Particle Flow Simulation of Failure Process of Defective Sandstone under Different Intermediate Principal Stress under True Triaxial Action

Abstract

In order to explore the mechanical response characteristics of fractured sandstone under true triaxial different medium principal stresses, matdem particle flow software was used to study the mechanical response characteristics, fracture mechanism and damage evolution characteristics of sandstone specimens under the conditions of 30 MPa, 40 MPa and 50 MPa respectively. The simulation results are verified by true triaxial test. The results show that under true triaxial stress, the increase of medium principal stress is beneficial to increase the strength of sandstone. The fracture degree of the specimen increases with the increase of the intermediate principal stress, and finally the interlacing macroscopic cracks are formed. When the intermediate principal stress is perpendicular to the fracture strike, the fracture mode of sandstone is that the macroscopic fracture plane is perpendicular to the fracture strike, and the fracture mechanism of sandstone under true triaxial compression is mainly shear failure, accompanied by tensile failure. With the increasing of the intermediate principal stress, the fractal dimension of the fracture of sandstone specimen increases significantly and the degree of fracture deepens. Combined with the true triaxial test results, the rationality of particle flow simulation test is proved.

1. Introduction

With the rapid development of economy, most of the mines in China have been transferred to deep mining since the end of the nineteenth century [1]. Under the complex geological environment and human long-term underground engineering activities, deep rocks form various forms of natural defects, such as faults, cracks, weak interlayers, etc. [2,3]. The failure process of rock under the action of external load is essentially the process of expansion and deformation to failure of various defects in rock. The fracture law and stability of these natural defects in rock have always been a hot spot in the field of rock mechanics, and the mechanical properties of fractured rock are the key issues [4]. Many domestic scholars analyze various characteristics of rock with fractures from different perspectives and methods. Liu Jie et al. [5] combined true triaxial loading and unloading failure test and three-dimensional particle flow simulation to explore the deformation and failure mechanism of granite under different stress paths. Wang Jie et al. [6] used particle flow software to construct a rock-filled body (BER) and a rock-filled body (REB) combination model with D of 10, 20, 30, 40 mm, and then carried out uniaxial compression experiments. Shan RL et al. [7] used the self-developed DRTS-500 rock triaxial test system to carry out conventional triaxial compression test and triaxial unloading creep test on ice-containing fractured red sandstone samples in Shilawusu mining area, Northwest China. The instantaneous strength and deformation characteristics of rock specimens were analyzed, and the creep deformation characteristics and damage evolution characteristics were discussed. Yao N et al. [8] studied the different mechanical behaviors of layered rock with different bedding angles during uniaxial compression test based on the numerical simulation model of layered rock mass in laboratory test. Liu Xinrong et al. [9] pre-peak cyclic numerical direct shear test considering second-order asperity rock joints is perfectly realized by using two-dimensional particle flow code, and the variation law of cumulative damage characteristics and shear strength of joints in pre-peak cyclic direct shear test is analyzed from macro and micro perspectives. The reliability of the calculation method is verified by comparing and analyzing the existing indoor test results. Faizi Saad Ali et al. [10] used discrete element method (DEM) study the response of anisotropic rocks under true triaxial testing. Numerical samples of seven different bedding orientations (beta = 0(o); 15(o); 30(o); 45(o); 60(o); 75(o); and 90(o)) are created. Lu Wenbin et al. [11] used simulate a columnar jointed rock mass with different dip angles (beta = 0 degrees–90 degrees) and uniaxial compression, triaxial compression and true triaxial unloading tests were conducted to determine the anisotropic properties; and unloading failure mechanism of jointed basalt rock.

There are many previous studies on various mechanisms and characteristics of rock with natural defects, but most of them are based on pseudo-triaxial studies with the same side pressure, which is difficult to accurately reflect the mechanical properties and crack propagation law of rock with fractures under true triaxial action. In view of this, many scholars observe the mesoscopic changes inside the rock by various experimental means. However, the process of sample sampling and preparation is tedious and the overall test cost is high. Based on this, this paper adopts matdem particle flow code, from mesoscopic view, reflect the three axis under the action of fractured Crack propagation law of fractured sandstone and the related mechanical properties, and true triaxial test to load of specimen, by analyzing the results, verifying the rationality of the simulation of particle flow, in order to provide references for related rock mechanics simulation test.

2. Determination of Mechanical Parameters of Specimens

Sandstone from a mining area in Inner Mongolia was selected as the raw material for the test, as shown in Figure 1a. The sampling place is shown in Figure 1b. The whole sandstone raw rock with cracks is selected, and the sandstone raw rock collected from the mining area is processed into the size that meets the test requirements with a stone saw machine according to the test design requirements. The raw materials are uniformly processed into rectangular samples of 100 mm × 100 mm × 150 mm, and the end faces are finally ground down with a stone grinder. The specimen made is shown in Figure 1c. In order to determine the mechanical parameters of sandstone specimens, uniaxial compression was used to test the specimens, and the mechanical parameters were determined as shown in

Table 1. Mechanical parameters of specimens.

Mechanical Parameters	Test Value
Poisson’s ratio	0.3
Compressive strength/MPa	48.24
Modulus of elasticity/Gpa	13.1

.

3. Model Building

3.1. Matdem and Bonding Theory

Matdem is a particle flow (matrix discrete element) software, which uses innovative GPU matrix calculation method and 3D contact algorithm to realize tens of millions of 3d unit motion calculations per second, and the number of calculation units and calculation speed reach the current world leading level. The software realizes automatic stacking modeling, layered material allocation, joint surface and load setting, etc. Due to its fast calculation speed, large number of calculation units and ability to accurately simulate the relationship between particle units under various conditions, it is widely used in civil engineering, mining, water conservancy and hydropower, and mechanical fields [12,13,14].

In many particle flow software, the parallel bonding model is a common bonding mode, which is similar to the bonding between two contact entities by injecting bonding materials of certain cross-section shape and size. The bonding model is shown in Figure 2. It shows that a block-like contact area is formed between two contact elements, which provides bonding between contact elements. The parallel bonding model can transfer not only force but also bending moment. When the contact position moves relative to each other, forces and torques are generated inside the bonding body. The resultant force and torque in the parallel bonding model are represented by F and M respectively, and the vector of force and torque can be decomposed into normal and tangential components, as shown in Equations (1) and (2):

F = Fⁿn + F^sj

(1)

M = Mⁿn + M^sj

(2)

Fⁿ, F^s, Mⁿ, M^s are the normal force, tangential force, normal moment and tangential moment respectively. n and t are normal and tangential unit vectors, respectively. When the stress between the bonding elements exceeds its bonding strength, the parallel bonding structure breaks, and the model turns into a conventional linear model. For the simulation of most rock materials, the parallel bonding model has good applicability, so the contact model selected in this paper is the parallel bonding model [15,16,17,18,19].

3.2. Establishment of Particle Flow Model

Matdem grain flow program can be an accurate reflection of real sandstone stress—strain and displacement and the internal microcracks extended the rule and process of formation, so the model with fracture mechanics characteristics of sandstone microscopic particles, the reasonable simulation of the sandstone specimen under true triaxial failure mechanism and mechanical characteristics is very important. According to relevant mechanical experimental literature in the laboratory, the strength of the cylinder specimen is slightly smaller than that of the cuboid specimen, but the difference between them is about 5%. In matdem particle flow program, the influence of upper and lower end constraints is small, and the difference between the two specimens can be ignored. In this paper, the uniaxial compression test of cylindrical specimen is used to determine the mesoscopic parameters of sandstone. The specimen used in this test is 100 mm × 100 mm×150 mm, and the size of the numerical model is set as 100 mm × 100 mm × 150 mm. The particle radius of this model is distributed within the range of 0.01 mm~0.025 mm. The model generated a total of 9172 particles. The numerical model of the specimen is shown in Figure 3. In order to approximate the loading process of the real test to the maximum extent, the upper and lower loading plates were added into the matdem program, and the loading rate was given as 0.03 m/s. The hydraulic servo mechanism was used to adjust the side wall movement to apply confining pressure on the specimen. The test stops when the macroscopic crack propagates to an observable stage. After the training test, the accuracy of the model reached about 95.4773%.

The uniaxial compression results of the specimen numerical model in the particle flow software are basically consistent with the laboratory results, and their coMParison curves are shown in Figure 4. It can be seen from Figure 4a that there is a slight difference between the stress-strain curves of numerical simulation and laboratory test. The stress-strain curves of all tests increase at first, reach peak value and then decrease. The strain values corresponding to the peak values of matdem and the modified test are smaller than those corresponding to the unmodified test. However, the peak values of matdem and the modified test are slightly larger than those of the unmodified test, which is considered to be caused by the existence of cracks in the natural sandstone specimen or the closure of the joints after compaction. The uniaxial compression test results show that there is an oblique main fracture in the sample. The mechanical parameters of sandstone are shown in Table 1. The cracks in the particle flow model are set by the joint properties, and the main cracks corresponding to the specimen are formed through the contact between adjacent particles [20,21,22,23,24,25]. In order to determine the microscopic parameters of the simulated particle flow, the microscopic parameters in the parallel bond model are continuously debugged by the trial and error method, and a set of reasonable microscopic parameters are finally determined, so that the basic macroscopic mechanical parameters (uniaxial compressive strength, elastic modulus) and stress-strain curves obtained by uniaxial compression of sandstone model samples in numerical simulation are basically consistent with the results obtained by laboratory tests. Determine the microscopic parameters of the sample model particle model as shown in

Table 2. Microscopic parameters of sample model particle model.

Mesoscopic Parameters	Taking Values
porosity	0.6
Elastic modulus of particles/GPa	18
Normal and tangential stiffness ratio of particles	4.8
solid friction factor	0.3
Parallel bond elastic modulus/GPa	16
parallel-bond stiffness ratio	4.5
Parallel bond normal strength/MPa	55
Parallel bond tangential strength/MPa	30

. In this case, the prefabricated crack is located in the center of the geometry and is 25 mm in length. The schematic diagram of sandstone sample under true triaxial action is shown in Figure 4b.

4. Result Analysis

4.1. Mechanical Response Characteristics of Sandstone

In order to study the effect of medium principal stress on mechanical properties and fracture propagation characteristics of fractured sandstone under true triaxial action, numerical simulation of true triaxial compression particle flow under different medium principal stress was carried out on specimens with inclination Angle of −45°, where σ₃ = 10 MPa and medium principal stress σ₂ = 30, 40 and 50 MPa respectively. Figure 5 shows that when other conditions remain unchanged, when the intermediate principal stress increases from 30 MPa to 50 MPa, the stress peak strength of sandstone specimen increases gradually, and so does the corresponding axial strain. When the principal stress σ₂ = 40 MPa, the peak stress strength of sandstone specimen is 54 MPa. When σ₂ = 50 MPa, the stress peak strength is 63 MPa. When the medium principal stress σ₂ increases from 40 MPa to 50 MPa, the stress peak strength increases by 16.67%, indicating that the medium principal stress can improve the strength of sandstone specimen to a certain extent. Figure 6 shows the cloud diagram of anisotropic strain changes of sandstone specimens with different medium principal stresses. When σ₂ = 40 MPa, the strain in the X direction changes obviously, and the maximum positive strain is 0.02 and the maximum negative strain is 0.12. When σ₂ = 50 MPa, the negative strain tends to decrease slightly, which may be caused by the influence of the intermediate principal stress on the strength of the specimen. When σ₂ = 50 MPa, the maximum value of positive strain is 0.08 and the maximum value of negative strain is 0.15. With the gradual increase of intermediate principal stress, the positive strain increases and more particles reach the maximum strain value in the specimen.

4.2. Sandstone Fracture Mode and Damage Characteristics

Figure 7 for the principal stress in the sigma 2 respectively 30,40,50 MPa when including fractured sandstone specimen under the condition of true triaxial eventually burst mode, when the normal stress and shear stress between the particles unit more than the normal and tangential compressive strength, the parallel bond structures occur between granular cell damage, and cracks, the blue for compressive shear stress produced by the crack, Red is the crack produced by tensile stress. It can be seen from Figure 7 that the fracture mode of sandstone specimen is a −45° macro fracture surface along the fracture development direction. When the medium principal stress is perpendicular to the fracture development direction, it is easy to form a macro fracture surface perpendicular to the fracture development direction. With the increase of the intermediate principal stress σ₂, the macroscopic cracks of the specimen increase gradually, and the fracture degree of the specimen deepens. When the principal stress σ₂ is 40 MPa, the macroscopic cracks have reached the maximum. When the intermediate stress increases to 50 MPa, the macroscopic cracks do not change. However, the cracks caused by tensile stress are still developing and reach the maximum when σ₂ = 50 MPa, finally forming the form of interlacing macroscopic cracks caused by tensile stress and compressive shear stress.

It can be seen from Figure 8 that both the axial strain and the number of cracks increase in general, and the number of cracks increases with the increase of the intermediate principal stress. When σ₂ = 50 MPa, the crack initiation of the specimen is earlier than that of the other two cases. According to Figure 6, when the medium principal stress increases from 30 MPa to 40 MPa, the peak stress increases from 47 MPa to 56 MPa, and the crack initiation stress also increases, indicating that the increase of the medium principal stress σ₂ is beneficial to the stability of fractured sandstone specimen. When the stress exceeds 85–90% of the peak stress, the number of cracks increases sharply. At this time, instability failure occurs in the sandstone specimen, and the number of tensile cracks is slightly larger than that of shear cracks. The intermediate principal stress increased from 40 MPa to 50 MPa, and the peak stress increased from 56 MPa to 64 MPa, but the crack initiation stress still increased, indicating that the excessive intermediate principal stress led to the accelerated propagation of internal cracks in the sandstone specimen, which eventually spread through and formed macroscopic cracks. At this time, the sandstone specimen was completely unstable and destroyed, with shear cracks playing a dominant role.

Under the action of true triaxial, sandstone fissure is produced, developed and extended continuously. The fracture field is constantly changing with the increase of intermediate principal stress, and its distribution surface looks irregular, but in fact, from another perspective, its graph has self-similarity, and fractal theory, as a new quantitative method, can well describe this nonlinear problem. In this paper, box-counting dimension is used to calculate the fractal dimension of sandstone specimens under different medium principal stresses. The grid of size W was used to cover the fracture maps under different medium principal stresses, and the number of grids containing cracks was calculated by statistical method and denoted as N (W). Logarithms of w and N (w) were taken respectively, and the obtained data were fitted. In order to further elaborate the influence of different intermediate principal stresses on fracture development, damage variable D is introduced and defined as:

$$D_n=\frac{d_0-d_n}{d_0-d_{\mathrm{f}}}$$

(3)

d₀ is the fractal dimension of specimen crack at the initial moment of damage; D_n is the fractal dimension after the action of different intermediate principal stresses. D_f is the fractal dimension at the time of damage and failure, and its damage and failure time is 2.

According to the above theory, the damaged specimens are processed by binarization method, and the results are shown in Figure 9. Based on the above data, fractal dimensions of fissure and damage under different intermediate principal stresses are drawn, as shown in Figure 9. It can be seen from Figure 10 that with the increase of intermediate principal stress, the overall trend of damage variable and fracture dimension is growth. In the initial state, there is a main crack in the specimen, its fracture dimension is 1.072, and its damage variable is 0.37. When the medium principal stress increases to 30 MPa and the medium principal stress is 15 MPa, the fracture dimension and damage variable increase sharply, the fracture dimension increases by 8.96% and the damage variable increases by 56.8%, indicating that under the action of true triaxial, the growth rate of fracture extension dimension and damage variable is the largest. There are many secondary cracks produced by the initial primary cracks in the specimen, and both ends of the primary cracks extend along the −45° Angle. When the intermediate principal stress reaches 20 MPa, the tensile stress begins to act and produce cracks; when the intermediate principal stress is 30 MPa, the macroscopic cracks increase significantly and the fracture degree deepens. When the medium principal stress increases from 30 MPa to 50 MPa, the fracture dimension and damage variable have a similar trend to that of the medium principal stress of 30 MPa, but their growth rates are slightly lower than that of the medium principal stress of 30 MPa. As the fracture dimension and damage variable increase gradually, the fracture grid becomes more complex and the time fracture degree deepens. Oolitic structure was formed on the surface of the specimen accompanied by rock particle shedding. At this time, the fracture degree of the specimen reached the maximum.

5. True Triaxial Verification Test

As mentioned above, a −45° crack was prefabricated for the 100 mm × 100 mm × 150 mm cuboid sample polished by a millstone machine. The finished specimen was put into the testing machine, and the loading rate of the upper and lower loading plates was set at 0.03 m/s for true triaxial test, as shown in Figure 11.

Considering that it is difficult to observe the crack development of the broken specimen after the true triaxial test, the method combining scanning and digital image is used to observe the crack situation. After triaxial damage the specimen using the elastic and plastic sealed paper packages, specimen after fixation with test marker will crack in paper, with the scanner will finally cracks with the requirments of paper to scan images and the output, as shown in Figure 12, the blue for compressive shear stress crack, red for the tensile stress crack, green for the initial crack. Around the initial cracks, many secondary cracks are produced by compression shear stress, and surface microcracks are formed. Due to the action of tensile stress, new inclined cracks are generated at the end of the initial crack and extend to both ends. Secondary cracks are formed by the joint action of tensile stress and compressive shear stress around the initial crack, resulting in the initial failure mode of the specimen. As the test continued, produced by the tensile stress of crack rate gradually reduced, produced by pressure shear stress crack expanding rapidly, through to the other end of the specimen, the formation of final failure mode of the specimens, and the compressive shear stress crack range is greater than the tensile stress in the range, the specimen is shear failure on the whole, accompany during loading in burst mode.

6. Conclusions

(1)

Under the condition of true triaxial stress, the intermediate principal stress helps to increase the strength of fractured sandstone. In the process of increasing the intermediate principal stress σ2 from 40 MPa to 50 MPa, when the intermediate principal stress exceeds 30% of the maximum principal stress, the peak stress intensity increases by 16.67%, and the excessive intermediate principal stress will cause the strength of fractured rock to decrease.

(2)

Under the influence of true triaxial action, the macro fracture mode of sandstone is -45O macro fracture surface along the fracture development direction, and when the principal stress and fracture development direction are -90O, it is easy to produce macro fracture surface perpendicular to the fracture trend. With the increase of intermediate principal stress, the fracture degree of specimen increases, and finally the specimen forms a form of interlacing cracks.

(3)

Under the action of true triaxial stress, the crack initiation stress increases with the increase of the medium principal stress, indicating that the increase of the medium principal stress is beneficial to the stability of the fractured sandstone specimen. When the specimen stress exceeds 85% to 90% of the peak stress, the specimen instability failure occurs. When the intermediate principal stress increases from 40 MPa to 50 MPa, the crack initiation stress decreases slightly, indicating that the excessive intermediate principal stress accelerates the propagation of cracks inside the sandstone specimen, and the sandstone specimen is completely unstable and shear cracks play a dominant role.

(4)

With the increase of the intermediate principal stress, the fractal dimension of the fracture presents an obvious phenomenon of increasing dimension, and the damage degree of the fracture deepens, the fracture grid is gradually complex, and the compression shear stress fracture and tensile stress fracture gradually expand and form macro fracture, and the fracture degree of the sandstone specimen continues to increase, resulting in sandstone deformation and failure.

(5)

The comparison between the true triaxial test and the particle flow simulation test shows that the results are basically similar, which can truly reflect the failure mechanism and failure mode of the specimen, and proves that the particle flow simulation test has certain reliability and rationality.