# Analysis of the Dynamic Impact Mechanical Characteristics of Prestressed Saturated Fractured Coal and Rock

Authors: Wen Wang; Shiwei Zhang; Huamin Li; Shuang Gong; Zhumeng Liu
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

The natural and water-saturated states of coal samples under static and static-dynamic loads were tested using the Split-Hopkinson pressure bar (SHPB) method and RMT-150 system, respectively. The differences in the strength reduction coefficient and elastic modulus reduction coefficient of water-saturated coal samples under static and static-dynamic loads were discussed. The experimental results for coal were compared with the corresponding characteristics of typical sandstone samples under static and static-dynamic loads. Furthermore, a fracture model of a hydrous wing branch fracture under static-dynamic loading was established based on the theory of fracture damage mechanics. The difference in dynamic strength between coal and sandstone samples for both the natural state and water-saturated state was analyzed. On this basis, the effect of water on the fracture surface of coal and the tensile strength and shear strength of the branch fracture surface were fully considered. In addition, criteria of the branch fracture surface for crack initiation and crack arrest were also established. Finally, the phenomenon of increasing elastic modulus in saturated coal samples was explained with this criterion.


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## Body

## 1. Introduction

Coal is a heterogeneous natural medium that contains a large number of pores and microcracks. In the water-saturated state, the large amount of free water in the pores and fractures of coal has an important influence on its physical, chemical, and mechanical properties. Owing to rock-water chemical interactions and the impact of pore water pressure, the decrease in stored water and reduction of the strength of coal rock materials under the condition of static loading is an indisputable fact in geologic engineering [1–3]. However, owing to differences in the properties and internal structure of coal and rock materials under dynamic loads, the dynamic strength of a water-saturated coal specimen can exhibit either strengthening or weakening compared to the natural state specimen. Pu and Ma [4, 5] analyzed Split-Hopkinson pressure bar (SHPB) experimental results for sandstone in a coalmine with different moisture contents and concluded that the dynamic uniaxial compressive strength of sandstone increased as a power function with increasing moisture content of the specimens. Wang et al. [6, 7] carried out static-dynamic load experiments on coal specimens with different moisture contents using the improved SHPB method and the RMT-150 test system. They found that the dynamic peak strength of the coal specimens decreases with increasing water-saturation time under static-dynamic loads (intermediate strain rate). Ding et al. [8] performed dynamic experiments on clay with four different moisture contents using the SHPB method. The results showed that the uniaxial compressive strength of the specimens gradually decreased with increasing moisture content and the water in the surface of unsaturated clay has a significant effect on its mechanical properties. Ding et al. [9] studied the effect of free water content on the dynamic mechanical properties of cement mortar under high strain rates and determined that the dynamic compressive strength of cement mortar decreases with increasing water content. The dynamic compressive strength of saturated cement mortar specimens was 23% lower than that of completely dry cement mortar specimens. Dynamic impact tests of argillaceous siltstone were conducted by Zhan and Zhang [10]. Zhao et al. [11] studied the saturated specimens have stronger loading rate dependence than the dry specimens. Saturated coal specimens have higher indirect tensile strength than dry ones. Petrov et al. [12] discovered that the temporal dependence of the dynamic compressive and split tensile strengths of dry and saturated limestone samples can be predicted by the incubation time criterion.The results showed that the peak strength of specimens decreases with increasing water content. It can be seen that, under the intermediate strain rates, the dynamic strength of different coal and rock materials varies from the natural to water-saturated state. However, this difference is difficult to explain from macroscopic mechanics experiments or theoretical analyses. Furthermore, coal and rock are made up of many types of mineral grains. There are many fractures and microstructures between these grains, which will expand under water-rock interaction, particularly for low strength materials.At present, the mechanism for strength improvement of water-saturated coal and rock specimens has been discussed under dynamic impact. However, the mechanism for coal and rock strength reduction and the criteria for coal and rock strength reduction and improvement have not been addressed. The study of the mesomechanical properties of saturated rock under dynamic impact is typically based on fracture mechanics. The mesostructure of water-saturated rock was analyzed by using a microfracture model [13]. During the analysis, the effect of varying stress on the main fracture surface is primarily considered. At the same time, the hydration corrosion of coal materials and stress distribution at the branch fracture surface, which have a significant impact on water-saturated low strength materials, are rarely taken into account. Therefore, integrating the above factors for coal materials, analyses of the strength properties of water-saturated coal and rock under static-dynamic loads have theoretical and practical significance.


## 2. Methods

### 2.1. Testing Equipment

The tests include two loading models: static load and static-dynamic load. The static load is carried out on a RMT-150 rock mechanics test system, which uses a displacement sensor (range: 5 mm), and pressure sensors measure axial deformation and axial loading, respectively. The loading process is displacement-controlled, and the load speed is 0.002 mm/s. The static-dynamic load test is conducted with a modified SHPB testing system, as shown in Figure1 [14, 15]. This system uses half-sine stress wave loading.Figure 1
SHPB test apparatus with bar.In these experiments, the features of a one-dimensional elastic wave are invariant as it propagates in a slender rod. The stress, strain, and strain rate of the specimen are calculated from the voltage values measured by a strain gauge on the pressure bar. The relationships can be expressed by the following equation:(1)σt=AeEe2AsεIt+εRt+εTt,εt=CeLs∫0tεIt−εRt−εTtdt,ε˙t=CeLsεIt−εRt−εTt,where As and Ae are the specimen area and cross-sectional area of the pressure bar, respectively; Ee is the elastic modulus of the pressure bar; εI, εR, and εT are the incident strain, reflection strain, and transmission strain, respectively, which are measured on the incident bar and transmission bar; and Ce and le are the wave velocity and length of the stress bar, respectively.


### 2.2. Preparation of Specimens

To reduce variation in the results caused by differences in the coal specimens, the coal specimens for the static-dynamic load tests were collected as a relatively homogeneous lump coal in the bottom of the coal seam. The specimens were then drilled to formφ50 mm × 30 mm cylinders, and the surface flatness of both ends of the specimens was <0.02 mm on a certain structural surface. These specimen parameters satisfied the requirements for the static and static-dynamic load tests. For preparation of the coal specimens (Figure 2), the specimens were randomly divided into two states: the natural state and the water-saturated state. Natural state specimens were prepared by placing the specimens on a dryer/water separator, and kept for 24 h, after which the specimens were sealed with reference to the 60–70% relative humidity in the coalmine. Water-saturated state specimens were prepared by the free water absorption method in accordance with methods for determining the physical and mechanical properties of coal and rock [16]. In this specific method, the surface of water is 1-2 cm above the surface of the natural coal specimens in the vessel. The specimens were then weighed every 24 h, and the weight of the coal specimens changed by ≤0.01 g between two successive weighings after immersion for 7 d. The specimens were thus in the water-saturated state. The range of moisture contents in the saturated-state specimens was 3.2–6.1%.Figure 2
Coal specimens for testing.


### 2.3. Test Scheme

Static load testing was conducted on the natural state and water-saturated coal specimens. Figure3 shows the stress-strain curves for specimens under static loads. Static-dynamic load tests were performed on an improved SHPB testing system. During the tests, the specimens were subjected to prestatic loads with a load stress of 12 MPa (approximately 30% of the specimen peak strength in the natural state). Dynamic loads were then applied to the specimens. The inflation pressure was used to control the speed of the bullet impact. To obtain similar strain rates for the specimens, natural state specimens (D1-1 to D1-3 and D3-2 to D3-4) and water-saturated specimens (D1-4 to D2-2 and D2-3 to D3-1) were used for the static and static-dynamic load tests, respectively. In these tests, coal specimens in each state were tested three times with each loading method. The experimental results are shown in Figure 4.Figure 3
Static stress-strain curves for water-saturated coal specimens.Figure 4
Dynamic stress-strain curves for coal specimens with different moisture states. (a) Natural state. (b) Water-saturated state.
(a)
(b)


### 2.4. Static and Static-Dynamic Test Results

According to the stress-strain curve for the uniaxial compression test (Figure3), the natural state specimens have uniaxial compressive strengths of 42.07–43.11 MPa, with an average of 42.71 MPa; the elastic modulus is 1.90–2.11 GPa, with an average of 2.65 GPa. The uniaxial compressive strength of water-saturated coal specimens is 20.40–25.30 MPa, with an average of 22.17 MPa; the elastic modulus is 1.28–1.66 GPa, with an average of 1.52 GPa. The physical parameters of the uniaxial compression test are summarized in Table 1. The water absorption of natural specimens is 3.2–6.1%. The average uniaxial compressive strength reduction coefficient for the water-saturated specimens is 0.52, and the average elastic modulus reduction coefficient is 0.66.Table 1
Physicomechanical parameters of natural and water-saturated specimens under uniaxial compression.
Specimen number
State
R
C (MPa)
w (%)
P (kg·m−3)
E (GPa)
D1-1
Natural
43.11
—
1.33
2.11
D1-2
Natural
42.07
—
1.35
1.90
D1-3
Natural
42.92
—
1.34
1.94
D1-4
Saturated
25.30
3.2
1.37
1.73
D2-1
Saturated
20.81
6.1
1.45
1.28
D2-2
Saturated
20.4
4.5
1.36
1.66
A21
Natural
137.3
—
2.61
34.3
A22
Natural
129.5
—
2.606
32.6
A23
Natural
162.4
—
2.625
35.9
A214
Saturated
111.2
0.23
2.623
31.5
A215
Saturated
130.0
0.77
2.608
36.4
A216
Saturated
97.7
0.40
2.603
29.5The compressive strength of sandstone is higher [17]. The water absorption rate of the saturated sandstone is 0.343–0.771%, with an average of 0.434%. Figure 5 shows the static stress-strain curves for natural (A21–A23) and water-saturated (A214–A216) sandstone specimens under uniaxial compression. The elastic modulus of natural state sandstone specimens is 32.6–35.9 GPa, with an average of 34.2 GPa; the peak strength is 129.5–162.4 MPa, with an average of 143.1 MPa. For saturated sandstone specimens, the elastic modulus is 29.5–36.4 GPa, with an average of 32.4 GPa; the peak strength is 97.7–130.0 MPa, with an average of 112.9 MPa. Thus, the compressive strength reduction coefficient for water-saturated sandstone is 0.79; the elastic modulus reduction coefficient is 0.94.Figure 5
Static stress-strain curves for natural and water-saturated sandstone specimens.In the static-dynamic load tests, a static load of 12 MPa is applied as preloading, after which the impact loads are applied. The dynamic strength characteristics of coal specimens in different moisture states are tested. The dynamic strain rate range is 90–155 s−1, and three samples for each moisture content are tested. Dynamic stress-strain curves for different aqueous coal specimens under static-dynamic loads are shown in Figure 4.The results show that the dynamic strength of coal specimens varies with different moisture states. Within the same moisture state, there is also some variation in the coal specimens. Figure4(a) shows the stress-strain curves for coal specimens in the natural state. The dynamic compressive strength is 32.2–40.6 MPa, with an average of 37.2 MPa; the elastic modulus is 7.14–7.89 GPa, with an average of 7.64 GPa. Figure 4(b) shows the stress-strain curves for saturated coal specimens. The dynamic compressive strength is 28.08–33.82 MPa, with an average of 31.53 MPa; the elastic modulus is 8.75–9.11 GPa, with an average of 8.98 GPa. The average dynamic strength reduction coefficient of water-saturated coal specimens is 0.85, and the dynamic elastic modulus is increased by 17%.Pu and Ma [5] carried out uniaxial impact compression tests for two coalmines with sandstone having four different moisture states. The results are shown in Figure 6. Under the medium strain rate, the dynamic uniaxial compressive strength of sandstone increases with the moisture content of specimens. The dynamic compressive strength of the specimens is highest for the forced water saturation and natural saturation states. Additionally, their compressive strengths are similar. The dynamic uniaxial compressive strength of sandstone is second in the water-saturated state and is the lowest in the dry state. When the two types of sandstone reached a maximum with the natural water absorption method, the dynamic uniaxial compressive strength of the saturated sandstone was increased by 18% and 29%, respectively, compared to the dry state. This is contrary to the trend reported by Chang for the peak strength of coal. Wang and Li [13, 18] concluded that water-saturated concrete and granite have similar mechanical properties.Figure 6
Dynamic stress-strain curves for sandstone specimens. (a) Yangzhuang coalmine (ε˙ = 198 s−1). (b) Hengyuan north auxiliary shaft (ε˙ = 200 s−1).
(a)
(b)


## 2.1. Testing Equipment

The tests include two loading models: static load and static-dynamic load. The static load is carried out on a RMT-150 rock mechanics test system, which uses a displacement sensor (range: 5 mm), and pressure sensors measure axial deformation and axial loading, respectively. The loading process is displacement-controlled, and the load speed is 0.002 mm/s. The static-dynamic load test is conducted with a modified SHPB testing system, as shown in Figure1 [14, 15]. This system uses half-sine stress wave loading.Figure 1
SHPB test apparatus with bar.In these experiments, the features of a one-dimensional elastic wave are invariant as it propagates in a slender rod. The stress, strain, and strain rate of the specimen are calculated from the voltage values measured by a strain gauge on the pressure bar. The relationships can be expressed by the following equation:(1)σt=AeEe2AsεIt+εRt+εTt,εt=CeLs∫0tεIt−εRt−εTtdt,ε˙t=CeLsεIt−εRt−εTt,where As and Ae are the specimen area and cross-sectional area of the pressure bar, respectively; Ee is the elastic modulus of the pressure bar; εI, εR, and εT are the incident strain, reflection strain, and transmission strain, respectively, which are measured on the incident bar and transmission bar; and Ce and le are the wave velocity and length of the stress bar, respectively.


## 2.2. Preparation of Specimens

To reduce variation in the results caused by differences in the coal specimens, the coal specimens for the static-dynamic load tests were collected as a relatively homogeneous lump coal in the bottom of the coal seam. The specimens were then drilled to formφ50 mm × 30 mm cylinders, and the surface flatness of both ends of the specimens was <0.02 mm on a certain structural surface. These specimen parameters satisfied the requirements for the static and static-dynamic load tests. For preparation of the coal specimens (Figure 2), the specimens were randomly divided into two states: the natural state and the water-saturated state. Natural state specimens were prepared by placing the specimens on a dryer/water separator, and kept for 24 h, after which the specimens were sealed with reference to the 60–70% relative humidity in the coalmine. Water-saturated state specimens were prepared by the free water absorption method in accordance with methods for determining the physical and mechanical properties of coal and rock [16]. In this specific method, the surface of water is 1-2 cm above the surface of the natural coal specimens in the vessel. The specimens were then weighed every 24 h, and the weight of the coal specimens changed by ≤0.01 g between two successive weighings after immersion for 7 d. The specimens were thus in the water-saturated state. The range of moisture contents in the saturated-state specimens was 3.2–6.1%.Figure 2
Coal specimens for testing.


## 2.3. Test Scheme

Static load testing was conducted on the natural state and water-saturated coal specimens. Figure3 shows the stress-strain curves for specimens under static loads. Static-dynamic load tests were performed on an improved SHPB testing system. During the tests, the specimens were subjected to prestatic loads with a load stress of 12 MPa (approximately 30% of the specimen peak strength in the natural state). Dynamic loads were then applied to the specimens. The inflation pressure was used to control the speed of the bullet impact. To obtain similar strain rates for the specimens, natural state specimens (D1-1 to D1-3 and D3-2 to D3-4) and water-saturated specimens (D1-4 to D2-2 and D2-3 to D3-1) were used for the static and static-dynamic load tests, respectively. In these tests, coal specimens in each state were tested three times with each loading method. The experimental results are shown in Figure 4.Figure 3
Static stress-strain curves for water-saturated coal specimens.Figure 4
Dynamic stress-strain curves for coal specimens with different moisture states. (a) Natural state. (b) Water-saturated state.
(a)
(b)


## 2.4. Static and Static-Dynamic Test Results

According to the stress-strain curve for the uniaxial compression test (Figure3), the natural state specimens have uniaxial compressive strengths of 42.07–43.11 MPa, with an average of 42.71 MPa; the elastic modulus is 1.90–2.11 GPa, with an average of 2.65 GPa. The uniaxial compressive strength of water-saturated coal specimens is 20.40–25.30 MPa, with an average of 22.17 MPa; the elastic modulus is 1.28–1.66 GPa, with an average of 1.52 GPa. The physical parameters of the uniaxial compression test are summarized in Table 1. The water absorption of natural specimens is 3.2–6.1%. The average uniaxial compressive strength reduction coefficient for the water-saturated specimens is 0.52, and the average elastic modulus reduction coefficient is 0.66.Table 1
Physicomechanical parameters of natural and water-saturated specimens under uniaxial compression.
Specimen number
State
R
C (MPa)
w (%)
P (kg·m−3)
E (GPa)
D1-1
Natural
43.11
—
1.33
2.11
D1-2
Natural
42.07
—
1.35
1.90
D1-3
Natural
42.92
—
1.34
1.94
D1-4
Saturated
25.30
3.2
1.37
1.73
D2-1
Saturated
20.81
6.1
1.45
1.28
D2-2
Saturated
20.4
4.5
1.36
1.66
A21
Natural
137.3
—
2.61
34.3
A22
Natural
129.5
—
2.606
32.6
A23
Natural
162.4
—
2.625
35.9
A214
Saturated
111.2
0.23
2.623
31.5
A215
Saturated
130.0
0.77
2.608
36.4
A216
Saturated
97.7
0.40
2.603
29.5The compressive strength of sandstone is higher [17]. The water absorption rate of the saturated sandstone is 0.343–0.771%, with an average of 0.434%. Figure 5 shows the static stress-strain curves for natural (A21–A23) and water-saturated (A214–A216) sandstone specimens under uniaxial compression. The elastic modulus of natural state sandstone specimens is 32.6–35.9 GPa, with an average of 34.2 GPa; the peak strength is 129.5–162.4 MPa, with an average of 143.1 MPa. For saturated sandstone specimens, the elastic modulus is 29.5–36.4 GPa, with an average of 32.4 GPa; the peak strength is 97.7–130.0 MPa, with an average of 112.9 MPa. Thus, the compressive strength reduction coefficient for water-saturated sandstone is 0.79; the elastic modulus reduction coefficient is 0.94.Figure 5
Static stress-strain curves for natural and water-saturated sandstone specimens.In the static-dynamic load tests, a static load of 12 MPa is applied as preloading, after which the impact loads are applied. The dynamic strength characteristics of coal specimens in different moisture states are tested. The dynamic strain rate range is 90–155 s−1, and three samples for each moisture content are tested. Dynamic stress-strain curves for different aqueous coal specimens under static-dynamic loads are shown in Figure 4.The results show that the dynamic strength of coal specimens varies with different moisture states. Within the same moisture state, there is also some variation in the coal specimens. Figure4(a) shows the stress-strain curves for coal specimens in the natural state. The dynamic compressive strength is 32.2–40.6 MPa, with an average of 37.2 MPa; the elastic modulus is 7.14–7.89 GPa, with an average of 7.64 GPa. Figure 4(b) shows the stress-strain curves for saturated coal specimens. The dynamic compressive strength is 28.08–33.82 MPa, with an average of 31.53 MPa; the elastic modulus is 8.75–9.11 GPa, with an average of 8.98 GPa. The average dynamic strength reduction coefficient of water-saturated coal specimens is 0.85, and the dynamic elastic modulus is increased by 17%.Pu and Ma [5] carried out uniaxial impact compression tests for two coalmines with sandstone having four different moisture states. The results are shown in Figure 6. Under the medium strain rate, the dynamic uniaxial compressive strength of sandstone increases with the moisture content of specimens. The dynamic compressive strength of the specimens is highest for the forced water saturation and natural saturation states. Additionally, their compressive strengths are similar. The dynamic uniaxial compressive strength of sandstone is second in the water-saturated state and is the lowest in the dry state. When the two types of sandstone reached a maximum with the natural water absorption method, the dynamic uniaxial compressive strength of the saturated sandstone was increased by 18% and 29%, respectively, compared to the dry state. This is contrary to the trend reported by Chang for the peak strength of coal. Wang and Li [13, 18] concluded that water-saturated concrete and granite have similar mechanical properties.Figure 6
Dynamic stress-strain curves for sandstone specimens. (a) Yangzhuang coalmine (ε˙ = 198 s−1). (b) Hengyuan north auxiliary shaft (ε˙ = 200 s−1).
(a)
(b)


## 3. Mechanism and Discussion

The analysis shows that, in addition to the influence of mineral composition, structure, and type of cementation, rock mechanical properties are also influenced by the water environment [19, 20]. Ogata et al. [21] show that the tensile strength of rocks with high porosity on the water saturation condition was decreased on both static and dynamic condition. Huang et al. [22] found that the tensile strength softening factor decreases with the loading rate. Moreover, static and dynamic loading strength tests of sandstone under water-bearing conditions were carried out by Zhou et al. [23, 24], revealing that the static and dynamic sandstone strengths decreased by 29.88% and 40.55%, following saturation. The permeability of coal and rock specimens is primarily determined by the presence of fractures and the mineral components of these fractures. The presence of a variety of mineral components directly affects the physical and chemical properties of coal and rock. Coal and rock materials contain pores and fractures of varying numbers and shapes. The fractures contain a variety of minerals, such as sulfides, oxides, carbonates, and silicates. As a type of solvent, water has a small corrosive effect on high strength materials, but a highly corrosive effect on low strength materials. The corrosive effect of water can readily create small weak parts in the fractures, resulting in the holes continuing to expand under stress. Erosion by corrosive molecules leads to the fractures and holes continuously increasing and enlarging. Moreover, water can dissolve some minerals in the coal and rock, causing water absorption expansion of montmorillonite in the mineral. This results in generation of uneven stress in the interior of coal and rock specimens. In addition, because the cementing material in the fractures and a portion of the cement between particles are dissolved, the cohesion force between particles and cement decreases. Furthermore, the microcomposition of coal and rock is changed, and the original microstructure can be broken, causing the strength of coal and rock materials to decrease.Comparing Figures3 and 4 leads to the following conclusions: the strength reduction coefficient and elastic modulus reduction coefficient of the saturated coal specimens with a high water absorption rate are 0.52 and 0.77, respectively. However, sandstone is relatively dense, and the strength reduction coefficient and elastic modulus reduction coefficient of the saturated sandstone specimens with a low water absorption rate are 0.79 and 0.94, respectively.


### 3.1. The Force between Particles

Compared to coal, there are few fractures and pores in the sandstone. Additionally, sandstone has a denser structure, greater content of high strength materials, and weaker water-rock interaction than coal. Therefore, the strength and antideformation ability of coal decreases significantly under static loads, but the water has little effect on the strength and antideformation ability of sandstone. A comparison of the natural state and water-saturated state of a coal specimen is shown in Figure7.Figure 7
Sketch of the coal specimen in the natural state and water-saturated state. (a) Coal specimen D1-2. (b) Natural state. (c) Water-saturated state.
(a)
(b)
(c)Furthermore, before the coal and rock specimens are soaked in water, some water is contained internally, occurring on the surface of particles in the form of crystal water, pore water, and fracture water. Thus, an attractive effect will occur between the water and particles; at the same time, capillary pressure will also be produced. Under the action of capillary stress, a bridge of water molecules can be formed, and a concave surface appears between the particles. This effect will bond the particles together and constitutes the internal bond strength of the rock, as shown in Figure8.Figure 8
Model of grain-water molecule interactions.The attractive force,F, between particles includes the surface tension and capillary pressure. The capillary pressure, Pc, can be expressed as follows:(2)Pc=cosϕ+θ1−cosϕσR,where R is the particle radius, σ is the surface tension of water in the air, and θ is the contact angle. With decreasing particle radius, Pc approaches infinity, and the attractive force between particles can be expressed as follows:(3)F=2πRσsinϕsinϕ+θ+πRm2pc,where Rm is the radius of the concave water droplet, which can be calculated geometrically. The reaction force between the particles is expressed as follows:(4)F=sinϕsinϕ+θ+sinϕ+θ+cosϕ−1−sinθ221−cosθcosϕ+θ2πRσ.Equation (4) indicates that, after the specimen is water-saturated, water molecules enter the pores between particles, causing Rm to increase; F gradually decreases, and the cohesion force between particles in the rock decreases. Furthermore, the strength also decreases.Owing to the high pore density of coal, the cement strength between particles is lower. Coal also has a higher water absorption rate (Table1), resulting in a larger Rm for the coal specimens than of the sandstone. In the saturated state, the forces acting among particles in the coal specimens are weaker than in sandstone, and the strength is decreased observably.


### 3.2. Fracture Propagation Characteristics under Dynamic Load

The propagation and aggregation of microfractures within coal and rock is the fundamental cause of macrodamage to the coal and rock under external loading. To analyze the effect of free water in the fractures of saturated coal and rock on crack propagation under static-dynamic loading, this study simplifies the three-dimensional hydrous fracture to a plane fracture, and a single fracture is taken as an example. For the parameters of the hydrous single fracture, the static load isσs, dynamic load is σd, fracture length is 2a, and angle is β, as shown in Figure 9.Figure 9
Pressure at the crack surface caused by free water under coupled static-dynamic loads.


#### 3.2.1. Fracture Propagation Characteristics of Saturated Coal and Rock under Static-Dynamic Loads

During static loading, wing fractures will occur with increasing static load. The fracture propagation velocity is faster than the static loading speed. In addition, the fracture tip will fill with free water due to occurrence of a siphon phenomenon in the fracture tip.This results in a decrease in the friction coefficient of the fracture contact surface, increase in the microrupture activity of the coal and rock specimens, acceleration of fracture propagation, and concatenation and transfixion between fractures. Furthermore, the macrodestructive force of the specimens decreases, and the compressive strength of the specimens is also reduced. Wang and Li [13] investigated the force of free water in fracture propagation in coal and rock under static loads.The fracture growth characteristics of coal and rock are basically consistent under static loads. In the natural state, the number of fractures further increases as the coal and rock specimens absorb water. This occurs because of the low intensity of coal and its dense fractures, coupled with the corrosive effects of water. At the same time, the reaction force between particles decreases significantly [13]. This leads the velocity of the new fracture to grow faster in coal specimens under static loads than in the saturated sandstone. The strength reduction coefficient and elastic modulus reduction coefficient of the coal and rock specimens are smaller.


#### 3.2.2. Fracture Propagation Characteristics of Saturated Coal and Rock under Dynamic Loads

The propagation characteristics of coal and rock fractures under static-dynamic loading can be obtained from the mesomechanical analytical method of concrete and rock mechanics [25, 26]. Under static-dynamic loads, a cohesive force, F, is formed from the surface tension of free water in the fractures, and resistance and F′ result from the Stefan effect and impede the fracture growth and further fracturing. The force that hinders fracture propagation, pdw, can be expressed as follows:(5)pdw=F+F′A=VR/2δ2cosφ+3ηr4/2πh3du/dtA,where V is the liquid volume, R is the surface energy, φ is the wetting angle, δ is the radius of the water meniscus, η is the liquid viscosity, r is the radius of two parallel circular plates filled with incompressible viscous fluids, u is the relative displacement corresponding to the separation of the two circular plates, h is the space between the two circular plates, and A is the area of the fracture containing water.


#### 3.2.3. Stress Balance on Both Ends of the Specimen during Dynamic Impact Processes

The stress state under static-dynamic loading is shown in Figure10. In the image, σI is the incident stress, σR is the reflected stress, σT is the transmission stress, and σs is the prestatic load.Figure 10
Loading diagram of a specimen under static-dynamic load.The stress balance on both ends of the specimen is a precondition for a reasonable equivalent of the dynamic load. Zncker and Closer proposed applying an equilibrium factor,μ, to measure the stress equilibrium state of specimens; μ was defined as the ratio of the stress difference between the two ends to the average stress in the specimen, as given by Equation (6). As the equilibrium factor approaches zero, the stress in the specimens becomes more uniform. The equilibrium factor is calculated as follows:(6)μ=2σSI−σSTσSI+σST,where σSI is the incident end stress and σST is the transmission end stress.During the impact process, the small size of the specimen and the complicated transient change in the stress mean that the present technique cannot achieve direct measurement of the stress distribution on both ends. According to the theory of SHPB tests, for the equivalent stress on both ends of the specimen, the equivalent formula is as follows:(7)σST=σT,σSI=σI+σR.The trend in the equilibrium factor during the dynamic shock process for a specimen under prestatic loads is shown in Figure11.Figure 11
Balance factor for specimens under static-dynamic loads.The reflection wave,σT, is a platform, which indicates that the experiment has achieved loading at a constant strain rate. During the whole impact loading process, the stress at the incident and transmission ends are almost equal. The equilibrium factor, μ, tends to approximately zero by 30 μs after loading and is maintained to 140 μs. Therefore, the dynamic impact force of coal specimens is a reasonable equivalent for the quasi-static stress, σd, coupled with the prestatic load, σs.During the static-dynamic loads, a prestatic load,σs, is first applied, followed by the dynamic load, σd. σs−d is the composite failure strength of the coal specimen hydrous fracture under a combination load. The friction coefficient of the branch fracture is divided by the dynamic load factor ,fd, of the branch fracture surface that is not in contact with water and the dynamic load factor, fdw, of the branch fracture surface in contact with water, owing to the low strength of natural saturated coal specimens at the fracture surface and the uneven distribution of water in the branch fracture. Therefore, considering the difference in stress between the areas in the branch fracture surface that are in contact with water and those that are not in contact with water is necessary.As shown in Figure9, surface ab–cd is the free water interface, and a mechanical analysis of the stress structure of a wing branch fracture was carried out under static-dynamic loading. The compressive stress is assumed to be positive in this analysis. Thus, the shear stress, τd−sw, and normal stress, σd−sw, of the wing branch fracture surface in contact with free water, and the shear stress, τd−s, and normal stress, σd−s, of the wing branch fracture not in contact with water (out of the ab–cd surface) can be obtained. The above parameters can be expressed by equations (8)–(11), as follows:(8)τs−dw=12σs−dsin2β+θ−fdwσs−dcos2β+θ−psw−pdw,(9)τs−d=12σs−dsin2β+θ−fdσs−dcos2β+θ,(10)σs−dw=σs−dcos2β+θ−psw−pdw,(11)σs−d=σs−dcos2β+θ,where fd−sw is the friction coefficient of the hydrous fracture in contact with free water under dynamic load, fd is the friction coefficient of the hydrous fracture not in contact with free water under dynamic load, pdw is the stress that inhibits fracture extension, and psw is the outward extrusion stress of free water on an airfoil fracture. Numerically, pdw is far larger than psw in the dynamic impact process. Both of these are involved in fracture development and breakthrough.From Equations (8)–(11), Equations (12) and (13) can be obtained for the relative shear stress, τl, generated normal to surface ab–cd, and the relative normal stress, σl:(12)τl=τs−d−τs−dw=fdσs−dcos2β+θ−fdwσs−dcos2β+θ−psw−pdw,(13)τa=σs−d−σs−dw=psw−pdw,where τl is the tensile stress at the branch fracture surface near the ab–cd interface and τa is the shear stress at the branch fracture surface near the ab–cd interface.At the coal and rock branch fracture surface, the setting tensile strength isτb, and the shear strength is τf. When the branch fracture surface of saturated coal and rock meets one of the conditions τb<τl or τf<τa, the branch fracture surface will be damaged and a new fracture will form. The branch fracture surface of saturated coal and rock can produce resistance due to the Stefan effect under static-dynamic loads. The extension of the fracture at the branch fracture surface in contact with water will result in a pressure difference at the branch fracture surface. Thus, the new fracture will generate owing to the low strength of the material. In other words, when the material meets one of the two conditions above, destruction of the fracture will result in more energy for generation and extension of the new fracture. Thus, the elastic energy storage in the tip of the initial fracture will decrease, and the coal and rock material strength will also decrease. During dynamic loading, the stress inhomogeneity causes the extension of the initial fracture to lag behind the new fracture. With breakthrough of the initial fracture and generation of the new fracture, fragments will be formed. At the same time, a stress wave will provide kinetic energy to the fragments, resulting in throwing and flying of fragments.When the branch fracture surface of saturated coal and rock simultaneously meets bothτb>τl and τf>τa, a new fracture will not be formed on the fracture surface under dynamic load conditions. The propagation velocity of the dynamic fracture is faster than the static, and the propagation velocity of the fracture is much lower than the load. Therefore, fracture water cannot diffuse into the fracture tip in a relatively short time. Under the action of the surface tension of free water, the water on the fracture surface will produce a cohesion force pdw−psw. Under dynamic loading, because the materials simultaneously meet τb>τl and τf>τa, the branch fracture surface will not generate new fractures. Thus, the entire branch fracture extension will be hampered, which increases the strength of the saturated coal and rock.As mentioned above, coal has low intensity, more developed fractures, a small strength reduction coefficient when saturated, and the reaction force between particles is also lower. Thus, under static-dynamic loads, the relative tensile stress and shear stress generated on the branch fracture surface of a saturated coal specimen is much higher than the tensile strength and shear strength of the fracture surface, i.e.,τb<τl and τf<τa. This can cause branch fracture tensile failure and shear failure. The damage of the coal specimen and the expansion of fractures are accelerated, and the dynamic strength of the coal specimen is weakened. In contrast, the effect of water-rock interactions on sandstone is smaller, the strength reduction coefficient is larger, and it has a high strength and relatively low fracture density. Under the same conditions, the relative tensile stress and shear stress generated in the fracture surface of saturated sandstone under static-dynamic loads are much higher than tensile strength and shear strength of the fracture surface, i.e., τb>τl and τf>τa. Therefore, the extension of whole branch fracture of naturally saturated sandstone will be hampered due to the cohesion force generated from the Stefan effect. Additionally, generation of new fractures on the branch fracture surface of sandstone is also suppressed. The dynamic peak strength of the specimen is improved.Furthermore, during dynamic impact, the diffusion of free water lags behind the expansion of the new fracture because of generation of new fractures in the branch fracture surface of saturated coal specimens. Thus, the diffusion speed of free water is largely determined by the growth speed of the new fracture, which means that each primary and new fracture will contain free water. The area of the branch fracture surface in contact with water will produce a cohesive force hindering the branch fracture from growing. This reduces the compressibility of free water, and the ability of the coal specimen to resist deformation is enhanced. At this time, the greater the fracture area in contact with water, the higher the stiffness of the water-rock common loads will be. This provides a reason for the increase in the deformation modulus of coal specimens with moisture content under medium strain rate conditions.


## 3.1. The Force between Particles

Compared to coal, there are few fractures and pores in the sandstone. Additionally, sandstone has a denser structure, greater content of high strength materials, and weaker water-rock interaction than coal. Therefore, the strength and antideformation ability of coal decreases significantly under static loads, but the water has little effect on the strength and antideformation ability of sandstone. A comparison of the natural state and water-saturated state of a coal specimen is shown in Figure7.Figure 7
Sketch of the coal specimen in the natural state and water-saturated state. (a) Coal specimen D1-2. (b) Natural state. (c) Water-saturated state.
(a)
(b)
(c)Furthermore, before the coal and rock specimens are soaked in water, some water is contained internally, occurring on the surface of particles in the form of crystal water, pore water, and fracture water. Thus, an attractive effect will occur between the water and particles; at the same time, capillary pressure will also be produced. Under the action of capillary stress, a bridge of water molecules can be formed, and a concave surface appears between the particles. This effect will bond the particles together and constitutes the internal bond strength of the rock, as shown in Figure8.Figure 8
Model of grain-water molecule interactions.The attractive force,F, between particles includes the surface tension and capillary pressure. The capillary pressure, Pc, can be expressed as follows:(2)Pc=cosϕ+θ1−cosϕσR,where R is the particle radius, σ is the surface tension of water in the air, and θ is the contact angle. With decreasing particle radius, Pc approaches infinity, and the attractive force between particles can be expressed as follows:(3)F=2πRσsinϕsinϕ+θ+πRm2pc,where Rm is the radius of the concave water droplet, which can be calculated geometrically. The reaction force between the particles is expressed as follows:(4)F=sinϕsinϕ+θ+sinϕ+θ+cosϕ−1−sinθ221−cosθcosϕ+θ2πRσ.Equation (4) indicates that, after the specimen is water-saturated, water molecules enter the pores between particles, causing Rm to increase; F gradually decreases, and the cohesion force between particles in the rock decreases. Furthermore, the strength also decreases.Owing to the high pore density of coal, the cement strength between particles is lower. Coal also has a higher water absorption rate (Table1), resulting in a larger Rm for the coal specimens than of the sandstone. In the saturated state, the forces acting among particles in the coal specimens are weaker than in sandstone, and the strength is decreased observably.


## 3.2. Fracture Propagation Characteristics under Dynamic Load

The propagation and aggregation of microfractures within coal and rock is the fundamental cause of macrodamage to the coal and rock under external loading. To analyze the effect of free water in the fractures of saturated coal and rock on crack propagation under static-dynamic loading, this study simplifies the three-dimensional hydrous fracture to a plane fracture, and a single fracture is taken as an example. For the parameters of the hydrous single fracture, the static load isσs, dynamic load is σd, fracture length is 2a, and angle is β, as shown in Figure 9.Figure 9
Pressure at the crack surface caused by free water under coupled static-dynamic loads.


### 3.2.1. Fracture Propagation Characteristics of Saturated Coal and Rock under Static-Dynamic Loads

During static loading, wing fractures will occur with increasing static load. The fracture propagation velocity is faster than the static loading speed. In addition, the fracture tip will fill with free water due to occurrence of a siphon phenomenon in the fracture tip.This results in a decrease in the friction coefficient of the fracture contact surface, increase in the microrupture activity of the coal and rock specimens, acceleration of fracture propagation, and concatenation and transfixion between fractures. Furthermore, the macrodestructive force of the specimens decreases, and the compressive strength of the specimens is also reduced. Wang and Li [13] investigated the force of free water in fracture propagation in coal and rock under static loads.The fracture growth characteristics of coal and rock are basically consistent under static loads. In the natural state, the number of fractures further increases as the coal and rock specimens absorb water. This occurs because of the low intensity of coal and its dense fractures, coupled with the corrosive effects of water. At the same time, the reaction force between particles decreases significantly [13]. This leads the velocity of the new fracture to grow faster in coal specimens under static loads than in the saturated sandstone. The strength reduction coefficient and elastic modulus reduction coefficient of the coal and rock specimens are smaller.


### 3.2.2. Fracture Propagation Characteristics of Saturated Coal and Rock under Dynamic Loads

The propagation characteristics of coal and rock fractures under static-dynamic loading can be obtained from the mesomechanical analytical method of concrete and rock mechanics [25, 26]. Under static-dynamic loads, a cohesive force, F, is formed from the surface tension of free water in the fractures, and resistance and F′ result from the Stefan effect and impede the fracture growth and further fracturing. The force that hinders fracture propagation, pdw, can be expressed as follows:(5)pdw=F+F′A=VR/2δ2cosφ+3ηr4/2πh3du/dtA,where V is the liquid volume, R is the surface energy, φ is the wetting angle, δ is the radius of the water meniscus, η is the liquid viscosity, r is the radius of two parallel circular plates filled with incompressible viscous fluids, u is the relative displacement corresponding to the separation of the two circular plates, h is the space between the two circular plates, and A is the area of the fracture containing water.


### 3.2.3. Stress Balance on Both Ends of the Specimen during Dynamic Impact Processes

The stress state under static-dynamic loading is shown in Figure10. In the image, σI is the incident stress, σR is the reflected stress, σT is the transmission stress, and σs is the prestatic load.Figure 10
Loading diagram of a specimen under static-dynamic load.The stress balance on both ends of the specimen is a precondition for a reasonable equivalent of the dynamic load. Zncker and Closer proposed applying an equilibrium factor,μ, to measure the stress equilibrium state of specimens; μ was defined as the ratio of the stress difference between the two ends to the average stress in the specimen, as given by Equation (6). As the equilibrium factor approaches zero, the stress in the specimens becomes more uniform. The equilibrium factor is calculated as follows:(6)μ=2σSI−σSTσSI+σST,where σSI is the incident end stress and σST is the transmission end stress.During the impact process, the small size of the specimen and the complicated transient change in the stress mean that the present technique cannot achieve direct measurement of the stress distribution on both ends. According to the theory of SHPB tests, for the equivalent stress on both ends of the specimen, the equivalent formula is as follows:(7)σST=σT,σSI=σI+σR.The trend in the equilibrium factor during the dynamic shock process for a specimen under prestatic loads is shown in Figure11.Figure 11
Balance factor for specimens under static-dynamic loads.The reflection wave,σT, is a platform, which indicates that the experiment has achieved loading at a constant strain rate. During the whole impact loading process, the stress at the incident and transmission ends are almost equal. The equilibrium factor, μ, tends to approximately zero by 30 μs after loading and is maintained to 140 μs. Therefore, the dynamic impact force of coal specimens is a reasonable equivalent for the quasi-static stress, σd, coupled with the prestatic load, σs.During the static-dynamic loads, a prestatic load,σs, is first applied, followed by the dynamic load, σd. σs−d is the composite failure strength of the coal specimen hydrous fracture under a combination load. The friction coefficient of the branch fracture is divided by the dynamic load factor ,fd, of the branch fracture surface that is not in contact with water and the dynamic load factor, fdw, of the branch fracture surface in contact with water, owing to the low strength of natural saturated coal specimens at the fracture surface and the uneven distribution of water in the branch fracture. Therefore, considering the difference in stress between the areas in the branch fracture surface that are in contact with water and those that are not in contact with water is necessary.As shown in Figure9, surface ab–cd is the free water interface, and a mechanical analysis of the stress structure of a wing branch fracture was carried out under static-dynamic loading. The compressive stress is assumed to be positive in this analysis. Thus, the shear stress, τd−sw, and normal stress, σd−sw, of the wing branch fracture surface in contact with free water, and the shear stress, τd−s, and normal stress, σd−s, of the wing branch fracture not in contact with water (out of the ab–cd surface) can be obtained. The above parameters can be expressed by equations (8)–(11), as follows:(8)τs−dw=12σs−dsin2β+θ−fdwσs−dcos2β+θ−psw−pdw,(9)τs−d=12σs−dsin2β+θ−fdσs−dcos2β+θ,(10)σs−dw=σs−dcos2β+θ−psw−pdw,(11)σs−d=σs−dcos2β+θ,where fd−sw is the friction coefficient of the hydrous fracture in contact with free water under dynamic load, fd is the friction coefficient of the hydrous fracture not in contact with free water under dynamic load, pdw is the stress that inhibits fracture extension, and psw is the outward extrusion stress of free water on an airfoil fracture. Numerically, pdw is far larger than psw in the dynamic impact process. Both of these are involved in fracture development and breakthrough.From Equations (8)–(11), Equations (12) and (13) can be obtained for the relative shear stress, τl, generated normal to surface ab–cd, and the relative normal stress, σl:(12)τl=τs−d−τs−dw=fdσs−dcos2β+θ−fdwσs−dcos2β+θ−psw−pdw,(13)τa=σs−d−σs−dw=psw−pdw,where τl is the tensile stress at the branch fracture surface near the ab–cd interface and τa is the shear stress at the branch fracture surface near the ab–cd interface.At the coal and rock branch fracture surface, the setting tensile strength isτb, and the shear strength is τf. When the branch fracture surface of saturated coal and rock meets one of the conditions τb<τl or τf<τa, the branch fracture surface will be damaged and a new fracture will form. The branch fracture surface of saturated coal and rock can produce resistance due to the Stefan effect under static-dynamic loads. The extension of the fracture at the branch fracture surface in contact with water will result in a pressure difference at the branch fracture surface. Thus, the new fracture will generate owing to the low strength of the material. In other words, when the material meets one of the two conditions above, destruction of the fracture will result in more energy for generation and extension of the new fracture. Thus, the elastic energy storage in the tip of the initial fracture will decrease, and the coal and rock material strength will also decrease. During dynamic loading, the stress inhomogeneity causes the extension of the initial fracture to lag behind the new fracture. With breakthrough of the initial fracture and generation of the new fracture, fragments will be formed. At the same time, a stress wave will provide kinetic energy to the fragments, resulting in throwing and flying of fragments.When the branch fracture surface of saturated coal and rock simultaneously meets bothτb>τl and τf>τa, a new fracture will not be formed on the fracture surface under dynamic load conditions. The propagation velocity of the dynamic fracture is faster than the static, and the propagation velocity of the fracture is much lower than the load. Therefore, fracture water cannot diffuse into the fracture tip in a relatively short time. Under the action of the surface tension of free water, the water on the fracture surface will produce a cohesion force pdw−psw. Under dynamic loading, because the materials simultaneously meet τb>τl and τf>τa, the branch fracture surface will not generate new fractures. Thus, the entire branch fracture extension will be hampered, which increases the strength of the saturated coal and rock.As mentioned above, coal has low intensity, more developed fractures, a small strength reduction coefficient when saturated, and the reaction force between particles is also lower. Thus, under static-dynamic loads, the relative tensile stress and shear stress generated on the branch fracture surface of a saturated coal specimen is much higher than the tensile strength and shear strength of the fracture surface, i.e.,τb<τl and τf<τa. This can cause branch fracture tensile failure and shear failure. The damage of the coal specimen and the expansion of fractures are accelerated, and the dynamic strength of the coal specimen is weakened. In contrast, the effect of water-rock interactions on sandstone is smaller, the strength reduction coefficient is larger, and it has a high strength and relatively low fracture density. Under the same conditions, the relative tensile stress and shear stress generated in the fracture surface of saturated sandstone under static-dynamic loads are much higher than tensile strength and shear strength of the fracture surface, i.e., τb>τl and τf>τa. Therefore, the extension of whole branch fracture of naturally saturated sandstone will be hampered due to the cohesion force generated from the Stefan effect. Additionally, generation of new fractures on the branch fracture surface of sandstone is also suppressed. The dynamic peak strength of the specimen is improved.Furthermore, during dynamic impact, the diffusion of free water lags behind the expansion of the new fracture because of generation of new fractures in the branch fracture surface of saturated coal specimens. Thus, the diffusion speed of free water is largely determined by the growth speed of the new fracture, which means that each primary and new fracture will contain free water. The area of the branch fracture surface in contact with water will produce a cohesive force hindering the branch fracture from growing. This reduces the compressibility of free water, and the ability of the coal specimen to resist deformation is enhanced. At this time, the greater the fracture area in contact with water, the higher the stiffness of the water-rock common loads will be. This provides a reason for the increase in the deformation modulus of coal specimens with moisture content under medium strain rate conditions.


## 3.2.1. Fracture Propagation Characteristics of Saturated Coal and Rock under Static-Dynamic Loads

During static loading, wing fractures will occur with increasing static load. The fracture propagation velocity is faster than the static loading speed. In addition, the fracture tip will fill with free water due to occurrence of a siphon phenomenon in the fracture tip.This results in a decrease in the friction coefficient of the fracture contact surface, increase in the microrupture activity of the coal and rock specimens, acceleration of fracture propagation, and concatenation and transfixion between fractures. Furthermore, the macrodestructive force of the specimens decreases, and the compressive strength of the specimens is also reduced. Wang and Li [13] investigated the force of free water in fracture propagation in coal and rock under static loads.The fracture growth characteristics of coal and rock are basically consistent under static loads. In the natural state, the number of fractures further increases as the coal and rock specimens absorb water. This occurs because of the low intensity of coal and its dense fractures, coupled with the corrosive effects of water. At the same time, the reaction force between particles decreases significantly [13]. This leads the velocity of the new fracture to grow faster in coal specimens under static loads than in the saturated sandstone. The strength reduction coefficient and elastic modulus reduction coefficient of the coal and rock specimens are smaller.


## 3.2.2. Fracture Propagation Characteristics of Saturated Coal and Rock under Dynamic Loads

The propagation characteristics of coal and rock fractures under static-dynamic loading can be obtained from the mesomechanical analytical method of concrete and rock mechanics [25, 26]. Under static-dynamic loads, a cohesive force, F, is formed from the surface tension of free water in the fractures, and resistance and F′ result from the Stefan effect and impede the fracture growth and further fracturing. The force that hinders fracture propagation, pdw, can be expressed as follows:(5)pdw=F+F′A=VR/2δ2cosφ+3ηr4/2πh3du/dtA,where V is the liquid volume, R is the surface energy, φ is the wetting angle, δ is the radius of the water meniscus, η is the liquid viscosity, r is the radius of two parallel circular plates filled with incompressible viscous fluids, u is the relative displacement corresponding to the separation of the two circular plates, h is the space between the two circular plates, and A is the area of the fracture containing water.


## 3.2.3. Stress Balance on Both Ends of the Specimen during Dynamic Impact Processes

The stress state under static-dynamic loading is shown in Figure10. In the image, σI is the incident stress, σR is the reflected stress, σT is the transmission stress, and σs is the prestatic load.Figure 10
Loading diagram of a specimen under static-dynamic load.The stress balance on both ends of the specimen is a precondition for a reasonable equivalent of the dynamic load. Zncker and Closer proposed applying an equilibrium factor,μ, to measure the stress equilibrium state of specimens; μ was defined as the ratio of the stress difference between the two ends to the average stress in the specimen, as given by Equation (6). As the equilibrium factor approaches zero, the stress in the specimens becomes more uniform. The equilibrium factor is calculated as follows:(6)μ=2σSI−σSTσSI+σST,where σSI is the incident end stress and σST is the transmission end stress.During the impact process, the small size of the specimen and the complicated transient change in the stress mean that the present technique cannot achieve direct measurement of the stress distribution on both ends. According to the theory of SHPB tests, for the equivalent stress on both ends of the specimen, the equivalent formula is as follows:(7)σST=σT,σSI=σI+σR.The trend in the equilibrium factor during the dynamic shock process for a specimen under prestatic loads is shown in Figure11.Figure 11
Balance factor for specimens under static-dynamic loads.The reflection wave,σT, is a platform, which indicates that the experiment has achieved loading at a constant strain rate. During the whole impact loading process, the stress at the incident and transmission ends are almost equal. The equilibrium factor, μ, tends to approximately zero by 30 μs after loading and is maintained to 140 μs. Therefore, the dynamic impact force of coal specimens is a reasonable equivalent for the quasi-static stress, σd, coupled with the prestatic load, σs.During the static-dynamic loads, a prestatic load,σs, is first applied, followed by the dynamic load, σd. σs−d is the composite failure strength of the coal specimen hydrous fracture under a combination load. The friction coefficient of the branch fracture is divided by the dynamic load factor ,fd, of the branch fracture surface that is not in contact with water and the dynamic load factor, fdw, of the branch fracture surface in contact with water, owing to the low strength of natural saturated coal specimens at the fracture surface and the uneven distribution of water in the branch fracture. Therefore, considering the difference in stress between the areas in the branch fracture surface that are in contact with water and those that are not in contact with water is necessary.As shown in Figure9, surface ab–cd is the free water interface, and a mechanical analysis of the stress structure of a wing branch fracture was carried out under static-dynamic loading. The compressive stress is assumed to be positive in this analysis. Thus, the shear stress, τd−sw, and normal stress, σd−sw, of the wing branch fracture surface in contact with free water, and the shear stress, τd−s, and normal stress, σd−s, of the wing branch fracture not in contact with water (out of the ab–cd surface) can be obtained. The above parameters can be expressed by equations (8)–(11), as follows:(8)τs−dw=12σs−dsin2β+θ−fdwσs−dcos2β+θ−psw−pdw,(9)τs−d=12σs−dsin2β+θ−fdσs−dcos2β+θ,(10)σs−dw=σs−dcos2β+θ−psw−pdw,(11)σs−d=σs−dcos2β+θ,where fd−sw is the friction coefficient of the hydrous fracture in contact with free water under dynamic load, fd is the friction coefficient of the hydrous fracture not in contact with free water under dynamic load, pdw is the stress that inhibits fracture extension, and psw is the outward extrusion stress of free water on an airfoil fracture. Numerically, pdw is far larger than psw in the dynamic impact process. Both of these are involved in fracture development and breakthrough.From Equations (8)–(11), Equations (12) and (13) can be obtained for the relative shear stress, τl, generated normal to surface ab–cd, and the relative normal stress, σl:(12)τl=τs−d−τs−dw=fdσs−dcos2β+θ−fdwσs−dcos2β+θ−psw−pdw,(13)τa=σs−d−σs−dw=psw−pdw,where τl is the tensile stress at the branch fracture surface near the ab–cd interface and τa is the shear stress at the branch fracture surface near the ab–cd interface.At the coal and rock branch fracture surface, the setting tensile strength isτb, and the shear strength is τf. When the branch fracture surface of saturated coal and rock meets one of the conditions τb<τl or τf<τa, the branch fracture surface will be damaged and a new fracture will form. The branch fracture surface of saturated coal and rock can produce resistance due to the Stefan effect under static-dynamic loads. The extension of the fracture at the branch fracture surface in contact with water will result in a pressure difference at the branch fracture surface. Thus, the new fracture will generate owing to the low strength of the material. In other words, when the material meets one of the two conditions above, destruction of the fracture will result in more energy for generation and extension of the new fracture. Thus, the elastic energy storage in the tip of the initial fracture will decrease, and the coal and rock material strength will also decrease. During dynamic loading, the stress inhomogeneity causes the extension of the initial fracture to lag behind the new fracture. With breakthrough of the initial fracture and generation of the new fracture, fragments will be formed. At the same time, a stress wave will provide kinetic energy to the fragments, resulting in throwing and flying of fragments.When the branch fracture surface of saturated coal and rock simultaneously meets bothτb>τl and τf>τa, a new fracture will not be formed on the fracture surface under dynamic load conditions. The propagation velocity of the dynamic fracture is faster than the static, and the propagation velocity of the fracture is much lower than the load. Therefore, fracture water cannot diffuse into the fracture tip in a relatively short time. Under the action of the surface tension of free water, the water on the fracture surface will produce a cohesion force pdw−psw. Under dynamic loading, because the materials simultaneously meet τb>τl and τf>τa, the branch fracture surface will not generate new fractures. Thus, the entire branch fracture extension will be hampered, which increases the strength of the saturated coal and rock.As mentioned above, coal has low intensity, more developed fractures, a small strength reduction coefficient when saturated, and the reaction force between particles is also lower. Thus, under static-dynamic loads, the relative tensile stress and shear stress generated on the branch fracture surface of a saturated coal specimen is much higher than the tensile strength and shear strength of the fracture surface, i.e.,τb<τl and τf<τa. This can cause branch fracture tensile failure and shear failure. The damage of the coal specimen and the expansion of fractures are accelerated, and the dynamic strength of the coal specimen is weakened. In contrast, the effect of water-rock interactions on sandstone is smaller, the strength reduction coefficient is larger, and it has a high strength and relatively low fracture density. Under the same conditions, the relative tensile stress and shear stress generated in the fracture surface of saturated sandstone under static-dynamic loads are much higher than tensile strength and shear strength of the fracture surface, i.e., τb>τl and τf>τa. Therefore, the extension of whole branch fracture of naturally saturated sandstone will be hampered due to the cohesion force generated from the Stefan effect. Additionally, generation of new fractures on the branch fracture surface of sandstone is also suppressed. The dynamic peak strength of the specimen is improved.Furthermore, during dynamic impact, the diffusion of free water lags behind the expansion of the new fracture because of generation of new fractures in the branch fracture surface of saturated coal specimens. Thus, the diffusion speed of free water is largely determined by the growth speed of the new fracture, which means that each primary and new fracture will contain free water. The area of the branch fracture surface in contact with water will produce a cohesive force hindering the branch fracture from growing. This reduces the compressibility of free water, and the ability of the coal specimen to resist deformation is enhanced. At this time, the greater the fracture area in contact with water, the higher the stiffness of the water-rock common loads will be. This provides a reason for the increase in the deformation modulus of coal specimens with moisture content under medium strain rate conditions.


## 4. Conclusion

(1)
Under static loading conditions, the saturated strength reduction coefficient and elastic modulus reduction coefficient of coal samples are lower than those of sandstone. Under dynamic loading conditions, the strength of naturally saturated coal samples decreases and the elastic modulus increases. This is the same strength variation trend and opposite elastic modulus variation trend as observed for sandstone.
(2)
The water-saturated coal specimens have greater water-rock interactions and strength reduction effect; its reduction in the ability to resist deformation is also very significant compared to saturated sandstone.
(3)
Under static-dynamic loads, owing to intensity differences in the saturated coal and rock materials, a stress difference results from the Stefan effect in the branch fracture surface, which is the main reason that the strength of coal and rock materials increases or decreases. Meanwhile, fracture propagation at saturated branch fracture surfaces can effectively explain the increase in the deformation modulus of coal with increasing moisture content.
(4)
According to a sliding model of wing branch fracture, the criteria for microfracture tensile stress and shear stress produced at the branch fracture surface were established. If the branch fracture surface simultaneously satisfies bothτb>τl and τf>τa, the branch fracture surface will not break down. The dynamic strength of water-saturated coal and rock will increase. If the branch fracture surface meets only one of the conditions τb<τl or τf<τa, the branch fracture surface will break down and the dynamic strength of water-saturated coal and rock will decrease.



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*Source: 5125923-2019-01-01.xml*