# A Damage Constitutive Model of Red Sandstone under Coupling of Wet-Dry Cycles and Impact Load

Authors: Bin Du; Haibo Bai
Journal: Shock and Vibration
 (2019)

---

## Abstract

In the engineering fields of mining, tunneling, slopes, and dams, rocks are usually subject to the coupling effect of impact load and wet-dry cycles. The deformation rule of rocks under the coupling effects is a symbolic mechanical property, which lays the foundation for the design and evaluation in the rock engineering. In this paper, the coupling damage was classified as mesodamage induced by wet-dry cycles and macrodamage induced by impact load, and the loading rate effect was considered as the load damage. Besides, a constitutive model of coupling damage was concluded based on Lemaitre’s strain equivalent assumption. Consequently, the validity of the model was verified by a series of dynamic compression tests of red sandstone. Results indicated that the proposed damage constitutive model can definitely describe the dynamic stress-strain curves of red sandstone after wet-dry cycles and impact load. The evolution of coupling damage curves showed that wet-dry cycle damage plays a dominant role in the elastic deformation stage, while the yield failure stage is controlled by the load damage in which the loading rate cannot be ignored. Parametric study was also performed to analyze the effect of parameters on dynamic stress-strain curves. The proposed mode has the simple and reliable operation with few parameters and can efficiently predict the long-term deformation behavior of rocks subject to multiple wet-dry cycles.


---

## Body

## 1. Introduction

Due to periodical moisture changes caused by the ground water level, rainfall, or other reasons, rocks are usually subject to cyclic wet-dry cycles treatment in various areas such as mining, tunneling, slopes, and dams. As a typical weathering, wet-dry cycles lead to rock deterioration, posing a threat to the safety and stability of rock engineering [1–4]. Therefore, the study of physical and mechanical properties of rocks against wet-dry cycles is vitally important.Recently, the effect of wet-dry cycles on physical and mechanical properties of rock materials has been emphasized in the relevant studies. In terms of physical properties, the density, porosity, slake durability, and P-wave velocity have been studied. For instance, the surface roughness, bulk density, and porosity of mudrock after 3 wet-dry cycles were measured by Pardini et al. [1]. It was concluded that the porosity increases with the increasing wet-dry cycles, while the surface roughness and bulk density are independent of wet-dry cycles. Özbek [3] carried out a systematic experiment and analyzed the specific gravity, apparent porosity, unit weight, water absorption by weight, P-wave velocity, and slake durability index of ignimbrite samples subject to wet-dry cycles. They claimed that cyclic wet-dry treatments have different negative effects on the physical parameters mentioned above, and the sensibility to wet-dry cycles depends on the mineral type of rocks. Zhou et al. [5] concluded that the porosity and water absorption of sandstone increase with the increasing wet and dry cycles, while SDI, density, and P-wave velocity decrease. In terms of mechanical properties, uniaxial compressive strength, shear strength, tensile strength, fracture toughness, and elastic modulus have been studied. Liu et al. [6] obtained that the elastic modulus and compressive strength of shaly sandstone decrease with the increasing wet-dry cycles, and the deterioration degree increases with the increase of wet-dry cycles. Hale and Shakoor [7] reported that the process of wet-dry treatment has no obvious effect on the unconfined compressive strength of sandstones. Khanlari and Abdilor [8] measured the uniaxial compressive strength of five different types of sandstone treated by wet-dry cycles. They found that there is no significant correlation between uniaxial compressive strength and wet-dry cycles, resulting from a variety of reasons, such as the mineral composition, anisotropic features, original defects, and artificial errors. Zhao et al. [9] reported that cyclic wet-dry cycles cannot reduce the tensile strength of sandstone with low clay mineral content because the weak performance mainly arises from the chemical and corrosive deterioration. Hua et al. [10] concluded that both the fracture toughness and tensile strength decrease with the increase of wet-dry cycles.The existing studies are mainly concentrated on static tests, while the effect of cyclic wet-dry on the dynamic response of rocks has not been clearly studied. In the practical rock engineering such as drilling, tunneling, and blasting, rocks may be broken under dynamic load induced by impact operation. After the cyclic wetting and drying treatment, rock samples are more close to the actual stress state of rock materials in geotechnical engineering [11–13]. Therefore, it is meaningful to investigate the dynamic properties of rocks subject to cyclic wet-dry cycles. Zhou et al. [5, 12] conducted a range of impact tests to investigate the dynamic tensile strength and dynamic compressive strength of sandstone after wet-dry cycles. It was concluded that both dynamic compression and tensile strength decrease with the increasing wet-dry cycles. According to the test results, a decay model was proposed to forecast the long-term dynamic tensile strength. However, the damage constitutive behaviors under the coupling effect of wet-dry cycles and dynamic load have not been studied yet. In natural environments, rock materials may subject to lots of wet-dry cycles, which could last for several years, and it is unpractical to obtain the dynamic properties of rock after every given wet-dry cycles via experiments. Thus, a damage constitutive model considering the loading rate and number of wet-dry cycles should be developed to predict the long-term mechanical properties of rock, which could be obtained without experiments.To characterize the evolution of dynamic stress-strain curves of rock under the coupling effect of wet-dry cycles and impact load, a damage constitutive model is developed from both the macro- and microscale in this paper. Based on Lemaitre’s strain equivalence hypothesis, the coupling damage variable is determined through the combination of wet-dry cycle damage and load damage. It should be noted that the loading rate is considered as the load damage. The evolution of coupling damage variable versus strain and parametric analysis of the model are discussed. In the end, the proposed model is verified through the comparison between measured and calculated stress-strain curves.


## 2. Statistical Damage Constitutive Model under the Coupling Effect of Wet-Dry Cycles and Load

To describe the stress-strain relationship of concrete, a multiscale damage constitutive model subject to the combined action of freeze-thaw cycles and load was proposed by Wang et al. [14] on the basis of Lemaitre’s strain equivalence hypothesis. In this model, the damage caused by freeze-thaw cycles was considered as the microscopic damage, which was determined by the variation of elastic modulus. Meanwhile, load damage was considered as the macroscopic damage described by Weibull distribution function. The validity of the proposed model was verified by experiments. However, the loading rate effect was ignored in this model, especially within the scope of dynamic tests. In this study, considering the coupling effect of wet-dry cycles and dynamic load with varying loading rates, an improved damage constitutive model is proposed on the basis of Wang’s model.


### 2.1. Damage Constitutive Relations of Red Sandstone under Wet-Dry Cycles and Load

The deterioration of red sandstone induced by wet-dry cycles is identified as the microscopic damage, and the damage caused by dynamic load is regarded as the macroscopic damage. It should be emphasized that the following analysis of coupling damage under wet-dry cycles and dynamic load all follow Lemaitre’s hypothesis.Figure1 shows the superposition of the damage variable. It is clearly seen that the damage status induced by the coupling treatment equals to the sum of status caused by macroscopic and microscopic damage and then minus the original status without any damage. The relation of strain among the three statuses can be expressed as follows:(1)ε12=ε1+ε2−ε0.Figure 1
Superposition of coupling damage variable. Samples subject to (a) both macroscopic and mesoscopic damage, (b) only mesoscopic damage, (c) only macroscopic damage, and (d) and free of any damage. The damage variable, strain, and elastic modulus under stress areD12, D1, D2, and D0; ɛ12, ɛ1, ɛ2, and ɛ0; and E12, E1, E2, and E0, respectively.When the stress remains the same, the relationship between stress and elastic modulus can be defined as follows:(2)σE12=σE1+σE2−σE0.According to Lemaitre’s strain equivalence hypothesis, the elastic modulus is defined as [15](3)E12=E01−D12,E1=E01−D1,E2=E01−D2.From equations (1)∼(3), D12, defined as the coupling damage variable, can be expressed as follows [16, 17]:(4)D12=1−1−D11−D21−D1D2.The damage variableD12 represents the coupling damage behavior under wet-dry cycles and load. On the one hand, when rocks are free of the wet-dry cycle, namely, D1 = 0 and then D12 = D2, unweathered rocks only show the load damage pattern under the stress. On the other hand, when rocks are only subject to wet-dry cycles, namely, D2 = 0 and then D12 = D1, the coupling damage variable equals to the macroscopic damage variable. Besides, with the increase of D1 or D2, the coupling damage variable always tends to increase, which is consistent with the actual situation. Accordingly, D12 reflects the actual evolution of damage subject to the coupling effect of wet-dry cycles and load. According to the theory of continuous damage, the constitutive relation of red sandstone under wet-dry cycles and dynamic load can be expressed as Lemaitre [18]:(5)σ=E1−D12ε=Eε1−D11−D21−D1D2.


### 2.2. Determination of Damage Variable

#### 2.2.1. Damage Variable of Red Sandstone Subject to Wet-Dry Cycles

When the red sandstone is subject to cyclic wet-dry treatment, the mechanical properties of rocks have a varying deterioration, especially the elastic modulus. Thus, the damage caused by wet-dry cycles can be measured by the relative variation of the elastic modulus. Therefore, the wet-dry cycle damage variable of red sandstone is described as Huang et al. [19]:(6)D1=1−EnE0,where E0 is the elastic modulus of the untreated specimen and En is the elastic modulus of the specimen after n cycles of wet-dry treatment.


#### 2.2.2. Damage Variable of Red Sandstone Subject to Load

As a typical heterogeneous material, there are large numbers of uniform and randomly distributed internal defects in the rock (e.g., flaws, microvoids, grain boundaries, and weak medium) [20]. Before the establishment of the load damage evolution equation, the following assumptions should be addressed: (1) for each microelement, the evolution of strain and stress shows a linear relationship before failure and (2) the strength of the microelement satisfies the Weibull distribution, and the probability density function p(ɛ) is defined as follows [21–24]:(7)pε=mF0εF0m−1exp−εF0m,where ɛ is the strain of red sandstone and F0 and m represent distribution parameters of the model. When t is the broken microelements after a certain load, N is the total number of microelements, the damage variable D2 is defined as follows:(8)D2=tN=N∫0εpxdxN=1−e−ε/F0m.According to the Drucker–Prager failure criterion and extremum method, the distribution parametersF0 and m of the model can be calculated as follows [25]:(9)m=−1lnσs/Eεs,(10)F0=εsmm,where σs and εs are the peak stress and peak strain of red sandstone under the stress and E is the elastic modulus. The elastic modulus, which is calculated as the slope of stress-strain curves between 40% and 60% of the peak stress, can be approximately calculated by the following equation [26]:(11)E=σ0.6−σ0.4ε0.6−ε0.4,where σ0.6 and σ0.4 are 60% and 40% of peak strength, respectively, and ε0.6 and ε0.4 are 60% and 40% of peak strain, respectively.When equation (10) is substituted into equation (8), the load damage variable is expressed as follows:(12)D2=1−e−1/mε/εsm.As discussed in Sections2.1 and 2.2, the coupling damage variable is divided into two parts and then combined by equation (5). Based on this principle, D1 can be calculated from the static compression test subject to different numbers of wet-dry cycles. Except for the basic variable ε, the others (εs, σs, E, and m) are all dependent parameters of loading rate and can be defined with the aid of the regression method. It is noted that the analysis of the above four parameters should be defined on the basis of dynamic compression tests on unweathered specimens.


## 2.1. Damage Constitutive Relations of Red Sandstone under Wet-Dry Cycles and Load

The deterioration of red sandstone induced by wet-dry cycles is identified as the microscopic damage, and the damage caused by dynamic load is regarded as the macroscopic damage. It should be emphasized that the following analysis of coupling damage under wet-dry cycles and dynamic load all follow Lemaitre’s hypothesis.Figure1 shows the superposition of the damage variable. It is clearly seen that the damage status induced by the coupling treatment equals to the sum of status caused by macroscopic and microscopic damage and then minus the original status without any damage. The relation of strain among the three statuses can be expressed as follows:(1)ε12=ε1+ε2−ε0.Figure 1
Superposition of coupling damage variable. Samples subject to (a) both macroscopic and mesoscopic damage, (b) only mesoscopic damage, (c) only macroscopic damage, and (d) and free of any damage. The damage variable, strain, and elastic modulus under stress areD12, D1, D2, and D0; ɛ12, ɛ1, ɛ2, and ɛ0; and E12, E1, E2, and E0, respectively.When the stress remains the same, the relationship between stress and elastic modulus can be defined as follows:(2)σE12=σE1+σE2−σE0.According to Lemaitre’s strain equivalence hypothesis, the elastic modulus is defined as [15](3)E12=E01−D12,E1=E01−D1,E2=E01−D2.From equations (1)∼(3), D12, defined as the coupling damage variable, can be expressed as follows [16, 17]:(4)D12=1−1−D11−D21−D1D2.The damage variableD12 represents the coupling damage behavior under wet-dry cycles and load. On the one hand, when rocks are free of the wet-dry cycle, namely, D1 = 0 and then D12 = D2, unweathered rocks only show the load damage pattern under the stress. On the other hand, when rocks are only subject to wet-dry cycles, namely, D2 = 0 and then D12 = D1, the coupling damage variable equals to the macroscopic damage variable. Besides, with the increase of D1 or D2, the coupling damage variable always tends to increase, which is consistent with the actual situation. Accordingly, D12 reflects the actual evolution of damage subject to the coupling effect of wet-dry cycles and load. According to the theory of continuous damage, the constitutive relation of red sandstone under wet-dry cycles and dynamic load can be expressed as Lemaitre [18]:(5)σ=E1−D12ε=Eε1−D11−D21−D1D2.


## 2.2. Determination of Damage Variable

### 2.2.1. Damage Variable of Red Sandstone Subject to Wet-Dry Cycles

When the red sandstone is subject to cyclic wet-dry treatment, the mechanical properties of rocks have a varying deterioration, especially the elastic modulus. Thus, the damage caused by wet-dry cycles can be measured by the relative variation of the elastic modulus. Therefore, the wet-dry cycle damage variable of red sandstone is described as Huang et al. [19]:(6)D1=1−EnE0,where E0 is the elastic modulus of the untreated specimen and En is the elastic modulus of the specimen after n cycles of wet-dry treatment.


### 2.2.2. Damage Variable of Red Sandstone Subject to Load

As a typical heterogeneous material, there are large numbers of uniform and randomly distributed internal defects in the rock (e.g., flaws, microvoids, grain boundaries, and weak medium) [20]. Before the establishment of the load damage evolution equation, the following assumptions should be addressed: (1) for each microelement, the evolution of strain and stress shows a linear relationship before failure and (2) the strength of the microelement satisfies the Weibull distribution, and the probability density function p(ɛ) is defined as follows [21–24]:(7)pε=mF0εF0m−1exp−εF0m,where ɛ is the strain of red sandstone and F0 and m represent distribution parameters of the model. When t is the broken microelements after a certain load, N is the total number of microelements, the damage variable D2 is defined as follows:(8)D2=tN=N∫0εpxdxN=1−e−ε/F0m.According to the Drucker–Prager failure criterion and extremum method, the distribution parametersF0 and m of the model can be calculated as follows [25]:(9)m=−1lnσs/Eεs,(10)F0=εsmm,where σs and εs are the peak stress and peak strain of red sandstone under the stress and E is the elastic modulus. The elastic modulus, which is calculated as the slope of stress-strain curves between 40% and 60% of the peak stress, can be approximately calculated by the following equation [26]:(11)E=σ0.6−σ0.4ε0.6−ε0.4,where σ0.6 and σ0.4 are 60% and 40% of peak strength, respectively, and ε0.6 and ε0.4 are 60% and 40% of peak strain, respectively.When equation (10) is substituted into equation (8), the load damage variable is expressed as follows:(12)D2=1−e−1/mε/εsm.As discussed in Sections2.1 and 2.2, the coupling damage variable is divided into two parts and then combined by equation (5). Based on this principle, D1 can be calculated from the static compression test subject to different numbers of wet-dry cycles. Except for the basic variable ε, the others (εs, σs, E, and m) are all dependent parameters of loading rate and can be defined with the aid of the regression method. It is noted that the analysis of the above four parameters should be defined on the basis of dynamic compression tests on unweathered specimens.


## 2.2.1. Damage Variable of Red Sandstone Subject to Wet-Dry Cycles

When the red sandstone is subject to cyclic wet-dry treatment, the mechanical properties of rocks have a varying deterioration, especially the elastic modulus. Thus, the damage caused by wet-dry cycles can be measured by the relative variation of the elastic modulus. Therefore, the wet-dry cycle damage variable of red sandstone is described as Huang et al. [19]:(6)D1=1−EnE0,where E0 is the elastic modulus of the untreated specimen and En is the elastic modulus of the specimen after n cycles of wet-dry treatment.


## 2.2.2. Damage Variable of Red Sandstone Subject to Load

As a typical heterogeneous material, there are large numbers of uniform and randomly distributed internal defects in the rock (e.g., flaws, microvoids, grain boundaries, and weak medium) [20]. Before the establishment of the load damage evolution equation, the following assumptions should be addressed: (1) for each microelement, the evolution of strain and stress shows a linear relationship before failure and (2) the strength of the microelement satisfies the Weibull distribution, and the probability density function p(ɛ) is defined as follows [21–24]:(7)pε=mF0εF0m−1exp−εF0m,where ɛ is the strain of red sandstone and F0 and m represent distribution parameters of the model. When t is the broken microelements after a certain load, N is the total number of microelements, the damage variable D2 is defined as follows:(8)D2=tN=N∫0εpxdxN=1−e−ε/F0m.According to the Drucker–Prager failure criterion and extremum method, the distribution parametersF0 and m of the model can be calculated as follows [25]:(9)m=−1lnσs/Eεs,(10)F0=εsmm,where σs and εs are the peak stress and peak strain of red sandstone under the stress and E is the elastic modulus. The elastic modulus, which is calculated as the slope of stress-strain curves between 40% and 60% of the peak stress, can be approximately calculated by the following equation [26]:(11)E=σ0.6−σ0.4ε0.6−ε0.4,where σ0.6 and σ0.4 are 60% and 40% of peak strength, respectively, and ε0.6 and ε0.4 are 60% and 40% of peak strain, respectively.When equation (10) is substituted into equation (8), the load damage variable is expressed as follows:(12)D2=1−e−1/mε/εsm.As discussed in Sections2.1 and 2.2, the coupling damage variable is divided into two parts and then combined by equation (5). Based on this principle, D1 can be calculated from the static compression test subject to different numbers of wet-dry cycles. Except for the basic variable ε, the others (εs, σs, E, and m) are all dependent parameters of loading rate and can be defined with the aid of the regression method. It is noted that the analysis of the above four parameters should be defined on the basis of dynamic compression tests on unweathered specimens.


## 3. Materials and Experiments

### 3.1. Specimen Preparation

The sampled rock was red sandstone taken from Linyi, Shandong province of China. According to the analysis of X-ray diffraction, the main mineral compositions of the red sandstone were quartz (66.4%), feldspar (18.3%), calcite (8.6%), hematite (6.3%), and other mineral components (<0.5%), such as chlorite and illite. The density, uniaxial compression strength, P-wave velocity, and Poisson’s ratio were 2478 kg/m3, 74.2 MPa, 2598 m/s, and 0.24, respectively. Red sandstone was processed into cylindrical specimens for static compression tests (diameter: 50 mm; length: 100 mm) and dynamic compression tests (diameter: 50 mm, length: 25 mm). All the specimens were carefully polished to satisfy the standard recommended by International Society for Rock Mechanics (ISRM) [27]. The specimens were extracted from the same block with good petrographic uniformity and geometrical integrity. Moreover, ultrasonic tests were conducted to achieve the good homogeneousness. Red sandstone specimens were divided into five groups; namely, specimens were subject to 0, 5, 10, 15, and 20 cycles. To decrease the dispersion and ensure the validity of test data, at least three samples were tested under each condition, and the test data were averaged in this paper.


### 3.2. Cyclic Wetting and Drying Setup

Generally, two procedures are involved in an intact wet-dry cycle: the process from dry to the saturated state and the process from saturated to the dry state. Based on this principle, several methods were conducted to simulate the process of wet-dry cycles by researchers [28–31], while the uniform standard has not been formed. In this study, the wet-dry cycle was implemented by natural submerging and oven drying. For each wet-dry cycle, specimens were submerged into the purified water for 24 hours. Then, they were dried in an oven for 24 hours at a constant temperature of 60°C and cooled to the room temperature. The number of the wet-dry cycle for the untreated specimens was 0.


### 3.3. Testing Equipment

Dynamic tests were performed by a split Hopkinson pressure bar (SHPB) system with 50 mm diameter. As shown in Figure2, the SHPB system includes a transmitted bar, an incident bar, and a striker bar, whose lengths are 1400 mm, 2400 mm, and 290 mm, respectively. All the bars are made of high strength 40Cr steel with an elastic modulus of 210 GPa and a density of 7800 kg/m3. Two strain gauges are, respectively, pasted on the incident bar and transmitted bar to gather strain signals through a dynamic signal acquisition instrument when the striker bar is launched by the gas gun. The specimen is sandwiched between the incident bar and the transmitted bar. The received incident wave, reflected wave, and transmitted wave are expressed as εI, εR, and εT, respectively.Figure 2
SHPB testing system.


## 3.1. Specimen Preparation

The sampled rock was red sandstone taken from Linyi, Shandong province of China. According to the analysis of X-ray diffraction, the main mineral compositions of the red sandstone were quartz (66.4%), feldspar (18.3%), calcite (8.6%), hematite (6.3%), and other mineral components (<0.5%), such as chlorite and illite. The density, uniaxial compression strength, P-wave velocity, and Poisson’s ratio were 2478 kg/m3, 74.2 MPa, 2598 m/s, and 0.24, respectively. Red sandstone was processed into cylindrical specimens for static compression tests (diameter: 50 mm; length: 100 mm) and dynamic compression tests (diameter: 50 mm, length: 25 mm). All the specimens were carefully polished to satisfy the standard recommended by International Society for Rock Mechanics (ISRM) [27]. The specimens were extracted from the same block with good petrographic uniformity and geometrical integrity. Moreover, ultrasonic tests were conducted to achieve the good homogeneousness. Red sandstone specimens were divided into five groups; namely, specimens were subject to 0, 5, 10, 15, and 20 cycles. To decrease the dispersion and ensure the validity of test data, at least three samples were tested under each condition, and the test data were averaged in this paper.


## 3.2. Cyclic Wetting and Drying Setup

Generally, two procedures are involved in an intact wet-dry cycle: the process from dry to the saturated state and the process from saturated to the dry state. Based on this principle, several methods were conducted to simulate the process of wet-dry cycles by researchers [28–31], while the uniform standard has not been formed. In this study, the wet-dry cycle was implemented by natural submerging and oven drying. For each wet-dry cycle, specimens were submerged into the purified water for 24 hours. Then, they were dried in an oven for 24 hours at a constant temperature of 60°C and cooled to the room temperature. The number of the wet-dry cycle for the untreated specimens was 0.


## 3.3. Testing Equipment

Dynamic tests were performed by a split Hopkinson pressure bar (SHPB) system with 50 mm diameter. As shown in Figure2, the SHPB system includes a transmitted bar, an incident bar, and a striker bar, whose lengths are 1400 mm, 2400 mm, and 290 mm, respectively. All the bars are made of high strength 40Cr steel with an elastic modulus of 210 GPa and a density of 7800 kg/m3. Two strain gauges are, respectively, pasted on the incident bar and transmitted bar to gather strain signals through a dynamic signal acquisition instrument when the striker bar is launched by the gas gun. The specimen is sandwiched between the incident bar and the transmitted bar. The received incident wave, reflected wave, and transmitted wave are expressed as εI, εR, and εT, respectively.Figure 2
SHPB testing system.


## 4. Experimental Results and Analysis

### 4.1. Results of Wet-Dry Cycles

Figure3 shows the surface feature of red sandstone subject to different numbers of wet-dry cycles. It can be seen that there is no obvious flaw or defect on the surface, and the microdeterioration plays a leading role in the process of cycles.Figure 3
Surface features of specimens under different wet-dry cycles: (a)n = 0; (b) n = 5; (c) n = 10; (d) n = 15; (e) n = 20.
(a)
(b)
(c)
(d)
(e)According to test results of static compression and equation (11), the elastic modulus of red sandstone after 0, 5, 10, 15, and 20 wet-dry cycles are calculated as 6.29 GPa, 5.60 GPa, 5.23 GPa, 4.82 GPa, and 4.53 GPa, respectively. Then, the wet-dry damage variable is obtained by equation (6). The relation of D1 with the number of wet-dry cycles is described in Figure 4. It is clear that the wet-dry damage D1 increases with the increase of wet-dry cycles, and the increasing speed appears to be faster in the first cycle than that in the last wet-dry cycles. It can be fitted with exponential function in equation (13). The damage mechanism can be explained as follows: cycling wet-dry cycles not only lead to the dissolution of soft granules, resulting in the increase of microvoids, but also lead to the formation, expansion, and connection of microcracks. Therefore, the integrality of microstructure is weakened. With the increasing number of wet-dry cycles, irreversible damage induced by wet-dry cycles accumulates in the rock, resulting in degradation of physical and mechanical properties:(13)D1=0.424−0.421e−0.053n.Figure 4
Variation ofD1 with the number of wet-dry cycles.


### 4.2. Results of the Dynamic Compression Test

#### 4.2.1. Dynamic Stress Equilibrium

In this study, a pulse-shaping device is designed by a soft rubber sheet of 10 mm in diameter and 1 mm in thickness placed on the free end of the incident bar. This device has been widely used by many researches. Figure5 shows the stress evolution of a typical dynamic compression test. It is found that the transmitted pulse almost equals to the sum of reflected pulse and incident pulse, namely, (εT=εI+εR). It suggests that the dynamic force balance on both ends of the specimen has been assessed successfully. In addition, the incident wave is shaped from a rectangular pulse to a ramp pulse, and thus, the axial inertial effect can be ignored [32–34]. For other specimens subject to this dynamic test, the dynamic force balance should be checked cautiously.Figure 5
Dynamic stress equilibrium of the specimen.


#### 4.2.2. Dynamic Stress-Strain Curves

Based on 1-D wave theory, the dynamic stress-strain curves of specimen are subject to varying wet-dry cycles and loading rates, as shown in Figure6. For each group with a certain number of wet-dry cycles, the dynamic tests were conducted under various loading rates which were controlled by the impact pressure of the gas gun varying from 0.25 MPa to 0.5 MPa. To achieve a better presentation, Figure 6 lists a few curves for each group. As illustrated in Figure 6, the dynamic stress-strain curves are influenced both by the number of wet-dry cycles and loading rate. Besides, there are some similar characteristics in the shape of curves; i.e., the compaction behavior is not obvious in the initial stage, and the elastic deformation stage lasts for a long time before the yielding of the specimen. When the wet-dry cycles remain unchanged, the dynamic compression stress (defined as the peak stress), dynamic elastic modulus, and peak strain all increase with the increasing loading rate, regardless of the number of wet-dry cycles. Furthermore, except for the ascending branch, the descending branch of curves is also influenced by the loading rate.Figure 6
Dynamic stress-strain curves of specimens subject to (a)n = 0; (b) n = 5; (c) n = 10; (d) n = 15; (e) n = 20 wet-dry cycles.
(a)
(b)
(c)
(d)
(e)


### 4.3. Theoretical Verification

In Section3.1, the damage (D1) caused by wet-dry cycles is discussed, and the loading effect should take into account the remaining parameters, namely, dynamic elastic modulus (E), peak strain (ɛs), and dynamic compression stress (σs). Loading rate can be affected by these parameters.Based on specimens free from the dynamic compression test, the measured parameters (E, ɛs, and σs) and calculated parameter (m) under different loading rates are listed in Table 1. Figure 7 illustrates the variation of parameters with the loading rate. The changing rules can be linearly fitted, as shown in the following equation:(14)E=6.224+0.0175ε˙,m=19.80−0.037ε˙,σs=95.37+0.602ε˙,εs=0.017+3.627×10−5ε˙.Table 1
Parameters of the load damage model.
No.
ε
˙(s−1)
σ
s
ε
s
E (GPa)
m
DC0-1
106
157.8
0.0213
7.90
15.88
DC0-2
138
180.2
0.0220
8.76
14.74
DC0-3
168
195.7
0.0226
9.35
13.39
DC0-4
206
221.0
0.0244
9.79
12.94
DC0-5
229
232.2
0.0250
10.21
10.51
DC0-6
267
255.7
0.0272
10.86
10.17Figure 7
Variation of parameters versus loading. (a)–(d) The variation ofE, m, σs, and ɛs versus loading rate, respectively.
(a)
(b)
(c)
(d)By combining the equations mentioned above, the statistical constitutive model of red sandstone under the coupling effect of the loading rate and wet-dry cycles can be obtained.


### 4.4. Coupling Damage Evolution of Red Sandstone

As shown in equation (12), the coupling damage variable is crucial for the evolution of deformation characteristics and determines the shape and changing laws of stress-strain curves. Concluded from equation (13), evolution curves of coupling damage under different numbers of wet-dry cycles at a certain loading rate (178 s−1) are drawn in Figure 8. It can be seen that, there are mainly three stages in damage evolution curves, namely, the stable stage, rapid growth stage, and completing stage. No significant change can be observed with the increase of wet-dry cycles. During the first stage, the mesoscopic damage induced by wet-dry cycles plays a leading role, and the value of coupling damage almost equals to the wet-dry damage part. At this stage, the microvoids inside the rock begin to squeeze under the load, and the squeezing distance increases with the increasing number of wet-dry cycles, resulting in the serious deterioration. In the initiation, expansion, and coalescence of internal cracks, coupling damage sharply increases within a short range of strain, namely, between 0.0225 and 0.0325, and then increases to 1 practically. When the coupling damage is close to 1, the specimen is broken completely. The more the number of wet-dry cycles, the shorter the distance of the rapid growth stage. Figure 9 presents the coupling damage evolution of red sandstone subject to varying loading rates with 10 wet-dry cycles. The evolution process is similar to that in Figure 8 and can be classified into three stages, namely, two stable stages and a rapid growth stage. The critical strain of the first two stages is about 0.02. It is concluded that the starting point of the rapid growth stage is determined by the number of wet-dry cycles through the comparison of Figures 8 and 9. Besides, the more the cycles of wet-dry treatment, the bigger the strain of the starting point. Under the same wet-dry cycles, a larger strain period is required in the second stage when the loading rate is greater. In summary, the wet-dry damage plays a dominant role in the elastic deformation stage, while the yield failure stage is controlled by the load damage part, in which the loading rate cannot be ignored.Figure 8
Damage evolution curves of red sandstone under different wet-dry cycles.Figure 9
Damage evolution curves of red sandstone under different loading rates.


### 4.5. Parameter Sensitivity Analysis

To study the effect of two key parameters (ε˙ and D1) on the stress-strain curves, the parametric study was carried out based on the above fitting equation. The stress-strain curves at ε˙ = 120 are selected for the comparison. D1 is taken as 0, 0.11, 0.168, 0.234, and 0.28, respectively, and five stress-strain curves are achieved, as shown in Figure 10. The peak strength and elastic modulus decrease with the adding up of D1, while the peak strain is almost invariable. Figure 11 presents the stress-strain curves at n = 5 when ε˙ is taken as 100, 150, 200, 250, and 300. It indicates that the peak stress, peak strain, and elastic modulus all increase with the increase of the loading rate.Figure 10
Effect ofD1 on stress-strain curves.Figure 11
Loading rate effect on stress-strain curves.


### 4.6. Validation of the Statistical Damage Constitutive Model

To validate the rationality of the proposed model, red sandstone specimens after 5, 10, 15, and 20 wet-dry cycles were tested at the varying loading rate. Figure12 shows the comparison of measured and calculated curves. As shown in Figure 12, the measured curves agree well with calculated curves in the ascending branch, while the consistency of the descending branch is not completely ideal. This is mainly caused by the change of the deformation characteristic in the descending branch. When the stress exceeds the peak point, the residual resistance of specimens after the destruction is mainly provided by the broken fragments rather than the rock itself. In fact, considering the practical value for rock engineering, the information provided by the ascending branch of stress-strain curves is more indispensable than that of the descending branch. Therefore, the proposed damage model can accurately reflect the mechanical response under the coupling effect of wet-dry cycles and dynamic load.Figure 12
Comparison of measured and calculated stress-strain curves: (a)n = 0; (b) n = 5; (c) n = 10; (d) n = 15; (e) n = 20.
(a)
(b)
(c)
(d)
(e)


## 4.1. Results of Wet-Dry Cycles

Figure3 shows the surface feature of red sandstone subject to different numbers of wet-dry cycles. It can be seen that there is no obvious flaw or defect on the surface, and the microdeterioration plays a leading role in the process of cycles.Figure 3
Surface features of specimens under different wet-dry cycles: (a)n = 0; (b) n = 5; (c) n = 10; (d) n = 15; (e) n = 20.
(a)
(b)
(c)
(d)
(e)According to test results of static compression and equation (11), the elastic modulus of red sandstone after 0, 5, 10, 15, and 20 wet-dry cycles are calculated as 6.29 GPa, 5.60 GPa, 5.23 GPa, 4.82 GPa, and 4.53 GPa, respectively. Then, the wet-dry damage variable is obtained by equation (6). The relation of D1 with the number of wet-dry cycles is described in Figure 4. It is clear that the wet-dry damage D1 increases with the increase of wet-dry cycles, and the increasing speed appears to be faster in the first cycle than that in the last wet-dry cycles. It can be fitted with exponential function in equation (13). The damage mechanism can be explained as follows: cycling wet-dry cycles not only lead to the dissolution of soft granules, resulting in the increase of microvoids, but also lead to the formation, expansion, and connection of microcracks. Therefore, the integrality of microstructure is weakened. With the increasing number of wet-dry cycles, irreversible damage induced by wet-dry cycles accumulates in the rock, resulting in degradation of physical and mechanical properties:(13)D1=0.424−0.421e−0.053n.Figure 4
Variation ofD1 with the number of wet-dry cycles.


## 4.2. Results of the Dynamic Compression Test

### 4.2.1. Dynamic Stress Equilibrium

In this study, a pulse-shaping device is designed by a soft rubber sheet of 10 mm in diameter and 1 mm in thickness placed on the free end of the incident bar. This device has been widely used by many researches. Figure5 shows the stress evolution of a typical dynamic compression test. It is found that the transmitted pulse almost equals to the sum of reflected pulse and incident pulse, namely, (εT=εI+εR). It suggests that the dynamic force balance on both ends of the specimen has been assessed successfully. In addition, the incident wave is shaped from a rectangular pulse to a ramp pulse, and thus, the axial inertial effect can be ignored [32–34]. For other specimens subject to this dynamic test, the dynamic force balance should be checked cautiously.Figure 5
Dynamic stress equilibrium of the specimen.


### 4.2.2. Dynamic Stress-Strain Curves

Based on 1-D wave theory, the dynamic stress-strain curves of specimen are subject to varying wet-dry cycles and loading rates, as shown in Figure6. For each group with a certain number of wet-dry cycles, the dynamic tests were conducted under various loading rates which were controlled by the impact pressure of the gas gun varying from 0.25 MPa to 0.5 MPa. To achieve a better presentation, Figure 6 lists a few curves for each group. As illustrated in Figure 6, the dynamic stress-strain curves are influenced both by the number of wet-dry cycles and loading rate. Besides, there are some similar characteristics in the shape of curves; i.e., the compaction behavior is not obvious in the initial stage, and the elastic deformation stage lasts for a long time before the yielding of the specimen. When the wet-dry cycles remain unchanged, the dynamic compression stress (defined as the peak stress), dynamic elastic modulus, and peak strain all increase with the increasing loading rate, regardless of the number of wet-dry cycles. Furthermore, except for the ascending branch, the descending branch of curves is also influenced by the loading rate.Figure 6
Dynamic stress-strain curves of specimens subject to (a)n = 0; (b) n = 5; (c) n = 10; (d) n = 15; (e) n = 20 wet-dry cycles.
(a)
(b)
(c)
(d)
(e)


## 4.2.1. Dynamic Stress Equilibrium

In this study, a pulse-shaping device is designed by a soft rubber sheet of 10 mm in diameter and 1 mm in thickness placed on the free end of the incident bar. This device has been widely used by many researches. Figure5 shows the stress evolution of a typical dynamic compression test. It is found that the transmitted pulse almost equals to the sum of reflected pulse and incident pulse, namely, (εT=εI+εR). It suggests that the dynamic force balance on both ends of the specimen has been assessed successfully. In addition, the incident wave is shaped from a rectangular pulse to a ramp pulse, and thus, the axial inertial effect can be ignored [32–34]. For other specimens subject to this dynamic test, the dynamic force balance should be checked cautiously.Figure 5
Dynamic stress equilibrium of the specimen.


## 4.2.2. Dynamic Stress-Strain Curves

Based on 1-D wave theory, the dynamic stress-strain curves of specimen are subject to varying wet-dry cycles and loading rates, as shown in Figure6. For each group with a certain number of wet-dry cycles, the dynamic tests were conducted under various loading rates which were controlled by the impact pressure of the gas gun varying from 0.25 MPa to 0.5 MPa. To achieve a better presentation, Figure 6 lists a few curves for each group. As illustrated in Figure 6, the dynamic stress-strain curves are influenced both by the number of wet-dry cycles and loading rate. Besides, there are some similar characteristics in the shape of curves; i.e., the compaction behavior is not obvious in the initial stage, and the elastic deformation stage lasts for a long time before the yielding of the specimen. When the wet-dry cycles remain unchanged, the dynamic compression stress (defined as the peak stress), dynamic elastic modulus, and peak strain all increase with the increasing loading rate, regardless of the number of wet-dry cycles. Furthermore, except for the ascending branch, the descending branch of curves is also influenced by the loading rate.Figure 6
Dynamic stress-strain curves of specimens subject to (a)n = 0; (b) n = 5; (c) n = 10; (d) n = 15; (e) n = 20 wet-dry cycles.
(a)
(b)
(c)
(d)
(e)


## 4.3. Theoretical Verification

In Section3.1, the damage (D1) caused by wet-dry cycles is discussed, and the loading effect should take into account the remaining parameters, namely, dynamic elastic modulus (E), peak strain (ɛs), and dynamic compression stress (σs). Loading rate can be affected by these parameters.Based on specimens free from the dynamic compression test, the measured parameters (E, ɛs, and σs) and calculated parameter (m) under different loading rates are listed in Table 1. Figure 7 illustrates the variation of parameters with the loading rate. The changing rules can be linearly fitted, as shown in the following equation:(14)E=6.224+0.0175ε˙,m=19.80−0.037ε˙,σs=95.37+0.602ε˙,εs=0.017+3.627×10−5ε˙.Table 1
Parameters of the load damage model.
No.
ε
˙(s−1)
σ
s
ε
s
E (GPa)
m
DC0-1
106
157.8
0.0213
7.90
15.88
DC0-2
138
180.2
0.0220
8.76
14.74
DC0-3
168
195.7
0.0226
9.35
13.39
DC0-4
206
221.0
0.0244
9.79
12.94
DC0-5
229
232.2
0.0250
10.21
10.51
DC0-6
267
255.7
0.0272
10.86
10.17Figure 7
Variation of parameters versus loading. (a)–(d) The variation ofE, m, σs, and ɛs versus loading rate, respectively.
(a)
(b)
(c)
(d)By combining the equations mentioned above, the statistical constitutive model of red sandstone under the coupling effect of the loading rate and wet-dry cycles can be obtained.


## 4.4. Coupling Damage Evolution of Red Sandstone

As shown in equation (12), the coupling damage variable is crucial for the evolution of deformation characteristics and determines the shape and changing laws of stress-strain curves. Concluded from equation (13), evolution curves of coupling damage under different numbers of wet-dry cycles at a certain loading rate (178 s−1) are drawn in Figure 8. It can be seen that, there are mainly three stages in damage evolution curves, namely, the stable stage, rapid growth stage, and completing stage. No significant change can be observed with the increase of wet-dry cycles. During the first stage, the mesoscopic damage induced by wet-dry cycles plays a leading role, and the value of coupling damage almost equals to the wet-dry damage part. At this stage, the microvoids inside the rock begin to squeeze under the load, and the squeezing distance increases with the increasing number of wet-dry cycles, resulting in the serious deterioration. In the initiation, expansion, and coalescence of internal cracks, coupling damage sharply increases within a short range of strain, namely, between 0.0225 and 0.0325, and then increases to 1 practically. When the coupling damage is close to 1, the specimen is broken completely. The more the number of wet-dry cycles, the shorter the distance of the rapid growth stage. Figure 9 presents the coupling damage evolution of red sandstone subject to varying loading rates with 10 wet-dry cycles. The evolution process is similar to that in Figure 8 and can be classified into three stages, namely, two stable stages and a rapid growth stage. The critical strain of the first two stages is about 0.02. It is concluded that the starting point of the rapid growth stage is determined by the number of wet-dry cycles through the comparison of Figures 8 and 9. Besides, the more the cycles of wet-dry treatment, the bigger the strain of the starting point. Under the same wet-dry cycles, a larger strain period is required in the second stage when the loading rate is greater. In summary, the wet-dry damage plays a dominant role in the elastic deformation stage, while the yield failure stage is controlled by the load damage part, in which the loading rate cannot be ignored.Figure 8
Damage evolution curves of red sandstone under different wet-dry cycles.Figure 9
Damage evolution curves of red sandstone under different loading rates.


## 4.5. Parameter Sensitivity Analysis

To study the effect of two key parameters (ε˙ and D1) on the stress-strain curves, the parametric study was carried out based on the above fitting equation. The stress-strain curves at ε˙ = 120 are selected for the comparison. D1 is taken as 0, 0.11, 0.168, 0.234, and 0.28, respectively, and five stress-strain curves are achieved, as shown in Figure 10. The peak strength and elastic modulus decrease with the adding up of D1, while the peak strain is almost invariable. Figure 11 presents the stress-strain curves at n = 5 when ε˙ is taken as 100, 150, 200, 250, and 300. It indicates that the peak stress, peak strain, and elastic modulus all increase with the increase of the loading rate.Figure 10
Effect ofD1 on stress-strain curves.Figure 11
Loading rate effect on stress-strain curves.


## 4.6. Validation of the Statistical Damage Constitutive Model

To validate the rationality of the proposed model, red sandstone specimens after 5, 10, 15, and 20 wet-dry cycles were tested at the varying loading rate. Figure12 shows the comparison of measured and calculated curves. As shown in Figure 12, the measured curves agree well with calculated curves in the ascending branch, while the consistency of the descending branch is not completely ideal. This is mainly caused by the change of the deformation characteristic in the descending branch. When the stress exceeds the peak point, the residual resistance of specimens after the destruction is mainly provided by the broken fragments rather than the rock itself. In fact, considering the practical value for rock engineering, the information provided by the ascending branch of stress-strain curves is more indispensable than that of the descending branch. Therefore, the proposed damage model can accurately reflect the mechanical response under the coupling effect of wet-dry cycles and dynamic load.Figure 12
Comparison of measured and calculated stress-strain curves: (a)n = 0; (b) n = 5; (c) n = 10; (d) n = 15; (e) n = 20.
(a)
(b)
(c)
(d)
(e)


## 5. Conclusions

(1)
The coupling damage caused by wet-dry cycles and impact load is classified as mesodamage induced by wet-dry cycles and macrodamage induced by impact load. According to Lemaitre’s strain equivalent assumption, a damage constitutive model is proposed and validated by experiments.
(2)
The evolution of the coupling damage can be classified into three stages: stable stage, rapid growth stage, and completing stage. The critical strain between the stable stage and rapid growth stage is defined by the number of wet-dry cycles, and the critical strain between rapid growth and completing stage was controlled by the loading rate. Wet-dry cycle damage plays a key role in the elastic deformation stage, while the yield failure stage is controlled by the load damage when considering the loading rate.
(3)
Parametric analysis shows that the peak strain is irrelevant toD1 under a fixed loading rate, while the dynamic compression stress, elastic modulus, and peak strain tend to increase with the increasing loading rate when the number of wet-dry cycles remains unchanged. These analysis results agree well with the experiments.



---

*Source: 7692424-2019-05-09.xml*