# Influence of Microscopic Parameters on the Stress-Strain Relation in Rocks

Authors: Yanhui Cheng; Weijun Yang
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

Macromaterial properties should correspond to the mesoscopic parameters simulated in practical engineering problems. Discrete element contains a variety of particle models and its corresponding mesoscopic parameters, and the one-to-one relationship between the mesoscopic parameters and macroscopic parameters is difficult to establish. This paper studies the influence of microscopical characteristic parameters, such as particle contact stiffness ratio, parallel bond stiffness ratio, particle contact modulus, and parallel bond elastic modulus, on the stress-strain relation in rocks, which shows that (1) The range of particle contact stiffness ratiokn/ks largely varies, but the stress-strain relation curve is relatively small. The particle contact stiffness has less influence on the elastic modulus of the simulated specimens than kn/ks. (2) Before the failure of the specimen, the axial strain corresponding to the peak compressive strength increases with the increase in the stiffness ratio kn¯/ks¯ of the parallel bond. (3) The particle contact modulus Ec has a great influence on the elastic modulus of sandstone and is characterized by the increase in the particle contact modulus Ec, corresponding axial strain for the peak compressive strength decreases, and the slope of the stress-strain relationship curves before damage increases. (4) The elastic modulus of the parallel bond greatly influences the uniaxial compressive strength, and the relationship between them is proportional.


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## Body

## 1. Introduction

Determination of the mechanical properties of rock is an important part of rock engineering design [1–4]. Rock mechanics parameters are usually obtained by laboratory tests and numerical analysis. Discrete element is a widely used numerical method in the study of rock micro- and macromechanical properties to solve the noncontinuous medium problem [5–8]. Scholars further used this technique to slope, mining, rock burst, debris flow, joint, and other practical projects where it obtained good simulation results [9–11]. First, macromaterial properties should correspond to the mesoscopic parameters simulated in practical engineering problems. However, discrete element contains a variety of particle models and its corresponding mesoscopic parameters, and the one-to-one relationship between the mesoscopic parameters and macroscopic parameters is difficult to establish [12–15]. Therefore, many scholars selected different particle bond models for macroscopic materials and studied the influence of microscopic parameters on macroproperties of materials [16–18]. For instance, the particles in PFC2D were used to contact the bond model by Huang and Detournay [19]. Yang et al. [16] adopted the parallel bond model and studied the effect of the microscopic parameters on Young’s modulus, Poisson’s ratio, and uniaxial compressive strength of the adhesive materials. The shear behavior of rock joints is numerically simulated using the discrete element code PFC2D by Bahaaddini et al. [17]. However, these studies mainly consider the parameters such as particle bonding strength, particle friction coefficient, and particle bond strength. Stiffness properties among mesoscopic particles affect the force between particles features affecting the macroscopic appearance of the material [20, 21]. These properties include parallel bond stiffness ratio, particle contact modulus, and parallel to the compressive modulus of elasticity [22–24]. Therefore, the influence of rigidity attribute of the particle on the macroscopic mechanical characteristics of the material must be further studied. Based on the considerations mentioned above, the use of the particle flow discrete element numerical simulation method (PFC), the condition of uniaxial compression, and material microstructures such as rock strength, stiffness parameters for the influence of the macroscopic mechanical behavior are studied in this paper.


## 2. Modeling

The mesoscopic parameters used in this paper are referred to the work by Bahaaddini et al. [5] and Cundall [25], as shown in Table 1. Figure 1 shows the numerical stimulation model with the height of 100 mm and diameter of 50 mm. The particle minimum diameter Dmin = 0.28 mm, the maximum particle size of the particle is Dmax = 0.42 mm, and the particle radius ratio is 1.5, which is randomly generated between the maximum radius and the minimum radius. Also, the uniform distribution can be followed. Moreover, embedded Fish language by the servo control method is used to control the model of the “wall” movement to realize the uniaxial and conventional triaxial and the numerical simulation of the direct shear test [26, 27]. A parallel bond model is used for particle bonding, and the parallel bonding radius is set to 1.0. The results of numerical simulation in this paper are compared with the indoor test results shown in Table 2 to verify the reliability of the model. In addition, the model used was 42 mm in diameter and 82 mm in height. Before calculating, the parameters should be verified [28–30]. Table 2 shows that the numerical simulation results of this paper are close to those of the laboratory test results. It can be seen that the compressive strength of the numerical simulation is slightly less than that of the test results, and the elastic modulus is slightly larger than the test results, but the difference between them is very small.Table 1
Mesoscopic parameters.
Particle microscopic parameters
Parallel bonding microscopic parameters
Particle density (kg/m3)
2205
Parallel bond elastic modulus (GPa)
2.8
Particle contact modulus (GPa)
2.8
Normal strength (MPa)
20 ± 4.5
Friction factorμ
0.6
Shear strength (MPa)
20 ± 4.5
Particle contact stiffness ratiokn/ks
1.45
Parallel bond stiffness ratiokn/ks
1.45Figure 1
Granular aggregate.Table 2
Comparison of the simulation model and the results of the laboratory model.
UCS (MPa)
E (GPa)
υ
Laboratory results
Average value
27.40
4.20
0.20
Numerical results
Average value
26.64
4.28
0.17


## 3. Influence of Microscopical Characteristic Parameters

### 3.1. Particle Contact Stiffness Ratio

The particle contact stiffness ratio is the ratio of the contact stiffness of particles to the normal stiffness and tangential stiffness ofkn/ks. The following steps should be performed to analyze the influence of kn/ks on macroscopic mechanical behavior: change the parameter kn/ks, keep the other microscopic parameters unchanged, and obtain the stress-strain relation curve as shown in Figure 2. The relationship between the particle contact stiffness ratio kn/ks and the macroscopic mechanical behavior of the simulated specimens was also investigated. Furthermore, the parameters of macroscopic mechanical properties under different particle contact stiffness ratio kn/ks simulation tests can be obtained by the numerical simulation results of discrete element as shown in Table 3.Figure 2
The stress-strain curves of different particle contact stiffness ratios are compared.Table 3
The relationship between the particle contact stiffness ratio and macroscopic mechanical properties.
Particle contact stiffness ratio
Modulus of elasticity (GPa)
Poisson’s ratio
Uniaxial compressive strength (MPa)
0.5
4.39
0.136
27.26
1
4.33
0.147
28.48
1.45
4.22
0.157
27.06
2
4.17
0.166
26.37
3
4.06
0.178
26.80
4
3.99
0.187
26.37
5
3.94
0.193
25.80
6
3.90
0.197
25.59
7
3.87
0.201
25.63
8
3.84
0.203
24.98
9
3.82
0.205
24.84
10
3.81
0.207
24.54The elastic deformation stage occurs before specimen failure and is the stage where the stress-strain relationship is in a straight line. However, materials called elastomers cause the sudden destruction of the simulated specimen with the deformation properties of rock because the plastic deformation stage is not obvious. The range of particle contact stiffness ratiokn/ks largely varies, but the stress-strain relation curve is relatively small. Uniaxial compressive strength change is small but can be seen when the particle contact stiffness kn/ks is 1.0. The particle contact stiffness kn/ks for the axial strain can also be seen when the peak compressive strength and elastic limit range are small and when the uniaxial compressive strength reaches the maximum value. The slope of the stress-strain relation curve is relatively small, although the contact stiffness of the particles is larger than the range of variation. The particle contact stiffness has less influence on the elastic modulus of the simulated specimens than kn/ks. In general, the effect of particle contact stiffness on uniaxial compressive strength and elastic modulus E in the macroscopic mechanical properties of the simulated specimens is small because the model used in this paper is a parallel bond model. Furthermore, the macroscopic mechanical properties of the specimens are mainly related to the microscopic parameters related to the parallel bonding.


### 3.2. Parallel Bond Stiffness Ratio

The parallel bonding stiffness ratiokn¯/ks¯ is a ratio of the normal stiffness kn¯ to the shear stiffness ks¯ of the two particles when the microscopic parameters are kept unchanged, and kn¯/ks¯ are the only ones changed. Numerical simulation of the uniaxial compression test is conducted to obtain the stress-strain relation curve of different parallel bond stiffness ratio kn¯/ks¯ situations as shown in Figure 3. The parameters of macroscopic mechanical properties are obtained based on the numerical simulation results of discrete elements as shown in Table 4.Figure 3
Stress-strain relation for different parallel bonding stiffness ratios.Table 4
Relation between the macromechanical properties with different parallel bonding stiffness ratios.
Parallel bonding stiffness ratio
Elastic modulus (GPa)
Poisson’s ratio
Uniaxial compressive strength (MPa)
0.5
5.20
0.030
23.73
1
4.70
0.107
25.51
1.45
4.22
0.157
27.06
2
3.90
0.190
28.00
3
3.52
0.228
27.49
4
3.26
0.251
26.92
5
3.07
0.268
26.60
6
2.93
0.280
24.97
7
2.81
0.290
24.72
8
2.72
0.297
24.06
9
2.64
0.304
23.73
10
2.57
0.310
23.38Before the failure of the specimen, the axial strain corresponding to the peak compressive strength increases with the increase in the stiffness ratiokn¯/ks¯ of the parallel bond. The variation range of axial strain is nearly 4 mm, and the elastic limit becomes smaller with the increase in the parallel bond stiffness ratio. The peak compressive strength reached a maximum of 28 MPa at the time of parallel bond stiffness ratio 2.0. Conversely, when the parallel bond stiffness ratio is less than 2.0, the uniaxial compressive strength increases as the parallel bond stiffness ratio increases. When the increasing parallel bond stiffness ratio is greater than 2.0, the uniaxial compressive strength has slowly small increase. When the parallel bond stiffness ratio is greater than 5.0, the peak compressive strength decreases significantly. This phenomenon is caused by the gradual increase in the parallel bond stiffness ratio. In addition, the parallel bond’s normal stiffness is greater than the tangential stiffness; specimen “rigid” enhanced by plastic deformation damage to brittle failure mode change gradually become a rigid body, and the uniaxial compressive strength decreases. The slope of the stress-strain relation curve before the sudden failure also decreases with the increase in the parallel bond stiffness ratio. The effect of the parallel bond stiffness on the elastic modulus of the specimens is relatively large and inversely proportional to each other. This phenomenon is caused by the high rigidity of the parallel bond between particles. A strong “rigidity” indicates small “elasticity” and small macroscopic elastic modulus.


### 3.3. Particle Contact Modulus

The particle contact modulusEc is the elastic modulus of contact between particles. Certain deformation occurs when the pellet is affected by the force. Figure 4 shows the stress-strain of the obtained relation of particle contact modulus Ec. The relationship between the particle contact modulus Ec and the macroscopic mechanical behavior of the simulated specimens is also discussed. In addition, Table 5 shows the parameters of macroscopic mechanical properties under different particle contact modulus Ec simulation tests obtained by the numerical simulation results of discrete elements.Figure 4
Stress-strain relation for different particle contact moduli.Table 5
Relation between the macromechanical properties with different particle contact moduli.
Particle contact modulus GPa)
Elastic modulus (GPa)
Poisson’s ratio
Uniaxial compressive strength (MPa)
1
3.09
0.121
24.85
1.5
3.40
0.130
26.25
2
3.73
0.141
25.95
2.5
4.06
0.149
26.32
2.8
4.22
0.157
27.06
3.5
4.64
0.170
26.81
4
4.89
0.180
27.54
4.5
5.24
0.187
29.15
5
5.49
0.194
27.53
6
6.00
0.207
28.44
7
6.52
0.224
29.86Figure4 shows that the particle contact modulus Ec has a great influence on the stress-strain relation curve. The particle contact modulus Ec has a great influence on the elastic modulus of sandstone and is characterized by the increase in the particle contact modulus Ec, corresponding axial strain for the peak compressive strength decreases, and the slope of the stress-strain relationship curves before damage increases. This phenomenon is also caused by the increase in the particle contact modulus Ec which increases the elastic modulus E of macroscopic materials. The uniaxial compressive strength has the general increasing trend with the increasing particle contact modulus Ec, but the amplitude of increase is relatively small. Hence, the particle contact modulus Ec minimally affects the uniaxial compressive strength, and the variation is not evident.


### 3.4. Parallel Bond Elastic Modulus

The elastic modulus of the parallel bondEc¯ is the elastic modulus of the material between two particles in each parallel bond model. Figure 5 shows the stress-strain relation curve. Table 6 shows the parameters of macroscopic mechanical properties obtained from the numerical simulation results of discrete elements.Figure 5
Stress-strain relation for different elastic moduli of the parallel bond.Table 6
Relation between the macromechanical properties with different elastic moduli of the parallel bond.
Elastic modulus of the parallel bond (GPa)
Elastic modulus (GPa)
Poisson’s ratio
Uniaxial compressive strength (MPa)
1
2.49
0.238
32.14
1.5
3.02
0.200
30.28
2
3.50
0.178
28.82
2.5
4.02
0.158
27.39
2.8
4.22
0.157
27.06
3.5
4.83
0.146
26.73
4
5.25
0.140
25.37
4.5
5.67
0.135
25.12
5
6.09
0.132
24.61
6
6.92
0.126
22.85
7
7.74
0.123
22.59Figure5 also shows that the parallel bonded elastic modulus greatly influences the stress-strain curve most notably with the parallel bond and when the increase in the elastic modulus and uniaxial compressive strength is large. The elastic modulus of the parallel bond greatly influences the uniaxial compressive strength, and the relationship between them is proportional. Before the failure of the test piece, the axial strain and elastic limit corresponding to the peak compressive strength decrease with the increase in the elastic modulus of the parallel bond. This phenomenon is caused by the increase in the slope of the stress-strain relation curve, which increases the elastic modulus of the parallel bond. Therefore, the elastic modulus of the parallel bond that greatly influences the elastic modulus E of the simulated specimens mainly affects the elastic modulus of the parallel bond.


## 3.1. Particle Contact Stiffness Ratio

The particle contact stiffness ratio is the ratio of the contact stiffness of particles to the normal stiffness and tangential stiffness ofkn/ks. The following steps should be performed to analyze the influence of kn/ks on macroscopic mechanical behavior: change the parameter kn/ks, keep the other microscopic parameters unchanged, and obtain the stress-strain relation curve as shown in Figure 2. The relationship between the particle contact stiffness ratio kn/ks and the macroscopic mechanical behavior of the simulated specimens was also investigated. Furthermore, the parameters of macroscopic mechanical properties under different particle contact stiffness ratio kn/ks simulation tests can be obtained by the numerical simulation results of discrete element as shown in Table 3.Figure 2
The stress-strain curves of different particle contact stiffness ratios are compared.Table 3
The relationship between the particle contact stiffness ratio and macroscopic mechanical properties.
Particle contact stiffness ratio
Modulus of elasticity (GPa)
Poisson’s ratio
Uniaxial compressive strength (MPa)
0.5
4.39
0.136
27.26
1
4.33
0.147
28.48
1.45
4.22
0.157
27.06
2
4.17
0.166
26.37
3
4.06
0.178
26.80
4
3.99
0.187
26.37
5
3.94
0.193
25.80
6
3.90
0.197
25.59
7
3.87
0.201
25.63
8
3.84
0.203
24.98
9
3.82
0.205
24.84
10
3.81
0.207
24.54The elastic deformation stage occurs before specimen failure and is the stage where the stress-strain relationship is in a straight line. However, materials called elastomers cause the sudden destruction of the simulated specimen with the deformation properties of rock because the plastic deformation stage is not obvious. The range of particle contact stiffness ratiokn/ks largely varies, but the stress-strain relation curve is relatively small. Uniaxial compressive strength change is small but can be seen when the particle contact stiffness kn/ks is 1.0. The particle contact stiffness kn/ks for the axial strain can also be seen when the peak compressive strength and elastic limit range are small and when the uniaxial compressive strength reaches the maximum value. The slope of the stress-strain relation curve is relatively small, although the contact stiffness of the particles is larger than the range of variation. The particle contact stiffness has less influence on the elastic modulus of the simulated specimens than kn/ks. In general, the effect of particle contact stiffness on uniaxial compressive strength and elastic modulus E in the macroscopic mechanical properties of the simulated specimens is small because the model used in this paper is a parallel bond model. Furthermore, the macroscopic mechanical properties of the specimens are mainly related to the microscopic parameters related to the parallel bonding.


## 3.2. Parallel Bond Stiffness Ratio

The parallel bonding stiffness ratiokn¯/ks¯ is a ratio of the normal stiffness kn¯ to the shear stiffness ks¯ of the two particles when the microscopic parameters are kept unchanged, and kn¯/ks¯ are the only ones changed. Numerical simulation of the uniaxial compression test is conducted to obtain the stress-strain relation curve of different parallel bond stiffness ratio kn¯/ks¯ situations as shown in Figure 3. The parameters of macroscopic mechanical properties are obtained based on the numerical simulation results of discrete elements as shown in Table 4.Figure 3
Stress-strain relation for different parallel bonding stiffness ratios.Table 4
Relation between the macromechanical properties with different parallel bonding stiffness ratios.
Parallel bonding stiffness ratio
Elastic modulus (GPa)
Poisson’s ratio
Uniaxial compressive strength (MPa)
0.5
5.20
0.030
23.73
1
4.70
0.107
25.51
1.45
4.22
0.157
27.06
2
3.90
0.190
28.00
3
3.52
0.228
27.49
4
3.26
0.251
26.92
5
3.07
0.268
26.60
6
2.93
0.280
24.97
7
2.81
0.290
24.72
8
2.72
0.297
24.06
9
2.64
0.304
23.73
10
2.57
0.310
23.38Before the failure of the specimen, the axial strain corresponding to the peak compressive strength increases with the increase in the stiffness ratiokn¯/ks¯ of the parallel bond. The variation range of axial strain is nearly 4 mm, and the elastic limit becomes smaller with the increase in the parallel bond stiffness ratio. The peak compressive strength reached a maximum of 28 MPa at the time of parallel bond stiffness ratio 2.0. Conversely, when the parallel bond stiffness ratio is less than 2.0, the uniaxial compressive strength increases as the parallel bond stiffness ratio increases. When the increasing parallel bond stiffness ratio is greater than 2.0, the uniaxial compressive strength has slowly small increase. When the parallel bond stiffness ratio is greater than 5.0, the peak compressive strength decreases significantly. This phenomenon is caused by the gradual increase in the parallel bond stiffness ratio. In addition, the parallel bond’s normal stiffness is greater than the tangential stiffness; specimen “rigid” enhanced by plastic deformation damage to brittle failure mode change gradually become a rigid body, and the uniaxial compressive strength decreases. The slope of the stress-strain relation curve before the sudden failure also decreases with the increase in the parallel bond stiffness ratio. The effect of the parallel bond stiffness on the elastic modulus of the specimens is relatively large and inversely proportional to each other. This phenomenon is caused by the high rigidity of the parallel bond between particles. A strong “rigidity” indicates small “elasticity” and small macroscopic elastic modulus.


## 3.3. Particle Contact Modulus

The particle contact modulusEc is the elastic modulus of contact between particles. Certain deformation occurs when the pellet is affected by the force. Figure 4 shows the stress-strain of the obtained relation of particle contact modulus Ec. The relationship between the particle contact modulus Ec and the macroscopic mechanical behavior of the simulated specimens is also discussed. In addition, Table 5 shows the parameters of macroscopic mechanical properties under different particle contact modulus Ec simulation tests obtained by the numerical simulation results of discrete elements.Figure 4
Stress-strain relation for different particle contact moduli.Table 5
Relation between the macromechanical properties with different particle contact moduli.
Particle contact modulus GPa)
Elastic modulus (GPa)
Poisson’s ratio
Uniaxial compressive strength (MPa)
1
3.09
0.121
24.85
1.5
3.40
0.130
26.25
2
3.73
0.141
25.95
2.5
4.06
0.149
26.32
2.8
4.22
0.157
27.06
3.5
4.64
0.170
26.81
4
4.89
0.180
27.54
4.5
5.24
0.187
29.15
5
5.49
0.194
27.53
6
6.00
0.207
28.44
7
6.52
0.224
29.86Figure4 shows that the particle contact modulus Ec has a great influence on the stress-strain relation curve. The particle contact modulus Ec has a great influence on the elastic modulus of sandstone and is characterized by the increase in the particle contact modulus Ec, corresponding axial strain for the peak compressive strength decreases, and the slope of the stress-strain relationship curves before damage increases. This phenomenon is also caused by the increase in the particle contact modulus Ec which increases the elastic modulus E of macroscopic materials. The uniaxial compressive strength has the general increasing trend with the increasing particle contact modulus Ec, but the amplitude of increase is relatively small. Hence, the particle contact modulus Ec minimally affects the uniaxial compressive strength, and the variation is not evident.


## 3.4. Parallel Bond Elastic Modulus

The elastic modulus of the parallel bondEc¯ is the elastic modulus of the material between two particles in each parallel bond model. Figure 5 shows the stress-strain relation curve. Table 6 shows the parameters of macroscopic mechanical properties obtained from the numerical simulation results of discrete elements.Figure 5
Stress-strain relation for different elastic moduli of the parallel bond.Table 6
Relation between the macromechanical properties with different elastic moduli of the parallel bond.
Elastic modulus of the parallel bond (GPa)
Elastic modulus (GPa)
Poisson’s ratio
Uniaxial compressive strength (MPa)
1
2.49
0.238
32.14
1.5
3.02
0.200
30.28
2
3.50
0.178
28.82
2.5
4.02
0.158
27.39
2.8
4.22
0.157
27.06
3.5
4.83
0.146
26.73
4
5.25
0.140
25.37
4.5
5.67
0.135
25.12
5
6.09
0.132
24.61
6
6.92
0.126
22.85
7
7.74
0.123
22.59Figure5 also shows that the parallel bonded elastic modulus greatly influences the stress-strain curve most notably with the parallel bond and when the increase in the elastic modulus and uniaxial compressive strength is large. The elastic modulus of the parallel bond greatly influences the uniaxial compressive strength, and the relationship between them is proportional. Before the failure of the test piece, the axial strain and elastic limit corresponding to the peak compressive strength decrease with the increase in the elastic modulus of the parallel bond. This phenomenon is caused by the increase in the slope of the stress-strain relation curve, which increases the elastic modulus of the parallel bond. Therefore, the elastic modulus of the parallel bond that greatly influences the elastic modulus E of the simulated specimens mainly affects the elastic modulus of the parallel bond.


## 4. Conclusions

(1)
The range of the particle contact stiffness ratiokn/ks largely varies, but the stress-strain relation curve is relatively small. Uniaxial compressive strength change is small. The slope of the stress-strain relation curve is relatively small, although the contact stiffness of the particles is larger than the range of variation. The particle contact stiffness has less influence on the elastic modulus of the simulated specimens than kn/ks.
(2)
Before the failure of the specimen, the axial strain corresponding to the peak compressive strength increases with the increase in the stiffness ratiokn¯/ks¯ of the parallel bond. Conversely, when the parallel bond stiffness ratio is less than 2.0, the uniaxial compressive strength increases as the parallel bond stiffness ratio increases. When the increasing parallel bond stiffness ratio is greater than 2.0, the uniaxial compressive strength has slowly small increase. When the parallel bond stiffness ratio is greater than 5.0, the peak compressive strength decreases significantly. The slope of the stress-strain relation curve before the sudden failure also decreases with the increase in the parallel bond stiffness ratio.
(3)
The particle contact modulusEc has a great influence on the elastic modulus of sandstone and is characterized by the increase in particle contact modulus Ec, corresponding axial strain for the peak compressive strength decreases, and the slope of the stress-strain relationship curves before damage increases. The uniaxial compressive strength has the general increasing trend with the increasing particle contact modulus Ec, but the amplitude of increase is relatively small.
(4)
The elastic modulus of the parallel bond greatly influences the uniaxial compressive strength, and the relationship between them is proportional. Before the failure of the test piece, the axial strain and elastic limit corresponding to the peak compressive strength decreases with the increase in the elastic modulus of the parallel bond.



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*Source: 7050468-2018-06-25.xml*