# Failure Behavior of Thin Disc Sandstone under Coupled Bending-Seepage Condition

Authors: Luzhen Wang; Hailing Kong
Journal: Geofluids
 (2020)

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## Abstract

A floor aquifuge usually bends and fails when mining above a confined aquifer, which may lead to water inrush disaster. The floor aquifuge was simplified as a thin disc model in this paper, and a series of coupled bending-seepage tests of sandstone were carried out by a patent test system. The variation characteristics of load-displacement, load-time, and permeability-time were analyzed. The deflection and stress in the thin disc rock samples were deduced; the initiation and propagation of cracks were analyzed. The failure behavior of the thin disc rock samples was described. It shows the following: (1) The bending failure behavior relates to the stress distribution and crack evolution inside the thin disc. (2) The main cause of crack initiation is a tension-shear failure. (3) The 5 mm thickness thin discs form petal-shaped cracks, due to tensile stress, while petal-shaped cracks only appear at the cap block of the 10 mm discs, which are sheared into two pieces along the conical surface with an inclination about 45°. (4) Water inrush occurs after bending failure in the floor aquifuge, and it is an opportune moment to grout along the crack propagation trend lines to prevent the water inrush disasters.


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## Body

## 1. Introduction

Due to the gradual depletion of shallow mineral resources, mining activities are shifted to the deep. In deep mining activities, geological conditions become complex; confined water pressure increased. Continuous deep mining may result in bending deformation and failure of the floor aquifuge, which may link the groundwater to the working face, and induce water inrush accidents [1–4]. Hence, the bending failure of the floor above a confined aquifer is one of the reasons to induce water inrush.Scholars began to study the mechanism of floor breakage and water inrush since the last century. Some important theories have been proposed successively, such as the key stratum theory, three-underlying belts theory, in situ rifting and zero failure theory, relative aquifuge thickness and water inrush coefficient method, and water inrush critical index method [5–9]. In recent years, due to the complex geological background and mining conditions, the floor breakage and water inrush mechanisms have been extensively and deeply studied based on the above theories, and new achievements emerged endlessly.A large-scale model test is one of the effective means to reproduce the process of water inrush from floor breakage. Zhao et al. [10, 11] developed a large-scale three-dimensional solid-fluid coupling simulation experimental equipment, analyzed the wall rock distortion and breakage process, and described the variation law of stress and water pressure and their influence on water inrush. Li et al. [12] divided the mining-induced floor stresses into three distinct stages with the help of a fluid-solid coupling model test system. Chen et al. [13] described the crack formation and evolution law in the floor aquifuge by the solid-fluid coupling similar simulation test system and found that most mining cracks were vertical tensile cracks, shear cracks, and layer cracks. Kang et al. [14] studied the failure law of the floor after mining using a large-scale simulation test system and divided the floor fracture belt into an open-off-cut fracture zone, central goaf compaction zone, and working-face fracture zone. Liu et al. [15] found that the failure range morphological map of the mining floor was approximately a spoon shape and obtained the maximum failure depth of the floor aquifuge by similar simulation tests. Based on the multifunctional simulation test apparatus, Hua et al. [16] pointed out that the floor heave was mainly composed of the compression bending type, shear dislocation type, and extrusion fluidity type, and the floor heave was mainly caused by the tensile failure or tensile-shear failure.The similar model test can reproduce the process of floor’s breakage and water inrush, but it still has some disadvantages; for example, the model sizes are usually too large, cost too much, and last for long test periods. Therefore, theoretical research and numerical calculation are essential.Miao et al. [17] simplified the floor aquifuge to a rock beam model with both ends fixed and loaded uniformly and analyzed the strength characteristics and water-resisting performance of the floor aquifuge. Sun and Wang [18] regarded the floor aquifuge as a square elastic thin plate with fixed constraints on four sides and deduced an instability mechanics criterion of the inclined floor water-resisting key strata using the elastic thin plate theory. Gao et al. [19] established a fracture mechanics model based on the thin plate theory and regarded that the essence of water inrush in the mining field was the result of the accumulation of microcracks and propagation of macrocracks in the floor aquifuge caused by mining-induced stress and confined water pressure. Wang et al. [20] established a plate model subjected to horizontal extrusion force, deduced the critical load of floor buckling failure based on the thin plate yield theory, and obtained that the floor thickness was the main factor affecting the critical load. Guo et al. [21] analyzed the distribution of displacement and pore water pressure on the floor using Comsol Multiphysics and pointed out that the key cause of water inrush was the release of energy from the floor above the confined water aquifer when mining. Pang [22] simplified the layered floor to homogeneous elastic plates, calculated the floor failure depth by means of the average modulus method, and considered that the lithology and composite structure of the floor are two important factors affecting the failure depth. Xie [23] used Flac3D to analyze the damage law of elastic modulus of the floor aquifuge caused by mining qualitatively and found that the floor crack rate controlled the stress distribution in the floor and the thickness of floor aquifuge played an important role in controlling the displacement and failure depth of the floor. Obviously, the mechanism of water inrush can be clarified by mastering the failure behavior in the floor aquifuge.The research above studied the stress distribution [10–12, 17, 23], displacement change [10, 11, 21, 23], crack evolution [13, 14, 19, 23], and failure law [15, 16, 18, 20, 22] in the floor aquifuge. The achievements reach a consensus that (1) the essence of water inrush from the floor aquifuge is the result of the combined action of mining-induced stress and confined water pressure, (2) the stress variation and crack evolution in the floor may lead to water inrush, and (3) the thickness and lithology of the floor aquifuge are two important factors affecting the failure behavior of the floor. However, the above studies ignored the correlation between the distribution of stress and the propagation of cracks when the floor aquifuge was bending, and the formation and propagation of cracks have not been described in-depth, so the bending failure characteristics and the induced water inrush behavior in the floor aquifuge need to be studied further.To this end, this paper established a thin disc model, conducted the bending-seepage tests of different thicknesses and lithology under the coupled bending-seepage condition, calculated the deflection and stresses inside the thin disc gray sandstone and red sandstone, revealed the formation and propagation mechanism of the cracks in the thin disc rock with different thicknesses and lithology, and described the bending failure behavior in the floor aquifuge, as well as the bending-failure-induced water inrush behavior. This research is expected to provide references for the description of floor failure and the prevention of water inrush.


## 2. Simplified Model and Experimental Process

### 2.1. Simplified Model

With coal extraction continuing, the confined water pressure and the mining-induced stress become the force sources. Subject to their joint action, the floor aquifuge undergoes bending deformation. In the bending floor aquifuge, the cracks are formed, developed, and penetrated, and water flow changes from seepage to turbulence and rushes into the working face, which may cause water inrush disaster.A simplified model is shown in Figure1, established based on the following assumptions:(1)
The floor aquifuge in practical engineering is assumed as a circular thin plate relative to the stratum; the self-weight of the thin disc rock sample is ignored.(2)
Owing to the strength and stiffness of the surrounding rock mass that are far greater than those of the floor aquifuge, the constraint of the thin disc rock sample is peripheral clamping.(3)
The floor aquifuge is assumed above the confined aquifer. The confined water pressure is regarded as a uniformly distributed pore pressure,q, distributed on the lower surface of the thin disc rock sample.(4)
Excavation disturbances induce floor heave; the upward bending deformation is applied to the thin disc rock sample, which is realized by applying an upward concentrated force,P, on the lower surface of the thin disc rock sample by a conical indenter.Figure 1
Simplified model for floor aquifuge.
(a)(b)(c)As shown in Figure1(c), the peripheral clamping is equivalent to a force couple, M, and a force, F/2πa. a is the radius of the thin disc rock sample, F is a circumferential load applied by the Material Testing Machine, and a retaining ring is used to act on the annular upper surface of the thin disc rock sample.


### 2.2. Testing System

In order to simulate the heave and breakage of the floor aquifuge and realize the coupled bending deformation and water flow applied on the floor aquifuge, a testing system that can conduct the coupling bending-seepage test is designed and manufactured [3, 24]. It consists of an axial loading subsystem, a pore pressure loading and controlling subsystem, a permeameter subsystem, and a data acquisition and analysis subsystem. The overall design of the testing system is shown in Figure 2.Figure 2
Testing system.As the core of the testing system, the permeameter subsystem contains a bottom plate, a cylinder, a permeable piston, a conical indenter, a retaining ring, seals, etc., shown in Figure3. The thin disc rock sample is fixed between the retaining ring and the conical indenter. It is critical to tightly seal the interspace between the outer boundary of the sample and the cylinder wall using high water materials and to ensure that the liquid only flows within the rock sample but not seeps out from surrounding voids before and during the loading process.Figure 3
Schematic of the permeameter.The test adopted the steady-state permeation method. Stable pore water pressure was provided by driving oil pressure in the pore pressure loading and controlling subsystem. Water flowed through the pressure transducer, flow transducer, and the inlet at the bottom plate, entered the conical indenter, and evenly distributed on the lower surface of the thin disc rock sample to simulate the action of the confined water pressure on the floor aquifuge. An axial load was applied onto the piston of the permeameter, then loaded on the upper surface of the thin disc rock sample through the retaining ring as circumferential load. Under the bidirectional actions of concentrated force from the conical indenter and circumferential load from the retaining ring, the rock sample underwent bending deformation, which was used to simulate the floor heave bending deformation of the floor aquifuge due to excavation disturbances.


### 2.3. Testing Scheme

#### 2.3.1. Material Characterization and Sample Preparation

The floor aquifuge in Pan mine in Sichuan is taken as the example in this paper, which is mainly gray sandstone and red sandstone. All samples were acquired from a single rock block to ensure their similar physical properties, respectively. The physical and mechanical properties are listed in Table1.Table 1
Physical and mechanical properties of gray sandstone and red sandstone.
LithologyGray sandstoneRed sandstoneUniaxial compressive strength (σc, MPa)54.751.2Tensile strength (σt, MPa)5.024.71Elastic modulus (E, GPa)32.029.2Poisson ratio (υ)0.230.16Density (ρ, kgm-3)24732574Internal friction angle (ϕ, °)47.843.0Cohesion (C, MPa)21.222.9Based on the simplified model of the floor aquifuge and the diameter of the permeameter cylinder, the rock sample was processed as a 50 mm diameter disc. According to the definition of the thin plate theory, the thickness-diameter ratio should be less than 1/5. Therefore, two thicknesses, 5 mm and 10 mm, are selected in this research.Samples were examined before testing to exclude those with obvious macroscopic defects and to ensure the testing values free from the impacts of macrojoints and fissures. The structures of the selected thin disc rock samples were compact, and there was no visible natural microfissure. Then, thin disc rock samples were polished to keep the surface smooth at two ends.Taking lithology and thickness as the influencing factors, four independent tests were carried out, marked GS05, GS10, RS05, and RS10. Three samples were tested in each independent test.


#### 2.3.2. Testing Design

At the beginning of testing, water was injected into the permeameter to saturation at least half an hour. Then, the pore pressure was loaded, and the sealing performance of the system was timely checked. According to relevant geological data, the pore pressure was set as 2 MPa in this test. When the pore pressure was completely stable at 2 MPa, the rock sample was loaded by the axial loading subsystem in the displacement control mode at a loading rate of 0.5 mm/min till the rock sample failed.In addition, the pore pressureq and water flow Q were simultaneously acquired from the beginning of loading; the load F and displacement u during bending deformation were real-time recorded.Permeability,k, can be calculated by Darcy’s law as below:(1)k=μvq/δ,where μ is the dynamic viscosity of water, v is the seepage speed calculated from v=Q/πa2, q is the pore pressure, and δ is the thickness of the thin disc rock sample.


## 2.1. Simplified Model

With coal extraction continuing, the confined water pressure and the mining-induced stress become the force sources. Subject to their joint action, the floor aquifuge undergoes bending deformation. In the bending floor aquifuge, the cracks are formed, developed, and penetrated, and water flow changes from seepage to turbulence and rushes into the working face, which may cause water inrush disaster.A simplified model is shown in Figure1, established based on the following assumptions:(1)
The floor aquifuge in practical engineering is assumed as a circular thin plate relative to the stratum; the self-weight of the thin disc rock sample is ignored.(2)
Owing to the strength and stiffness of the surrounding rock mass that are far greater than those of the floor aquifuge, the constraint of the thin disc rock sample is peripheral clamping.(3)
The floor aquifuge is assumed above the confined aquifer. The confined water pressure is regarded as a uniformly distributed pore pressure,q, distributed on the lower surface of the thin disc rock sample.(4)
Excavation disturbances induce floor heave; the upward bending deformation is applied to the thin disc rock sample, which is realized by applying an upward concentrated force,P, on the lower surface of the thin disc rock sample by a conical indenter.Figure 1
Simplified model for floor aquifuge.
(a)(b)(c)As shown in Figure1(c), the peripheral clamping is equivalent to a force couple, M, and a force, F/2πa. a is the radius of the thin disc rock sample, F is a circumferential load applied by the Material Testing Machine, and a retaining ring is used to act on the annular upper surface of the thin disc rock sample.


## 2.2. Testing System

In order to simulate the heave and breakage of the floor aquifuge and realize the coupled bending deformation and water flow applied on the floor aquifuge, a testing system that can conduct the coupling bending-seepage test is designed and manufactured [3, 24]. It consists of an axial loading subsystem, a pore pressure loading and controlling subsystem, a permeameter subsystem, and a data acquisition and analysis subsystem. The overall design of the testing system is shown in Figure 2.Figure 2
Testing system.As the core of the testing system, the permeameter subsystem contains a bottom plate, a cylinder, a permeable piston, a conical indenter, a retaining ring, seals, etc., shown in Figure3. The thin disc rock sample is fixed between the retaining ring and the conical indenter. It is critical to tightly seal the interspace between the outer boundary of the sample and the cylinder wall using high water materials and to ensure that the liquid only flows within the rock sample but not seeps out from surrounding voids before and during the loading process.Figure 3
Schematic of the permeameter.The test adopted the steady-state permeation method. Stable pore water pressure was provided by driving oil pressure in the pore pressure loading and controlling subsystem. Water flowed through the pressure transducer, flow transducer, and the inlet at the bottom plate, entered the conical indenter, and evenly distributed on the lower surface of the thin disc rock sample to simulate the action of the confined water pressure on the floor aquifuge. An axial load was applied onto the piston of the permeameter, then loaded on the upper surface of the thin disc rock sample through the retaining ring as circumferential load. Under the bidirectional actions of concentrated force from the conical indenter and circumferential load from the retaining ring, the rock sample underwent bending deformation, which was used to simulate the floor heave bending deformation of the floor aquifuge due to excavation disturbances.


## 2.3. Testing Scheme

### 2.3.1. Material Characterization and Sample Preparation

The floor aquifuge in Pan mine in Sichuan is taken as the example in this paper, which is mainly gray sandstone and red sandstone. All samples were acquired from a single rock block to ensure their similar physical properties, respectively. The physical and mechanical properties are listed in Table1.Table 1
Physical and mechanical properties of gray sandstone and red sandstone.
LithologyGray sandstoneRed sandstoneUniaxial compressive strength (σc, MPa)54.751.2Tensile strength (σt, MPa)5.024.71Elastic modulus (E, GPa)32.029.2Poisson ratio (υ)0.230.16Density (ρ, kgm-3)24732574Internal friction angle (ϕ, °)47.843.0Cohesion (C, MPa)21.222.9Based on the simplified model of the floor aquifuge and the diameter of the permeameter cylinder, the rock sample was processed as a 50 mm diameter disc. According to the definition of the thin plate theory, the thickness-diameter ratio should be less than 1/5. Therefore, two thicknesses, 5 mm and 10 mm, are selected in this research.Samples were examined before testing to exclude those with obvious macroscopic defects and to ensure the testing values free from the impacts of macrojoints and fissures. The structures of the selected thin disc rock samples were compact, and there was no visible natural microfissure. Then, thin disc rock samples were polished to keep the surface smooth at two ends.Taking lithology and thickness as the influencing factors, four independent tests were carried out, marked GS05, GS10, RS05, and RS10. Three samples were tested in each independent test.


### 2.3.2. Testing Design

At the beginning of testing, water was injected into the permeameter to saturation at least half an hour. Then, the pore pressure was loaded, and the sealing performance of the system was timely checked. According to relevant geological data, the pore pressure was set as 2 MPa in this test. When the pore pressure was completely stable at 2 MPa, the rock sample was loaded by the axial loading subsystem in the displacement control mode at a loading rate of 0.5 mm/min till the rock sample failed.In addition, the pore pressureq and water flow Q were simultaneously acquired from the beginning of loading; the load F and displacement u during bending deformation were real-time recorded.Permeability,k, can be calculated by Darcy’s law as below:(1)k=μvq/δ,where μ is the dynamic viscosity of water, v is the seepage speed calculated from v=Q/πa2, q is the pore pressure, and δ is the thickness of the thin disc rock sample.


## 2.3.1. Material Characterization and Sample Preparation

The floor aquifuge in Pan mine in Sichuan is taken as the example in this paper, which is mainly gray sandstone and red sandstone. All samples were acquired from a single rock block to ensure their similar physical properties, respectively. The physical and mechanical properties are listed in Table1.Table 1
Physical and mechanical properties of gray sandstone and red sandstone.
LithologyGray sandstoneRed sandstoneUniaxial compressive strength (σc, MPa)54.751.2Tensile strength (σt, MPa)5.024.71Elastic modulus (E, GPa)32.029.2Poisson ratio (υ)0.230.16Density (ρ, kgm-3)24732574Internal friction angle (ϕ, °)47.843.0Cohesion (C, MPa)21.222.9Based on the simplified model of the floor aquifuge and the diameter of the permeameter cylinder, the rock sample was processed as a 50 mm diameter disc. According to the definition of the thin plate theory, the thickness-diameter ratio should be less than 1/5. Therefore, two thicknesses, 5 mm and 10 mm, are selected in this research.Samples were examined before testing to exclude those with obvious macroscopic defects and to ensure the testing values free from the impacts of macrojoints and fissures. The structures of the selected thin disc rock samples were compact, and there was no visible natural microfissure. Then, thin disc rock samples were polished to keep the surface smooth at two ends.Taking lithology and thickness as the influencing factors, four independent tests were carried out, marked GS05, GS10, RS05, and RS10. Three samples were tested in each independent test.


## 2.3.2. Testing Design

At the beginning of testing, water was injected into the permeameter to saturation at least half an hour. Then, the pore pressure was loaded, and the sealing performance of the system was timely checked. According to relevant geological data, the pore pressure was set as 2 MPa in this test. When the pore pressure was completely stable at 2 MPa, the rock sample was loaded by the axial loading subsystem in the displacement control mode at a loading rate of 0.5 mm/min till the rock sample failed.In addition, the pore pressureq and water flow Q were simultaneously acquired from the beginning of loading; the load F and displacement u during bending deformation were real-time recorded.Permeability,k, can be calculated by Darcy’s law as below:(1)k=μvq/δ,where μ is the dynamic viscosity of water, v is the seepage speed calculated from v=Q/πa2, q is the pore pressure, and δ is the thickness of the thin disc rock sample.


## 3. Experimental Results

### 3.1. Variation Characteristics of Load-Displacement Curves

Figure4 shows the load-displacement (F‐u) curves of the gray sandstone and red sandstone with two different thicknesses. They have similar variation characteristics, which can be divided into four stages OA‐AB‐BC‐CDE, as shown in Figure 5.Figure 4
Load-displacement curves of four different sandstone samples.Figure 5
Four-stage dividing sketch of load-displacement curves.The section of theOA segment is shorter and steeper. This section contains adaptive adjustment and elastic deformation of the thin disc structure under uniform water pressure and bending load. Obviously, the thin disc samples have low elastic deformation.TheAB segment is a line with a very small slope; the deformation increases significantly while the load-carrying capacity increases very small in this stage. The structure produces large plastic deformation, and microcracks in the disc germinate and develop; floor heave occurs in the AB section [16, 25, 26].TheBC section has a great slope. With the increase of deformation, the load increases sharply, and the bearing capacity of the thin disc structure is strengthened. In this stage, the crack propagation inside the thin disc changes qualitatively, microcracks develop and grow continuously, forming macrocracks, and the internal structure of the thin disc is damaged gradually. Although the thin disc structure is still in the overall state, the internal cracks penetrate rapidly. The bearing capacity of the thin disc structure reaches its peak value at point C, which is listed in Table 2. It presents that the thicker disc rock sample has a greater peak load, the peak value of red sandstone with the same thickness is larger than that of gray sandstone, and the displacement of gray sandstone is less than that of red sandstone.Table 2
The peak values for different samples.
Sample numberGS05GS10RS05RS10Peak load (kN)4.0656.4195.9157.283Displacement on the peak load (mm)0.6090.6031.1621.363After the peak value,CDE is the postfracture stage. In this stage, cracks intersect each other, under the action of splitting expansion of pore water pressure, the macrofracture surface is united and formed, and rock blocks slide along the fracture surface, which may cause water inrush [27, 28]. The bearing capacity of the sample decreases slowly with the increase of deformation in CD and then drops rapidly in DE, and the thin disc structure collapses till its bearing capacity loses completely. Generally, if the deformation and heave behavior of the floor have not been dealt with in time before the peak value in the project, it can be remedied in the CD section. Because the floor still has its bearing capacity, it is not too late to control the floor deformation and heave behavior by means of grouting. Mastering the characteristics of crack initiation and propagation can better guide grouting.


### 3.2. Time-Varying Characteristics of Load and Permeability

Figure6 shows the load-time (F‐t) curves and permeability-time (k‐t) curves of the gray sandstone and red sandstone with two different thicknesses.Figure 6
Time-varying characteristics of load and permeability.
(a)
GS05(b)
GS10(c)
RS05(d)
RS10The permeability of the intact thin disc rock sample is about 10-17 m2 at the beginning of the experiments, which is consistent with the testing result of standard intact rock samples [29–37]. It increases rapidly after bending failure occurred several seconds, water flow changes from seepage to turbulence simultaneously, and then, water inrush may occur. In this process, permeability evolved from 10-17 m2 up to 10-11 m2, increasing six orders of magnitude. Its evolution rule is obviously different from standard rock samples [32], in which permeability usually increases three to four orders of magnitude after the postpeak strength. Obviously, the permeability of the thin disc structure changes greatly with stronger mutation, because the stress distribution and crack evolution of the thin disc are different from that of the standard rock sample.As shown in Figure6, the peak permeability always lags the peak load, indicating that it still has time and opportunity to take measures to prevent water inrush disaster, which is consistent with the CD section in Figure 5. Table 3 lists the occurrence time of peak load and peak permeability, as well as the lagging time.Table 3
The lagging times for different samples.
Sample numberGS05GS10RS05RS10Time on the peak load occurrence (s)310118193186Time on the peak permeability occurrence (s)356174203190Lagging time (s)4656104The sample GS05 experienced the longest time; this is because the sealing material between the outer boundary of the sample and the cylinder wall was compacted for a long time. The compression of the sealing material only prolongs the test time but does not affect the deformation measurement of the sample. Because of the difference in experimental operation, the peak load and permeability occurrence times of four samples cannot be compared separately. Only the lagging time can be analyzed; it shows that the lagging time of gray sandstone is longer than that of red sandstone. It is because the brittle behaviors of the thin disc in the postfracture stage are different, which relates to the crack propagation. The longer the lagging time is, the longer the time that can be used for water inrush prevention and control is, the more effective the water inrush risk can be reduced.


### 3.3. Bending Failure Pattern

Figure7 shows the bending failure patterns of the four thin disc rock samples. The 5 mm thickness disc rock samples formed petal-shaped cracks after bending failure, as shown in Figures 7(a) and 7(c). For the 10 mm thickness disc samples, a cap block was cut out along a conical surface with an inclination about 45°, as shown in Figures 7(b) and 7(d), and petal-shaped cracks also appear in the cap block.Figure 7
Bending failure patterns.
(a)
GS05(b)
GS10(c)
RS05(d)
RS10Obviously, when the thin disc rock samples are under the coupled bending-seepage condition, their failure behaviors belong to the problem of structural failure. It is related not only to material properties but also to the structural properties and external force characteristics. In order to analyze the bending failure behavior in-depth, the stress distribution and crack evolution in the thin disc rock structures should be studied further.


## 3.1. Variation Characteristics of Load-Displacement Curves

Figure4 shows the load-displacement (F‐u) curves of the gray sandstone and red sandstone with two different thicknesses. They have similar variation characteristics, which can be divided into four stages OA‐AB‐BC‐CDE, as shown in Figure 5.Figure 4
Load-displacement curves of four different sandstone samples.Figure 5
Four-stage dividing sketch of load-displacement curves.The section of theOA segment is shorter and steeper. This section contains adaptive adjustment and elastic deformation of the thin disc structure under uniform water pressure and bending load. Obviously, the thin disc samples have low elastic deformation.TheAB segment is a line with a very small slope; the deformation increases significantly while the load-carrying capacity increases very small in this stage. The structure produces large plastic deformation, and microcracks in the disc germinate and develop; floor heave occurs in the AB section [16, 25, 26].TheBC section has a great slope. With the increase of deformation, the load increases sharply, and the bearing capacity of the thin disc structure is strengthened. In this stage, the crack propagation inside the thin disc changes qualitatively, microcracks develop and grow continuously, forming macrocracks, and the internal structure of the thin disc is damaged gradually. Although the thin disc structure is still in the overall state, the internal cracks penetrate rapidly. The bearing capacity of the thin disc structure reaches its peak value at point C, which is listed in Table 2. It presents that the thicker disc rock sample has a greater peak load, the peak value of red sandstone with the same thickness is larger than that of gray sandstone, and the displacement of gray sandstone is less than that of red sandstone.Table 2
The peak values for different samples.
Sample numberGS05GS10RS05RS10Peak load (kN)4.0656.4195.9157.283Displacement on the peak load (mm)0.6090.6031.1621.363After the peak value,CDE is the postfracture stage. In this stage, cracks intersect each other, under the action of splitting expansion of pore water pressure, the macrofracture surface is united and formed, and rock blocks slide along the fracture surface, which may cause water inrush [27, 28]. The bearing capacity of the sample decreases slowly with the increase of deformation in CD and then drops rapidly in DE, and the thin disc structure collapses till its bearing capacity loses completely. Generally, if the deformation and heave behavior of the floor have not been dealt with in time before the peak value in the project, it can be remedied in the CD section. Because the floor still has its bearing capacity, it is not too late to control the floor deformation and heave behavior by means of grouting. Mastering the characteristics of crack initiation and propagation can better guide grouting.


## 3.2. Time-Varying Characteristics of Load and Permeability

Figure6 shows the load-time (F‐t) curves and permeability-time (k‐t) curves of the gray sandstone and red sandstone with two different thicknesses.Figure 6
Time-varying characteristics of load and permeability.
(a)
GS05(b)
GS10(c)
RS05(d)
RS10The permeability of the intact thin disc rock sample is about 10-17 m2 at the beginning of the experiments, which is consistent with the testing result of standard intact rock samples [29–37]. It increases rapidly after bending failure occurred several seconds, water flow changes from seepage to turbulence simultaneously, and then, water inrush may occur. In this process, permeability evolved from 10-17 m2 up to 10-11 m2, increasing six orders of magnitude. Its evolution rule is obviously different from standard rock samples [32], in which permeability usually increases three to four orders of magnitude after the postpeak strength. Obviously, the permeability of the thin disc structure changes greatly with stronger mutation, because the stress distribution and crack evolution of the thin disc are different from that of the standard rock sample.As shown in Figure6, the peak permeability always lags the peak load, indicating that it still has time and opportunity to take measures to prevent water inrush disaster, which is consistent with the CD section in Figure 5. Table 3 lists the occurrence time of peak load and peak permeability, as well as the lagging time.Table 3
The lagging times for different samples.
Sample numberGS05GS10RS05RS10Time on the peak load occurrence (s)310118193186Time on the peak permeability occurrence (s)356174203190Lagging time (s)4656104The sample GS05 experienced the longest time; this is because the sealing material between the outer boundary of the sample and the cylinder wall was compacted for a long time. The compression of the sealing material only prolongs the test time but does not affect the deformation measurement of the sample. Because of the difference in experimental operation, the peak load and permeability occurrence times of four samples cannot be compared separately. Only the lagging time can be analyzed; it shows that the lagging time of gray sandstone is longer than that of red sandstone. It is because the brittle behaviors of the thin disc in the postfracture stage are different, which relates to the crack propagation. The longer the lagging time is, the longer the time that can be used for water inrush prevention and control is, the more effective the water inrush risk can be reduced.


## 3.3. Bending Failure Pattern

Figure7 shows the bending failure patterns of the four thin disc rock samples. The 5 mm thickness disc rock samples formed petal-shaped cracks after bending failure, as shown in Figures 7(a) and 7(c). For the 10 mm thickness disc samples, a cap block was cut out along a conical surface with an inclination about 45°, as shown in Figures 7(b) and 7(d), and petal-shaped cracks also appear in the cap block.Figure 7
Bending failure patterns.
(a)
GS05(b)
GS10(c)
RS05(d)
RS10Obviously, when the thin disc rock samples are under the coupled bending-seepage condition, their failure behaviors belong to the problem of structural failure. It is related not only to material properties but also to the structural properties and external force characteristics. In order to analyze the bending failure behavior in-depth, the stress distribution and crack evolution in the thin disc rock structures should be studied further.


## 4. Discussion

The hard rock floor aquifuge is mainly a local plastic failure, and the stability coefficient is generally high; it can be solved with the elastic solution [38].


### 4.1. Deflection Calculation and Distribution in Thin Disc Rock Samples

Based on the simplified model in Figure1(c), circumferential load, F, and pore pressure, q, can be measured timely during the test; the concentrated force, P, and the force couple, M, can be calculated as(2)P=F−qπa2,M=a2F2π−qa.In polar coordinates, as shown in Figure1, the transverse loads on the thin disc rock sample are symmetrical around the z-axis, which is a vertical plate face up. The deflection, w, of the elastic thin disc is also symmetrical around the z-axis, and it is a function of r but does not change with θ. When the thin disc rock sample bends axial-symmetrically, the differential equation of the deflection is(3)d2dr2+1rddrd2wdr2+1rdwdr=qD,where D=Eδ3/121−υ2 is the bending stiffness of thin disc and E and υ are elastic modulus and Poisson’s ratio, respectively.A semi-inverse method is used to solve the differential Equation (3); the general solution for the deflection of the thin disc rock sample is(4)w=q64Dr4+A1r2+A2r2lnra+A3lnra+A4,where A1, A2, A3, and A4 are coefficients, determined by the boundary conditions.Based on the boundary conditions in the mechanical model, the deflection is expressed as(5)w=q64Da2−r22+P8πD12a2−r2+r2lnra.Applying Equation (5), the deflection distribution inside the thin disc rock sample along the radius direction at different loading times could be obtained.The deflection distributions along the radius direction whenP=0 are shown in Figure 8(a). It shows that when the thin disc rock sample is only under the action of pore pressure, the maximum deflections of samples GS05, GS10, RS05, and RS10 are 0.0191, 0.00374, 0.0264, and 0.0046 mm, respectively. It can be inferred that the thinner the floor aquifuge, the larger the deformation caused by the confined water pressure.Figure 8
Deflection distribution curves.
(a)
WhenP=0(b)
WhenP=PmaxFigure8(b) shows the deflection distributions at the time of peak load appearance, P=Pmax, the maximum deflections of samples GS05, GS10, RS05, and RS10 are 0.0218, 0.0132, 0.0797, and 0.0203 mm, respectively. Mining disturbance induces further bending deformation of the floor. Under the combined action of mining-induced stress and confined water pressure, the deflection of the floor aquifuge is 1.2 to 4 times of that under the alone confined water pressure.It is worth noting that these deflections are the elastic ultimate load-bearing displacements. The displacements are different from the values in Table2, which includes not only the deformation of the thin disc rock sample but also the deformation of the sealing material and the displacement of the sample after being fractured.The maximum deflections both occur at the center of the thin disc rock sample whenP=0 and P=Pmax; as a result, the center of the disc is the dangerous position that the disc may break from here.Sandstone is a typical brittle material at room temperature; its elastic deformation is weak. The bending failure of the thin disc rock sample mainly results from the strength, but not the stiffness. Therefore, in addition to analyzing the deformation in the disc, stress distribution should be emphasized.


### 4.2. Stress Calculation and Distribution in Thin Disc Rock Samples

As Figure9 shows, through the analysis of the microelement in the thin disc rock sample, the internal force equation of the element is calculated as follows:(6)Mr=−Dd2wdr2+υrdwdr=−P4π1+lnra+υlnra+q161+υa2−r2−2r2,Mθ=−D1rdwdr+υd2wdr2=−P4πυ+lnra+υlnra+q16υa2−3r2+a2−r2,Mrθ=0,FSr=−Dd3wdr3+1rd2wdr2=−P4πr2+lnra+q16ra2−9r2,FSθ=0.Figure 9
Internal force distribution of microelement in the thin disc rock sample.Using Equation (6), the distribution of internal forces along the radius direction can be solved and drawn, as shown in Figure 10.Figure 10
Internal force distribution curves.
(a)
Mr distribution curves(b)
Mθ distribution curves(c)
FSr distribution curvesAs seen in Figure10(a), the reverse bending point of Mr is about 0.01 m away from the center of the disc. When r<0.01 m, the upper surface of the disc is in tension and the lower surface is in compression, and when r>0.01 m, it is just the opposite. All the samples except for GS05 have the maximum value of Mr at the center, where it is the most dangerous cross-section. Particularly, the maximum value of Mr occurs at the edge in the sample of GS05. The value in the center is slightly less than that in the edge; both the center and edge cross-sections are dangerous sections.It can be seen from Figure10(b) that the Mθ distribution curves are similar to the distribution curves of Mr, but the reverse bending point is different, which is about 0.022 m away from the center. When r<0.022 m, the upper surface of the disc rock sample is in tension and the lower surface is in compression, and when r>0.022 m, the upper surface is compressed and the lower surface is tensed. The maximum value of Mθ occurs at the center, so the central cross-section is the most dangerous point.It can be seen from Figure10(c) that when r≈0.004 m, the shear force FSr in the disc changes from positive to negative. The maximum value of FSr occurs at the center of the disc, so it is the most dangerous cross-section; the direction of the shear force at the dangerous section is downward.From Equations (5) and (6), the stress at every cross-section of the thin disc rock sample can be calculated as follows:(7)σr=−Ez1−υ2d2wdr2+υrdwdr=12Mrδ3z=−3δ3Pπ1+lnra+υlnra−q41+υa2−r2−2r2z,σθ=−Ez1−υ21rdwdr+υd2wdr2=12Mθδ3z=−3δ3Pπυ+lnra+υlnra−q4υa2−3r2+a2−r2z,τrθ=τθr=0,τrz=6FSrδ3δ24−z2=−3δ3P2πr2+lnra−q8ra2−9r2δ24−z2,τθz=τzθ=0.Using Equation (7), the stress distribution at the central dangerous cross-section along the direction of the disc thickness (i.e., the z-axis direction) is shown in Figure 11(a). For the sample of GS05, the edge cross-section is dangerous too; its stress distribution along the direction of the disc thickness is shown in Figure 11(b).Figure 11
Stress distribution at the dangerous cross-section.
(a)
Stress distribution at the central dangerous cross-section along the thickness direction(b)
Stress distribution at the edge dangerous cross-section along the thickness direction for the sample of GS05As seen in Figure11, both σr and σθ are linearly distributed along the thickness direction, forming the bending moments Mr and Mθ, respectively. The maximum values of σr and σθ occur at the upper and lower points of the dangerous cross-sections. τrz is parabolically distributed along the thickness direction, constituting the shear force FSr. The maximum value τrz occurs at the neutral point of the dangerous cross-section.Summing up, the location and stresses of the dangerous points in the thin disc rock samples can be calculated and are shown in Table4.Table 4
The location and stresses of the dangerous points in the thin disc rock samples.
SamplesLocation/stressesGS05The upper and lower points of the central cross-section/σrmax=22.3MPaThe upper and lower points of the edge cross-section/σrmax=27.0MPaThe upper and lower points of the central cross-section/σθmax=23.7MPaThe neutral point of the central cross-section/τrzmax=48.8MPaGS10The upper and lower points of the central cross-section/σrmax=46.4MPaThe upper and lower points of the central cross-section/σθmax=54.7MPaThe neutral point of the central cross-section/τrzmax=130.5MPaRS05The upper and lower points of the central cross-section/σrmax=120.2MPaThe upper and lower points of the central cross-section/σθmax=144.8MPaThe neutral point of the central cross-section/τrzmax=199.9MPaRS10The upper and lower points of the central cross-section/σrmax=59.8MPaThe upper and lower points of the central cross-section/σθmax=72.7MPaThe neutral point of the central cross-section/τrzmax=173.3MPaAs illustrated in Table4, for sample GS05, σrmax at the upper and lower points of the edge cross-section is very large, and the thin disc rock sample may be damaged from the edge. For samples GS05, GS10, RS05, and RS10, both σrmax and σθmax occur at the upper and lower points of the central cross-section, and τrzmax occurs at the neutral point of the central cross-section. σθmax is larger than σrmax at the upper and lower points of the central cross-section, and the thin disc rock sample may be damaged mainly by σθmax from the center.It also shows that the sample RS05 has the largestσrmax, σθmax, and τrzmax, followed by RS10, GS10, and GS05. The thin disc red sandstone has higher stress than the gray one. The greater the stress is, the faster the cracks develop in the thin disc, and the shorter the time of water inrush passage through. This is consistent with the experimental phenomenon that the lagging time of red sandstone is shorter than that of gray sandstone in Table 3.The stress state of dangerous points in the thin disc rock samples can be described as shown in Figure12.Figure 12
Stress state of dangerous points.
(a)
The biaxial tensile stress state(b)
The biaxial compressive stress state(c)
The uniaxial tensile stress state(d)
The uniaxial compressive stress state(e)
The pure shear stress stateFigure13 and Table 5 show the specific locations of the dangerous points on the thin discs and the corresponding stress states.Figure 13
The location of the danger points on thin discs.Table 5
The details of the stress state at the dangerous points.
Stress stateLocation of the dangerous pointsa: the biaxial tensile stress stateThe upper points of the central cross-section in samples of GS05, GS10, RS05, and RS10.b: the biaxial compressive stress stateThe lower points of the central cross-section in samples of GS05, GS10, RS05, and RS10.c: the uniaxial tensile stress stateThe lower points of the edge cross-section in the sample of GS05.d: the uniaxial compressive stress stateThe upper points of the edge cross-section in the sample of GS05.e: the pure shear stress stateThe neutral points of the central cross-section in samples of GS05, GS10, RS05, and RS10.Based on the description for the location and stress state of the dangerous points on the thin discs, these dangerous points are in the biaxial tensile/compressive stress state, the uniaxial tensile/compressive stress state, and the pure shear stress state, respectively. The structural failure of the thin disc rock sample is mainly due to the stress reaching or exceeding its strength limit, and therefore, the strength condition should be established to reveal the failure mechanism.


### 4.3. Crack Initiation and Propagation in Thin Disc Rock Samples

Sandstone is a kind of brittle material so that we use the Maximum Tensile Stress Theory and Mohr-Coulomb Strength Theory to analyze the bending failure behavior.


#### 4.3.1. The Cause of the Crack Initiation

Combining Figures12 and 13, the upper points of the central cross-section in the four samples are the dangerous points. They are in the biaxial tensile stress state, and the principal stresses of the dangerous points are σθmax, σrmax, and 0. According to the Maximum Tensile Stress Theory, when bending failure occurs, the maximum tangential tensile stress reaches and exceeds the allowable tensile stress. So, the upper points of the central cross-section in the disc, i.e., the points a in Figure 13, are the points of crack initiation, and the disc rock sample is tensed at these points along the tangential direction.Meanwhile, the lower points of the edge cross-section in sample GS05 are also the dangerous points. They are in the uniaxial tensile stress state, and the principal stresses of the dangerous points areσrmax, 0, and 0. When bending failure occurs in sample GS05, the maximum radial tensile stress reaches and exceeds the allowable tensile stress. Therefore, for the sample GS05, besides the upper points of the central cross-section, the lower points of the edge cross-section, i.e., the points c in Figure 13, are the points of crack initiation, too. That is to say, the sample GS05 is tensed simultaneously at the location a along the tangential direction and at location c along the radial direction.In addition, the neutral points of the central cross-section in the four thin disc rock samples are also dangerous points, which are in the pure shear stress state, and the principal stresses of the dangerous points areτrzmax, 0, and -τrzmax. According to the Mohr-Coulomb Strength Theory, it is obviously obtained that shear is the main cause of the bending failure. It shows that the four kinds of discs also suffer from shear failure from the neutral points of the central cross-section, i.e., the points e in Figure 13; this location is also the points of crack initiation.In summary, the crack initiation points for the samples GS10, RS05, and RS10 area and e as shown in Figure 13; the crack develops resulting from the upper point a of the central cross-section which is tensed along the tangential direction and the neutral point e of the central cross-section which is sheared. The crack initiation points for sample GS05 are a, c, and e in Figure 13; they are caused by the upper points a of the central cross-section and the lower points c of the edge cross-section that are tensed simultaneously along the tangential direction at the center and radial direction at the edge and the neutral points e of the central cross-section that is sheared. Briefly, tension-shear failure is the main reason for crack initiation.


#### 4.3.2. The Cause of Crack Propagation

As seen in Table4, there are two dangerous locations in sample GS05. One is at the disc center, where τrzmax and σθmax play major roles, and τrzmax is about 2.06 times of σθmax. The other is at the lower points of the disc edge, which is the uniaxial tensile stress state dominated by σrmax. Crack propagation in this disc is caused by the tangential tensile stress and shear stress at the center and the radial tensile stress at the edge. The cracks are developed oppositely by the tension-shear failure at the center and tension failure at the edge, producing the petal-shaped cracks shown in Figure 14(a).Figure 14
The mechanism of crack propagation.
(a)
GS05(b)
RS05(c)
GS10 and RS10For sample RS05, the cracks propagate from the center, whereτrzmax is only 1.38 times σθmax. Crack propagation of this disc is mainly due to the tension failure, which is perpendicular to the radius direction and caused by the tangential tensile stress. The cracks develop from the center to the edge; petal-shaped cracks are formed, as shown in Figure 14(b).For samples GS10 and RS10,τrzmax is about 2.38 times σθmax. Due to the thicker thickness, crack propagation in the disc is mainly caused by shear failure; meanwhile, the tension failure plays a secondary role. As shown in Figure 14(c), the cracks rise along a conical surface with an inclination about 45° from the center to the edge, and the disc rock sample is sheared into two pieces. The cap block is accompanied by petals with tension cracks caused by the tangential tensile stress.


### 4.4. Bending-Failure-Induced Water Inrush Behavior in the Floor Aquifuge

During mining activities, the floor aquifuge is bending; the deflection and stress are changing timely. It is safe when the deflection and stress are far less than the allowable values. The cracks initiate and propagate with the increase of stress, and the confined water pressure split and expand the cracks continuously. When bending failure occurs, the maximum tensile stress exceeds the allowable values; the cracks propagate to penetrate.Although the occurrence of bending failure is really very dangerous, the water inrush disaster does not happen immediately, because of the water inrush lagging. There is a short time to take measures to grout the cracks and to prevent the water inrush accident. According to the crack initiation and propagation (Figure14), the grouting should be at the center or the edge of the floor aquifuge along the crack propagation trend lines.


## 4.1. Deflection Calculation and Distribution in Thin Disc Rock Samples

Based on the simplified model in Figure1(c), circumferential load, F, and pore pressure, q, can be measured timely during the test; the concentrated force, P, and the force couple, M, can be calculated as(2)P=F−qπa2,M=a2F2π−qa.In polar coordinates, as shown in Figure1, the transverse loads on the thin disc rock sample are symmetrical around the z-axis, which is a vertical plate face up. The deflection, w, of the elastic thin disc is also symmetrical around the z-axis, and it is a function of r but does not change with θ. When the thin disc rock sample bends axial-symmetrically, the differential equation of the deflection is(3)d2dr2+1rddrd2wdr2+1rdwdr=qD,where D=Eδ3/121−υ2 is the bending stiffness of thin disc and E and υ are elastic modulus and Poisson’s ratio, respectively.A semi-inverse method is used to solve the differential Equation (3); the general solution for the deflection of the thin disc rock sample is(4)w=q64Dr4+A1r2+A2r2lnra+A3lnra+A4,where A1, A2, A3, and A4 are coefficients, determined by the boundary conditions.Based on the boundary conditions in the mechanical model, the deflection is expressed as(5)w=q64Da2−r22+P8πD12a2−r2+r2lnra.Applying Equation (5), the deflection distribution inside the thin disc rock sample along the radius direction at different loading times could be obtained.The deflection distributions along the radius direction whenP=0 are shown in Figure 8(a). It shows that when the thin disc rock sample is only under the action of pore pressure, the maximum deflections of samples GS05, GS10, RS05, and RS10 are 0.0191, 0.00374, 0.0264, and 0.0046 mm, respectively. It can be inferred that the thinner the floor aquifuge, the larger the deformation caused by the confined water pressure.Figure 8
Deflection distribution curves.
(a)
WhenP=0(b)
WhenP=PmaxFigure8(b) shows the deflection distributions at the time of peak load appearance, P=Pmax, the maximum deflections of samples GS05, GS10, RS05, and RS10 are 0.0218, 0.0132, 0.0797, and 0.0203 mm, respectively. Mining disturbance induces further bending deformation of the floor. Under the combined action of mining-induced stress and confined water pressure, the deflection of the floor aquifuge is 1.2 to 4 times of that under the alone confined water pressure.It is worth noting that these deflections are the elastic ultimate load-bearing displacements. The displacements are different from the values in Table2, which includes not only the deformation of the thin disc rock sample but also the deformation of the sealing material and the displacement of the sample after being fractured.The maximum deflections both occur at the center of the thin disc rock sample whenP=0 and P=Pmax; as a result, the center of the disc is the dangerous position that the disc may break from here.Sandstone is a typical brittle material at room temperature; its elastic deformation is weak. The bending failure of the thin disc rock sample mainly results from the strength, but not the stiffness. Therefore, in addition to analyzing the deformation in the disc, stress distribution should be emphasized.


## 4.2. Stress Calculation and Distribution in Thin Disc Rock Samples

As Figure9 shows, through the analysis of the microelement in the thin disc rock sample, the internal force equation of the element is calculated as follows:(6)Mr=−Dd2wdr2+υrdwdr=−P4π1+lnra+υlnra+q161+υa2−r2−2r2,Mθ=−D1rdwdr+υd2wdr2=−P4πυ+lnra+υlnra+q16υa2−3r2+a2−r2,Mrθ=0,FSr=−Dd3wdr3+1rd2wdr2=−P4πr2+lnra+q16ra2−9r2,FSθ=0.Figure 9
Internal force distribution of microelement in the thin disc rock sample.Using Equation (6), the distribution of internal forces along the radius direction can be solved and drawn, as shown in Figure 10.Figure 10
Internal force distribution curves.
(a)
Mr distribution curves(b)
Mθ distribution curves(c)
FSr distribution curvesAs seen in Figure10(a), the reverse bending point of Mr is about 0.01 m away from the center of the disc. When r<0.01 m, the upper surface of the disc is in tension and the lower surface is in compression, and when r>0.01 m, it is just the opposite. All the samples except for GS05 have the maximum value of Mr at the center, where it is the most dangerous cross-section. Particularly, the maximum value of Mr occurs at the edge in the sample of GS05. The value in the center is slightly less than that in the edge; both the center and edge cross-sections are dangerous sections.It can be seen from Figure10(b) that the Mθ distribution curves are similar to the distribution curves of Mr, but the reverse bending point is different, which is about 0.022 m away from the center. When r<0.022 m, the upper surface of the disc rock sample is in tension and the lower surface is in compression, and when r>0.022 m, the upper surface is compressed and the lower surface is tensed. The maximum value of Mθ occurs at the center, so the central cross-section is the most dangerous point.It can be seen from Figure10(c) that when r≈0.004 m, the shear force FSr in the disc changes from positive to negative. The maximum value of FSr occurs at the center of the disc, so it is the most dangerous cross-section; the direction of the shear force at the dangerous section is downward.From Equations (5) and (6), the stress at every cross-section of the thin disc rock sample can be calculated as follows:(7)σr=−Ez1−υ2d2wdr2+υrdwdr=12Mrδ3z=−3δ3Pπ1+lnra+υlnra−q41+υa2−r2−2r2z,σθ=−Ez1−υ21rdwdr+υd2wdr2=12Mθδ3z=−3δ3Pπυ+lnra+υlnra−q4υa2−3r2+a2−r2z,τrθ=τθr=0,τrz=6FSrδ3δ24−z2=−3δ3P2πr2+lnra−q8ra2−9r2δ24−z2,τθz=τzθ=0.Using Equation (7), the stress distribution at the central dangerous cross-section along the direction of the disc thickness (i.e., the z-axis direction) is shown in Figure 11(a). For the sample of GS05, the edge cross-section is dangerous too; its stress distribution along the direction of the disc thickness is shown in Figure 11(b).Figure 11
Stress distribution at the dangerous cross-section.
(a)
Stress distribution at the central dangerous cross-section along the thickness direction(b)
Stress distribution at the edge dangerous cross-section along the thickness direction for the sample of GS05As seen in Figure11, both σr and σθ are linearly distributed along the thickness direction, forming the bending moments Mr and Mθ, respectively. The maximum values of σr and σθ occur at the upper and lower points of the dangerous cross-sections. τrz is parabolically distributed along the thickness direction, constituting the shear force FSr. The maximum value τrz occurs at the neutral point of the dangerous cross-section.Summing up, the location and stresses of the dangerous points in the thin disc rock samples can be calculated and are shown in Table4.Table 4
The location and stresses of the dangerous points in the thin disc rock samples.
SamplesLocation/stressesGS05The upper and lower points of the central cross-section/σrmax=22.3MPaThe upper and lower points of the edge cross-section/σrmax=27.0MPaThe upper and lower points of the central cross-section/σθmax=23.7MPaThe neutral point of the central cross-section/τrzmax=48.8MPaGS10The upper and lower points of the central cross-section/σrmax=46.4MPaThe upper and lower points of the central cross-section/σθmax=54.7MPaThe neutral point of the central cross-section/τrzmax=130.5MPaRS05The upper and lower points of the central cross-section/σrmax=120.2MPaThe upper and lower points of the central cross-section/σθmax=144.8MPaThe neutral point of the central cross-section/τrzmax=199.9MPaRS10The upper and lower points of the central cross-section/σrmax=59.8MPaThe upper and lower points of the central cross-section/σθmax=72.7MPaThe neutral point of the central cross-section/τrzmax=173.3MPaAs illustrated in Table4, for sample GS05, σrmax at the upper and lower points of the edge cross-section is very large, and the thin disc rock sample may be damaged from the edge. For samples GS05, GS10, RS05, and RS10, both σrmax and σθmax occur at the upper and lower points of the central cross-section, and τrzmax occurs at the neutral point of the central cross-section. σθmax is larger than σrmax at the upper and lower points of the central cross-section, and the thin disc rock sample may be damaged mainly by σθmax from the center.It also shows that the sample RS05 has the largestσrmax, σθmax, and τrzmax, followed by RS10, GS10, and GS05. The thin disc red sandstone has higher stress than the gray one. The greater the stress is, the faster the cracks develop in the thin disc, and the shorter the time of water inrush passage through. This is consistent with the experimental phenomenon that the lagging time of red sandstone is shorter than that of gray sandstone in Table 3.The stress state of dangerous points in the thin disc rock samples can be described as shown in Figure12.Figure 12
Stress state of dangerous points.
(a)
The biaxial tensile stress state(b)
The biaxial compressive stress state(c)
The uniaxial tensile stress state(d)
The uniaxial compressive stress state(e)
The pure shear stress stateFigure13 and Table 5 show the specific locations of the dangerous points on the thin discs and the corresponding stress states.Figure 13
The location of the danger points on thin discs.Table 5
The details of the stress state at the dangerous points.
Stress stateLocation of the dangerous pointsa: the biaxial tensile stress stateThe upper points of the central cross-section in samples of GS05, GS10, RS05, and RS10.b: the biaxial compressive stress stateThe lower points of the central cross-section in samples of GS05, GS10, RS05, and RS10.c: the uniaxial tensile stress stateThe lower points of the edge cross-section in the sample of GS05.d: the uniaxial compressive stress stateThe upper points of the edge cross-section in the sample of GS05.e: the pure shear stress stateThe neutral points of the central cross-section in samples of GS05, GS10, RS05, and RS10.Based on the description for the location and stress state of the dangerous points on the thin discs, these dangerous points are in the biaxial tensile/compressive stress state, the uniaxial tensile/compressive stress state, and the pure shear stress state, respectively. The structural failure of the thin disc rock sample is mainly due to the stress reaching or exceeding its strength limit, and therefore, the strength condition should be established to reveal the failure mechanism.


## 4.3. Crack Initiation and Propagation in Thin Disc Rock Samples

Sandstone is a kind of brittle material so that we use the Maximum Tensile Stress Theory and Mohr-Coulomb Strength Theory to analyze the bending failure behavior.


### 4.3.1. The Cause of the Crack Initiation

Combining Figures12 and 13, the upper points of the central cross-section in the four samples are the dangerous points. They are in the biaxial tensile stress state, and the principal stresses of the dangerous points are σθmax, σrmax, and 0. According to the Maximum Tensile Stress Theory, when bending failure occurs, the maximum tangential tensile stress reaches and exceeds the allowable tensile stress. So, the upper points of the central cross-section in the disc, i.e., the points a in Figure 13, are the points of crack initiation, and the disc rock sample is tensed at these points along the tangential direction.Meanwhile, the lower points of the edge cross-section in sample GS05 are also the dangerous points. They are in the uniaxial tensile stress state, and the principal stresses of the dangerous points areσrmax, 0, and 0. When bending failure occurs in sample GS05, the maximum radial tensile stress reaches and exceeds the allowable tensile stress. Therefore, for the sample GS05, besides the upper points of the central cross-section, the lower points of the edge cross-section, i.e., the points c in Figure 13, are the points of crack initiation, too. That is to say, the sample GS05 is tensed simultaneously at the location a along the tangential direction and at location c along the radial direction.In addition, the neutral points of the central cross-section in the four thin disc rock samples are also dangerous points, which are in the pure shear stress state, and the principal stresses of the dangerous points areτrzmax, 0, and -τrzmax. According to the Mohr-Coulomb Strength Theory, it is obviously obtained that shear is the main cause of the bending failure. It shows that the four kinds of discs also suffer from shear failure from the neutral points of the central cross-section, i.e., the points e in Figure 13; this location is also the points of crack initiation.In summary, the crack initiation points for the samples GS10, RS05, and RS10 area and e as shown in Figure 13; the crack develops resulting from the upper point a of the central cross-section which is tensed along the tangential direction and the neutral point e of the central cross-section which is sheared. The crack initiation points for sample GS05 are a, c, and e in Figure 13; they are caused by the upper points a of the central cross-section and the lower points c of the edge cross-section that are tensed simultaneously along the tangential direction at the center and radial direction at the edge and the neutral points e of the central cross-section that is sheared. Briefly, tension-shear failure is the main reason for crack initiation.


### 4.3.2. The Cause of Crack Propagation

As seen in Table4, there are two dangerous locations in sample GS05. One is at the disc center, where τrzmax and σθmax play major roles, and τrzmax is about 2.06 times of σθmax. The other is at the lower points of the disc edge, which is the uniaxial tensile stress state dominated by σrmax. Crack propagation in this disc is caused by the tangential tensile stress and shear stress at the center and the radial tensile stress at the edge. The cracks are developed oppositely by the tension-shear failure at the center and tension failure at the edge, producing the petal-shaped cracks shown in Figure 14(a).Figure 14
The mechanism of crack propagation.
(a)
GS05(b)
RS05(c)
GS10 and RS10For sample RS05, the cracks propagate from the center, whereτrzmax is only 1.38 times σθmax. Crack propagation of this disc is mainly due to the tension failure, which is perpendicular to the radius direction and caused by the tangential tensile stress. The cracks develop from the center to the edge; petal-shaped cracks are formed, as shown in Figure 14(b).For samples GS10 and RS10,τrzmax is about 2.38 times σθmax. Due to the thicker thickness, crack propagation in the disc is mainly caused by shear failure; meanwhile, the tension failure plays a secondary role. As shown in Figure 14(c), the cracks rise along a conical surface with an inclination about 45° from the center to the edge, and the disc rock sample is sheared into two pieces. The cap block is accompanied by petals with tension cracks caused by the tangential tensile stress.


## 4.3.1. The Cause of the Crack Initiation

Combining Figures12 and 13, the upper points of the central cross-section in the four samples are the dangerous points. They are in the biaxial tensile stress state, and the principal stresses of the dangerous points are σθmax, σrmax, and 0. According to the Maximum Tensile Stress Theory, when bending failure occurs, the maximum tangential tensile stress reaches and exceeds the allowable tensile stress. So, the upper points of the central cross-section in the disc, i.e., the points a in Figure 13, are the points of crack initiation, and the disc rock sample is tensed at these points along the tangential direction.Meanwhile, the lower points of the edge cross-section in sample GS05 are also the dangerous points. They are in the uniaxial tensile stress state, and the principal stresses of the dangerous points areσrmax, 0, and 0. When bending failure occurs in sample GS05, the maximum radial tensile stress reaches and exceeds the allowable tensile stress. Therefore, for the sample GS05, besides the upper points of the central cross-section, the lower points of the edge cross-section, i.e., the points c in Figure 13, are the points of crack initiation, too. That is to say, the sample GS05 is tensed simultaneously at the location a along the tangential direction and at location c along the radial direction.In addition, the neutral points of the central cross-section in the four thin disc rock samples are also dangerous points, which are in the pure shear stress state, and the principal stresses of the dangerous points areτrzmax, 0, and -τrzmax. According to the Mohr-Coulomb Strength Theory, it is obviously obtained that shear is the main cause of the bending failure. It shows that the four kinds of discs also suffer from shear failure from the neutral points of the central cross-section, i.e., the points e in Figure 13; this location is also the points of crack initiation.In summary, the crack initiation points for the samples GS10, RS05, and RS10 area and e as shown in Figure 13; the crack develops resulting from the upper point a of the central cross-section which is tensed along the tangential direction and the neutral point e of the central cross-section which is sheared. The crack initiation points for sample GS05 are a, c, and e in Figure 13; they are caused by the upper points a of the central cross-section and the lower points c of the edge cross-section that are tensed simultaneously along the tangential direction at the center and radial direction at the edge and the neutral points e of the central cross-section that is sheared. Briefly, tension-shear failure is the main reason for crack initiation.


## 4.3.2. The Cause of Crack Propagation

As seen in Table4, there are two dangerous locations in sample GS05. One is at the disc center, where τrzmax and σθmax play major roles, and τrzmax is about 2.06 times of σθmax. The other is at the lower points of the disc edge, which is the uniaxial tensile stress state dominated by σrmax. Crack propagation in this disc is caused by the tangential tensile stress and shear stress at the center and the radial tensile stress at the edge. The cracks are developed oppositely by the tension-shear failure at the center and tension failure at the edge, producing the petal-shaped cracks shown in Figure 14(a).Figure 14
The mechanism of crack propagation.
(a)
GS05(b)
RS05(c)
GS10 and RS10For sample RS05, the cracks propagate from the center, whereτrzmax is only 1.38 times σθmax. Crack propagation of this disc is mainly due to the tension failure, which is perpendicular to the radius direction and caused by the tangential tensile stress. The cracks develop from the center to the edge; petal-shaped cracks are formed, as shown in Figure 14(b).For samples GS10 and RS10,τrzmax is about 2.38 times σθmax. Due to the thicker thickness, crack propagation in the disc is mainly caused by shear failure; meanwhile, the tension failure plays a secondary role. As shown in Figure 14(c), the cracks rise along a conical surface with an inclination about 45° from the center to the edge, and the disc rock sample is sheared into two pieces. The cap block is accompanied by petals with tension cracks caused by the tangential tensile stress.


## 4.4. Bending-Failure-Induced Water Inrush Behavior in the Floor Aquifuge

During mining activities, the floor aquifuge is bending; the deflection and stress are changing timely. It is safe when the deflection and stress are far less than the allowable values. The cracks initiate and propagate with the increase of stress, and the confined water pressure split and expand the cracks continuously. When bending failure occurs, the maximum tensile stress exceeds the allowable values; the cracks propagate to penetrate.Although the occurrence of bending failure is really very dangerous, the water inrush disaster does not happen immediately, because of the water inrush lagging. There is a short time to take measures to grout the cracks and to prevent the water inrush accident. According to the crack initiation and propagation (Figure14), the grouting should be at the center or the edge of the floor aquifuge along the crack propagation trend lines.


## 5. Conclusions

A simplified thin disc model was introduced to study the bending failure of the floor aquifuge. Based on a self-designed experimental system, thin disc gray and red sandstone samples were tested under coupled bending-seepage condition to study the failure behavior. The failure behavior in thin disc sandstone with two different lithologies and thicknesses were analyzed. The main conclusions can be drawn as follows:(1)
The failure process of thin disc sandstone can be divided into four stages: adaptive adjustment and elastic deformation stage, plastic deformation and microcrack development stage, bearing capacity strengthening and macrocrack formation stage, and postfracture stage, of which water inrush disaster occurs at the postfracture stage.(2)
The permeability changes from 10-17 m2 to 10-11 m2 in the thin disc structure, and the sharp increase leads to more sudden and intense water inrush. The peak permeability always lags the peak load, and the lagging time of water inrush in gray sandstone is longer than that in red sandstone owing to the difference of crack propagation.(3)
The crack initiation point occurred at the center because of the tangential tensile stress and shear stress. The crack propagation is related to disc thickness and lithology.(4)
Water inrush accident occurs after the bending failure in the floor aquifuge. It is an opportune moment to grout along the crack propagation trend lines to prevent the water inrush disasters.



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*Source: 8830358-2020-08-28.xml*