw cycles.

#### 3.1.2. The Fractal Dimension Characteristics of Fractured Freeze–Thawed Sandstone under Impact Loading

_{T}represents the total mass of all fragments, x denotes grain size, x

_{m}is the maximum size of fractured fragments and D is fractal dimension.

_{T}]-lg[x/x

_{m}] coordinate is established, and the slope, K, can be determined through linear fitting. The fractal dimension is derived according to Equation (3).

_{T}] and lg(x/x

_{m}). In the case of different strain rates, the fracturing of the sandstone specimens subjected to different freeze–thaw cycles shows similar fractal dimension characteristics, while the fractal dimension of fractured sandstone changes with freeze–thaw cycles and strain rate. As illustrated in Figure 11, fractal dimension increases with growing strain rate, which agrees with the experimental results in paper [39]. Moreover, the fractal dimension rises with an increasing number of freeze–thaw cycles, which is further illustrated in Figure 12.

^{−1}, the fractal dimensions of the sandstone specimens subjected to different numbers of freeze–thaw cycles are all increased significantly. When the strain rate is less than 95 s

^{−1}, the impact load is relatively small, so the size of the crushed blocks is larger than that under a larger strain rate. The fractal dimension under an impact strain rate of 95 s

^{−1}is between 1.265 and 1.897; while the fractal dimension over an impact strain rate of 95 s

^{−1}is significantly increased to between 1.955 and 2.764.

#### 3.2. Law of Energy Dissipation of Freeze–Thawed Sandstone under Impact Loading

^{2}), C is the P-wave velocity (m/s) of the bars, and ${\epsilon}_{i}\left(t\right)$,${\epsilon}_{r}\left(t\right)$ and ${\epsilon}_{t}\left(t\right)$ refer to strain signals of the incident wave, reflected wave and transmitted wave, respectively.

^{3}).

#### 3.2.1. Change Law of Energy Dissipation Rate of Freeze–Thawed Sandstone under Different Strain Rates

^{−1}, the stress–strain curve is shown to be a closed curve, indicating the damage degree in the rock is low. However, when the number of freeze–thaw cycles is greater than 50 and strain rate is higher than 91 s

^{−1}, more irreversible damage occurs to the rock and the stress–strain curve is found to be an open-ended curve.

^{−1}, the energy dissipation rate curve is also presented as a closed curve; as the number of freeze–thaw cycles and strain rate increase, the energy dissipation rate curve becomes an open-ended curve, which is consistent with changes in the stress–strain curve and the energy dissipation rate curve. The energy dissipation rate near the peak stress point reaches its maximum value. After that, stress and energy dissipation rate are largely reduced due to crack propagation and the occurrence of slipping on the crack surface [29].

#### 3.2.2. Relationship between Peak Strength and Peak Energy Dissipation Rate for Freeze–Thawed Sandstone

#### 3.2.3. Relationship between Peak Energy Dissipation Rate and Fractal Dimension for Freeze–Thawed Sandstone

#### 3.2.4. Analysis of Freeze–Thaw Damage to Sandstone Based on Peak Energy Dissipation Rate

_{n}is the freeze–thaw damage variable, ${\eta}_{d}^{n}$ denotes the peak energy dissipation rate of sandstone subjected to the Nth freeze–thaw cycling and ${\eta}_{d}^{0}$ represents the peak energy dissipation rate of sandstone without freeze–thaw cycling treatment.

_{n}), defined based on energy dissipation rate, is able to favorably elucidate the influence of freeze–thaw cycling on the strength of sandstone.

## 4. Conclusions

- (1)
- The peak strength of sandstone subjected to different numbers of freeze–thaw cycles increases exponentially with growing strain rate, indicating a distinct strain rate effect.
- (2)
- The DIF of the freeze–thawed sandstone increases with the increase in strain rate. There is a strain rate threshold: when strain rate is smaller than 105.96 s
^{−1}, the increasing rate of the DIF slows down with the increase in the number of freeze–thaw cycles; when strain rate is higher than 105.96 s^{−1}, the increasing rate of the DIF grows with the increase in the number of freeze–thaw cycles. - (3)
- In the case of the same strain rate, the fractal dimension of fractured sandstone increases with the increase in the number of freeze–thaw cycles; in the case of the same number of freeze–thaw cycles, the fractal dimension of fractured sandstone gradually increases with increasing strain rate.
- (4)
- The freeze–thaw damage variable established based on peak energy dissipation rate can be used to elucidate the damage degree of sandstone under freeze–thaw cycling, and the peak dynamic compressive strength of freeze–thawed sandstone at different strain rates is reduced with the increase in the freeze–thaw damage variable.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Xu, G.M.; Liu, Q.S. Analysis of mechanism of rock failure due to freeze-thaw cycling and mechanical testing study on frozen-thawed rocks. Chin. J. Rock Mech. Eng.
**2005**, 24, 3076–3082. (In Chinese) [Google Scholar] - Hori, M.; Morihiro, H. Micromechanical analysis on deterioration due to freezing and thawing in porous brittle materials. Int. J. Eng. Sci.
**1998**, 36, 511–522. [Google Scholar] [CrossRef] - Yang, G.; Shen, Y.; Jia, H.; Wei, R.; Zhang, H.; Liu, H. Research progress and tendency in characteristics of multi-scale damage mechanics of rock under freezing-thawing. Chin. J. Rock Mech. Eng.
**2018**, 37, 545–563. (In Chinese) [Google Scholar] [CrossRef] - Jia, H.-L.; Zhao, S.-Q.; Ding, S.; Wang, T.; Dong, Y.-H.; Tan, X.-J. Study on the evolution and influencing factors of frost heaving force of water bearing cracks during freezing-thawing process. Chin. J. Rock Mech. Eng.
**2022**, 41, 1832–1845. (In Chinese) [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] - Lan, Y.; Gao, H.; Zhao, Y. Pore Structure Characteristics and Strength Variation of Red Sandstone under Freeze–Thaw Cycles. Materials
**2022**, 15, 3856. [Google Scholar] [CrossRef] - Huang, S.; Yu, S. Effect of water saturation on the strength of sandstones: Experimental investigation and statistical analysis. Bull. Eng. Geol. Environ.
**2022**, 81, 323. [Google Scholar] [CrossRef] - Wang, P.; Xu, J.-Y.; Fang, X.-Y.; Wang, P.-X.; Liu, S.-H.; Wang, H.-Y. Water softening and freeze-thaw cycling induced decay of red-sandstone. Rock Soil Mech.
**2018**, 39, 2065–2072. (In Chinese) [Google Scholar] [CrossRef] - Tan, X.; Chen, W.; Yang, J.; Cao, J. Laboratory investigations on the mechanical properties degradation of granite under freeze-thaw cycles. Cold Reg. Sci. Technol.
**2011**, 68, 130–138. [Google Scholar] [CrossRef] - Shi, L.; Liu, Y.; Meng, X.; Zhang, H. Study on mechanical properties and damage characteristics of red sandstone under freeze-thaw and load. Adv. Civ. Eng.
**2021**, 2021, 8867489. [Google Scholar] [CrossRef] - Zhou, Y.X.; Xia, K.; Li, X.B.; Li, H.B.; Ma, G.W.; Zhao, J.; Zhou, Z.; Dai, F. Suggested methods for determining the dynamic strength parameters and mode-I fracture toughness of rock materials. Int. J. Rock Mech. Min. Sci.
**2012**, 49, 105–112. [Google Scholar] [CrossRef] - Wang, K.; Feng, G.; Bai, J.; Guo, J.; Yang, X.; Cui, B.; Shi, X.; Song, C. Experimental study on dynamic mechanical characteristics and fracture behaviors of coal under water–gas-temperature coupling conditions. Theor. Appl. Fract. Mech.
**2022**, 122, 103609. [Google Scholar] [CrossRef] - Zhou, K.-P.; Li, B.; Li, J.; Deng, H.-W.; 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] - Zhao, R.; Zhai, Y.; Meng, F.; Li, Y.; Li, Y. Research on interactions among parameters affecting dynamic mechanical properties of sandstone after freeze-thaw cycles. Eng. Geol.
**2021**, 293, 106332. [Google Scholar] [CrossRef] - Takarli, M.; Prince, W.; Siddique, R. Damage in granite under heating/cooling cycles and water freeze–thaw condition. Int. J. Rock Mech. Min. Sci.
**2008**, 45, 1164–1175. [Google Scholar] [CrossRef] - Chen, T.; Yeung, M.; Mori, N. Effect of water saturation on deterioration of welded tuff due to freeze-thaw action. Cold Reg. Sci. Technol.
**2004**, 38, 127–136. [Google Scholar] [CrossRef] - Weng, L.; Wu, Z.; Taheri, A.; Liu, Q.; Lu, H. Deterioration of dynamic mechanical properties of granite due to freeze-thaw weathering: Considering the effects of moisture conditions. Cold Reg. Sci. Technol.
**2020**, 176, 103092. [Google Scholar] [CrossRef] - Chen, C.; Zheng, Y.; Sun, C. Experimental study on dynamic tensile properties and energy evolution of sandstone after freeze-thaw cycles. Chin. J. Rock Mech. Eng.
**2021**, 40, 2445–2453. (In Chinese) [Google Scholar] [CrossRef] - Liu, X.-H.; Xue, Y.; Zheng, Y.; Gui, X. Research on energy release in coal rock fragmentation process under impact load. Chin. J. Rock Mech. Eng.
**2021**, 40, 3201–3211. (In Chinese) [Google Scholar] [CrossRef] - Weng, L.; Wu, Z.; Liu, Q.; Wang, Z. Energy dissipation and dynamic fragmentation of dry and water-saturated siltstones under sub-zero temperatures. Eng. Fract. Mech.
**2019**, 220, 106659. [Google Scholar] [CrossRef] - Han, Z.; Li, D.; Zhou, T.; Zhu, Q.; Ranjith, P. Experimental study of stress wave propagation and energy characteristics across rock specimens containing cemented mortar joint with various thicknesses. Int. J. Rock Mech. Min. Sci.
**2020**, 131, 104352. [Google Scholar] [CrossRef] - Feng, J.; Wang, E.; Shen, R.; Chen, L.; Li, X.; Xu, Z. Investigation on energy dissipation and its mechanism of coal under dynamic loads. Geomech. Eng.
**2016**, 11, 657–670. [Google Scholar] [CrossRef] - Wang, P.; Xu, J.; Fang, X.; Wang, P. Energy dissipation and damage evolution analyses for the dynamic compression failure process of red-sandstone after freeze-thaw cycles. Eng. Geol.
**2017**, 221, 104–113. [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] - Ping, Q.; Luo, X.; Ma, Q.-Y.; Yuan, P. Broken energy dissipation characteristics of sandstone specimens under impact loads. Chin. J. Rock Mech. Eng.
**2015**, 34, 4197–4203. (In Chinese) [Google Scholar] [CrossRef] - Hong, L.; Zhou, Z.L.; Yin, T.B.; Liao, G.Y.; Ye, Z.Y. Energy consumption in rock fragmentation at intermediate strain rate. J. Cent. South Univ. Technol.
**2009**, 16, 677–682. [Google Scholar] [CrossRef] - Liu, S.; Xu, J.; Liu, S.; Wang, P. Fractal study on the dynamic fracture of red sandstone after F-T cycles. Environ. Earth Sci.
**2022**, 81, 152. [Google Scholar] [CrossRef] - Wang, K.; Feng, G.; Bai, J.; Guo, J.; Shi, X.; Cui, B.; Song, C. Dynamic behaviour and failure mechanism of coal subjected to coupled water-static-dynamic loads. Soil Dyn. Earthq. Eng.
**2022**, 153, 107084. [Google Scholar] [CrossRef] - Feng, J.; Wang, E.; Chen, X.; Ding, H. Energy dissipation rate: An indicator of coal deformation and failure under static and dynamic compressive loads. Int. J. Min. Sci. Technol.
**2018**, 28, 397–406. [Google Scholar] [CrossRef] - Jia, H.; Ding, S.; Zi, F.; Dong, Y.; Shen, Y. Evolution in sandstone pore structures with freeze-thaw cycling and interpretation of damage mechanisms in saturated porous rocks. Catena
**2020**, 195, 104915. [Google Scholar] [CrossRef] - Liping, W.; Ning, L.; Jilin, Q.; Yanzhe, T.; Shuanhai, X. A study on the physical index change and triaxial compression test of intact hard rock subjected to freeze-thaw cycles. Cold Reg. Sci. Technol.
**2019**, 160, 39–47. [Google Scholar] [CrossRef] - Liu, J.; Zhang, H.; Wang, R.-H.; Wang, F.; He, Z.-W. Investigation of progressive damage and deterioration of sandstone under freezing-thawing cycle. Rock Soil Mech.
**2021**, 42, 1381–1394. (In Chinese) [Google Scholar] [CrossRef] - Mousavi, S.Z.S.; Tavakoli, H.; Moarefvand, P.; Rezaei, M. Micro-structural, petro-graphical and mechanical studies of schist rocks under the freezing-thawing cycles. Cold Reg. Sci. Technol.
**2020**, 174, 103039. [Google Scholar] [CrossRef] - Gao, F.; Xiong, X.; Zhou, K.-P.; Li, J.-L.; Shi, W.-C. Strength deterioration model of saturated sandstone under freeze-thaw cycles. Rock Soil Mech.
**2019**, 40, 926–932. (In Chinese) [Google Scholar] [CrossRef] - GB/T 50266—2013; Standard for Tests Method of Engineering Rock Masses. China Planning Press: Beijing, China, 2013. (In Chinese)
- Shen, Y.J.; Yang, G.S.; Rong, T.L.; Liu, H.; Lv, W.Y. Proposed scheme for freeze-thaw cycle tests on rock. Chin. J. Geotech. Eng.
**2016**, 38, 1775–1782. (In Chinese) [Google Scholar] [CrossRef] - Wang, P.; Xu, J.; Liu, S.; Wang, H. Dynamic mechanical properties and deterioration of red-sandstone subjected to repeated thermal shocks. Eng. Geol.
**2016**, 212, 44–52. [Google Scholar] [CrossRef] - Xie, H.-P. Fractal geometry and its application to rock and soil materials. Chin. J. Geotech. Eng.
**1992**, 14, 14–24. (In Chinese) [Google Scholar] - Xu, J.-Y.; Liu, S. Research on fractal characteristics of marble fragments subjected to impact loading. Rock Soil Mech.
**2012**, 33, 3225–3229. (In Chinese) [Google Scholar] [CrossRef] - Zhang, Z.; Kou, S.; Jiang, L.; Lindqvist, P.-A. Effects of loading rate on rock fracture: Fracture characteristics and energy partitioning. Int. J. Rock Mech. Min. Sci.
**2000**, 37, 745–762. [Google Scholar] [CrossRef]

**Figure 1.**Sandstone specimens. (

**a**) The sandstone specimen under dynamic loading; (

**b**) the sandstone specimen under static loading.

**Figure 5.**Influences of the number of freeze–thaw cycles and strain rate on peak strength of sandstone.

**Figure 6.**Influences of the number of freeze–thaw cycles and strain rate on peak strength loss rate of sandstone.

**Figure 7.**Surface morphology of the sandstone subjected to 50 freeze–thaw cycles. (

**a**) The sandstone specimen under dynamic loading; (

**b**) the sandstone specimen under static loading.

**Figure 8.**The curve for the relationship between strain rate and peak strength of freeze–thawed sandstone. Note: FT*-# refers to the rule of specimen numbering, “FT” represents freeze–thaw cycling, “*” denotes the number of freeze–thaw cycles and “#”is strain rate. These definitions are applicable to the rest of the figures hereinafter.

**Figure 12.**The failure modes of freeze–thawed sandstone under different strain rates (specimen diameter 100 mm, height 50 mm).

**Figure 13.**The curves for the strain–stress relationship of freeze–thawed sandstone at different strain rates and energy dissipation rates.

**Figure 14.**Relationship between peak energy dissipation rate and peak strength of freeze–thawed sandstone at different strain rates. Note: ${\eta}_{d}$ and ${\sigma}_{d}$ denote peak energy dissipation rate and dynamic compressive strength; the units are MJ/m

^{3}and MPa, respectively.

**Figure 15.**Relationship between strain rate and peak energy dissipation rate of freeze–thawed sandstone.

**Figure 16.**Relationship between peak energy dissipation rate and fractal dimension of freeze–thawed sandstone at different strain rates.

**Figure 17.**Changes in freeze–thaw damage to sandstone under different numbers of freeze–thaw cycles and different strain rates.

**Figure 18.**Relationship between average freeze–thaw damage variable and peak dynamic compressive strength.

Dry Density (g/cm ^{3}) | P-Wave Velocity (m/s) | Porosity (%) | Uniaxial Compressive Strength (MPa) | Young’s Modulus (GPa) |
---|---|---|---|---|

2.37 | 2900 ± 50 | 9.74 | 65.54 | 9.08 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jia, P.; Mao, S.; Qian, Y.; Wang, Q.; Lu, J. The Dynamic Compressive Properties and Energy Dissipation Law of Sandstone Subjected to Freeze–Thaw Damage. *Water* **2022**, *14*, 3632.
https://doi.org/10.3390/w14223632

**AMA Style**

Jia P, Mao S, Qian Y, Wang Q, Lu J. The Dynamic Compressive Properties and Energy Dissipation Law of Sandstone Subjected to Freeze–Thaw Damage. *Water*. 2022; 14(22):3632.
https://doi.org/10.3390/w14223632

**Chicago/Turabian Style**

Jia, Peng, Songze Mao, Yijin Qian, Qiwei Wang, and Jialiang Lu. 2022. "The Dynamic Compressive Properties and Energy Dissipation Law of Sandstone Subjected to Freeze–Thaw Damage" *Water* 14, no. 22: 3632.
https://doi.org/10.3390/w14223632