# Influence of Drying-Wetting Cycles on Dynamic Tensile Properties of Red-Sandstone Using the Brazilian Test

Authors: Bin Du; Wu Sun; Qiangqiang Cheng
Journal: Advances in Materials Science and Engineering
 (2023)

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## Abstract

The phenomenon of the drying-wetting cycle is a component of the weathering process and has a significant impact on the characteristics of rock materials. In order to examine the physical and dynamic tensile properties of red sandstone subjected to drying-wetting cycles, physical tests and dynamic splitting tensile tests were performed on red sandstone specimens under varying drying-wetting conditions. First, the split Hopkins pressure bar was used to perform dynamic tensile tests on red sandstone samples after varying numbers of drying-wetting cycles (0, 20, 40, 60, and 80). Second, the effects of the loading rate and drying-wetting cycles on dynamic tensile strength and energy dissipation of red sandstone were studied. Finally, taking into account the effect of the loading rate, a model was developed to predict the trend of decay in the dynamic tensile strength of red sandstone following drying-wetting cycles. The findings indicate that as the number of drying-wetting cycles increases, there is a decrease in longitudinal wave velocity and a marked degradation of the red sandstone due to the effects of the drying-wetting cycles. The dynamic tensile strength and dissipation energy density of red sandstone samples increase with the increasing loading rate. Under consistent loading rate conditions, the dynamic tensile strength and dissipative energy density of red sandstone samples exhibit a decrease as the number of drying-wetting cycles increases.


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## Body

## 1. Introduction

A number of rock engineering applications, including mines, dams, and building foundations, are subject to varying levels of wetting-drying cycles throughout their service life due to the occurrence of periodic changes in water conditions, including fluctuations in rainfall, tidal ebb and flow, and variations in groundwater levels [1–3]. Under the long-term drying-wetting cycles, the physical and mechanical properties of the rock mass will degrade to varying degrees. If the degradation of rock materials in the field is not monitored or intervened timely, geological disasters (such as landslides, foundation deformation, and pillar degradation) can be caused [4, 5]. Consequently, a substantial amount of research has been conducted on the physical and mechanical properties of rock masses subjected to drying-wetting cycles.Hale and Shakoor [6] conducted a study examining the impact of drying-wetting cycles on the unconfined compressive strength of six different types of sandstone. The findings indicated that the drying-wetting cycles had a slight effect on reducing the compressive strength of sandstone. Doostmohammadi et al. [7] investigated the influence of drying-wetting cycles on the expansion behaviors of the mudstone. It was found that as the number of drying-wetting cycles increases, the time required for the mudstone to reach ultimate expansion gradually decreases. Özbek et al. [8] summarized that with the increasing number of drying-wetting cycles, the unit weight, uniaxial compressive strength, and P-wave velocity of ignimbrites samples show a decreasing trend, while the polarity and water absorption by weight show an increasing trend. Zhou et al. [9] determined the variation in the physical and mechanical properties of sandstone subjected to drying-wetting cycles through experimental testing. The results indicated that an increase in the number of drying-wetting cycles leads to an increase in porosity and water absorption and a decrease in density, longitudinal wave velocity, slake durability index, compressive strength, and elastic modulus. Besides, it was also pointed out that the growth and propagation of microcracks in rock mass are the main reasons for the decline of rock mechanical properties. Lin et al. [10] discovered that during drying-wetting cycles, calcite and chlorite in the rock matrix can undergo dissolution, leading to an increase in porosity and a decrease in the strength of the rock specimens.Rock is a prototypical brittle material, characterized by tensile strength that is typically one order of magnitude lower than its compressive strength. In underground projects, tensile failure of the rock mass can readily occur when the rock is subjected to dynamic loads such as blasting, earthquakes, or rock bursts [11]. In addition, excessive local or overall tensile stress can lead to the cracking failure of split sidewalls and floors in engineering. Therefore, obtaining a thorough understanding of the dynamic tensile mechanical properties of the rock mass is of substantial importance in engineering applications. The International Society of Rock Mechanics (ISRM) recommends two test methods for accurate measurement of dynamic tensile strength: direct tensile test and indirect tensile test [12]. Direct tensile testing has some disadvantages, such as high operation difficulty, high requirements for sample processing precision, and great economic investment. To address the limitations of the direct tensile testing method, various indirect testing techniques, including Brazilian disk testing, ring testing, and three-point bending testing, have gained widespread adoption in laboratory settings [13].Karakul and Ulusay [14] conducted experiments and discovered that the tensile strength of saturated Turkish sandstone exhibits a reduction of 63% as compared to that of dry samples. Furthermore, their findings reveal that the variation patterns of both tensile strength and compressive strength in the saturated samples are comparable. According to the study conducted by Erguler and Ulusay [15], it was found that the tensile strength of saturated rock is reduced by 90% compared to dry rock. Furthermore, they established an exponential relationship between the tensile strength of clay-bearing rock and its water content. As reported by Zhao et al. [16], the impact of the drying-wetting cycle on the tensile strength of sandstone was studied. The results indicated that the tensile strength of sandstone with a low clay mineral content decreased slightly. This was attributed to the limited role of the expansion and dissolution of clay minerals in the weakening of sandstone. The dynamic tensile properties of granite under both dry and saturated conditions were investigated by Lou [17] using Split Hopkinson Pressure Bar (SHPB) testing equipment. The findings revealed that the strain rate had a significant effect on the dynamic tensile strength of the sample. The results showed that compared to the dry sample, the strain rate sensitivity of the water-saturated sample was stronger. Zhou et al. [18] explored the impact of drying-wetting cycles on the dynamic tensile properties of sandstone. They employed scanning electron microscopy (SEM) to uncover the deterioration mechanism of drying-wetting cycles and eventually developed a decay model that incorporates the loading rate and the number of drying-wetting cycles.The aforementioned study primarily focuses on the static properties of the rock mass during the drying-wetting cycle. However, there is limited research on the dynamic properties of rock after drying-wetting cycles, particularly with regard to the dynamic tensile properties. This work aims to investigate the influence of drying-wetting cycles on the dynamic tensile properties of red sandstone. Red sandstone samples were subjected to dynamic splitting tensile tests using the split Hopkinson pressure bar (SHPB) test equipment following 0, 20, 40, 60, and 80 drying-wetting cycles at various loading rates, and the degradation of the tensile properties of red sandstone after long-term drying-wetting cycles was predicted. The dynamic tensile strength and energy consumption density of red sandstone under the loading rates and drying-wetting cycles were compared and analyzed.


## 2. Experimental Materials and Methodology

### 2.1. Sample Preparation

The red sandstone block was dark red without significant fractures. According to the recommendations of ISRM on the determination of dynamic tensile strength of materials by the Brazilian split test [19], rock blocks with good integrity and homogeneity were cored and cut into a disk sample, with a diameter of 50 mm and a thickness of 25 mm. According to previous studies [9], the maximum deviation of the end face in the vertical axis direction is required to be ≤0.25°, the diameter error is required to be ≤0.1 mm, and the end face parallelism is required to be ≤0.02 mm. The obtained rock samples were further screened to ensure good integrity and uniformity of rock samples. First, macro observation of rock samples was conducted, and the samples with obvious cracks or defects were removed. Second, the density comparison method was used to remove the samples with large density dispersion. Finally, ultrasonic techniques were utilized to measure the longitudinal wave velocity of the samples. Samples with large dispersion in wave velocity were then excluded.A complete drying-wetting cycle includes water saturation and oven drying. First, the selected samples were immersed in pure water for 48 hours at a room temperature to saturation, and then these saturated samples were dried in an oven for 24 hours, and the oven temperature was set to 50°C. In this test, five levels of drying-wetting cycles were set, namely,n = 0, 20, 40, 60, and 80.


### 2.2. Dynamic Impact Test

The diameter of the SHPB is 50 mm. The waveform shaping technology was utilized, which involved affixing a circular rubber sheet, with a diameter of 1 cm and a thickness of 1 mm, to the end face of both the incident rod and the punch. As depicted in Figure1, the strain gauges were positioned 1 m from the input rod and output rod to the sample. The improved impact waveform was a sine wave (Figure 2).Figure 1
Schematic diagram of the SHPB test system.Figure 2
Dynamic stress balance of the red sandstone sample.In the dynamic Brazilian split test, the indirect tensile strength was calculated using the three-wave method [20] as described in the following equation:(1)σt=2PtmaxπDL,where σt is the tensile strength (MPa), Ptmax is the maximum effective load of compression bar at both ends of the rock sample (kN), D is the diameter of the rock sample (m), and L is the thickness of the rock sample (m).The effective loadPt under the compression bar at both ends of the rock sample can be expressed as [21] follows:(2)Pt=p1t+p2t2=EAεitεrt+εtt2,where A and E represent the cross-sectional area (m2) and elastic modulus (GPa) of the bars, respectively. εi,εr, and εt represent the pulse signals.In the Brazilian split test under impact load, the loading rate was commonly employed as an indicator of the dynamic mechanical properties of rock samples. Figure3 illustrates the time history curve of the dynamic tensile stress of red sandstone specimens. The figure demonstrates that the stress-time curve is approximately linear for a period prior to the attainment of the peak stress of the rock sample. The slope of this linear segment is defined as the loading rate for the test [22].Figure 3
History of dynamic tensile strength in the center of the red sandstone sample.


## 2.1. Sample Preparation

The red sandstone block was dark red without significant fractures. According to the recommendations of ISRM on the determination of dynamic tensile strength of materials by the Brazilian split test [19], rock blocks with good integrity and homogeneity were cored and cut into a disk sample, with a diameter of 50 mm and a thickness of 25 mm. According to previous studies [9], the maximum deviation of the end face in the vertical axis direction is required to be ≤0.25°, the diameter error is required to be ≤0.1 mm, and the end face parallelism is required to be ≤0.02 mm. The obtained rock samples were further screened to ensure good integrity and uniformity of rock samples. First, macro observation of rock samples was conducted, and the samples with obvious cracks or defects were removed. Second, the density comparison method was used to remove the samples with large density dispersion. Finally, ultrasonic techniques were utilized to measure the longitudinal wave velocity of the samples. Samples with large dispersion in wave velocity were then excluded.A complete drying-wetting cycle includes water saturation and oven drying. First, the selected samples were immersed in pure water for 48 hours at a room temperature to saturation, and then these saturated samples were dried in an oven for 24 hours, and the oven temperature was set to 50°C. In this test, five levels of drying-wetting cycles were set, namely,n = 0, 20, 40, 60, and 80.


## 2.2. Dynamic Impact Test

The diameter of the SHPB is 50 mm. The waveform shaping technology was utilized, which involved affixing a circular rubber sheet, with a diameter of 1 cm and a thickness of 1 mm, to the end face of both the incident rod and the punch. As depicted in Figure1, the strain gauges were positioned 1 m from the input rod and output rod to the sample. The improved impact waveform was a sine wave (Figure 2).Figure 1
Schematic diagram of the SHPB test system.Figure 2
Dynamic stress balance of the red sandstone sample.In the dynamic Brazilian split test, the indirect tensile strength was calculated using the three-wave method [20] as described in the following equation:(1)σt=2PtmaxπDL,where σt is the tensile strength (MPa), Ptmax is the maximum effective load of compression bar at both ends of the rock sample (kN), D is the diameter of the rock sample (m), and L is the thickness of the rock sample (m).The effective loadPt under the compression bar at both ends of the rock sample can be expressed as [21] follows:(2)Pt=p1t+p2t2=EAεitεrt+εtt2,where A and E represent the cross-sectional area (m2) and elastic modulus (GPa) of the bars, respectively. εi,εr, and εt represent the pulse signals.In the Brazilian split test under impact load, the loading rate was commonly employed as an indicator of the dynamic mechanical properties of rock samples. Figure3 illustrates the time history curve of the dynamic tensile stress of red sandstone specimens. The figure demonstrates that the stress-time curve is approximately linear for a period prior to the attainment of the peak stress of the rock sample. The slope of this linear segment is defined as the loading rate for the test [22].Figure 3
History of dynamic tensile strength in the center of the red sandstone sample.


## 3. Test Results and Discussion

### 3.1. Longitudinal Wave Velocity

The damage characteristics of rock samples can be effectively described by measuring the longitudinal wave velocity of rock samples. The RSM-SY6 intelligent sonic instrument was used to test the wave velocity of red sandstone samples. Takingn as the abscissa and the longitudinal wave velocity of red sandstone as the ordinate, the scatter diagram of test data is drawn in Figure 4. It can be seen that the longitudinal wave velocity of the red sandstone gradually decreases with the increase of n. After undergoing twenty cycles of drying and wetting, the decay amplitude of wave velocity exhibits the greatest reduction, and thereafter, the rate of decay gradually decreases and eventually levels off into a gentle decline.Figure 4
Change law of longitudinal wave velocity of the red-sandstone versusn.The reasons for the decay of the wave velocity can be explained as follows: (1) under the action of the drying-wetting cycles, the initial cracks in the rock sample expand to varying degrees, which hinders the wave propagation and weakens the wave energy propagation; (2) due to the nonuniformity of mineral components in rock samples, the difference in mineral particle shrinkage results in the generation of new cracks, thus prolonging the wave propagation time. Figure5 shows the typical SEM photos of the sections of red-sandstone specimen when n = 40.Figure 5
SEM photos of the sections of the red-sandstone specimen (whenn = 40).


### 3.2. Dynamic Tensile Strength

#### 3.2.1. Loading Rate Effect

After a series of stress balance verification, the dynamic tensile strengthσt of red sandstone under different combination conditions can be calculated by equation (1), as shown in Table 1. On this basis, the curve of σt of the red sandstone changing with the loading rate under different drying-wetting cycles is drawn, as shown in Figure 6.Table 1
Dynamic tensile test results of samples.
No.nLoading rate (GPa/s)Tensile strength (MPa)STS 0-10011.3STS 0–2019512.14STS 0–3027611.05STS 0–4038912STS 0–5052312.76STS 0–6059712.78STS 0–7066912.87STS 0–8084514.72STS 0–9099316.23STS 0–100102216.34STS 0–110112617.48STS 0–120120519.63STS 20–12008.50STS 20–2202268.43STS 20–3203178.76STS 20–4204389.53STS 20–4207188.05STS 20–52082710.97STS 20–72093311.06STS 20–8209369.17STS 20–920101012.85STS 20–1020109813.69STS 20–1120115614.19STS 40–14006.84STS 40–2402387.34STS 40–3403147.11STS 40–4404537.22STS 40–5405767.99STS 40–6407188.05STS 40–7409369.17STS 40–840102111.45STS 40–940115812.12STS 40–1040121613.18STS 60–16005.57STS 60–2602415.62STS 60–3603335.95STS 60–4603875.81STS 60–5605106.57STS 60–6606036.23STS 60–7607276.82STS 60–8608297.53STS 60–9609057.64STS 60–106010258.34STS 60–116011139.95STS 80–18004.3STS 80–2802014.81STS 80–3802954.56STS 80–4803814.37STS 80–5804924.98STS 80–6806185.29STS 80–7807365.28STS 80–8809146.64STS 80–98010056.88STS 80–108011388.84STS 80–118012009.32Figure 6
Change law ofσt of red sandstone samples versus the loading rate.It can be seen that there is a significant correlation between the loading rate andσt of rock samples free from drying-wetting cycles (n = 0) and subjected to drying-wetting cycles. The higher the loading rate, the greater the σt of the sample.Theσt of the sample increases more quickly as the loading rate is increased. The expression between σt of red sandstone and the loading rate can be obtained through the fitting function [23]. Figure 6 shows the detailed fitting curve of σt of red sandstone and the loading rate, and equation (3) shows the fitting equation.(3)σt=10.81+0.481e0.0024σ˙R2=0.979n=0,8.06+0.347e0.0025σ˙R2=0.972n=20,6.58+0.256e0.0027σ˙R2=0.962n=40,5.46+0.154e0.003σ˙R2=0.964n=60,4.26+0.132e0.003σ˙R2=0.980n=80.The loading rate effect of red sandstone is related ton; thus, the parameters in σt of the red sandstone are also affected by n. The fitting equation can be uniformly expressed as follows: σt=B+CeDσ˙. Figure 7 shows the change of parameters B, C, and D in the unified equation with n.Figure 7
Change law of each parameter versusn: (a) parameter B, (b) parameter C, and (c) parameter D.
(a)(b)(c)As shown in Figure7, each parameter has a good linear relationship with n. The σt of red sandstone can be expressed as follows:(4)σt=−0.0785n+10.174+−0.00445n+0.4522e8.5×10−6n+0.0024σ˙.


#### 3.2.2. Drying-Wetting Cycle Effect

To analyze the influence ofn on the σt of red sandstone, the loading rates of 200, 400, 600, 800, and 1000 GPa/s are substituted into equation (3), and variation curves of the σt of red sandstone with n under different loading rates are drawn, as shown in Figure 8. Under a given loading rate, the σt decreases gradually with the increase of n. Additionally, the red sandstone experiences the greatest decrease in σt following the initial 20 drying-wetting cycles.Figure 8
Change law ofσt of red sandstone samples versus n.Although the previous research results contribute to understanding the evolution law ofσt of the rock mass with n, test results also have some limitations. Actually, the rock may experience hundreds of thousands of drying-wetting cycles in the natural environment, and the long-term residual strength of the rock mass should be analyzed and predicted. Therefore, a reliable prediction model for the residual strength of the rock mass is required. To verify the overall performance of rocks after freeze-thaw cycles and heating-cooling cycles, Mutlutürk et al. [24] proposed an attenuation model and verified its effectiveness through the experimental results of various types of rocks, as listed in equation.(5)IN=I0e−λN.To introduce the decay model to the field of dynamics, Wang et al. [25, 26] considered the influence of the strain rate on the damage factor. The results demonstrate that the modified model can effectively predict the dynamic tensile strength of red sandstone specimens subjected to multiple thermal shocks. However, the applicability of this equation in predicting the dynamic tensile strength of rock specimens after drying-wetting cycles requires additional investigation. Considering the influence of n and the loading rate in this study, the damage equation can be rewritten in the form of equation as follows:(6)σtn=σt0e−λσ˙n,where n is the number of drying-wetting cycles and σt0 is the dynamic tensile strength of the sample without drying-wetting cycles (n = 0). σtn is the dynamic tensile strength after n drying-wetting cycles, and λσ˙ is the decay factor corresponding to different loading rates.Figure9 shows the scatter plot and fitting curves of the decay factor versus n at loading rates of 200, 500, 800, and 1100 GPa/s. It has been observed that, with an increase in the value of n, the decay factor decreases at a given loading rate. This implies that the extent of damage to the rock sample increases, which aligns with the findings from previous studies. Besides, a numerical fitting method is used to obtain decay factors at different loading rates, and the curves of decay factor changing with loading rates are drawn in Figure 10. It can be seen that the higher the loading rate, the smaller the decay factor λ. The relationship between decay factors and loading rates is fitted with an exponential function, as expressed in equation (7). By substituting equation (3) (when n = 0) and equation (7) into equation (6), the prediction model of σt of the red sandstone under the coupling action of the wetting drying cycle and loading rate can be obtained as follows:(7)λ=0.014−0.001e0.0012σ˙R2=0.980,(8)σtn=10.81+0.481e0.0024σ˙e−0.014−0.001e0.0012σ˙n.Figure 9
Change law of the decay factor versusn.Figure 10
Change law of the decay factor versus the loading rate.


### 3.3. Energy Dissipation Law

The natural rock exhibits numerous imperfections, and the alteration and disintegration of this rock can be conceptualized as the exchange of energy and matter with the external surroundings [27, 28]. The rock undergoes energy emission, including heat energy and radiation energy, as a result of the mechanical and thermal energy exerted by the external environment [29–31]. The energy absorbed by the rock specimen during an impact test can be calculated if the energy dissipation due to friction between the elastic rod and the specimen during the propagation of stress waves is disregarded [32].(9)WS=AECB∫0Tεit2−εrtt2−εtt2dt,where A, E, CB, As, and ls represent the cross-sectional area of the rod, the elastic modulus of the rod, the stress wave velocity of the rod, the cross-sectional area of the sample, and the length of the sample, respectively; εit, εrt, and εtt represent the strain corresponding to the incident wave, reflected wave, and transmitted wave on the pressure rod, respectively; T is the time at which the sample is completely destroyed.The calculation equation of energy consumption density is as follows:(10)ws=WSV,where ws is the energy consumption density and V is the sample volume.Figure11 shows the time history curve of energy change when the impact pressure is 0.25 MPa, and n = 0. Under an impact load, the generation of stress waves can lead to the propagation of energy. When the incident stress wave energy is low, the specimen is in an elastic deformation phase. Conversely, as the incident energy increases, the specimen enters a plastic phase, where the primary cracks broaden and multiple new cracks emerge, thus reducing the propagation of the transmitted wave [33]. When the incident energy is large, the sample breaks into powder instantly, the stress wave propagation stops, and the energy value of the sample tends to a maximum. Based on the dynamic tensile stress-strain characteristics and failure morphology of the sample, the energy change curve of the sample can be divided into three stages:(1)
Stage 1: 0–50μs. At this stage, the red sandstone sample undergoes elastic deformation, and the absorbed energy is retained as elastic potential.(2)
Stage 2: 50–160μs. At this point, the transmission energy of the sample changes little as the stress wave propagates, whereas the input energy, reflection energy, and absorption energy change significantly. The absorption energy has the largest increased amplitude, followed by reflection energy, and incident energy has the least increased amplitude. The previous results show that upon the arrival of the incident wave at the interface between the sample and the incident rod, a significant portion of the incident energy is reflected in a backward direction through the incident rod. As the residual energy traverses the red sandstone sample, a substantial amount of energy is absorbed by the specimen for dissipation through damage evolution and deformation failure, with only a minimal portion redirected to the transmission rod.(3)
Stage 3: 160–220μs. Under the impact stress, the primary cracks within the interior of the sample rapidly expand and a substantial number of additional cracks are formed. The cracks continuously spread until the sample is ultimately destroyed. The highest value of incident energy is observed, followed by absorbed energy, with the transmitted energy having the lowest value, as the energy value eventually tends towards a steady state.Figure 11
Energy-time curve of the red sandstone sample in the SHPB tensile test.Figure12 displays the evolution of the dissipation energy density of samples subjected to varying drying-wetting cycles with the increment in the loading rate. As the loading rate increases, the dissipation energy density also increases, as a result of the acceleration of rock deformation, the amplification of activated cracks, and the enhancement of energy absorption by the rock. The dissipation energy density of red sandstone samples increases linearly with the loading rate under different n, which can be described ws=aiσ˙+bi. In this equation, ws refers to the dissipation energy density, and a and b are function coefficients. Table 2 presents the pattern of change in parameters a and b. For a constant loading rate, as n increases, the energy dissipation density decreases. This can be attributed to the heightened severity of deterioration in the red sandstone samples, resulting in a lower energy expenditure during rock failure.Figure 12
Energy consumption density of red sandstone samples versus the loading rate.Table 2
Changing rules of parametersa and b.
na (×10−4)bR202.23−0.0190.936202.10−0.0370.976401.91−0.0340.970601.83−0.0450.976801.54−0.0330.979


## 3.1. Longitudinal Wave Velocity

The damage characteristics of rock samples can be effectively described by measuring the longitudinal wave velocity of rock samples. The RSM-SY6 intelligent sonic instrument was used to test the wave velocity of red sandstone samples. Takingn as the abscissa and the longitudinal wave velocity of red sandstone as the ordinate, the scatter diagram of test data is drawn in Figure 4. It can be seen that the longitudinal wave velocity of the red sandstone gradually decreases with the increase of n. After undergoing twenty cycles of drying and wetting, the decay amplitude of wave velocity exhibits the greatest reduction, and thereafter, the rate of decay gradually decreases and eventually levels off into a gentle decline.Figure 4
Change law of longitudinal wave velocity of the red-sandstone versusn.The reasons for the decay of the wave velocity can be explained as follows: (1) under the action of the drying-wetting cycles, the initial cracks in the rock sample expand to varying degrees, which hinders the wave propagation and weakens the wave energy propagation; (2) due to the nonuniformity of mineral components in rock samples, the difference in mineral particle shrinkage results in the generation of new cracks, thus prolonging the wave propagation time. Figure5 shows the typical SEM photos of the sections of red-sandstone specimen when n = 40.Figure 5
SEM photos of the sections of the red-sandstone specimen (whenn = 40).


## 3.2. Dynamic Tensile Strength

### 3.2.1. Loading Rate Effect

After a series of stress balance verification, the dynamic tensile strengthσt of red sandstone under different combination conditions can be calculated by equation (1), as shown in Table 1. On this basis, the curve of σt of the red sandstone changing with the loading rate under different drying-wetting cycles is drawn, as shown in Figure 6.Table 1
Dynamic tensile test results of samples.
No.nLoading rate (GPa/s)Tensile strength (MPa)STS 0-10011.3STS 0–2019512.14STS 0–3027611.05STS 0–4038912STS 0–5052312.76STS 0–6059712.78STS 0–7066912.87STS 0–8084514.72STS 0–9099316.23STS 0–100102216.34STS 0–110112617.48STS 0–120120519.63STS 20–12008.50STS 20–2202268.43STS 20–3203178.76STS 20–4204389.53STS 20–4207188.05STS 20–52082710.97STS 20–72093311.06STS 20–8209369.17STS 20–920101012.85STS 20–1020109813.69STS 20–1120115614.19STS 40–14006.84STS 40–2402387.34STS 40–3403147.11STS 40–4404537.22STS 40–5405767.99STS 40–6407188.05STS 40–7409369.17STS 40–840102111.45STS 40–940115812.12STS 40–1040121613.18STS 60–16005.57STS 60–2602415.62STS 60–3603335.95STS 60–4603875.81STS 60–5605106.57STS 60–6606036.23STS 60–7607276.82STS 60–8608297.53STS 60–9609057.64STS 60–106010258.34STS 60–116011139.95STS 80–18004.3STS 80–2802014.81STS 80–3802954.56STS 80–4803814.37STS 80–5804924.98STS 80–6806185.29STS 80–7807365.28STS 80–8809146.64STS 80–98010056.88STS 80–108011388.84STS 80–118012009.32Figure 6
Change law ofσt of red sandstone samples versus the loading rate.It can be seen that there is a significant correlation between the loading rate andσt of rock samples free from drying-wetting cycles (n = 0) and subjected to drying-wetting cycles. The higher the loading rate, the greater the σt of the sample.Theσt of the sample increases more quickly as the loading rate is increased. The expression between σt of red sandstone and the loading rate can be obtained through the fitting function [23]. Figure 6 shows the detailed fitting curve of σt of red sandstone and the loading rate, and equation (3) shows the fitting equation.(3)σt=10.81+0.481e0.0024σ˙R2=0.979n=0,8.06+0.347e0.0025σ˙R2=0.972n=20,6.58+0.256e0.0027σ˙R2=0.962n=40,5.46+0.154e0.003σ˙R2=0.964n=60,4.26+0.132e0.003σ˙R2=0.980n=80.The loading rate effect of red sandstone is related ton; thus, the parameters in σt of the red sandstone are also affected by n. The fitting equation can be uniformly expressed as follows: σt=B+CeDσ˙. Figure 7 shows the change of parameters B, C, and D in the unified equation with n.Figure 7
Change law of each parameter versusn: (a) parameter B, (b) parameter C, and (c) parameter D.
(a)(b)(c)As shown in Figure7, each parameter has a good linear relationship with n. The σt of red sandstone can be expressed as follows:(4)σt=−0.0785n+10.174+−0.00445n+0.4522e8.5×10−6n+0.0024σ˙.


### 3.2.2. Drying-Wetting Cycle Effect

To analyze the influence ofn on the σt of red sandstone, the loading rates of 200, 400, 600, 800, and 1000 GPa/s are substituted into equation (3), and variation curves of the σt of red sandstone with n under different loading rates are drawn, as shown in Figure 8. Under a given loading rate, the σt decreases gradually with the increase of n. Additionally, the red sandstone experiences the greatest decrease in σt following the initial 20 drying-wetting cycles.Figure 8
Change law ofσt of red sandstone samples versus n.Although the previous research results contribute to understanding the evolution law ofσt of the rock mass with n, test results also have some limitations. Actually, the rock may experience hundreds of thousands of drying-wetting cycles in the natural environment, and the long-term residual strength of the rock mass should be analyzed and predicted. Therefore, a reliable prediction model for the residual strength of the rock mass is required. To verify the overall performance of rocks after freeze-thaw cycles and heating-cooling cycles, Mutlutürk et al. [24] proposed an attenuation model and verified its effectiveness through the experimental results of various types of rocks, as listed in equation.(5)IN=I0e−λN.To introduce the decay model to the field of dynamics, Wang et al. [25, 26] considered the influence of the strain rate on the damage factor. The results demonstrate that the modified model can effectively predict the dynamic tensile strength of red sandstone specimens subjected to multiple thermal shocks. However, the applicability of this equation in predicting the dynamic tensile strength of rock specimens after drying-wetting cycles requires additional investigation. Considering the influence of n and the loading rate in this study, the damage equation can be rewritten in the form of equation as follows:(6)σtn=σt0e−λσ˙n,where n is the number of drying-wetting cycles and σt0 is the dynamic tensile strength of the sample without drying-wetting cycles (n = 0). σtn is the dynamic tensile strength after n drying-wetting cycles, and λσ˙ is the decay factor corresponding to different loading rates.Figure9 shows the scatter plot and fitting curves of the decay factor versus n at loading rates of 200, 500, 800, and 1100 GPa/s. It has been observed that, with an increase in the value of n, the decay factor decreases at a given loading rate. This implies that the extent of damage to the rock sample increases, which aligns with the findings from previous studies. Besides, a numerical fitting method is used to obtain decay factors at different loading rates, and the curves of decay factor changing with loading rates are drawn in Figure 10. It can be seen that the higher the loading rate, the smaller the decay factor λ. The relationship between decay factors and loading rates is fitted with an exponential function, as expressed in equation (7). By substituting equation (3) (when n = 0) and equation (7) into equation (6), the prediction model of σt of the red sandstone under the coupling action of the wetting drying cycle and loading rate can be obtained as follows:(7)λ=0.014−0.001e0.0012σ˙R2=0.980,(8)σtn=10.81+0.481e0.0024σ˙e−0.014−0.001e0.0012σ˙n.Figure 9
Change law of the decay factor versusn.Figure 10
Change law of the decay factor versus the loading rate.


## 3.2.1. Loading Rate Effect

After a series of stress balance verification, the dynamic tensile strengthσt of red sandstone under different combination conditions can be calculated by equation (1), as shown in Table 1. On this basis, the curve of σt of the red sandstone changing with the loading rate under different drying-wetting cycles is drawn, as shown in Figure 6.Table 1
Dynamic tensile test results of samples.
No.nLoading rate (GPa/s)Tensile strength (MPa)STS 0-10011.3STS 0–2019512.14STS 0–3027611.05STS 0–4038912STS 0–5052312.76STS 0–6059712.78STS 0–7066912.87STS 0–8084514.72STS 0–9099316.23STS 0–100102216.34STS 0–110112617.48STS 0–120120519.63STS 20–12008.50STS 20–2202268.43STS 20–3203178.76STS 20–4204389.53STS 20–4207188.05STS 20–52082710.97STS 20–72093311.06STS 20–8209369.17STS 20–920101012.85STS 20–1020109813.69STS 20–1120115614.19STS 40–14006.84STS 40–2402387.34STS 40–3403147.11STS 40–4404537.22STS 40–5405767.99STS 40–6407188.05STS 40–7409369.17STS 40–840102111.45STS 40–940115812.12STS 40–1040121613.18STS 60–16005.57STS 60–2602415.62STS 60–3603335.95STS 60–4603875.81STS 60–5605106.57STS 60–6606036.23STS 60–7607276.82STS 60–8608297.53STS 60–9609057.64STS 60–106010258.34STS 60–116011139.95STS 80–18004.3STS 80–2802014.81STS 80–3802954.56STS 80–4803814.37STS 80–5804924.98STS 80–6806185.29STS 80–7807365.28STS 80–8809146.64STS 80–98010056.88STS 80–108011388.84STS 80–118012009.32Figure 6
Change law ofσt of red sandstone samples versus the loading rate.It can be seen that there is a significant correlation between the loading rate andσt of rock samples free from drying-wetting cycles (n = 0) and subjected to drying-wetting cycles. The higher the loading rate, the greater the σt of the sample.Theσt of the sample increases more quickly as the loading rate is increased. The expression between σt of red sandstone and the loading rate can be obtained through the fitting function [23]. Figure 6 shows the detailed fitting curve of σt of red sandstone and the loading rate, and equation (3) shows the fitting equation.(3)σt=10.81+0.481e0.0024σ˙R2=0.979n=0,8.06+0.347e0.0025σ˙R2=0.972n=20,6.58+0.256e0.0027σ˙R2=0.962n=40,5.46+0.154e0.003σ˙R2=0.964n=60,4.26+0.132e0.003σ˙R2=0.980n=80.The loading rate effect of red sandstone is related ton; thus, the parameters in σt of the red sandstone are also affected by n. The fitting equation can be uniformly expressed as follows: σt=B+CeDσ˙. Figure 7 shows the change of parameters B, C, and D in the unified equation with n.Figure 7
Change law of each parameter versusn: (a) parameter B, (b) parameter C, and (c) parameter D.
(a)(b)(c)As shown in Figure7, each parameter has a good linear relationship with n. The σt of red sandstone can be expressed as follows:(4)σt=−0.0785n+10.174+−0.00445n+0.4522e8.5×10−6n+0.0024σ˙.


## 3.2.2. Drying-Wetting Cycle Effect

To analyze the influence ofn on the σt of red sandstone, the loading rates of 200, 400, 600, 800, and 1000 GPa/s are substituted into equation (3), and variation curves of the σt of red sandstone with n under different loading rates are drawn, as shown in Figure 8. Under a given loading rate, the σt decreases gradually with the increase of n. Additionally, the red sandstone experiences the greatest decrease in σt following the initial 20 drying-wetting cycles.Figure 8
Change law ofσt of red sandstone samples versus n.Although the previous research results contribute to understanding the evolution law ofσt of the rock mass with n, test results also have some limitations. Actually, the rock may experience hundreds of thousands of drying-wetting cycles in the natural environment, and the long-term residual strength of the rock mass should be analyzed and predicted. Therefore, a reliable prediction model for the residual strength of the rock mass is required. To verify the overall performance of rocks after freeze-thaw cycles and heating-cooling cycles, Mutlutürk et al. [24] proposed an attenuation model and verified its effectiveness through the experimental results of various types of rocks, as listed in equation.(5)IN=I0e−λN.To introduce the decay model to the field of dynamics, Wang et al. [25, 26] considered the influence of the strain rate on the damage factor. The results demonstrate that the modified model can effectively predict the dynamic tensile strength of red sandstone specimens subjected to multiple thermal shocks. However, the applicability of this equation in predicting the dynamic tensile strength of rock specimens after drying-wetting cycles requires additional investigation. Considering the influence of n and the loading rate in this study, the damage equation can be rewritten in the form of equation as follows:(6)σtn=σt0e−λσ˙n,where n is the number of drying-wetting cycles and σt0 is the dynamic tensile strength of the sample without drying-wetting cycles (n = 0). σtn is the dynamic tensile strength after n drying-wetting cycles, and λσ˙ is the decay factor corresponding to different loading rates.Figure9 shows the scatter plot and fitting curves of the decay factor versus n at loading rates of 200, 500, 800, and 1100 GPa/s. It has been observed that, with an increase in the value of n, the decay factor decreases at a given loading rate. This implies that the extent of damage to the rock sample increases, which aligns with the findings from previous studies. Besides, a numerical fitting method is used to obtain decay factors at different loading rates, and the curves of decay factor changing with loading rates are drawn in Figure 10. It can be seen that the higher the loading rate, the smaller the decay factor λ. The relationship between decay factors and loading rates is fitted with an exponential function, as expressed in equation (7). By substituting equation (3) (when n = 0) and equation (7) into equation (6), the prediction model of σt of the red sandstone under the coupling action of the wetting drying cycle and loading rate can be obtained as follows:(7)λ=0.014−0.001e0.0012σ˙R2=0.980,(8)σtn=10.81+0.481e0.0024σ˙e−0.014−0.001e0.0012σ˙n.Figure 9
Change law of the decay factor versusn.Figure 10
Change law of the decay factor versus the loading rate.


## 3.3. Energy Dissipation Law

The natural rock exhibits numerous imperfections, and the alteration and disintegration of this rock can be conceptualized as the exchange of energy and matter with the external surroundings [27, 28]. The rock undergoes energy emission, including heat energy and radiation energy, as a result of the mechanical and thermal energy exerted by the external environment [29–31]. The energy absorbed by the rock specimen during an impact test can be calculated if the energy dissipation due to friction between the elastic rod and the specimen during the propagation of stress waves is disregarded [32].(9)WS=AECB∫0Tεit2−εrtt2−εtt2dt,where A, E, CB, As, and ls represent the cross-sectional area of the rod, the elastic modulus of the rod, the stress wave velocity of the rod, the cross-sectional area of the sample, and the length of the sample, respectively; εit, εrt, and εtt represent the strain corresponding to the incident wave, reflected wave, and transmitted wave on the pressure rod, respectively; T is the time at which the sample is completely destroyed.The calculation equation of energy consumption density is as follows:(10)ws=WSV,where ws is the energy consumption density and V is the sample volume.Figure11 shows the time history curve of energy change when the impact pressure is 0.25 MPa, and n = 0. Under an impact load, the generation of stress waves can lead to the propagation of energy. When the incident stress wave energy is low, the specimen is in an elastic deformation phase. Conversely, as the incident energy increases, the specimen enters a plastic phase, where the primary cracks broaden and multiple new cracks emerge, thus reducing the propagation of the transmitted wave [33]. When the incident energy is large, the sample breaks into powder instantly, the stress wave propagation stops, and the energy value of the sample tends to a maximum. Based on the dynamic tensile stress-strain characteristics and failure morphology of the sample, the energy change curve of the sample can be divided into three stages:(1)
Stage 1: 0–50μs. At this stage, the red sandstone sample undergoes elastic deformation, and the absorbed energy is retained as elastic potential.(2)
Stage 2: 50–160μs. At this point, the transmission energy of the sample changes little as the stress wave propagates, whereas the input energy, reflection energy, and absorption energy change significantly. The absorption energy has the largest increased amplitude, followed by reflection energy, and incident energy has the least increased amplitude. The previous results show that upon the arrival of the incident wave at the interface between the sample and the incident rod, a significant portion of the incident energy is reflected in a backward direction through the incident rod. As the residual energy traverses the red sandstone sample, a substantial amount of energy is absorbed by the specimen for dissipation through damage evolution and deformation failure, with only a minimal portion redirected to the transmission rod.(3)
Stage 3: 160–220μs. Under the impact stress, the primary cracks within the interior of the sample rapidly expand and a substantial number of additional cracks are formed. The cracks continuously spread until the sample is ultimately destroyed. The highest value of incident energy is observed, followed by absorbed energy, with the transmitted energy having the lowest value, as the energy value eventually tends towards a steady state.Figure 11
Energy-time curve of the red sandstone sample in the SHPB tensile test.Figure12 displays the evolution of the dissipation energy density of samples subjected to varying drying-wetting cycles with the increment in the loading rate. As the loading rate increases, the dissipation energy density also increases, as a result of the acceleration of rock deformation, the amplification of activated cracks, and the enhancement of energy absorption by the rock. The dissipation energy density of red sandstone samples increases linearly with the loading rate under different n, which can be described ws=aiσ˙+bi. In this equation, ws refers to the dissipation energy density, and a and b are function coefficients. Table 2 presents the pattern of change in parameters a and b. For a constant loading rate, as n increases, the energy dissipation density decreases. This can be attributed to the heightened severity of deterioration in the red sandstone samples, resulting in a lower energy expenditure during rock failure.Figure 12
Energy consumption density of red sandstone samples versus the loading rate.Table 2
Changing rules of parametersa and b.
na (×10−4)bR202.23−0.0190.936202.10−0.0370.976401.91−0.0340.970601.83−0.0450.976801.54−0.0330.979


## 4. Conclusion

(1)
The red sandstone is significantly degraded as a result of drying-wetting cycles. As the number of drying-wetting cycles increases, the longitudinal wave velocity of the red sandstone decreases in a gradual manner.(2)
When the number of drying-wetting cycles is fixed, the dynamic tensile strength of the red sandstone increases exponentially as the loading rate increases. Conversely, for a constant loading rate, an increase in the number of drying-wetting cycles results in a decrease in the dynamic tensile strength of the red sandstone, indicating that drying-wetting cycles have a discernible effect on the degradation of the dynamic tensile properties of the red sandstone.(3)
Based on the effect of the loading rate on the decay factor, a dynamic tensile strength decay model for red sandstone has been established that takes into account both the loading rate and the number of drying-wetting cycles. The model data align well with the experimental results, making the proposed model a viable tool for forecasting the long-term dynamic tensile strength of red sandstone samples.(4)
The dissipation energy density of red sandstone under dynamic tension experiences a significant impact from drying-wetting cycles. As the number of drying-wetting cycles increases at a constant loading rate, the dissipation energy density decreases.



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*Source: 8330326-2023-02-20.xml*