Dynamic Mechanical Properties of Heat-Treated Shale under Different Temperatures

Abstract

As a typical rock, shale’s reservoir depth is about 1500–4000 m, and the temperature of the shale reservoir at this depth is 150 °C. Therefore, in order to study the dynamic strength of shale at this temperature, it is necessary to consider the effects of temperature and strain rate on the dynamic strength of shale, and then establish the damage constitutive model of shale. This paper took black shale from the Sichuan Basin as the research object, combined it with the separated Hopkinson bar experiment and temperature control system, and conducted the Hopkinson bar experiment on shale at room temperature, 60 °C, 90 °C, 120 °C, and 150 °C, and at three groups of air pressures of 0.2 MPa, 0.3 MPa, and 0.4 MPa. The stress–strain curves of shale at the same strain rate and different temperature and at the same temperature and different strain rate were obtained. In the temperature difference range of this experiment, the dynamic strength of the sample presented two opposite trends (increasing and decreasing) with the increase in temperature, which was determined via the direction of the bedding. The peak strength linearly increased with the increase in strain rate. Based on the Weibull statistical distribution and the D–P failure criterion, a statistical damage constitutive model of shale dynamic strength considering the effects of temperature and strain rate was obtained. By modifying the parameters F₀ and m, the dynamic strength statistical damage constitutive model of shale was in good agreement with the experimental results.

1. Introduction

Deflagration fracturing technology [1], also known as high-energy gas fracturing (HEGF) technology, is a unique oil and gas field stimulation technology [2]. The advantages of HEGF technology applied to rocks are as follows: First, stress waves can lead to the formation of multiple radial fractures that communicate with natural fractures. High temperatures and high-pressure gases can effectively extend the fractures and increase the content of shale gas flowing into the wellbore to a large extent. Second, deflagration fracturing is suitable for fracturing stimulations of liquid-sensitive oil and gas reservoirs. Third, the fracture itself is cut, resulting in the dislocation between layers to form support, so the fracture is not easy to close; fourth, in the actual project, deflagrant compression cracking does not require too much large equipment, which can meet the requirements of economic practicality; fifth, as a waterless fracturing technology, detonation fracturing has achieved a better environmental protection effect.

HEGF generally forms a pre-fracture through perforation, and then injects a liquid drug into the perforation section to generate a large amount of high-temperature and high-pressure gas after ignition, which exerts an effect on the pre-fracture and further expands and extends it, as shown in Figure 1a. According to the physical model, the geometric model of fracture propagation in high-energy gas fracturing of horizontal wells is defined, as shown in Figure 1b. σ_H is the maximum principal stress, σ_h is the minimum principal stress, L is the reservoir length, W is the reservoir width, and P_t is the gas pressure distribution in the fracture [3].

In terms of shale mechanics research, Horsrud et al. [4,5,6] introduced a series of shale formations and analyzed their series mechanical parameters via heating. It was found that the shale has strong anisotropy and that the temperature has an obvious influence on the anisotropy, porosity, and permeability of the shale. The heating of the shale significantly reduces the strength, stirring speed, and sound velocity of the shale, and tests from other relevant scientific procedures have concluded that the shale may cause changes in the interlayer structure due to pressure and temperature unloading. Based on this, a physics-based micromechanical model was created in order to quantify the damaged state of shale under loading through macroscopic measurements. Niandou et al. [7,8] found that the volume strain of shale mainly comes from compression deformation through the experiments of compressibility and triaxial compression tests of shale under different confining pressures and loads. The transition to dilatancy only occurs in the region close to the peak stress. Increasing the magnitude of the isotropic stress field has little effect on the velocity anisotropy. Luo et al. [9] studied shale with different bedding (0°, 30°, 45°, 60°, and 90°) and conducted a study through the triaxial SHPB test with a confining pressure of 0~15 MPa. It was found that the dynamic mechanical properties of shale mainly depend on the interaction between bedding dip angle and various confining pressures. Huang et al. [10] used the split Hopkinson pressure bar test to study the dynamic compressive mechanical response of shale specimens with five different bed dip angles after heat treatment at 25–200 °C. It was found that with the increase in heat treatment temperature, the failure mode of samples with bedding dip angles of 45° and 60° were significantly affected, but the bedding dip angles of 0°, 30, and 90° were not fundamentally changed.

Since hydraulic fracturing technology is mostly used in the exploitation of shale gas abroad, research on the mechanical properties of shale is mainly focused on static research, and the exploration on the dynamic mechanical behavior of shale is relatively rare. Although our country started late in shale research, some substantial progress has been made. Dong et al. [11,12,13,14] conducted uniaxial and triaxial compression tests on shale, analyzed the mechanical properties, strength, and failure mode of shale, and revealed the anisotropic failure mechanism of shale. Kang et al. [15] divided the pyrolysis weight loss of oil shale into three stages based on the new technology of in situ thermal injection mining of oil shale. When the temperature exceeds 200 °C, the influence of temperature on bulk density, specific gravity, porosity, etc., increases. When the temperature exceeds 300 °C, there is a significant increase in the number, length, and width of shale fractures. At the same time, microcracks are formed in the direction perpendicular to the bedding plane. Deng [16] et al. studied the anisotropy of shale generated by layered waves and gave the P-wave and S-wave velocities of samples in different directions under different conditions. Wang Yi [17] found that fracturing fluid intrudes into rock mass along with fractures, which is the main reason for the decline in shale strength. Liu et al. [18,19] carried out uniaxial compression mechanical experiments of shale under four kinds of low strain rates and analyzed the influence of shale strain rates on the relevant mechanical parameters of shale. Zhang [20] conducted the separation Hopkinson bar compression experiment for the high strain rate of shale, and preliminarily explored the dynamic constitutive relationship of shale. Yang et al. [21] used the separated Hopkinson pressure bar system to carry out impact tests on shale. It was concluded that the damage degree of shale significantly increases with the increase in impact velocity under different strain rates, and that the ductility and failure resistance of rock materials are improved under passive confining pressure.

Crack initiation and its effects on the dynamic behavior of rock, particularly regarding ultrasonic wave velocity, have been studied extensively in recent years. Numerical manifold method (NMM) simulations have been used to investigate the propagation of stress waves through the fractured rock mass [22]. A study by Fan [23] demonstrated that the stress wave velocity is affected by the size and orientation of the fractures, as well as the properties of the surrounding rock’s matrix. Micro-fractures have also been found to impact the dynamic behavior of rocks significantly. A study by Zhou [24] on wave propagation through rock mass with micro-fractures found that the velocity and amplitude of the waves are affected by the orientation and distribution of the micro-fractures. Thermal shock-induced damage is another factor that can affect the dynamic behavior of rocks. The spatial gradient distribution of damage to granite induced via thermal shock has been studied. It was found that the thermal shock-induced damage significantly affects the ultrasonic wave velocity and attenuation [25]. Finally, experimental investigations of thermal effects on the dynamic behavior of granite have demonstrated that changes in temperature can cause significant changes in the ultrasonic wave velocity, attenuation, and dispersion [26]. These studies have contributed to a better understanding of the effects of crack initiation and thermal shock-induced damage on the dynamic behavior of rocks. Different material grains have different anisotropy and thermophysical properties, which leads to incongruous deformation across grain borders and thermal stress between or within grains [27]. When the thermal stress is greater than the material’s strength, cracking takes place, causing a lot of cracks and a corresponding drop in strength. However, the mechanical properties of various rocks vary with temperature as a result of variations in mineral composition, particle size, and microstructure. Research has shown that Yangbajing geothermal wells in China and larderello geothermal wells in Italy both have temperatures above 300 °C [28]. As a result, the issue of rock engineering under high-temperature environments has emerged as a key area for research in the field of rock mechanics [29].

The physical and mechanical properties of shale exhibit significant anisotropy, impacting the design and construction of high geothermal tunnels and deep resource mining. The research on the stress–strain relationship of shale under different conditions and different strain rates is not perfect domestically and abroad. Therefore, this paper took shale from the Sichuan Basin as the research object and conducted Hopkinson bar compression experiments at normal temperature, 60 °C, 90 °C, 120 °C, and 150 °C, and at three groups of air pressures of 0.2 MPa, 0.3 MPa, and 0.4 MPa. The differences and similarities of the stress–strain effects of shale under the same strain rate with different temperatures, and the same temperature with different strain rates, were analyzed in depth. Based on the experimental results, a constitutive model considering shale temperature was established. The research results can provide an important theoretical basis for the strength, fracture, and deformation theory of shale. The research process is shown in Figure 2.

2. Preparation of Shale Specimen

2.1. Sampling and Processing

The samples were taken from Liutang Township, Shizhu County, Chongqing, on the edge of the Sichuan Basin, which can better represent the rock profiles of black shale. Engineering bricks with an inner diameter of 56 mm were used for core sampling and a core length of 150 mm. The cores were quickly sealed with cling film and kraft paper and sampled and numbered after being taken out for use and subsequent processing later. The sampled shale is shown in Figure 3. The rough sampling method described above leads to various defects in the cores and therefore requires secondary machining to meet the strict requirements of the SHPB test that the specimen must be in close contact with the incident bar and transmission bar. The finishing process ensured that the final experimental specimens have a more uniform length, a flatter section, good parallelism, and no bending of the specimens. The processing apparatus used was a C6132 machine Ltd. (Tengzhou, China) [30] of Yunnan Yunji Group Import & Export Co., Ltd. (Yuxi City, China) to finely process the sample. The cores were machined into standard rock samples with a height of 50 mm and a diameter of 50 mm, and the tolerance of dimensions, parallelism, and diameter were ± 0.02 mm.

2.2. Shale CT Scanning and Material Composition

CT scanning technology has been widely used in industry, including X-ray tomography, Compton scattering tomography, and Mossbauer effect tomography, which are mainly used to detect the interior of industrial products. In this paper, X-ray scanning technology was employed in the CT scanning experiments, which has the characteristics of high spatial resolution, non-destructive detection, thin scanning layer, good image quality, and fast speed, which can better reconstruct the cross-section of the shale sample and detect the various defects, such as cracks, delamination, and impurities, in the samples.

Five groups of CT scans of the shale were analyzed. A comparison of the original shale samples and CT results is shown in Figure 4, showing the original and scanned images of samples 1# to 5#. The 1# original sample has obvious impurities and small radiating cracks, which could be seen clearly through the CT scan. No cracks were observed in the original sample of 2#, but the CT scanning results showed that the cracks were located to the right of the center of the shale circle and did not penetrate the sample. The original samples of 3# and 4# were observed via visual and CT scans to have distinct joints’ planes, and the joints’ planes were roughly parallel. The 5# sample was relatively dense overall, with little tramp mass. Based on the CT scan results of the shale, there was a high probability of defects, such as joints and fractures, in the shale itself, but the shale was relatively dense throughout. It can be inferred that the existence of defects in the shale itself has a certain impact on its mechanical properties.

The structure of natural shale is complex, and the composition and content of the material directly affect the mechanical properties of the shale. The shales were quantitatively and qualitatively analyzed via XRD diffraction tests. The composition of each substance is outlined in

Table 1. Material composition of shale.

Number	Geological Age	Clay	Quartz	Plagioclase	Calcite	Dolomite	Pyrite	Other	Clay Mineral	Non-Clay Minerals	Lithology
Dz01	S11	37	46	1	-	-	-	1	37	63	Gray shale
Dz02	S1s	43	37	7	3	2	3	1	43	57	Gray calcareous shale
Dz03	S1s	43	22	7	1	2	1	1	43	57	Gray calcareous shale
Lt01	S11	32	56	3	2	6	2	-	32	68	Black shale
Lt02	S11	38	45	1	7	1	7	1	38	62	Black shale
Lt03	O3w	33	46	-	1	16	1	1	33	67	Linxiang Formation of Upper Ordovician
Ltc1	S11	31	51	5	2	-	2	6	31	69	Black shale

[31].

3. Splitting and Stretching Test

The static splitting test of three kinds of loading angle specimens was carried out using an electronic universal testing machine and a gray-dot camera. The main crack propagation, secondary crack generation, and final rupture of the specimens were different during the bad form loading process.

Figure 5 shows the final failure diagram of the specimen with a loading angle of 0°. The test starts from the initial crack tip during the loading process, and gradually expands to form a “one” shape main crack with the increase in the load. The principal crack expansion morphology of the three specimens was similar. With the increase in load in the initial crack during loading, stress concentration occurs in the crack tip region. When the initial crack tip reaches the condition of cracking, the initial crack tip begins to crack and extends to the upper and lower ends of the loading along the diameter loading direction, forming a through principal crack that divides the specimen into two parts. Due to the Poisson effect in the loading process of the specimen, when the effective tensile stress exceeds the tensile strength of the specimen, the splitting tensile failure occurs along the diameter of the loading plane. Figure 5 shows the final failure diagram of the specimen with a loading angle of 0°.

Table 2. Splitting tensile strength.

Specimen Number	Failure Load Value (kN)	Splitting Strength (MPa)	Average Value (MPa)
H10	39.99	18.76	18.36
H11	39.91	18.72
H12	37.56	17.62

shows the splitting tensile strength of shale.

4. Shale SHPB Experiment

At present, a split Hopkinson pressure bar is commonly used to test the dynamic characteristics of brittle materials, such as concrete and ceramics [32,33,34]. The impact experiment of the shale was carried out using a SHPB device with a diameter of 50 mm. The test was carried out on the split Hopkinson experimental system in the structural dynamics laboratory of Southwest University of Science and Technology. It is mainly composed of a bullet, an incident bar, a specimen, a transmission bar, an absorption bar, a measuring device, and a data processing system. The schematic diagram of the experimental device is shown in Figure 6, where the lengths of the bullet, incident bar, and transmission bar are 600 mm, 3000 mm, and 3000 mm, respectively. Each bar has a diameter of 50 mm and is made of a steel alloy with a modulus of elasticity (E) of 72 gPa, a yield strength of 300 MPa, and an ultimate strength of 460 MPa. The shale specimen was placed between the incident bar and the transfer bar, where the contact surfaces are parallel. Strain gauges were installed on the incident bar and the transmission bar to collect waveform data. The actual experiment is shown in Figure 7. The position of the strain gauge was two times the length of the bullet from the incident end face. The strain gauge of the incident bar and the strain gauge of the transmission bar were symmetrical with respect to the specimen.

The pulse-shaping technique was used in SHPB testing to promote dynamic stress equilibrium in the sample to reduce the effects of the heterogeneity of the shale material and the effects of high-frequency impacts, which can improve the stress homogeneity of the sample. In this paper, by comparing the effects of different materials as forming sheets, an appropriate material was selected so that the rise time of the incident wave was 300 µs or more, i.e., the stress of the specimen was much less than the peak stress when the stress was uniform. It can be assumed that the experimental data obtained achieve a constant strain rate loading process of shale materials and satisfy the assumption of stress uniformity.

In the process of the waveform shaping experiment, according to the experience of predecessors [35], butter, copper, brass, rubber, and other materials were added to the front end of the incident bar to shape the curves. The circular sheets with a diameter of 15 mm and a thickness of 2 mm were prepared, and the SHPB empty bar experiment was completed, and the incident waveform diagrams of the different materials are shown in Figure 8.

According to the experiment, the loading time reached 400 ms when rubber was added as the shaping piece, and the waveform was similar to a sine wave. After adding the rubber shaping sheet, high-frequency impacts were effectively avoided. After analyzing the incident pulses of shaping pieces of different materials, rubber was selected as the shaping piece in this experiment, and its average loading time reached more than 300 ms. By calculating the stress balance time required by the shale, it can be seen that the stress balance time of the test pieces is far less than the time required to reach the peak stress. Therefore, the selected rubber shaping sheet can sufficiently meet the requirements of stress balance in the later analysis.

At present, there are few SHPB compression experiments on shale with temperature. High-temperature dynamic mechanical experiments have been proposed much earlier. As research on the dynamic mechanical properties of brittle materials has increased, SHPB experiments on concrete and rock at high temperatures have gradually increased, but SHPB experiments on shale with temperature are very rare. Previous research has indicated that the shale reservoir in the Sichuan Basin is approximately 1500–4000 m, and that the temperature of the shale reservoir reaches 150 °C at this depth. Therefore, several groups of tests at different temperatures have been designed to investigate the dynamic mechanical properties of the shale. The shale heating device is equipped with the SHPB experimental procedure described above. The heating furnace and temperature controller are shown in Figure 9.

In the SHPB experiment on the shale with temperature, the temperature of the equipment was first adjusted to the specified temperature. When the temperature reached the specified temperature, the high-temperature device was held at the specified temperature for one hour before testing to ensure that the internal temperature of the shale also reached the specified temperature, which was closer to the actual environment of the shale.

5. Analysis of the Experimental Results

5.1. Stress Balance Check and Typical Failure Modes of Shale Specimens

In shale SHPB compression bar tests, it is an effective measure to ensure the validity of the experimental results by verifying that a state of stress equilibrium is reached before the shale is damaged. Formula (1) can be used to verify whether the stress of the SHPB test meets the balance requirement:

$$W_i+W_r ={ W}_t$$

(1)

where $W_i, W_r$, and $W_t$ represent the incident stress wave, reflected stress wave, and transmitted stress wave, respectively. When the data were processed, the corresponding incidence curve, reflection curve, and transmission curve were used to replace the calibration, and the results are shown in Figure 10.

As can be seen from Figure 10, the stress balance at both ends of the test specimen was well achieved at different impact pressures corresponding to the three shale failure modes, i.e., this test can meet the stress balance requirements at different strain rates. However, this experiment requires a high level of accuracy, and the stress equilibrium state of the specimen needs to be verified each time during the subsequent data processing stage. The results that reach the equilibrium state can be used, while the failed data can be abandoned, to ensure the effectiveness of the dynamic mechanical analysis of shale materials.

The failure mode of shale specimens at different strain rates were obtained from the tests by adjusting the experimental air pressure. When the pressure was changed, the impact velocity of the bullet changed, resulting in a change in the test strain rate and, thus, a significant difference in the final damage pattern of the shale specimens. In the process of loading, a large number of small cracks will appear in shale due to the increasing load, and small cracks will form large cracks. Shale is a kind of highly brittle rock mass, and three different failure modes are produced with different impact velocities. As shown in Figure 11, the damage degree of the shale under different impact velocities and different strain rates can be divided into a simple fraction. Based on the reference to the concrete damage modes, the shale can also be divided into three damage forms: “cracking and spalling”, “fracturing and crushing” [36]. Through a large number of experiments, waveforms that are representative of the shale damage patterns were selected for analysis to verify the validity of the experiments. The pressures were 0.2 MPa, 0.3 MPa, and 0.4 MPa, and the velocities were 4.025 m/s, 6.126 m/s, and 8.219 m/s, respectively. The corresponding average strain rates were 44.62/s, 73.6/s, and 104.07/s, respectively.

When the impact pressure of the test was 0.2 MPa, the bullet velocity was 4.025 m/s, and the average strain rate was 44.62/s. As shown in Figure 10a, the specimen is slightly spalled and cracked locally under the impact pressure, but the specimen is not damaged. This phenomenon is called “cracking and spalling”. When the impact pressure was 0.3 MPa, as shown in Figure 10b, the average strain rate was 73.6/s, and the whole specimen was divided into several large pieces, which can be considered damaged; we called this “rupture”. When the impact pressure was 0.4 MPa, the average strain rate reached 104.07/s, as shown in Figure 10c; the specimen was completely crushed under the impact pressure, and the main body of the specimen was completely divided into powder and blocks. This kind of phenomenon is called “crushing”. Therefore, it can be concluded that the damage degree of the shale intensifies with the increase in impact pressure.

5.2. Stress–Strain Curve of Shale Material at Room Temperature

The validity of the experimental data was verified through a large number of SHPB experimental tests, and the stress–strain curves of the shale at room temperature were obtained, as shown in Figure 12.

Due to the difference in strain rates, and the influence of internal voids, fractures, joints, and other defects, as well as their discontinuity, anisotropy, and heterogeneity, the stress–strain curves of shale are quite different, but the types of stress–strain curves are generally the same under the strain rates, with little difference in their values. The effects of internal defects, discontinuities, anisotropy, and heterogeneity of the shale on the shale SHPB bar compression experiments decrease with increasing strain rates, and the stress–strain curves of the shale are similar. The shale stress–strain curves begin to show strong characteristics of linear elastic. When the strain rate is low, the corresponding damage mode indicates that the shale is only locally spalled, which does not affect the overall damage. Therefore, the unloading section of the shale can still bear a degree of compressive pressure. However, the high strain rate shows two modes of damage, which are rupture and fragmentation. In this case, the shale has shown overall damage, and the compressive strength of the shale drops sharply to zero. Consistent with the findings of Abbas et al. [22], the destruction process can be divided into four stages. The first stage: when the stress gradually increases with the strain, the load seat and the natural micro-crack close. The second stage: with further compression, the shale gradually densifies, compacts, or consolidates. The third stage: the deformation of shale can be widely recovered in the elastic compression and micro-crack stage. The fourth stage: the accelerated micro-crack or macro-crack stage, whereby the stress–strain curve basically deviates from the linear elastic deformation until it reaches its peak stress and failure. The peak strength of shale increases significantly with an increase in strain rate, indicating that shale is a brittle material with a strain rate effect.

5.3. Dynamic Compressive Strength Analysis of Shale

The ability of a material to resist failure under impact loading is called dynamic compressive strength. The maximum stress (peak stress) in the stress–strain curve, i.e., when the material is subjected to an impact load, is usually taken as the strength value.

The relationship between the dynamic compressive strength of shale and the strain rate is shown in Figure 13 by integrating the relationship between the peak strength and the average strain rate of the shale at normal temperature.

Concrete materials are sensitive to strain rates, and it is an important task to study the dynamic properties of concrete. Current studies generally agree that the dynamic tensile and compressive strength of concrete increases with increasing strain rates [37,38,39]. According to the peak strength of the shale under different strain rates, it can be found that the dynamic compressive strength of the shale has a strong strain rate effect, especially when the strain rates are between 10/s and 100/s; moreover, the peak strength of shale presents a linearly increasing trend with the increase in strain rates, which is similar to that of concrete materials, indicating that shale is also a rate-sensitive material.

5.4. Analysis of Shale under the Same Temperature and Different Strain Rates

Figure 14a–d show the stress–strain curves of shale under the different strain rates at 60 °C, 90 °C, 120 °C, and 150 °C, respectively. It can be seen that the peak stress and ultimate strain of the specimens increase with the increase in strain rates at the same temperature condition, showing the correlation. This phenomenon is called the strengthening effect of strain rate on the impact compression mechanical properties of shale materials.

However, due to the limited number of specimens, the maximum strain rate of shale at these temperatures did not exceed 100/s, and the stress–strain curves for high strain rates of shale at these temperatures were not able to be obtained, meaning that the experimental results are somewhat limited.

5.5. Analysis of Shale under the Same Pressure and Different Temperatures

Figure 15 shows the stress–strain curves of shale specimens under the same pressure at different temperatures. It can be seen from

Table 3. Values of each group at 0.2 MPa.

Temperature	60 °C	90 °C	120 °C	150 °C
Peak stress	112 MPa	121 MPa	123 MPa	118 MPa
Ultimate strain	0.0038	0.0033	0.0042	0.0075

and

Table 4. Values of each group at 0.3 MPa.

Temperature	60 °C	90 °C	120 °C	150 °C
Peak stress	151 MPa	109 MPa	146 MPa	115 MPa
Ultimate strain	0.0073	0.0068	0.0061	0.0056

that under the same impact pressure condition, within the temperature difference range of this experiment, the dynamic strength of the specimen presents two opposite trends (increase and decrease) with the increase in temperature, which is determined by the direction of bedding [10]. As shown in Figure 14a, when the strain rates are close, the strain of shale will increase with the increase in temperature, especially when the temperature is 150 °C, and the corresponding strain value is much larger than that at 60 °C, 90 °C, and 120 °C. This indicates that shale will soften when the temperature exceeds 150 °C, resulting in an increase in strain, with the corresponding elastic modulus and hardness decreasing. Through calculation, it can be found that the elastic modulus of shale at 150 °C is 2–3 times smaller than that at 60 °C, 90 °C, and 120 °C, indicating that temperature will greatly affect the elastic modulus of shale and, thus, affect the hardness of shale.

6. Dynamic Damage Constitutive Model

6.1. Definition of Damage Variable

Krajcinovic and Silva [40] started with the random distribution of flaws within the rock material and used the rock’s microelement strength to follow the Weibull distribution to build the statistical damage constitutive equation of the rock fracture process. A granite thermal–mechanical coupling damage model was developed by Xu et al. [41] using the Drucker–Prager criterion and the Weibull distribution. Zhu et al. [42] developed a statistical damage model using the modified Mohr–Coulomb criterion as the failure condition for the microelement strength. An enhanced statistical damage constitutive model of rock thermal treatment was put out by Wang et al. [43] and is based on the longitudinal velocity correction of the thermal damage variable.

As a typical non-homogeneous material, shale contains a large number of randomly distributed microcracks, pores, and other defects. Assuming that shale consists of many microbodies, whose dimensions are large enough in a spatial sense to contain many microdefects but small enough in a mechanical sense to be considered as particles, the microbodies have the following properties: (1) The shale is isotropic, i.e., having an isotropic-damaged body. (2) The shale conforms to Hooke’s law before failure but has no bearing capacity after failure. (3) The strength of each microbody follows the Weibull distribution. The expression of its probability density function is as follows [44]:

$$P\left(F\right)=\frac{m}{F_0}{\left(\frac{F}{F_0}\right)}^{m-1}\exp \left[-{\left(\frac{F}{F_0}\right)}^m\right]$$

(2)

where $F$ is the distribution variable of micro-body strength, and $F_0$ and $m$ are Weibull distribution parameters, reflecting the mechanical properties of shale materials.

When shale is subjected to impact loading, microelements are gradually damaged, and a statistical damage variable, $D$ can be introduced to represent this process. Under a certain load, $D$ can be expressed as:

$$D=\frac{N^f}{N}$$

(3)

where $N$ is the total number of microelements, and $N_f$ is the number of destroyed microelements. When loading to a specific load level, $F$:

$$N_f={\int }_0^{F}NP\left(x\right)dx$$

(4)

By substituting Equations (2) and (4) into Equation (3), the expression of the damage variable, $D$ can be obtained as:

$$D=1-\exp \left[{\left(-\frac{F}{F_0}\right)}^m\right]$$

(5)

6.2. Micro-Body Strength

According to Equation (5), the damage variable, $D$ is affected by the micro-body strength, $F$ which is related to the stress state of the rock. To express the effect of complex stress states on shale, the failure criterion of the microelement is considered to be consistent with the Drucker-Prager (D-P) failure criterion. The microbody strength, $F$ based on the D-P failure criterion is as follows [40]:

$$F=\alpha I_1+\sqrt{J_2}$$

(6)

$$\begin{gathered} \alpha = \frac{\sin \phi }{\sqrt{9+3{\sin }^2\phi }} \\ {I}_1 = \frac{({\sigma }_1+2{\sigma }_3)E{\epsilon }_1}{{\sigma }_1-2\mu {\sigma }_3} \\ \sqrt{J_2} = \frac{({\sigma }_1-{\sigma }_3)E{\epsilon }_1}{\sqrt{3}({\sigma }_1-2\mu {\sigma }_3)} \end{gathered}$$

(7)

where $\alpha$ is the strength parameter of the microelement, and $\phi$ is the internal friction angle of the rock. The internal friction angle of the shale selected in this paper was 28°, and the value of $\alpha$ was 0.1268. $I_1$ is the first invariant of the stress tensor, and $J_2$ is the second invariant of the stress deviation. Substituting Equation (7) into Equation (6), considering the uniaxial state of the shale in this paper, so ${\sigma }_2={\sigma }_3=0$ and ${\epsilon }_1=\epsilon$, then the expression of microelement strength is:

$$F=\left(\alpha +\frac{1}{\sqrt{3}}\right)E\epsilon$$

(8)

6.3. Establishment and Modification of the Constitutive Model

Using the strain equivalence assumption, the expressions of the statistical damage intrinsic model of the intensity of the uniaxial shale under impact loading are obtained as follows:

$$\sigma =E\epsilon \left(1-D\right)=E\epsilon \exp \left[-{\left(\frac{F}{F_0}\right)}^m\right]$$

(9)

According to Equation (9), the key to establish the constitutive model is to determine the two unknown parameters $F_0$ and $m$. Considering that the peak stress and corresponding strain of the stress–strain curves are easily obtained in the uniaxial impact test, this paper used the extremum method to determine the parameters $F_0$ and $m$. Since the derivative of the multivariate function at the extreme value point (${\epsilon }_m$,${\sigma }_{\max }$) is 0, both sides of Equations (8) and (9) are derived and simplified simultaneously as follows:

$$F_0=\left(\alpha +\frac{1}{\sqrt{3}}\right)E{\epsilon }_m{m}^{\frac{1}{m}}$$

(10)

$$m=-\frac{1}{\ln \left[{\sigma }_{\max }/\left(E{\epsilon }_m\right)\right]}$$

(11)

The constitutive model parameters calculated using Equations (10) and (11) are shown in

Table 5. Results of the constitutive model parameter calculation.

Temperature	Strain Rate (s−1)	${\mathit{F}}_0$ (MPa)	$\mathit{m}$
60 °C	20.63	141.767	4.902
90 °C	19.33	153.223	5.599
120 °C	20.33	144.413	4.878
150 °C	23.94	112.608	12.565

:

The relationship between $F_0$,$m$ and the strain rate can be established, which can effectively modify the failure statistical constitutive model of shale. Scatter plots were drawn with $F_0$ and $m$ as the ordinates and strain rate as the abscissa, respectively, and nonlinear fitting was carried out, as shown in Figure 16 and Figure 17.

The fit shows the following relationship between $F_0$,$m$ and the strain rate:

$$F_0=-1.70563{\stackrel{•}{\epsilon }}^2+65.6271\stackrel{•}{\epsilon }-482.51175$$

(12)

$$m=0.61848{\stackrel{•}{\epsilon }}^2-25.25049\stackrel{•}{\epsilon }+262.5967$$

(13)

The fitting correlation coefficients were 0.960 and 0.989, respectively. Equations (12) and (13) were substituted into Equation (9) to obtain the modified equation of the statistical intensity constitutive damage model of the shale under different strain rates:

$$\sigma =E\epsilon \exp \left[{\left(\frac{0.728E\epsilon }{-1.70563\stackrel{}{{\stackrel{•}{\epsilon }}^2+65.6271\stackrel{}{\stackrel{•}{\epsilon }-482.51175}}}\right)}^{0.61848\stackrel{•}{{\epsilon }^2}-25.25049\stackrel{•}{\epsilon }+262.5967}\right]$$

(14)

6.4. Verification of the Constitutive Model

The theoretical dynamic stress–strain curves with close strain rates of the shale were calculated via Equation (14) using the modified constitutive model, and the compared results with the experimental curves are shown in Figure 18.

From the model validation results, the theoretical curves are consistent with the experimental curves at the same strain rate. Based on the effect of the Weibull distribution parameters $F_0$ and $m$, a reasonable correction of the theoretical curve can better reflect the insensitivity of the peak stress of the shale to temperature. The modulus of elasticity decreases with increasing temperature. Notably, the shale modulus of elasticity significantly decreases at 150 °C. Of course, the model established in this paper cannot fully express the changing process of the dynamic stress–strain curves of the shale at each loading stage, and there are some limitations due to the limited data available. The errors caused via the determination of the model parameters and the choice of elastic modulus need to be further improved in future studies.

7. Conclusions

Through the shale SHPB compression bar test, it was found that the shale showed three typical damage forms of “cracking and spalling”, “breaking and crushing” at different impact velocities, and the degree of damage increases with the increase in impact pressure.

The Hopkinson bar compression tests of shale at room temperature, 60 °C, 90 °C, 120 °C, and 150 °C, and at 0.2 MPa, 0.3 MPa, and 0.4 MPa were carried out. The stress–strain curves of shale at the same strain rate with different temperatures and the same temperature with different strain rates were obtained. Under the same impact pressure condition, in the temperature difference range of this experiment, the dynamic strength of the specimen presents two opposite trends (increase and decrease) with the increase in temperature, which was determined via the direction of bedding.

Based on the Weibull statistical distribution and the D-P failure criterion, a statistical damage constitutive model of the shale dynamic strength considering temperature and strain rate effects was obtained. By modifying the parameters $F_0$ and $m$, the established dynamic statistical intensity damage constitutive model of the shale is in good agreement with the experimental results.