Experiments on the Coal Measures Sandstone’s Dynamic Mechanical Parameter Characteristics under the Combined Action of Temperature and Dynamic Load

Abstract

This paper conducts impact dynamics experiments on coal measures sandstone in a deep mine via the established dynamic load and temperature split Hopkinson pressure bar (SHPB) experimental system. Firstly, the experimental conditions for the impact dynamics of fine sandstone were determined, with temperatures of 18, 40, 60, 80, and 100 °C, an axial static load range of 1–9 MPa, and a preset bullet incidence velocity of 1.0–5.0 m/s. Secondly, based on the analysis of the basic parameters and physical composition, the dynamic stress and strain responses of fine sandstone under different experimental conditions were obtained, and the change mechanism of its dynamic mechanical process was theoretically analyzed. When the temperature rose from 18 °C to 100 °C, the dynamic peak stress of fine sandstone increased from 36.04 MPa to 73.41 MPa, with an increase of 103.7%. At a temperature of 40 °C, when the axial static load increased from 1 MPa to 9 MPa, the dynamic stress peak of fine sandstone increased from 57.25 MPa to 80.01 MPa, and the corresponding peak strain also showed an increasing trend. The experiment analyzed the variation characteristics of dynamic stress in fine sandstone under the combined action of different strain rates or bullet incidence velocities and different temperatures. In the strain rate range of 47.1 s⁻¹ to 140.9 s⁻¹, there was a significant strain rate effect on the dynamic peak stress and peak strain of fine sandstone, which increased with the increase of strain rate. The study found a polynomial relationship between the dynamic mechanical parameters of fine sandstone and the impact of experimental parameters, with a coefficient of determination greater than 0.9. A dynamic stress constitutive model for fine sandstone under one-dimensional stress state under dynamic load and temperature action was established, and the model validation and parameter determination of dynamic stress changes in fine sandstone under different temperature conditions were carried out. The research results provide a new experimental method for the static and dynamic mechanical analysis of coal and rock masses under complex geological conditions and can provide a basic reference for the prevention and control of dynamic disasters in deep mining processes.

1. Introduction

The deep mine’s coal and rock mechanical environment is intricate, and rock shows different mechanical properties from the shallow part [1,2]. During the exploitation and utilization of coal resources, there are many rock dynamic problems in the surrounding rock system of the mine, such as roof caving, deformation and fracture, and even rock burst disasters, which seriously affect the safety production of the coal mine. For example, in the Huainan mining area of China, the geothermal gradient reaches 3.2 °C/100 m, and when the mining depth of some mines reaches 1000 m, the mine temperature can reach 48 °C [3]. Therefore, the influence of temperature and dynamic and static load coupling should be considered in the study of rock dynamics in deep mines.

Taking temperature into account when studying the dynamic behavior of rocks, Xin et al. [4] first carried out impact experiments of sandstone by using an explosive plane wave generator combined with a high-speed rotating mirror camera light gap method. The dynamic tensile characteristics of marble were investigated by Liu et al. [5,6,7] in relation to various high temperature conditions and impact loading rates. Marble will have its dynamic tensile strength increased by a high temperature chilling procedure. Marble’s dynamic tensile strength increases initially and then falls with increasing temperature when the loading rate is sustained. Xu et al. [8] studied the correlation between marble’s dynamic mechanical parameters and loading speed at different temperatures. Peak strain and stress in the marble exhibit clear loading rate impacts, but after the temperature reaches 800 °C, the loading rate and the peak stress becomes insignificant. Yin et al. [9,10,11] studied the sandstone’s physical and mechanical parameter characteristics after cooling at temperatures of 25, 200, 400, 600, and 800 °C. The increase in temperature leads to a gradual decrease in the sandstone’s density, longitudinal wave velocity, and peak stress, and the decrease in its longitudinal wave velocity increases after 200 °C. Li et al. [12] experimentally studied the effect of different strain rates on the mechanical properties of sandstone after high temperature cooling at 800 °C. Peak strain decreases logarithmically with increasing strain rate, yet the dynamic elastic modulus and peak stress of sandstone rise logarithmically. Lu et al. [13] studied the dynamic mechanical parameters of granite at different temperatures. There are clear strain rate impacts for granite’s dynamic mechanical properties, including dynamic compressible strength and peak strain. The temperature boundary points are 300 °C and 500 °C, and there is little change in the dynamic mechanical parameters from room temperature to 300 °C. When the temperature rises to 500 °C, the dynamic mechanical parameters of granite significantly decrease. Wong et al. [14] studied the marble’s mechanical parameter changes under dynamic load at 250, 500, and 750 °C. The increase in temperature causes a decrease in the marble’s dynamic strength, mainly due to the increase in density and length of microcracks inside the sample caused by thermal stress. Wang et al. [15] studied the effect of temperature and strain rate on the dynamic mechanical properties of granite. The influence of temperature on the dynamic strength and elastic modulus of granite is relatively complex. Additionally, the strain rate and granite’s dynamic compressive strength have a favorable correlation at high temperatures. Ping et al. [16] analyzed the sandstone’s mechanical properties under different loading rates and six temperature conditions (25, 200, 400, 600, 800, 1000 °C). The correlation between dynamic peak stress and peak strain and loading rate is positively correlated under the same temperature conditions, while the correlation between dynamic elastic modulus and temperature is opposite. Shi et al. [17] carried out experiments and numerical simulation analysis on the dynamic mechanical properties of granite at different temperatures (20, 100, 200 °C). Rock has a maximum dynamic compressive strength of 20 °C, and at that temperature, the loading rate increases, increasing peak stress and peak strain. Zhang [18] experimentally studied the sandstone’s dynamic mechanical properties under hydro-thermal coupling in an air-dried state and at different temperatures. Sandstone experiences an increase in peak strain and a decrease in peak stress as a result of temperature increases. Zou et al. [19] analyzed the dynamic parameter variation characteristics of siltstone under the combined action of thermal and hydraulic forces. Axial pressure, osmotic water pressure, and temperature increases cause peak stress and strain to rise, brittleness to decrease, and ductility to increase. An et al. [20] studied the changes in dynamic mechanical parameters of marble under different impact rates at temperatures ranging from 25 to 800 °C. Marble’s dynamic peak strength decreases with increasing temperature, whereas its peak strain increases at the same impact rate. Li et al. [21] studied different cooling methods effects on preheated granite’s dynamic mechanical properties. As the preheated temperature and cooling rate increased and the size and quantity of internal pores increased, the dynamic compressive strength declined. Zhao et al. [22] experimentally studied the physical and mechanical properties of preheated granite at 100, 200, and 300 °C after rapid cooling in water at 0, 20, and 60 °C. Qi et al. [23] studied marble’s dynamic stress, static stress, and strain properties after heating and cooling at six temperatures (25, 100, 200, 400, 600, and 800 °C). The peak strain under both static and dynamic load increases with temperature, with the peak strain under dynamic load typically being greater than that under static load. Under static load, the peak stress decreased by 58.3% with an increase in temperature, and under dynamic load, the peak stress decreased by 80.8%. Li et al. [24] analyzed the influence granite’s dynamic tensile mechanical properties under different cooling rates at temperatures (200, 400, and 600 °C), and that granite’s dynamic tensile strength decreases as heating temperature and cooling rate rise. Zhou et al. [25] experimentally studied the mechanical parameters of granite at different impact speeds at nine temperature levels (25–800 °C). Dynamic peak stress increases with increasing impact speeds under the same temperature conditions, and the peak stress increase gradient of preheated samples undergoes a change in the same impact speeds. Zhang [26] studied the impact tensile mechanical behavior of coal measures sandstone under thermal damage (25, 100, 200, 400, 600, 800 °C). Sandstone’s dynamic strain and dynamic tensile strength gradually rise as strain rate increases. Zhu et al. [27] conducted the dynamic impact experiments on granite at 25–600 °C, and proposed that the increase in temperature leads to a gradual decrease in the dynamic stress of granite, while the elastic modulus exhibits a characteristic of first increasing and then decreasing. Yao et al. [28] experimentally studied the effects of water content (0%, 20%, 100%) and temperature (20, −20, −30 °C) on the dynamic impact mechanical behavior of rock. The dynamic impact strength of rock decreases at 20 °C as the water content increases. At −20 °C and −30 °C, the increase of water content leads to the dynamic impact strength of rock increases first and then decreases. However, under the same moisture content, with the decrease of temperature, the peak stress of the test block increases gradually. Yao and XU [29] experimentally studied the Longyou sandstone’s dynamic tensile strength under different dynamic load impacts at different temperatures (150, 250, 350, 450, 600, and 850 °C). At the same preheated temperature, the Longyou sandstone’s dynamic tensile strength of increased with the increase of loading speeds. At the same loading speeds, except 450 °C, with an increase in preheated temperature, the tensile strength falls. Yang et al. [30] analyzed the changes of sandstone’s dynamic mechanical parameters at different temperatures (200–1000 °C). In this temperature range, the sandstone’s dynamic uniaxial compressive strength is negatively correlated with the preheated temperature. Gao et al. [31] found that the sandstone’s dynamic elastic modulus and dynamic Poisson’s ratio decreases with the preheated temperature increase from room temperature to 700 °C, in which 400 °C is the boundary point of sandstone’s physical parameters. Qian et al. studied the fracture morphology and crack propagation of granite under triaxial stress conditions under high temperature [32]. Xie et al. systematically studied the dynamic behavior and response under rock engineering disturbances to establish a three-dimensional rock dynamics theory [33]. Yang et al. [34] studied sandstone’s dynamic mechanical parameter characteristics in coalfield fire area sandstone under different impact pressures at 25–1000 °C. While the peak stress drops with an increase in preheated temperature, the strain rate of the sandstone increases under the same impact velocity. Akdag et al. [35] carried out dynamic mode I fracture toughness experiments on granite with prefabricated cracks at (25–250 °C). Under all temperature conditions, the increase of loading rate leads to an increase in the dynamic peak stress of Australian granite and the dynamic fracture toughness.

Table 1. Progress of rock dynamics experiments based on temperature and strain rate.

Sources	Rock Type	Temperature	Strain-Rate Regimes (s−1)	Loading Rate (m/s)	Peak Incident Stress (MPa)	Chamber Pressure (MPa)
Yin, T.B.; Li, X.B.; Wang, Z.Q.; Jin, J.F. (2011) [9]	Sandstone	25, 200, 400, 600, 800 °C	79.05–100.26	-	-	-
Liu, S.; Xu, J.Y. (2013) [6]	Marble	25, 200, 400, 600, 800, 1000 °C	-	3, 5, 7, 11, 12, 13, 14, 15	-	-
Xu, J.Y.; Liu, S. (2013) [8]	Marble	25, 200, 400, 600, 800, 1000 °C	-	11, 12, 13, 14, 15	-	-
Li, M.; Mao, X.B.; Cao, L.L.; Mao, R.R.; Tao, J. (2014) [12]	Sandstone	Room temperature, 800 °C	14.17–107.02	9.54–22.30	-	-
Wong, L.N.Y.; Li, Z.H.; Kang, H.M. (2017) [14]	Carrara marble	250, 500, 750 °C	100–102	-	-	-
Shi, H.; Wang, Z.L.; Shi, G.Y.; Hao, S.Y. (2019) [17]	Granite	20, 100, 200 °C	-	-	160, 180, 205, 250, 300	-
Ping, Q.; Wu, M.J.; Yuan, P.; Zhang, H. (2019) [16]	Sandstone	25, 200, 400, 600, 800, 1000 °C	-	4.5, 5.5, 6.5, 7.5, 8.5, 9.5	-	-
Li, X.; Li, B.; Li, X.; Yin, T.; Wang, Y.; Dang, W. (2020) [21]	Granite	200, 400, 600 °C	60–220	-	-	-
Qi, H.Y.; Liu, L.; Meng, X.; Li, R.; Zhang, H.; Wang, Y.; Liu, Y.; Dong, L. (2021) [23]	Marble	25, 100, 200, 400, 600, 800 °C	95.12–236.73	10, 12, 14	-	-
Zhou, S.Q.; Wang, R.; Tian, N.C.; Li, D.W. (2022) [25]	Granite	25, 100, 200, 300, 400, 500, 600, 700, 800 °C	101–102	8.1, 12.5, 16.8	-	-
Yao, W.; Chen, P.Y.; Ge, J.J.; Lu, L.G.; Gu, K.K.; Xu, Y. (2022) [28]	Sandstone	20, −20, −30 °C	-	-	-	0.6
Akdag, S.; Karakus, M.; Nguyen, G.D.; Taheri, A.; Zhang, Q.B.; Zhao, J. (2023) [35]	Granite	25, 100, 175, 250 °C	-	2, 3, 5, 7, 8	-	-

summarizes the experimental research progress of relevant scholars on rocks under temperature and dynamic loads. Sandstone, granite, marble, and other rocks have a wide range of temperature effects (−20–1000 °C). When studying the relationship between rock dynamic stress, strain, and experimental parameters, most scholars use strain rates (10⁰–10² s⁻¹) and loading rates for analysis, while others use peak incident stress and chamber pressure for quantitative analysis.

The above research focuses on rock’s dynamic mechanical parameter characteristics under the combined influence of both dynamic and static stresses and a high warmed temperature. However, under current coal mining conditions, the temperature range of the surrounding rock system is below 100 °C, especially from temperature to 100 °C. The research on the rock dynamic mechanical parameter characteristics and failure mechanisms is not in-depth enough. Based on this, this paper uses the variable cross-section SHPB experimental system to conduct the fine sandstone impact dynamics experiment between 18 °C and 100 °C, and analyzes the characteristics of the dynamic mechanical parameters of the damage process and the microscopic morphological changes, which provides a reference for the surrounding rock constancy in the process of deep mine mining.

2. Test Method

2.1. Experimental System

In this experiment, the SHPB experimental system developed by the China University of Mining and Technology, which is composed of SHPB test system, ultra high-speed strain acquisition system, DIC image analysis system, and temperature control system [36]. The high-speed strain acquisition system consists of an ultra-dynamic strain gauge (LK2107 B, 16 bit, 8-channel), strain gauge junction boxes, and strain gauges, with a sampling rate of 40 MHz. The heating system consists of a heating furnace and a SHIMAX MAC3 temperature controller, with a temperature heating range of room temperature to 200 °C and an accuracy of ±1 °C.

2.2. Test Preparation

The experimental fine sandstone were taken from Da‘anshan Coal Mine, and the samples were a rectangular shape and had a size of 40 mm× 40 mm× 80 mm. The two ends of the samples were polished smooth, and the unevenness was less than 0.02 mm. Less than 0.25° separated the two endpoints of the sample’s angle. The samples are shown in Figure 1.

During the fine sandstone dynamics experiment, a total of 5 temperature levels were set at 18, 40, 60, 80, and 100 °C. The axial static load was set at 1.0, 3.0, 5.0, 7.0, and 9.0 MPa, with preset bullet incidence velocities of 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, and 5 m/s. To maintain a stable temperature field environment for the sample, the temperature was raised by 10 °C every 3 min and maintained at a constant temperature for 4 h after reaching the predetermined temperature.

2.3. Phase Change Characteristics of fine Sandstone

The phase composition of fine sandstone and the mass distribution of principal components were analyzed by X-ray diffractometer and MDI Jade 5 diffraction analysis software.

Figure 2 shows the analysis results of the physical composition of fine sandstone. The mineral components in this group of fine sandstone samples were quartz (Quartz, SiO₂), orthoclase (KAlSi₃Ox), muscovite (KAl₂[AlSi₃O₁₀](OH)₂), kaolinite (Kaolinite, Al₂Si₂O₅(OH)₄), and albite (Albite, NaAlSi₃O₈). When combined with the data from the X-ray diffraction pattern, the material composition of fine sandstone remains largely constant across a range of temperatures. In the temperature range of 18 °C~100 °C, the minerals in fine sandstone mainly change physically, with the proportion of quartz increasing from 32.6% to 34.6%, albite rising from 29.4% to 38.8%, orthoclase and muscovite fluctuating, and kaolinite rising from 12.3% to 8.9%. The free water in kaolinite and muscovite eventually leaks out as the temperature rises, and the friction between fine sandstone particles increases, which improves its macro-deformation resistance. At the same time, quartz grains will expand, promote the closure of primary cracks and pores inside them, and strengthen their macro-mechanical behavior. The appearance of thermal stress will make mineral grains dislocate, which will produce a stress field at the crack tip and increase the crack propagation power.

3. Experimental Results

3.1. Dynamic Mechanical Characteristics of Fine Sandstone at Different Temperatures

The testing temperature range for this group of fine sandstone was 18–100 °C, with an axial static load setting of 3.0 MPa. The preset bullet incidence velocity was 1.0 m/s, and the gas chamber pressure setting was 0.06 MPa. The density of fine sandstone was between 2.32 g/cm³ and 2.42 g/cm³, and the bullet incidence velocity during the experiment was 1.00 to 1.31 m/s [36]. Figure 3 characterizes the variation of fine sandstone’s dynamic stress and strain under seismic loading and different temperature conditions. Based on the macroscopic features of fine sandstone’s dynamic stress and strain response at various temperature conditions, the fine sandstone’s dynamic elastic modulus in the elastic section increased sharply from 7.4 GPa to 15.9 GPa. But, under the same temperature conditions, when the dynamic strain increased, the gradient of dynamic stress increase continuously decreased. At room temperature (18 °C) and 40 °C, the sample undergoes an initial compaction stage under dynamic loading. However, at temperatures of 60 °C, 80 °C, and 100 °C, the initial compaction stage of fine sandstone undergoes a shorter duration and rapidly enters the elastic deformation stage. From the macroscopic characteristics of fine sandstone’s dynamic stress and strain response under different temperature conditions, when the temperature is at 40 °C and 60 °C, there exists a plastic stress plateau section for dynamic stress (Figure 3 (S2, S3)). About 95% of the peak stress is represented by the stress value of the plastic plateau section. The stress plateau causes a “stress bimodal” pattern on the stress–strain curve, and stress rebound happens after the stress reaches its peak value.

Figure 4 shows the fitting relationship between fine sandstone’s dynamic peak stress, peak strain, and temperature. The dynamic peak stress of the fine sandstone rises from 36.04 MPa to 73.41 MPa as a result of the increase in temperature. In the low temperature range (room temperature, 40 °C), the change in peak stress is relatively small (S1—36.04 MPa, S2—35.16 MPa). When entering the high temperature range (60 °C to 100 °C), the fine sandstone’s peak stress increases rapidly. The relative increase in dynamic peak stress is introduced to describe the variation pattern of fine sandstone’s dynamic peak stress. When the real-time temperature rises from 40 °C to 60 °C, the relative increase in peak stress is 1.14 MPa per degree C. When the temperature increases from 60 °C to 80 °C in real time, the relative increase in peak stress is 0.15 MPa per degree C. When the temperature rises from 80 °C to 100 °C, the relative increase in peak stress is 0.62 MPa per degree C. Temperature and peak strain have a somewhat complicated relationship. From room temperature to 80 °C, the increase in temperature leads to a decrease in fine sandstone’s peak strain. When the temperature rises to 100 °C, the value of peak strain increases to 0.008.

Through the analysis of fitting relationship, the fine sandstone’s dynamical peak stress and dynamical peak strain at different temperatures have a polynomial relationship with temperature, and the fitting relationship is shown in Formulas (1) and (2).

$$\sigma =-1E-04T^3+0.0232T^2-0.754T+40.995R^2=0.94$$

(1)

$$\epsilon =-2E-08T^3+9E-06T^2-0.0008T+0.0275R^2=0.98$$

(2)

In the formula: $\sigma$ is the dynamic peak stress, $\epsilon$ is the strain at the dynamic peak stress point, and T is the temperature.

3.2. Dynamic Mechanical Characteristics of Fine Sandstone under Different Axial Static Load

Table 2. Experimental parameter design of dynamic experiment of fine sandstone under different axial static load.

Sampe No.	Air Chamber Pressure/MPa	Axial Static Load/MPa	Temperature /°C	Density /g/cm3	Bullet Incidence Velocity/m/s
S6	0.07	1.0	40	2.36	1.98
S7	0.07	3.0	40	2.41	1.86
S8	0.07	5.0	40	2.40	2.02
S9	0.07	7.0	40	2.39	2.02
S10	0.07	9.0	40	2.34	2.05

shows the initial design parameters of impact experiments of fine sandstone under different axial compression, including the preset air chamber pressure and axial static load. The sample density is between 2.34 and 2.41 g/cm³, and the bullet incident velocity is between 1.86 and 2.05 m/s during dynamic load test.

Figure 5 characterizes the fine sandstone’s dynamic stress and strain response under various seismic and axial static load scenarios. From the figure, it can be seen that under the condition that other influencing factors remain unchanged, an increase in axial static load will lead to an increasing trend in the fine sandstone’s dynamic stress. When the axial static load increases from 1.0 MPa to 9.0 MPa, the fine sandstone’s dynamic peak stress increases from 57.25 MPa to 80.01 MPa, with an increase in dynamic peak stress ranging from 0.87 to 6.17. During the compaction stage, the fine sandstone’s dynamic tangent elastic modulus is significantly higher than that of the linear elastic stage. From the dynamic stress and strain response of single sample, as the loading process progresses, the dynamic stress of fine sandstone exhibits two phenomena: (1) under low axial pressure conditions, the fine sandstone’s dynamic stress of increases to its peak value, leading to stress rebound (Figure 5 (S7)); (2) under high axial pressure conditions, the fine sandstone’s dynamic stress increases to its peak, then decreases, and then exhibits stress rebound (Figure 5 (S6, S8–S10)).

The fitting relationship between the axial static load, dynamic maximal stress, and maximum strain of fine sandstone is shown in Figure 6. The fine sandstone’s dynamic peak stress, peak strain, and axial static load are positively correlated, and the fitting relationship follows a cubic polynomial fitting relationship, as shown in Formulas (3) and (4). The dynamic peak stress increase gradient of fine sandstone ranges from 0.87 to 6.17, and the gradient of the stress rise increases in proportion to the axial static load. The bearing capacity of fine sandstone is significantly improved by axial pressure.

$$\sigma =0.0986T^3-1.1166T^2+5.0365T+53.244R^2=0.99$$

(3)

$$\epsilon =1E-05T^3-0.0002T^2+0.0011T+0.0022R^2=0.99$$

(4)

3.3. Dynamic Mechanical Characteristics of Fine Sandstone under Different Strain Rates

Table 3. Experimental parameter design of dynamic experiment of fine sandstone under different strain rate.

Sampe No.	Air Chamber Pressure/MPa	Axial Static Load/MPa	Temperature /°C	Density /g/cm3	Bullet Incidence Velocity/m/s	Srain Rate/s−1
S11	0.06	3.0	18	2.31	1.31	53.1
S12	0.07			2.40	2.79	101.5
S13	0.07			2.42	3.28	118.1
S14	0.07			2.37	4.97	137.7
S15	0.06	3.0	40	2.38	1.02	47.1
S16	0.07			2.35	1.98	72.7
S17	0.07			2.43	2.96	107.5
S18	0.06	3.0	60	2.42	1.31	53.1
S19	0.06			2.32	1.72	64.5
S20	0.07			2.30	3.27	117.6
S21	0.07			2.31	4.58	140.9
S22	0.06	3.0	80	2.43	1.12	49.1
S23	0.06			2.41	1.67	62.8
S24	0.07			2.38	2.08	76.3
S25	0.07			2.36	2.41	87.9
S26	0.06	3.0	100	2.35	1.18	50.2
S27	0.06			2.38	1.68	63.3
S28	0.07			2.41	2.10	77.1
S29	0.07			2.40	3.17	114.4

shows the initial design parameters of impact experiments of fine sandstone at different temperatures and different bullet incident velocities, including the preset air chamber pressure and axial static load. The sample density was between 2.30 and 2.43 g/cm³. The strain rate was between 49.1 s⁻¹ and 140.9 s⁻¹. And, when the dynamic load experiment was carried out, the bullet preset growth gradient was 0.5 m/s. The actual bullet incident velocity is shown in the column of

Table 4. Parameters of dynamic stress–strain one-dimensional constitutive model of fine sandstone at different temperatures.

Temperature/°C	E1/GPa	E2/GPa	F0	m
18	9.12	17.43	85.33	1.0
40	9.57	15.95	74.25	2.0
60	9.86	19.87	67.89	1.0
80	9.41	26.77	82.46	1.2
100	9.24	24.12	143.60	1.2

(bullet incidence velocity).

Figure 7 shows the characteristics of fine sandstone’s dynamic stress and strain response at different temperatures and different strain rate. When the temperature were 18 °C, 40 °C, 60 °C, 80 °C, and 100 °C, the increase in strain rate led to an increase in the fine sandstone’s dynamic stress and strain. For example, when the temperature was 18 °C, the fine sandstone’s peak dynamic stress increased from 35.14 MPa to 110.91 MPa. When the temperature was 40 °C, the peak dynamic stress increased from 54.86 MPa to 87.93 MPa. When the temperature was 60 °C, the peak dynamic stress increased from 54.57 MPa to 118.72 MPa. When the temperature was 80 °C, the peak dynamic stress increased from 34.47 MPa to 50.42 MPa. When the temperature was 100 °C, the peak dynamic stress increased from 87.53 MPa to 136.24 MPa. According to the macro change response of the stress–strain curve, under the different strain rates, fine sandstone quickly enters the elastic deformation period, and the stress increases rapidly. Once the stress peak has been reached, the stress changes in two states: (1) the stress decreases rapidly and rebounds directly; (2) the stress slowly drops to a low value (Figure 7b (107.5 s⁻¹), Figure 7c (140.9 s⁻¹), Figure 7d (87.9 s⁻¹) and Figure 7e (114.4 s⁻¹)), the phenomenon of stress rebound occurs, and the ratio of the stress inflection point value to its peak value is 60.7%, 88.7%, 78.6%, and 65.1%, respectively.

We statistically analyzed the relationship between fine sandstone’s peak stress, peak strain, and bullet incident velocity at the same temperature and different bullet incident velocities, as shown in Figure 8. The fine sandstone’s peak stress and strain were positively correlated with the bullet incident velocity under different temperatures. Through regression analysis, their response relationships were in accordance with the following formula:

$${\sigma }_i=a{v}_i^2+b{v}_i+c$$

(5)

$${\epsilon }_i=a{v}_i^2+b{v}_i+c$$

(6)

In the formula: σ_i is the fine sandstone’s peak stress under different bullet incident velocities, ε_i is the fine sandstone’s peak strain under different bullet incident velocities, v_i is the bullet incident velocity, and a, b, and c are the influence coefficients of incident velocity on dynamic strength.

4. Discussion

Under different temperatures, axial static load, and bullet incident velocity, the fine sandstone’s dynamic mechanical parameters are intrinsically related to various factors, and its stress law is obvious, while its strain expression law is complex. The fine sandstone’s dynamic stress and strain response indicates that its stress and strain curve is smooth under dynamic load, with occasional stress bimodal and obvious stress rebound. This is for two reasons: (1) the heterogeneity of the sample itself and the influence of the development state of primary pores and fractures; (2) thermal stress and external factors (axial static load, bullet incident speed) that affect the internal “skeleton” and crack growth and development of the sample.

4.1. Characteristics of Surface Strain Field of Fine Sandstone during Impact Process

The variation law of surface displacement acceleration during the impact test of fine sandstone was analyzed by using MatchID-2D software and DIC analysis digital image technology, and the results are shown in Figure 9.

Figure 9 is the graph showing the variation of surface displacement acceleration of fine sandstone under different dynamic stress levels, in which Figure 9a,c,e,g,i,k are horizontal displacements and Figure 9b,d,f,h,j,l are vertical displacements. The surface displacement in the horizontal direction shows a banded distribution at the initial stage, and shows an increasing trend from left to right. The magnitude difference is close to 2. The increase rate of surface displacement decreases and changes from the growth zone to the non-equivalent growth zone under the dynamic stress increases. The contact region between the sample’s left side and the transmission bar is where the rapid changing area of surface displacement is visible (Figure 9e). Then, the surface displacement change rate decreases as a whole, and the deceleration zone appears on the right and upper sides (Figure 9g). With the increase of dynamic load stress, the deceleration zone expands and forms a banded distribution on the right side of the fine sandstone (Figure 9i). The left-hand portion of the sample is covered by the deceleration change zone. The surface displacement in the vertical direction is generally decelerated (Figure 9b, blue/green area). The accelerated deformation zone is located in the left and lower sporadic areas of the sample. The surface displacement deceleration zone of the sample is expanded, and the accelerated deformation zone appears on the upper right side. Subsequently, the accelerated deformation zone expands and divides the decelerated deformation zone as the dynamic stress increases. From the numerical analysis of the color card, the acceleration is increased by nearly five times (Figure 9h,j). At the post-peak stress stage, the vertical surface displacement of the sample is obviously decelerated, and almost completely covered by the deceleration zone, showing a phenomenon of high at both ends and low in the middle, which is consistent with the ‘rebound’ phenomenon of fine sandstone after the peak.

4.2. Constitutive Relation

Based on Terzaghi’s effective stress principle, rock mechanical strength theory, and statistical damage theory, the constitutive model of fine sandstone impact dynamics under the coupling of dynamic load and temperature was derived. The assumptions are as follows:

(1)

The Nishihara model was used to reflect the elastic–viscoelastic–viscoplastic deformation characteristics of fine sandstone, which is expressed as a composite structure composed of three elements (viscosity mass and damaged body), and its stress process is shown in Figure 10.

(2)

Micro-elements exhibit linear elastic change in the stress–strain relationship, follow the generalized Hooke’s law prior to failure, and have the macroscopic isotropic damage characteristics.

(3)

The micro-element strength parameters obey Weibull probability statistical distribution, and its distribution pattern conforms to the following equation [37,38]:

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

(7)

where formula: F is the distribution variable of fine sandstone strength, and m and F₀ are constants to characterize the materials mechanical properties.

(4)

The viscous cylinder has no damage characteristics, and the constitutive relationship [39,40] is:

$${\sigma }_b=\eta d{\epsilon }_b/dt$$

(8)

In the formula: η is the viscosity parameter of the viscous cylinder.

(5)

In the constitutive relation modeling, the influence of inertia effect is ignored [41].

The dynamic stress and strain of the damaged body and the viscous cylinder in the combination satisfy the following formulas:

$$\left.\begin{array}{l}\sigma ={\sigma }_s+{\sigma }_i={\sigma }_a+{\sigma }_b={\sigma }_c\\ \epsilon ={\epsilon }_a+{\epsilon }_c\\ {\epsilon }_a={\epsilon }_b\end{array}\right\}$$

(9)

According to the strain equivalence principle [42], the constitutive relation of damaged materials can be expressed as:

$$\sigma =E(1-D)\epsilon$$

(10)

where E is the elastic modulus of the fine sandstone when it is lossless, D is the damage variable, and ε is the strain.

Drawing from the theories of continuous damage and statistical strength, A novel statistical model was suggested by Krajcinovic et al. [43] to characterize the damage of specimens:

$$\sigma =E\epsilon (1-\frac{n}{N})$$

(11)

where n is the number of micro-units that have been destroyed, N is the total number of micro-units.

Combining Formulas (8) and (9), we can get:

$$D=\frac{n}{N}$$

(12)

When the fine sandstone is subjected to load, the micro-elements are destroyed, and the number of micro-elements destroyed in any interval [F, F + dF] can be calculated by the probability density function of the strength of the micro-element. That is, when the external load changes from 0 to a certain load f, the number of micro-elements destroyed is: $N\varphi (F)dF$. When the external load changes from 0 to a certain load F, the number of damaged micro-elements is:

$$n(F)=N{\displaystyle {\int }_0^F\varphi (F)dF=N\left\{1-\exp [-{(\frac{F}{F_0})}^m]\right\}}{|}_0^F$$

(13)

Substitute Formula (13) into Formula (12):

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

(14)

When the dynamic and static load and temperature act together on coal and rock mass, in addition to the dynamic load stress deterioration sample, the temperature rise also increases the instantaneous elastic modulus damage degree. According to the previous research results [44], the instantaneous thermal damage variable D_T can be expressed as:

$$D_T=1-E_T/E_0$$

(15)

where E_T is the instantaneous elastic modulus of coal and rock at temperature T, and E₀ is the instantaneous elastic modulus at normal temperature.

Under the action of temperature and load, the total damage of rock [45] can be expressed as:

$$D_s=D+D_T-D{D}_T$$

(16)

Substitute Formulas (14) and (15) into (16) to get the expression of total damage variable:

$$D_s=1-\frac{E_T}{E}\exp [-{(\frac{F}{F_0})}^m]$$

(17)

Substituting Formulas (8), (10), and (17) into Formula (9), the constitutive relation [46] of the assembly can be obtained as follows:

$$\eta \dot{\sigma }+[E_1(1-D_s)+E_2(1-D_s)]\sigma =E_2(1-D_s)\cdot [\eta \dot{\epsilon }+E_1(1-D_s)\epsilon ]$$

(18)

Without considering the damage characteristics, replace the effective elastic modulus E (1 − D_s) with the elastic modulus E of the damaged body before the damage, and solve the Formula (18):

$$\eta \dot{\sigma }+(E_1+E_2)\sigma =E_2(\eta \dot{\epsilon }+E_1\epsilon )$$

(19)

When the dynamic load moment t = 0, the sample is subjected to static axial load, that is, ${\sigma }_0={\sigma }_s$, $\epsilon (0)={\epsilon }_0$.

According to Laplace transform principle:

$$\sigma (t+t_0)=E_2({\epsilon }_0+Ct)-\frac{E_2}{E_1+E_2}(E_2\eta C-E_2{\epsilon }_0)e^{-\frac{E_1+E_2}{\eta }t}-\frac{E_2}{E_1+E_2}(Ct-\frac{E_2\eta C}{E_1+E_2}+E_2{\epsilon }_0)$$

(20)

Substitute Formula (16) into (20) to get:

$$\begin{array}{l}\sigma (t)=\frac{E_T}{E}\exp [-{(\frac{F}{F_0})}^m]({\epsilon }_0+Ct)E_2-\frac{E_T}{E}\exp [-{(\frac{F}{F_0})}^m]\frac{E_2}{E_1+E_2}(E_2\eta C-\\ E_2{\epsilon }_0)e^{-\frac{\frac{E_T}{E}\exp [-{(\frac{F}{F_0})}^m](E_1+E_2)}{\eta }t}-\frac{E_2}{E_1+E_2}(Ct-\frac{E_2\eta C}{E_1+E_2}+\frac{E_T}{E}\exp [-{(\frac{F}{F_0})}^m]E_2{\epsilon }_0)\end{array}$$

(21)

A one-dimensional constitutive equation of fine sandstone in a one-dimensional stress state under the combined influence of temperature and dynamic load is found in Formula (21). When applying the above model, six parameters of E_T, E₁, E₂, F, F₀, and m need to be determined. The basis for determining the parameters is as follows: According to the previous research results, the static elastic modulus of the coal rock mass can be used to replace the constitutive model’s E₁, and the value of E₂ is comparable to the linear elastic modulus of the coal rock mass’s stress–strain curve under various levels of strain. It can be replaced by the dynamic elastic modulus, and the variation range of viscosity coefficient η is generally 0–1000 GPa·s. The strain ${\epsilon }_r\left(t\right)$, the average strain rate C and the time t₀ when the static load begins to load are measured data.

Based on the above model, simulation analysis was conducted on the dynamic stress process of fine sandstone under different temperature conditions. The simulation results are shown in Figure 11.

From Table 4, the static elastic modulus of fine sandstone varies from 9.12 to 9.86 GPa, the dynamic elastic modulus varies from 15.95 to 26.77 GPa, and F₀ varies from 67.89 to 143.60. The established theoretical model has a good response to the stress–strain curve (before the rebound stage) of fine sandstone and coal samples during the impact process, and cannot explain the stress mutation and post peak rebound phenomenon of damage to fine sandstone and coal samples. This is also a common problem in constructing a one-dimensional coal rock damage impact dynamic model, and some scholars have explored the stress rebound segment using segmented simulation analysis methods.

5. Conclusions

The paper experimentally studied the changes in dynamic stress–strain of fine sandstone under different temperatures (18–100 °C), axial static loads (1–9 MPa), bullet incidence velocities, or strain rates, and analyzed the changes in physical properties and surface displacement fields of fine sandstone. A one-dimensional stress state dynamic stress–strain constitutive model of fine sandstone was established and verified. The main conclusions are as follows:

(1)

The increase in temperature and axial static load leads to an increase in the dynamic peak stress of coal bearing fine sandstone. When the temperature changes from 18 degrees to 100 °C, the dynamic peak stress increases by 103.7%. When the axial static load changes from 1 MPA to 9 MPa, the dynamic peak stress increases by 39.7%. Within the temperature range studied in this article (18–100 °C), the dynamic peak stress and peak strain of fine sandstone are positively correlated with strain rate or bullet incidence velocity. At a temperature of 18 °C, when the strain rate changes from 53.1 s⁻¹ to 137.4 s⁻¹, the dynamic peak stress increases by 215.6%. When the temperature is 40 °C and the strain rate changes from 47.1 s⁻¹ to 107.5 s⁻¹, the dynamic peak stress increases by 60.3%. When the temperature is 60 °C and the strain rate changes from 53.1 s⁻¹ to 140.9 s⁻¹, the dynamic peak stress increases by 117.6%. When the temperature is 80 °C and the strain rate changes from 49.1 s⁻¹ to 87.9 s⁻¹, the dynamic peak stress increases by 46.3%. When the temperature is 100 °C and the strain rate changes from 50.2 s⁻¹ to 114.4 s⁻¹, the dynamic peak stress increases by 55.6%.

(2)

The influence of external experimental parameters on the dynamic mechanical properties of fine sandstone was analyzed, and it was found that the dynamic stress, strain, temperature, axial static load, strain rate, or bullet incidence velocity of fine sandstone exhibit a polynomial relationship, with a determination coefficient greater than 0.9.

(3)

The main components of this group of fine sandstone are quartz, sodium feldspar, muscovite, and kaolinite. The continuous effect of temperature causes free water to precipitate from muscovite and kaolinite, increasing the friction force between fine sandstone particles and improving their macroscopic bearing capacity.

(4)

Based on Terzaghi’s effective stress principle, rock mechanics strength theory, and statistical damage theory, a dynamic stress constitutive model of fine sandstone under a one-dimensional stress state was established. The values of model parameters were theoretically determined, and the model was validated through dynamic stress change curves of fine sandstone at different temperatures.