# Investigation on Damage Characteristic and Constitutive Model of Deep Sandstone under Coupled High Temperature and Impact Loads

Authors: Zhang Rongrong; Yang Yi; Ma Dongdong
Journal: Geofluids
 (2021)

---

## Abstract

To investigate the coupling damage characteristics of rock after high-temperature treatment under impact load, dynamic uniaxial compression tests for deep sandstone specimen under laboratory conditions varying with high temperature (i.e., 25°C, 100°C, 300°C, 500°C, 700°C, and 900°C) and strain rate (i.e., 170 s-1, 205 s-1, and 240 s-1) were performed using splitting Hopkinson pressure bar (SHPB) system. Coupling damage variable of deep sandstone was deduced based on the Lemaitre equivalent strain theory. Moreover, the damage parameters of deep sandstone were systematically determined according to the test data, and the effects of high temperature and strain rate on damage growth curves were investigated. Finally, a dynamic compound damage constitutive model, which could consider the coupling damage, was established and verified to describe the dynamic mechanical characteristic of deep sandstone. Theoretical and experimental results indicated that the simulated stress-strain curves matched the test data well and the proposed coupling damage constitutive model could reflect the high temperature-induced weakening and strain rate strengthening effect.


---

## Body

## 1. Introduction

With the depletion of shallow resources in the earth, the proportion of deep resource mining is increasing gradually, and the effect of high temperature on the physical and mechanical properties of rock materials is becoming more and more prominent [1–3]. Moreover, the utilization and stability analysis of deep rock engineering, such as nuclear waste disposal, core drilling, geothermal resource development, and postfire reconstruction of rock mass, is closely related to the physical and mechanical behavior of rocks after high temperature [4]. In addition to high temperature, deep rock structures are inevitably subject to impact loads derived from rock burst, blasting excavation, and earthquake [5–7]. Therefore, it is of great practical importance to study the dynamic behavior of rock after different high-temperature treatments for rational design and rapid excavation in deep rock engineering.Aiming at the influences of high temperature and impact load on physical and mechanical properties of deep rock, substantial effort has been performed by heating rock specimens to analyze the variation in macroscopic mechanical properties and microstructure and mineral composition. Many investigations have yielded valuable achievements. Sandstone has always been a hot research object in deep rock field because of its extensive existence; for instance, Li et al. [8] investigated the dynamic strength and deformation properties of coal measure sandstone after high-temperature treatment, and the thermal effect mechanism was studied using X-Ray Diffraction (XRD) and Scanning Electron Microscopy (SEM) technologies. Huang and Xia [9] heated Longyou sandstone at four treatment temperatures (25°C, 250°C, 450°C, and 600°C), and the Computed Tomography (CT) technique was utilized to quantify the thermal damage degree of sandstone specimens. Finally, the correlation between damage variable and dynamic peak stress was established. Additionally, a larger number of impact tests were carried out to investigate the mechanical properties of post heated rock with dynamic loads exerted by the splitting Hopkinson pressure bar (SHPB) system, which had been considered an invaluable device providing dynamic loading on various materials [10–13]. Yao et al. [14] carried out dynamic impact tests on Longyou sandstone specimens subjected to temperatures with various strain rates, and test results illustrated that loading rate presented positive effect on dynamic tensile strength, while the high temperature showed negative effect. To compare the thermal-induced damage for different rock types, four types of rock were selected and subjected to high temperature from 25°C to 800°C [15], and the XRD, Differential Scanning Calorimeter (DSC), and Derivative Thermogravimetry (DTG) techniques were adopted to study their mineral compositions and microscopic damage of rock specimens. Chen et al. [16] heated sandstone specimens at the rising temperature rate of 10°C/min from 25°C to 1000°C and then naturally cooled them to room temperature. The dynamic stress-strain curves and failure modes of sandstone were systematically investigated, and test results demonstrated that the break degree increased with increasing heat temperature and impact velocity. Xu and Liu [17] found that strain rate showed enhancement effect on both peak stress and peak strain of marble specimens after high-temperature treatment, while the relation between loading rate and elastic modulus was not obvious. Finally, the mechanism of strain rate enhancement effect on dynamic mechanical properties of rock was discussed by combining with the microstructure, energy absorption, and stress state. Zhang et al. [18] studied the compound damage growth characteristic of rock materials considering the freeze-thaw and external loads. Wang et al. [19] established the statistical damage constitutive model of saturated fine-grained sandstone based on damage mechanics, and the damage growth regular of the saturated fine-grained sandstone under different confining pressures was discussed. Shan et al. [20] analyzed the characteristics of dynamic stress-strain curves of artificial frozen red sandstone and found that frozen red sandstone can be considered as a nonuniform particle composed of elastic, damage, plastic, and viscous properties based on the damage evolution and the component model theory. Finally, a time-dependent damage model which included the damaged body element, the clay pot, and the spring was established and verified. Research on damage characteristic and constitutive relationship of rock subjected to coupling high temperature and dynamic loads can help comprehend its dynamic behaviors and fracture process and hence improve stability analysis of underground excavation in deep rock engineering. However, by summarizing the experimental and numerical simulation achievements, it can be noticed that the previous works on mechanical characteristic of rock bearing coupling dynamic load and high temperature mainly give attention to strength and deformation characteristic, energy dissipation, and failure mode, while limited attempts have been made to study the damage characteristic and constitutive mode of deep rock considering the coupling thermal and dynamic damage.This paper is organized as follows. Introduction presents the research status and engineering significance. Section2 describes the test procedure and effects of high temperature and strain rate on physical and dynamic mechanical parameters of deep sandstone specimen according to laboratory test results. The defined method for the coupled damage variable considering both temperature and impact load is detailed in Section 3; moreover, the effects of strain rate and temperature on damage growth curves are investigated and discussed. Finally, a compound damage constitutive model based on Zhu-Wang-Tang (ZWT) model is established and verified in Section 4. Section 5 summarizes the main conclusions of this research.


## 2. Test Device and Result

### 2.1. Sample Preparation

Sandstone is the most typical rock type in the deep mining area of Huainan. In this research, deep sandstone with fine grain is collected from a deep roadway in Huainan, China. The coring machine, cutting machine, and grinding machine are used to process deep sandstone, and a diameter of 50 mm with a height of 25 mm is adopted to conduct SHPB tests [21]. Before dynamic tests, the P-wave velocity of deep sandstone specimens is tested by using a nonmetallic ultrasonic wave velocity testing system, and the samples with similar wave velocities are selected to reduce the discretion of test data. The physical and static mechanical parameters of deep sandstone are listed in Table 1.Table 1
The physical and static mechanical parameters of deep sandstone.
Density (g/cm3)PorosityP-wave velocity (m/s)Compressive strength (MPa)Tensile strength (MPa)2.121.58%398289.111.3


### 2.2. High-Temperature Treatment and SHPB Test

The high-temperature resistance box is utilized in this test for hearing the deep sandstone specimens, and its highest value can reach 950°C. Hence, after processing and selection, 54 deep sandstone specimens are randomly separated into 6 groups and heated from room temperature to the desired temperature (i.e., 25°C, 100°C, 300°C, 500°C, 700°C, and 900°C) with the rising rate of 6°C/min, keeping the desired temperature for 4 h and then cooling naturally to room temperature. Therefore, each group has 9 specimens for conducting dynamic impact tests under various strain rates. Dynamic impact tests are conducted in State Key Laboratory of Mining Response and Disaster Prevention and Control in Deep Coal Mine, Anhui University of Science and Technology. The SHPB device is shown in Figure1. In this test, a front pulse shaper is used to generate loading wave with longer rising time, which is beneficial for achieving stress balance inside deep sandstone specimens. After the sandstone specimens are treated well, appropriate amounts of Vaseline are applied to both sides of specimens as well as the surfaces of bars. Rock specimen is placed between the incident and transmission bars, and then, the desired impact pressure is set. Open the pressure control switch after the above steps are completed. The dynamic mechanical parameters of deep sandstone specimens are calculated using “three-wave” method [22–24].Figure 1
SHPB device.


### 2.3. Analysis of Test Results

For deep sandstone specimens treated with various high temperatures, the P-wave velocity and porosity are shown in Figure2. When temperature increases from 25°C to 900°C, the P-wave velocity decreases from 3982 m/s to 2646 m/s, with a larger decrease of 33.56%, while its porosity increases by 182% after 900°C treatment compared with that at 25°C. This phenomenon is attributed to the thermal expansion of mineral particles and water vaporization of deep sandstone after high-temperature treatment, which is treated as thermal damage [25, 26]. Particularly, Jin et al. [27] investigated the inner structure and composition change of sandstone after different high-temperature treatments and found that when the temperature was higher than 500°C, the propagation of the original cracks at the boundary of different mineral particles increases and more new cracks are formed significantly due to the internal thermal stress caused by the nonuniform expansion of mineral particles. In addition, when the temperature further increased to 800°C, the crack development degree increased significantly and formed obvious crack network. Moreover, the variation rate of both wave velocity and porosity of deep sandstone specimen increases significantly after 300°C, which is caused by the loss of component and crystal water.Figure 2
Variation in P-wave velocity and porosity with temperature.In SHPB test, we adjust the impact pressure to guarantee that the strain rate of specimens fluctuates in a small range. The typical dynamic stress-strain curves of deep sandstone with various high temperatures and strain rates are shown in Figure3. It can be observed that under the same strain rate, similar characteristics are found for curves after various high temperatures, and the curve can be divided into three stages: elastic stage, plastic stage, and the postpeak failure stage. Under the same strain rate, the curve slope gradually decreases in the postpeak descent stage with the increase of temperature, indicating that the increase of temperature leads to rock failure gradually changing from brittleness to plasticity during the loading process. After the same treatment temperature, the peak stress, final strain, and peak strain increase with increasing strain rate.Figure 3
Dynamic stress-strain curves of deep sandstone with various strain rates.
(a)
170 s-1(b)
205 s-1(c)
240 s-1In this research, the dynamic strength of deep sandstone is defined as the peak stress in the dynamic stress-strain curves, and the variation in dynamic strength of deep sandstone with strain rate and temperature is shown in Figure4. After the same high-temperature treatment, the dynamic strength increases with the increase of strain rate, which shows obvious strain rate enhancement effect. Moreover, with increasing temperature treatment, the dynamic strengths of deep sandstone decrease gradually under the same strain rate. For instance, the dynamic strength of deep sandstone specimen after 900°C treatment is 84.45 MPa, which is much lower than the value of 179.87 MPa after 25°C. The main mechanism for this phenomenon is likely that the thermal damage causes increases in the porosity and decreases in the wave impedance of deep sandstone specimen. Therefore, less waves can be transmitted to the transmission bar through the rock specimen under the same incident wave, resulting in the decrease of dynamic strength.Figure 4
Variation in dynamic strength of deep sandstone with strain rate and temperature.


## 2.1. Sample Preparation

Sandstone is the most typical rock type in the deep mining area of Huainan. In this research, deep sandstone with fine grain is collected from a deep roadway in Huainan, China. The coring machine, cutting machine, and grinding machine are used to process deep sandstone, and a diameter of 50 mm with a height of 25 mm is adopted to conduct SHPB tests [21]. Before dynamic tests, the P-wave velocity of deep sandstone specimens is tested by using a nonmetallic ultrasonic wave velocity testing system, and the samples with similar wave velocities are selected to reduce the discretion of test data. The physical and static mechanical parameters of deep sandstone are listed in Table 1.Table 1
The physical and static mechanical parameters of deep sandstone.
Density (g/cm3)PorosityP-wave velocity (m/s)Compressive strength (MPa)Tensile strength (MPa)2.121.58%398289.111.3


## 2.2. High-Temperature Treatment and SHPB Test

The high-temperature resistance box is utilized in this test for hearing the deep sandstone specimens, and its highest value can reach 950°C. Hence, after processing and selection, 54 deep sandstone specimens are randomly separated into 6 groups and heated from room temperature to the desired temperature (i.e., 25°C, 100°C, 300°C, 500°C, 700°C, and 900°C) with the rising rate of 6°C/min, keeping the desired temperature for 4 h and then cooling naturally to room temperature. Therefore, each group has 9 specimens for conducting dynamic impact tests under various strain rates. Dynamic impact tests are conducted in State Key Laboratory of Mining Response and Disaster Prevention and Control in Deep Coal Mine, Anhui University of Science and Technology. The SHPB device is shown in Figure1. In this test, a front pulse shaper is used to generate loading wave with longer rising time, which is beneficial for achieving stress balance inside deep sandstone specimens. After the sandstone specimens are treated well, appropriate amounts of Vaseline are applied to both sides of specimens as well as the surfaces of bars. Rock specimen is placed between the incident and transmission bars, and then, the desired impact pressure is set. Open the pressure control switch after the above steps are completed. The dynamic mechanical parameters of deep sandstone specimens are calculated using “three-wave” method [22–24].Figure 1
SHPB device.


## 2.3. Analysis of Test Results

For deep sandstone specimens treated with various high temperatures, the P-wave velocity and porosity are shown in Figure2. When temperature increases from 25°C to 900°C, the P-wave velocity decreases from 3982 m/s to 2646 m/s, with a larger decrease of 33.56%, while its porosity increases by 182% after 900°C treatment compared with that at 25°C. This phenomenon is attributed to the thermal expansion of mineral particles and water vaporization of deep sandstone after high-temperature treatment, which is treated as thermal damage [25, 26]. Particularly, Jin et al. [27] investigated the inner structure and composition change of sandstone after different high-temperature treatments and found that when the temperature was higher than 500°C, the propagation of the original cracks at the boundary of different mineral particles increases and more new cracks are formed significantly due to the internal thermal stress caused by the nonuniform expansion of mineral particles. In addition, when the temperature further increased to 800°C, the crack development degree increased significantly and formed obvious crack network. Moreover, the variation rate of both wave velocity and porosity of deep sandstone specimen increases significantly after 300°C, which is caused by the loss of component and crystal water.Figure 2
Variation in P-wave velocity and porosity with temperature.In SHPB test, we adjust the impact pressure to guarantee that the strain rate of specimens fluctuates in a small range. The typical dynamic stress-strain curves of deep sandstone with various high temperatures and strain rates are shown in Figure3. It can be observed that under the same strain rate, similar characteristics are found for curves after various high temperatures, and the curve can be divided into three stages: elastic stage, plastic stage, and the postpeak failure stage. Under the same strain rate, the curve slope gradually decreases in the postpeak descent stage with the increase of temperature, indicating that the increase of temperature leads to rock failure gradually changing from brittleness to plasticity during the loading process. After the same treatment temperature, the peak stress, final strain, and peak strain increase with increasing strain rate.Figure 3
Dynamic stress-strain curves of deep sandstone with various strain rates.
(a)
170 s-1(b)
205 s-1(c)
240 s-1In this research, the dynamic strength of deep sandstone is defined as the peak stress in the dynamic stress-strain curves, and the variation in dynamic strength of deep sandstone with strain rate and temperature is shown in Figure4. After the same high-temperature treatment, the dynamic strength increases with the increase of strain rate, which shows obvious strain rate enhancement effect. Moreover, with increasing temperature treatment, the dynamic strengths of deep sandstone decrease gradually under the same strain rate. For instance, the dynamic strength of deep sandstone specimen after 900°C treatment is 84.45 MPa, which is much lower than the value of 179.87 MPa after 25°C. The main mechanism for this phenomenon is likely that the thermal damage causes increases in the porosity and decreases in the wave impedance of deep sandstone specimen. Therefore, less waves can be transmitted to the transmission bar through the rock specimen under the same incident wave, resulting in the decrease of dynamic strength.Figure 4
Variation in dynamic strength of deep sandstone with strain rate and temperature.


## 3. Damage Characteristic under Coupled High Temperature and Impact Loads

### 3.1. Definition Method of Coupling Damage Variable

Rock mass is cemented by a variety of mineral particles, which exists a large number of microcracks, microholes, and other defects. In addition, high temperature will inevitably lead to the generation of new cracks and the expansion of existing cracks and holes inside rock specimens. Therefore, deep sandstone specimen after high-temperature treatment is a composite damage geological material containing both thermal and native cracks, which should be considered in the definition of damage variable. In this research, the coupling damage variable is calculated based on the Lemaitre equivalent strain theory [28]. The calculation method of equivalent strain of rock specimen considering thermal damage and native damage is shown in Figure 5.Figure 5
The calculation method of equivalent strain of rock specimen.The coupling damage strain of rock specimen after high-temperature treatment under external load can be obtained as follows [29]:
(1)εc=εt+εn−ε0,where εc, εt, εn, and ε0 are the coupling damage strain, thermal damage strain, native damage strain, and no damage strain of rock specimen, respectively.From the Lemaitre equivalent strain theory,(2)σER1−Dc=σER1−Dt+σER1−Dn−σER,where σ is the stress of deep sandstone specimen; Dc, Dt, and Dn are the coupling damage variable, thermal damage variable, and native damage variable of deep sandstone specimen, respectively; and ER is the deformation modulus of deep sandstone specimen with no damage.From Equation (2), it can be obtained that
(3)Dc=1−1−Dt1−Dn1−DtDn.Previous investigation on rock damage indicates that the variation in P-wave velocity reflects the thermal-induced damage [30, 31], and the corresponding damage variable can be calculated based on the following:
(4)Dt=1−vnv02,where v0 is the P-wave velocity of deep sandstone specimen after 25°C and vn is the P-wave velocity of deep sandstone after different high temperatures.For the native damage, many theoretical and laboratory test results show that the Weibull distribution can reflect the native damage evolution process of rock under dynamic load [32, 33]; the native damage variable is defined as
(5)Dn=1−exp−ελn,where λ and n are the material parameters of deep sandstone.Therefore, the coupling damage variable can be obtained by substituting Equations (4) and (5) into Equation (3):
(6)Dc=1−vn/v02exp−ε/λn1−1−vn/v021−exp−ε/λn.


### 3.2. Damage Growth Characteristic of Deep Sandstone

From Equation (6), it can be noted that the damage growth curve of deep sandstone can be obtained after determining the parameters v0, vn, n, and λ. The values of v0 and vn can be confirmed from test results, as shown in Figure 2. In addition, parameters n and λ control the peak strain and the failure speed of dynamic stress-strain curves, which can be obtained by curve fitting in dynamic stress-strain curves. The damage growth curves of deep sandstone specimen with various temperatures are shown in Figure 6.Figure 6
Dynamic damage growth curves of deep sandstone specimen with various temperatures.
(a)
170 s-1(b)
205 s-1(c)
240 s-1Figure6 illustrates that there exists initial damage before dynamic loads in the curves, and its value increases with increasing temperature treatment. This phenomenon indicates that the damage growth curves have considered the thermal-induced damage caused by high-temperature treatment. Moreover, from the proportion of thermal damage, it can be found that the high-temperature thermal damage to deep sandstone specimen is relatively small when the temperature is less than 300°C, which has been observed and investigated in previous research for rock materials [34]. After 900°C treatment, the thermal-induced damage accounts for 55.8% of the total damage; the value shows that the thermal damage of deep sandstone specimen is great and its bearing capacity is greatly reduced after subjecting to relatively high temperature. With the increase of the deformation of deep sandstone specimen, the coupling damage of deep sandstone specimen gradually accumulates, and it presents “slow-fast-slow” three-stage characteristic. Moreover, its growth rate decreases with the increase of thermal damage under various strain rates.Figure7 displays the dynamic damage growth curves of deep sandstone specimen under different strain rates. After the same high-temperature treatment, the damage growth process of deep sandstone shows significant strain rate sensitivity. Specifically, with the increase of strain rate, the increasing rate of dynamic damage decreases gradually. This phenomenon indicates that the strain required for reaching failure state at relatively low strain rate is smaller compared with that at high strain rate. A large number of laboratory experiment date show that rock materials have obvious strain rate effect, and both the peak strain and failure strain increase with increasing strain rate [35–37], which causes the occurrence of the above phenomenon.Figure 7
Dynamic damage growth curves of deep sandstone specimen with various strain rates.


## 3.1. Definition Method of Coupling Damage Variable

Rock mass is cemented by a variety of mineral particles, which exists a large number of microcracks, microholes, and other defects. In addition, high temperature will inevitably lead to the generation of new cracks and the expansion of existing cracks and holes inside rock specimens. Therefore, deep sandstone specimen after high-temperature treatment is a composite damage geological material containing both thermal and native cracks, which should be considered in the definition of damage variable. In this research, the coupling damage variable is calculated based on the Lemaitre equivalent strain theory [28]. The calculation method of equivalent strain of rock specimen considering thermal damage and native damage is shown in Figure 5.Figure 5
The calculation method of equivalent strain of rock specimen.The coupling damage strain of rock specimen after high-temperature treatment under external load can be obtained as follows [29]:
(1)εc=εt+εn−ε0,where εc, εt, εn, and ε0 are the coupling damage strain, thermal damage strain, native damage strain, and no damage strain of rock specimen, respectively.From the Lemaitre equivalent strain theory,(2)σER1−Dc=σER1−Dt+σER1−Dn−σER,where σ is the stress of deep sandstone specimen; Dc, Dt, and Dn are the coupling damage variable, thermal damage variable, and native damage variable of deep sandstone specimen, respectively; and ER is the deformation modulus of deep sandstone specimen with no damage.From Equation (2), it can be obtained that
(3)Dc=1−1−Dt1−Dn1−DtDn.Previous investigation on rock damage indicates that the variation in P-wave velocity reflects the thermal-induced damage [30, 31], and the corresponding damage variable can be calculated based on the following:
(4)Dt=1−vnv02,where v0 is the P-wave velocity of deep sandstone specimen after 25°C and vn is the P-wave velocity of deep sandstone after different high temperatures.For the native damage, many theoretical and laboratory test results show that the Weibull distribution can reflect the native damage evolution process of rock under dynamic load [32, 33]; the native damage variable is defined as
(5)Dn=1−exp−ελn,where λ and n are the material parameters of deep sandstone.Therefore, the coupling damage variable can be obtained by substituting Equations (4) and (5) into Equation (3):
(6)Dc=1−vn/v02exp−ε/λn1−1−vn/v021−exp−ε/λn.


## 3.2. Damage Growth Characteristic of Deep Sandstone

From Equation (6), it can be noted that the damage growth curve of deep sandstone can be obtained after determining the parameters v0, vn, n, and λ. The values of v0 and vn can be confirmed from test results, as shown in Figure 2. In addition, parameters n and λ control the peak strain and the failure speed of dynamic stress-strain curves, which can be obtained by curve fitting in dynamic stress-strain curves. The damage growth curves of deep sandstone specimen with various temperatures are shown in Figure 6.Figure 6
Dynamic damage growth curves of deep sandstone specimen with various temperatures.
(a)
170 s-1(b)
205 s-1(c)
240 s-1Figure6 illustrates that there exists initial damage before dynamic loads in the curves, and its value increases with increasing temperature treatment. This phenomenon indicates that the damage growth curves have considered the thermal-induced damage caused by high-temperature treatment. Moreover, from the proportion of thermal damage, it can be found that the high-temperature thermal damage to deep sandstone specimen is relatively small when the temperature is less than 300°C, which has been observed and investigated in previous research for rock materials [34]. After 900°C treatment, the thermal-induced damage accounts for 55.8% of the total damage; the value shows that the thermal damage of deep sandstone specimen is great and its bearing capacity is greatly reduced after subjecting to relatively high temperature. With the increase of the deformation of deep sandstone specimen, the coupling damage of deep sandstone specimen gradually accumulates, and it presents “slow-fast-slow” three-stage characteristic. Moreover, its growth rate decreases with the increase of thermal damage under various strain rates.Figure7 displays the dynamic damage growth curves of deep sandstone specimen under different strain rates. After the same high-temperature treatment, the damage growth process of deep sandstone shows significant strain rate sensitivity. Specifically, with the increase of strain rate, the increasing rate of dynamic damage decreases gradually. This phenomenon indicates that the strain required for reaching failure state at relatively low strain rate is smaller compared with that at high strain rate. A large number of laboratory experiment date show that rock materials have obvious strain rate effect, and both the peak strain and failure strain increase with increasing strain rate [35–37], which causes the occurrence of the above phenomenon.Figure 7
Dynamic damage growth curves of deep sandstone specimen with various strain rates.


## 4. Dynamic Compound Damage Constitutive Model

### 4.1. Establishment of Dynamic Compound Damage Constitutive Model

The dynamic constitutive relationship of rock has always been a hot topic in rock dynamics research. Theoretical and test result indicate that the ZWT constitutive model can well reflect the dynamic strength and deformation response of rock materials at high strain rate [38], as shown in Figure 8. It includes a nonlinear spring, a Maxwell element with low frequency, and a Maxwell element with high frequency. In addition, theoretical analysis indicates that the Maxwell element with low frequency has no enough time to relax, which will degenerate into a spring. Hence, the simplified ZWT model is expressed as
(7)σ=E0ε+αε2+βε3+E2∫0tε•exp−t−τθ2dτ,where E0, α, and β are elastic constants; E1 and θ1 are the elastic constant and relaxation time of low-frequency Maxwell element, respectively; E2 and θ2 are the elastic constant and relaxation time of high-frequency Maxwell element, respectively; t is the loading time; and τ is the time during stress wave.Figure 8
The ZWT model.A large number of previous theoretical and experimental results show that at high strain rate, the damage inside rock specimens is a gradual accumulation process [21]. Therefore, the damage evolution process should be considered in the simplified ZWT model:
(8)σ=1−DcE0ε+αε2+βε3+E2∫0tε•exp−t−τθ2dτ.The dynamic damage constitutive model of deep sandstone can be obtained by substituting Equation (6) into Equation (8):
(9)σ=vn/v02exp−ε/λn1−1−vn/v021−exp−ε/λnE0ε+αε2+βε3+E2∫0tε•exp−t−τθ2dτ.


### 4.2. Verification of Dynamic Damage Compound Constitutive Model

The calculation method for parametersn and λ has been introduced in Section 3.2. The parameters of nonlinear spring (i.e., E0, α, and β) control the shape of increasing stage of dynamic stress-strain curves. In addition, the approximate scope of E2 and θ2 can be estimated by subtracting two dynamic stress-strain curves of deep sandstone specimens under various strain rates [39]. The determined parameters for dynamic constitutive model are listed in Table 2. The comparison between theoretical and test dynamic strengths is shown in Figure 9. The theoretical dynamic stress-strain curves are shown in Figure 10. The above results demonstrate that the theoretical curves can well predict the weakening effect of high temperature on deep sandstone strength. Moreover, the effects of strain rate and temperature on dynamic strengths of deep sandstone specimen can be considered in the prediction results. However, the proposed theoretical results cannot well predict the experimental data in some test conditions.Table 2
The dynamic constitutive model parameters of deep sandstone.
Strain rate (s-1)Temperature (°C)λnE0 (GPa)α (GPa)β (GPa)E2 (GPa)θ2 (s-1)170250.0061.760.3−1.22×1041.83×10611.18.2×10−41000.0081.735.6−1.13×1022.55×1057.78.5×10−43000.0091.739.2−2.85×1022.28×1058.18.7×10−45000.0101.743.2−3.99×1021.83×1057.98.2×10−47000.0111.742.2−3.87×1021.55×1058.18.3×10−49000.0121.748.6−4.92×1021.81×1057.38.4×10−4205250.0111.728.2−1.07×1021.58×1057.98.3×10−41000.0111.731.5−2.66×1022.35×1058.78.8×10−43000.0111.728.4−1.16×1021.38×1058.18.9×10−45000.0121.740.5−3.57×1021.68×1058.59.2×10−47000.0131.727.2−1.38×1026.29×1048.17.8×10−49000.0131.723.9−1.42×1025.5×1048.47.9×10−4240250.0111.762.9−6.97×1024.32×1058.37.2×10−41000.0121.754.2−5.29×1022.88×1058.38.9×10−43000.0121.745.3−4.33×1022.75×1058.18.2×10−45000.0131.756.6−4.89×1022.23×1057.88.6×10−47000.0131.734.8−2.98×1021.37×1058.18.5×10−49000.0171.714.8−2.18×1017.87×1038.28.1×10−4Figure 9
The comparison between theoretical and test dynamic strengths of deep sandstone specimen.Figure 10
The theoretical dynamic stress-strain curves of deep sandstone specimen with various temperatures.
(a)
170 s-1(b)
205 s-1(c)
240 s-1


## 4.1. Establishment of Dynamic Compound Damage Constitutive Model

The dynamic constitutive relationship of rock has always been a hot topic in rock dynamics research. Theoretical and test result indicate that the ZWT constitutive model can well reflect the dynamic strength and deformation response of rock materials at high strain rate [38], as shown in Figure 8. It includes a nonlinear spring, a Maxwell element with low frequency, and a Maxwell element with high frequency. In addition, theoretical analysis indicates that the Maxwell element with low frequency has no enough time to relax, which will degenerate into a spring. Hence, the simplified ZWT model is expressed as
(7)σ=E0ε+αε2+βε3+E2∫0tε•exp−t−τθ2dτ,where E0, α, and β are elastic constants; E1 and θ1 are the elastic constant and relaxation time of low-frequency Maxwell element, respectively; E2 and θ2 are the elastic constant and relaxation time of high-frequency Maxwell element, respectively; t is the loading time; and τ is the time during stress wave.Figure 8
The ZWT model.A large number of previous theoretical and experimental results show that at high strain rate, the damage inside rock specimens is a gradual accumulation process [21]. Therefore, the damage evolution process should be considered in the simplified ZWT model:
(8)σ=1−DcE0ε+αε2+βε3+E2∫0tε•exp−t−τθ2dτ.The dynamic damage constitutive model of deep sandstone can be obtained by substituting Equation (6) into Equation (8):
(9)σ=vn/v02exp−ε/λn1−1−vn/v021−exp−ε/λnE0ε+αε2+βε3+E2∫0tε•exp−t−τθ2dτ.


## 4.2. Verification of Dynamic Damage Compound Constitutive Model

The calculation method for parametersn and λ has been introduced in Section 3.2. The parameters of nonlinear spring (i.e., E0, α, and β) control the shape of increasing stage of dynamic stress-strain curves. In addition, the approximate scope of E2 and θ2 can be estimated by subtracting two dynamic stress-strain curves of deep sandstone specimens under various strain rates [39]. The determined parameters for dynamic constitutive model are listed in Table 2. The comparison between theoretical and test dynamic strengths is shown in Figure 9. The theoretical dynamic stress-strain curves are shown in Figure 10. The above results demonstrate that the theoretical curves can well predict the weakening effect of high temperature on deep sandstone strength. Moreover, the effects of strain rate and temperature on dynamic strengths of deep sandstone specimen can be considered in the prediction results. However, the proposed theoretical results cannot well predict the experimental data in some test conditions.Table 2
The dynamic constitutive model parameters of deep sandstone.
Strain rate (s-1)Temperature (°C)λnE0 (GPa)α (GPa)β (GPa)E2 (GPa)θ2 (s-1)170250.0061.760.3−1.22×1041.83×10611.18.2×10−41000.0081.735.6−1.13×1022.55×1057.78.5×10−43000.0091.739.2−2.85×1022.28×1058.18.7×10−45000.0101.743.2−3.99×1021.83×1057.98.2×10−47000.0111.742.2−3.87×1021.55×1058.18.3×10−49000.0121.748.6−4.92×1021.81×1057.38.4×10−4205250.0111.728.2−1.07×1021.58×1057.98.3×10−41000.0111.731.5−2.66×1022.35×1058.78.8×10−43000.0111.728.4−1.16×1021.38×1058.18.9×10−45000.0121.740.5−3.57×1021.68×1058.59.2×10−47000.0131.727.2−1.38×1026.29×1048.17.8×10−49000.0131.723.9−1.42×1025.5×1048.47.9×10−4240250.0111.762.9−6.97×1024.32×1058.37.2×10−41000.0121.754.2−5.29×1022.88×1058.38.9×10−43000.0121.745.3−4.33×1022.75×1058.18.2×10−45000.0131.756.6−4.89×1022.23×1057.88.6×10−47000.0131.734.8−2.98×1021.37×1058.18.5×10−49000.0171.714.8−2.18×1017.87×1038.28.1×10−4Figure 9
The comparison between theoretical and test dynamic strengths of deep sandstone specimen.Figure 10
The theoretical dynamic stress-strain curves of deep sandstone specimen with various temperatures.
(a)
170 s-1(b)
205 s-1(c)
240 s-1


## 5. Conclusions

(1)
With the increase of high-temperature treatment, both the P-wave velocity and the dynamic strength of deep sandstone specimen decrease gradually, while its porosity increases and the variation rate of P-wave velocity and porosity accelerate after 300°C treatment(2)
With the increase of strain, the coupling damage of deep sandstone specimen gradually accumulates and presents “slow-fast-slow” three-stage characteristic. In addition, its growth rate decreases with the increase of thermal damage under various strain rates. With the increase of strain rate, the dynamic damage increase rate of deep sandstone specimen decreases gradually(3)
The dynamic constitutive model stress-strain curves are verified to provide well description of the dynamic strength and deformation behavior of deep sandstone under different high temperatures and strain rates. Moreover, the predicted values of dynamic peak strength with various conditions are found to essentially match the trends of laboratory data



---

*Source: 9468290-2021-07-30.xml*