# Study of Post-Peak Strain Softening Mechanical Behaviour of Rock Material Based on Hoek–Brown Criterion

Authors: Qibin Lin; Ping Cao; Peixin Wang
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

In order to build the post-peak strain softening model of rock, the evolution laws of rock parametersm,s were obtained by using the evolutionary mode of piecewise linear function regarding the maximum principle stress. Based on the nonlinear Hoek–Brown criterion, the analytical relationship of the rock strength parameters m,s, cohesion c, and friction angle φ has been developed by theoretical derivation. According to the analysis on the four different types of rock, it is found that, within the range from 0 to σ3min, the peak hardness of the rock becomes smaller as the confining pressure increases and the degree of rock fragmentation decreases as well. The post-peak stress-strain curves obtained from the developed softening model are in good agreement with the laboratory test results under different confining pressures. In conclusion, the analytical method is reasonable, and it can predict the post-peak mechanical behaviour of rock well, which provides a new thought for the rock-softening simulation.


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## Body

## 1. Introduction

It is well known that the rock material is one type of heterogeneous brittle material with complex mechanical properties, which is significantly different from the metal material. In terms of post-peak strain softening characteristics, it is impossible to describe by the classical strength theory [1]. Many researchers have studied from different aspects regarding the post-peak strain softening behaviour. Rummel and Fairhurst [2] were one of the earliest researchers to study the general capabilities of post-peak compression tests of the rock. Tutluoglu et al. [3] gathered a lot of data related to the postfailure part of the stress-strain curve to study the relationship between prefailure and postfailure mechanical properties of the rock material. But the confining pressure on the post-peak mechanical behaviour of rock was not considered. Furthermore, the laboratory tests [4–8] show that the confining pressure greatly affected the post-peak mechanical behaviour of rock. Meng et al. [9] proposed a new evaluation method based on the magnitude and velocity of the post-peak stress drop, which accurately accounts for the influence of the confining pressure on strain softening. Alejano and Alonso [10] proposed a conveniently simple formulation of the dilatancy angle, which is capable of representing the rock sample’s strain softening behaviour in compressive tests. Walton et al. [11] performed a series of uniaxial and triaxial tests and studied post-peak behaviour about the strength, deformability, and dilatancy at low confinement. Fang and Harrison [12] utilized the concept of a degradation index to describe the variation in degradation that rock exhibits when it is subjected to various confining pressures. Zhao et al. [13] established evolution laws of the strength parameters of soft mudstone at the post-peak stage by considering stiffness degradation. By introducing the concept of generalized friction angle and generalized cohesion, Lu et al. [14] constructed the post-peak computing model combined with FLAC code. Leontiev and Huacasi [15] and Filho and Leontiev [16] have conducted research from different angles on the post-peak mechanics of rock.In order to study the post-peak strain softening mechanical properties of rock, the mechanical models described above are based on the Mohr–Coulomb criterion. However, the extent of rock damage was not studied in their studies. It is seldom to establish the post-peak strain softening model based on the Hoek–Brown criterion. Tan et al. [17] considered degradation and expansion behaviour of the rock constructing the post-peak strain softening model based on the Hoek–Brown criterion. The rock parameters m,s in the Hoek–Brown criterion reflect the failure condition of rock reasonably. In this paper, piecewise linear function was used to obtain the evolution of rock strength parameters m,s. Accordingly, the analytical relationships among rock strength parameters m,s, friction angle φ, and cohesion c were developed, and a new post-peak strain softening model was proposed. The verification with laboratory test results shows that the analytical model is reasonable to predict the post-peak rock behaviour.


## 2. Simplification of Strain Softening Process

Figure1 shows a series of stress-strain curves of Hawkesbury sandstone specimens under different confining pressures [18]. It can be observed that the rock failure is a staged development process. With the increase of confining pressure, the rock post-peak strain softening behaviour is gradually transformed into ideal elastic-plastic behaviour. In order to process the main deformation and failure characteristics more informative in mathematics [19, 20], Figure 1 has been simplified to Figure 2. In Figure 2, the dash line indicates the stress-strain curve obtained by the experiment and the solid line represents the ideal stress-strain curve. The strain softening model is typically divided into three stages: pre-peak elastic deformation stage, post-peak strain softening stage, and residual stage.Figure 1
Complete stress-strain curves of Hawkesbury sandstone specimens under different confining stresses [18].Figure 2
Test curves and idealized stress-strain curves.


## 3. Analysis of the Rock Parameters of the Hoek–Brown Criterion and the Post-Peak Constitutive Model

In the post-peak deformation stage, the development of strain softening is induced by the change of the rock strength parameters [12–14].


### 3.1. Strength Criterion

According to the Hoek–Brown criterion [21],(1)σ1=σ3+mησ3σci+sησci2,where σ1 is the effective major principal stress, σ3 is the effective minor principal stress, η is the strain softening parameter, and σci is the uniaxial compressive strength of intact rock. m and s are rock strength parameters; s represents the degree of fragmentation of rock ranging from 0.0 to 1.0. For unfractured rock mass (i.e., rock), s=1.0. m represents the hardness of rock ranging from 0.0000001 to 25. For intact hard rock, m=2.5.


### 3.2. Strain Softening Parameters

There are usually two methods for selecting strain softening parameters. The first one is based on the increment method:(2)γ˙=23ε˙1pε˙1p+ε˙2pε˙2p+ε˙3pε˙3p,wherein ε˙ipi=1,2,3 represents the changing rate of the ith principal plastic strain.In another method, the strain softening parameter is considered as a function of the internal variable. The major principal strainε1, the major principal plastic strain ε1p, the plastic shear strain γ1, and the equivalent plastic shear strain γ2 are frequently used in this method:(3)γ1=ε1p−ε3p,γ2=23ε1pε1p+ε2pε2p+ε3pε3p.As the rock parametersm,s vary with the increase of the major principal strain, the major principal strain ε1 from the above method is used as the softening parameter in this paper, which leads to ease of analytical calculation.


### 3.3. Analysis of Rock Strength Parametersm,s and the Post-Peak Constitutive Model

The relationship between post-peak strength parameters and strain softening parameter, namely, the evolution of strength parameters, can be obtained by the laboratory experiment and numerical simulation. For simplicity, the mentioned relationship is often assumed as the form of a piecewise linear function [22–25], which can be expressed as(4)ηγ=ηp−ηp−ηrγγ∗,γp<γ<γ∗ηr,γ≥γ∗,where η, ηp, and ηr represent the strength parameter, strength parameter at the peak, and strength parameter at the starting point of the residual stage, respectively; γp and γ∗ are the critical softening parameter at the peak and strength parameter at the starting point of the residual stage, as shown in Figure 3.Figure 3
Variation of the strength parameterη in the whole stress-strain curve.According to (4), the evolution of rock strength parameters m,s can be obtained:(5)mε=mp−ε1−εrmp−mrεr−εp,εp<ε<εrmr,ε≥εr,(6)sε=sp−ε1−εrsp−srεr−εp,εp<ε<εrsr,ε≥εr,where mpandsp are the peak values of mands; mrandsr are the peak residual values of mands; and εpandεr are the major principal strain values at the peak and at the starting point of the residual stage, respectively.By substituting (5) and (6) into (1), the integration in (1) may be carried out analytically, and the post-peak strain softening constitutive model can be obtained accordingly.


## 3.1. Strength Criterion

According to the Hoek–Brown criterion [21],(1)σ1=σ3+mησ3σci+sησci2,where σ1 is the effective major principal stress, σ3 is the effective minor principal stress, η is the strain softening parameter, and σci is the uniaxial compressive strength of intact rock. m and s are rock strength parameters; s represents the degree of fragmentation of rock ranging from 0.0 to 1.0. For unfractured rock mass (i.e., rock), s=1.0. m represents the hardness of rock ranging from 0.0000001 to 25. For intact hard rock, m=2.5.


## 3.2. Strain Softening Parameters

There are usually two methods for selecting strain softening parameters. The first one is based on the increment method:(2)γ˙=23ε˙1pε˙1p+ε˙2pε˙2p+ε˙3pε˙3p,wherein ε˙ipi=1,2,3 represents the changing rate of the ith principal plastic strain.In another method, the strain softening parameter is considered as a function of the internal variable. The major principal strainε1, the major principal plastic strain ε1p, the plastic shear strain γ1, and the equivalent plastic shear strain γ2 are frequently used in this method:(3)γ1=ε1p−ε3p,γ2=23ε1pε1p+ε2pε2p+ε3pε3p.As the rock parametersm,s vary with the increase of the major principal strain, the major principal strain ε1 from the above method is used as the softening parameter in this paper, which leads to ease of analytical calculation.


## 3.3. Analysis of Rock Strength Parametersm,s and the Post-Peak Constitutive Model

The relationship between post-peak strength parameters and strain softening parameter, namely, the evolution of strength parameters, can be obtained by the laboratory experiment and numerical simulation. For simplicity, the mentioned relationship is often assumed as the form of a piecewise linear function [22–25], which can be expressed as(4)ηγ=ηp−ηp−ηrγγ∗,γp<γ<γ∗ηr,γ≥γ∗,where η, ηp, and ηr represent the strength parameter, strength parameter at the peak, and strength parameter at the starting point of the residual stage, respectively; γp and γ∗ are the critical softening parameter at the peak and strength parameter at the starting point of the residual stage, as shown in Figure 3.Figure 3
Variation of the strength parameterη in the whole stress-strain curve.According to (4), the evolution of rock strength parameters m,s can be obtained:(5)mε=mp−ε1−εrmp−mrεr−εp,εp<ε<εrmr,ε≥εr,(6)sε=sp−ε1−εrsp−srεr−εp,εp<ε<εrsr,ε≥εr,where mpandsp are the peak values of mands; mrandsr are the peak residual values of mands; and εpandεr are the major principal strain values at the peak and at the starting point of the residual stage, respectively.By substituting (5) and (6) into (1), the integration in (1) may be carried out analytically, and the post-peak strain softening constitutive model can be obtained accordingly.


## 4. Analytical Study of Hoek–Brown Parameters

Before calculation by (5) and (6), the strength parameters m,s at the peak and the starting point of the residual stage should be determined. In rock engineering, strength parameters m,s are generally determined by Geological Strength Index (GSI). However, it is difficult to use such a method to indicate strength parameters for rock specimens. Therefore, the paper proposes to use the Mohr–Coulomb strength criterion to obtain strength parameters c,φ at the peak and at the starting point of the residual stage under different confining pressures. The strength parameters m,s can be obtained via the relationship between themselves and the strength parameters c,φ afterwards.


### 4.1. Strength Parametersc,φ at the Peak and the Starting Point of the Residual Stage

Cohesionc and friction angle φ at the peak and the starting point of the residual stage under different confining pressures are various [26]. The paper assumes that any point in the post-peak stage is satisfied with the Mohr–Coulomb ultimate failure condition during the elastic unloading process. In this case, the unloading path L is parallel with the secant M at the peak, as shown in Figure 4. The same plastic shear strain εps, under different confining pressures and the same unloading path, corresponds to a series of different ultimate stress states: σ1′,σ3′, σ1″,σ3″, and σ1‴,σ3‴. By using the above method, the groups of different ultimate stress at the peak and the starting point of the residual stage can be obtained and displayed as Mohr stress circles in Figure 5. According to several Mohr stress circles, the envelope can be drawn. Thus, the corresponding cohesion c and friction angle φ at the peak and the starting point of the residual stage under different confining pressures can be obtained by analysis of the Mohr strength straight lines through the tangent points of envelopes.Figure 4
Post-peak strain softening simplified model stress states under the same unloading path.Figure 5
Mohr diagram for the tested specimens at the same plastic shear strainεps under different confining pressures [9].


### 4.2. Analytical Derivation

Uniaxial compressive strength of rock mass may be expressed as [27](7)σcm=sσci.Uniaxial tensile strength of rock mass may be written as [12](8)σtm=12σcim−m2+4s.By combining with the Mohr–Coulomb criterion, cohesionc and friction angle φ can be obtained [28]:(9)c=12−σcm⁡⋅⁡σtm,(10)φ=tan−1σcm+σtm2−σcm⋅σtm.By combining (7)–(10), the relationship among m,s, σci, c, and φ can be obtained:(11)c2=18sσci2m2+4s−m,(12)tan2φ=2s+m−m2+4s28sm2+4s−m.Equations (11) and (12) can be modified to the following equations:(13)c2=18ABσci2,(14)tan2φ=2B−A28AB,where(15)A=m2+4s−m,(16)B=s.By combining (13)–(16), the following results can be expressed:(17)A=4cσcitan2⁡φ+1−tan⁡φ,(18)B=2cσcitan2⁡φ+1+tan⁡φ.Substituting (17) and (18) into (15) and (16), the following equations are obtained:(19)s=4c2σci2tan2⁡φ+1+tan⁡φ2,(20)m=8cσcitan⁡φ⋅tan2⁡φ+1⋅tan2⁡φ+1+tan⁡φ.After substituting the strength parametersc,φ at the peak and the starting point of the residual stage into (19) and (20), the rock strength parameters s,m at the peak and the starting point of the residual stage can be calculated accordingly.Equations (19) and (20) show that the rock strength parameters s,m are related to the uniaxial compressive strength of intact rock σci, friction angle φ, and cohesion c, which can be expressed as(21)s=fσci,c,φ,m=fσci,c,ϕ.In terms of several groups ofσ1,σ3 under different confining pressures, σci is calculated [10]:(22)σci=∑yn−∑xy−∑x∑y/n∑x2−∑x2/n∑xn,where x=σ3,y=σ1−σ32, and n is the number of groups of data.


## 4.1. Strength Parametersc,φ at the Peak and the Starting Point of the Residual Stage

Cohesionc and friction angle φ at the peak and the starting point of the residual stage under different confining pressures are various [26]. The paper assumes that any point in the post-peak stage is satisfied with the Mohr–Coulomb ultimate failure condition during the elastic unloading process. In this case, the unloading path L is parallel with the secant M at the peak, as shown in Figure 4. The same plastic shear strain εps, under different confining pressures and the same unloading path, corresponds to a series of different ultimate stress states: σ1′,σ3′, σ1″,σ3″, and σ1‴,σ3‴. By using the above method, the groups of different ultimate stress at the peak and the starting point of the residual stage can be obtained and displayed as Mohr stress circles in Figure 5. According to several Mohr stress circles, the envelope can be drawn. Thus, the corresponding cohesion c and friction angle φ at the peak and the starting point of the residual stage under different confining pressures can be obtained by analysis of the Mohr strength straight lines through the tangent points of envelopes.Figure 4
Post-peak strain softening simplified model stress states under the same unloading path.Figure 5
Mohr diagram for the tested specimens at the same plastic shear strainεps under different confining pressures [9].


## 4.2. Analytical Derivation

Uniaxial compressive strength of rock mass may be expressed as [27](7)σcm=sσci.Uniaxial tensile strength of rock mass may be written as [12](8)σtm=12σcim−m2+4s.By combining with the Mohr–Coulomb criterion, cohesionc and friction angle φ can be obtained [28]:(9)c=12−σcm⁡⋅⁡σtm,(10)φ=tan−1σcm+σtm2−σcm⋅σtm.By combining (7)–(10), the relationship among m,s, σci, c, and φ can be obtained:(11)c2=18sσci2m2+4s−m,(12)tan2φ=2s+m−m2+4s28sm2+4s−m.Equations (11) and (12) can be modified to the following equations:(13)c2=18ABσci2,(14)tan2φ=2B−A28AB,where(15)A=m2+4s−m,(16)B=s.By combining (13)–(16), the following results can be expressed:(17)A=4cσcitan2⁡φ+1−tan⁡φ,(18)B=2cσcitan2⁡φ+1+tan⁡φ.Substituting (17) and (18) into (15) and (16), the following equations are obtained:(19)s=4c2σci2tan2⁡φ+1+tan⁡φ2,(20)m=8cσcitan⁡φ⋅tan2⁡φ+1⋅tan2⁡φ+1+tan⁡φ.After substituting the strength parametersc,φ at the peak and the starting point of the residual stage into (19) and (20), the rock strength parameters s,m at the peak and the starting point of the residual stage can be calculated accordingly.Equations (19) and (20) show that the rock strength parameters s,m are related to the uniaxial compressive strength of intact rock σci, friction angle φ, and cohesion c, which can be expressed as(21)s=fσci,c,φ,m=fσci,c,ϕ.In terms of several groups ofσ1,σ3 under different confining pressures, σci is calculated [10]:(22)σci=∑yn−∑xy−∑x∑y/n∑x2−∑x2/n∑xn,where x=σ3,y=σ1−σ32, and n is the number of groups of data.


## 5. Verification and Discussion

### 5.1. Application of the Developed Analytical Method

In order to verify the developed post-peak strain softening model, the triaxial compression test results of the Hawkesbury sandstone [18] were used. The strength parameters c,φ at the peak and the starting point of the residual stage for the Hawkesbury sandstone were calculated via the above analytical method, and the results are shown in Table 1. According to (22), σci of the Hawkesbury sandstone is 21.21 MPa.Table 1
Data of the Hawkesbury sandstone in triaxial compression tests.
Confining pressure (MPa)
At the peak
At the starting point of the residual stage
ε
p (10−3)
c
p (Mpa)
φ
p (°)
ε
r (10−3)
c
r (Mpa)
φ
r (°)
2
5.5
5.66
31.9
6.3
0.79
32.7
5
6.5
6.46
27.4
8.0
1.74
28.7
8
7.7
10.28
19.2
9.0
4.89
21.5
12
8.8
12.19
14.2
10.0
7.05
15.3
18
12.3
14.17
9.6
13
8.61
9.9Substituting the data in Table1 into (19) and (20), the corresponding rock strength parameters m,s at the peak and the starting point of the residual stage are obtained and listed in Table 2.Table 2
Rock strength parametersm,s at the peak position and at the starting residual stage.
Confining pressure (MPa)
At the peak
At the starting point of the residual stage
S
p
/
S
r
ε
p (10−3)
m
p
s
p
ε
r (10−3)
m
r
s
r
2
5.5
1.99
0.46
6.3
0.29
0.01
48.00
5
6.5
1.65
0.50
8.0
0.49
0.04
13.09
8
7.7
1.42
0.93
9.0
0.98
0.33
4.06
12
8.8
1.09
0.99
10.0
0.70
0.38
2.87
18
12.3
0.77
1.25
13.0
0.48
0.47
—It can be observed from Table2 that the rock strength parameter s is greater than 1.0, which is contradicted with the defined range of s (from 0 to 1.0). The reason is that the cohesion c and the friction angle φ are obtained based on the Mohr–Coulomb strength criterion. There is a maximum limit of confining pressure σ3max for linear fitting between the Mohr–Coulomb criterion and Hoek–Brown criterion [29], as shown in Figure 6.Figure 6
Relationships between the minimum and maximum principal stresses for equivalent Hoek–Brown and Mohr–Coulomb criteria [29].In terms of the upper limit of the maximum confining pressureσ3max obtained in rock engineering, it is mainly influenced by the engineering experience and classification. It is suggested in this paper that the value of the maximum confining pressure σ3max can be reached when the strength parameter s value is equal to 1.0. In this elastic state, the rock specimen is in peak strength under the confining pressure. Furthermore, the internal structure of the specimen is unfractured; therefore, Hoek–Brown and Mohr–Coulomb criteria could be used to fit its stress-strain curve. On the contrary, once this upper limit of the maximum confining pressure is exceeded, it is unmeaningful to use the Hoek–Brown criterion or Mohr–Coulomb criterion since the calculated results display deviation from the real ones. In other words, it is no necessary to define a value to s.As indicated in Tables1 and 2, the cohesion and friction angle at the peak and the starting point of the residual stage change with the confining pressure. It may be concluded that cohesion c and friction angle φ are a function of the confining pressure σ3. By conducting secondary fitting of c, ϕ, and σ3 in Table 1, the fitting result is shown in Figures 7 and 8, and the fitting functions are(23)cpσ3=−0.01σ32+0.77σ3+3.67,R2=0.95,(24)φpσ3=0.04σ32−2.23σ3+36.75,R2=0.98,(25)crσ3=−0.02σ32+0.89σ3−1.39,R2=0.95,(26)φrσ3=−0.02σ32−1.88σ3+36.90,R2=0.96.Figure 7
Relationship ofc or φ changing with confining pressure at the peak.Figure 8
Relationship ofc or φ changing with confining pressure at the starting point of the residual stage.Substituting (23) and (24) into (19), the maximum confining pressure σ3max is calculated to be 12.1 MPa when s=1 in (19).It can be found from Table2 that when the specimen reaches the peak strength, mp2MPa>mp5MPa>mp8MPa>mp12MPa and sp2MPa<sp5MPa<sp8MPa<sp12MPa.Within a certain range of confining pressure and increasing confining pressure at the peak,mp becomes smaller while sp becomes greater. It can be concluded that the increase of confining pressure results in smaller hardness at the peak and better rock quality. With the increase of confining pressure, rock develops gradually from brittle to ductile, which results in increased plasticity, reduced brittleness, and reduced hardness. It is interesting to note that as a result of the increase of confining pressure, the internal extent of the damage becomes smaller and the fragmentation influences less when the rock reaches the peak.By substituting the corresponding values ofm⁡and⁡p for the confining pressures of 2, 5, 8, and 12 MPa into (5) and (6), the analytical results of such two equations can be obtained and substituted into (1). Thus, the post-peak strain softening model of the Hawkesbury sandstone can be established. Figure 9 displays the comparison between the whole stress-strain curves of numerical simulation and triaxial compression tests for the Hawkesbury sandstone.Figure 9
Comparison between Hawkesbury sandstone triaxial compression test data and numerical simulation curves.


### 5.2. Discussion of Results

In order to verify the analytical results, the same analytical method was carried out with laboratory results of the representative rock types: Indiana limestone [26], Tennessee marble [7], and Hegang granite [7]. Table 3 presents the axial strains and the rock parameters m,s of different rock types corresponding to the peak strength and residual strength. The values of Sp/Sr are related to different confining pressures. Figures 9–11 illustrate the comparison chart between the numerical simulation and experimental data for the Hawkesbury sandstone, Tennessee marble, and Hegang granite, respectively.Table 3
Rock strength parametersm,s of different rock types at the peak position and at the starting point of the residual stage.
Rock type
Confining pressure (MPa)
ε
p (10−3)
m
p
s
p
ε
r (10−3)
m
r
s
r
S
p
/
S
r
Indiana limestone
2
3.4
7.67
0.47
5.5
1.17
0.03
16.94
4
4.2
3.81
0.71
6.3
1.25
0.05
16.60
8
5.0
2.86
0.86
8.1
2.15
0.19
4.79
12
5.9
2.70
0.98
8.0
2.25
0.40
2.52
Tennessee marble
3
2.4
4.1
0.84
3.2
2.54
0.12
7.00
7
2.8
4.0
0.85
3.8
2.50
0.15
5.67
14
3.2
3.9
0.92
4.9
2.42
0.22
4.18
21
3.5
3.2
1.00
6.0
1.80
0.31
3.23
Hegang granite
2
4.2
18.83
0.37
5.6
2.36
0.02
24.18
5
5.5
14.44
0.44
8.8
2.45
0.02
19.82
10
6.7
13.50
0.64
12.6
2.53
0.09
7.29
15
7.5
11.99
0.98
20.0
2.95
0.15
6.63Figure 10
Comparison between Indiana limestone triaxial compression test data and numerical simulation curves.Figure 11
Comparison between Tennessee marble triaxial compression test data and numerical simulation curves.As can be seen from Figures9–12, numerical simulation is consistent with experimental data trends within a certain range of confining pressure, and the numerical simulation value agrees well with actual experimental values, especially for peak strength and residual strength. It indicates that the post-peak strain softening model established in the paper is reasonable, and it can accurately describe the mechanical post-peak behaviour of rock.Figure 12
Comparison between Hegang granite triaxial compression test data and numerical simulation curves.For better understanding of the post-peak softening characteristics and the degree of fragmentation and the degree of brittleness of rock, the brittleness index is proposed forSp/Sr. Figure 13 illustrates the relationship among Sp/Sr, confining pressure, and rock type.Figure 13
Relationships betweenSp/Sr and confining pressure for different types of the rock.As the confining pressure decreases,Sp/Sr gradually decreases and tends to become stable finally. In other words, as the confining pressure increases, the rock gradually develops brittleness to ductility. Influenced by the confining pressure, the performance of brittleness of different types of rock is varying. The results also show that the reaction of the Hawkesbury sandstone is the most violent, followed by the Hegang granite and Tennessee marble.


## 5.1. Application of the Developed Analytical Method

In order to verify the developed post-peak strain softening model, the triaxial compression test results of the Hawkesbury sandstone [18] were used. The strength parameters c,φ at the peak and the starting point of the residual stage for the Hawkesbury sandstone were calculated via the above analytical method, and the results are shown in Table 1. According to (22), σci of the Hawkesbury sandstone is 21.21 MPa.Table 1
Data of the Hawkesbury sandstone in triaxial compression tests.
Confining pressure (MPa)
At the peak
At the starting point of the residual stage
ε
p (10−3)
c
p (Mpa)
φ
p (°)
ε
r (10−3)
c
r (Mpa)
φ
r (°)
2
5.5
5.66
31.9
6.3
0.79
32.7
5
6.5
6.46
27.4
8.0
1.74
28.7
8
7.7
10.28
19.2
9.0
4.89
21.5
12
8.8
12.19
14.2
10.0
7.05
15.3
18
12.3
14.17
9.6
13
8.61
9.9Substituting the data in Table1 into (19) and (20), the corresponding rock strength parameters m,s at the peak and the starting point of the residual stage are obtained and listed in Table 2.Table 2
Rock strength parametersm,s at the peak position and at the starting residual stage.
Confining pressure (MPa)
At the peak
At the starting point of the residual stage
S
p
/
S
r
ε
p (10−3)
m
p
s
p
ε
r (10−3)
m
r
s
r
2
5.5
1.99
0.46
6.3
0.29
0.01
48.00
5
6.5
1.65
0.50
8.0
0.49
0.04
13.09
8
7.7
1.42
0.93
9.0
0.98
0.33
4.06
12
8.8
1.09
0.99
10.0
0.70
0.38
2.87
18
12.3
0.77
1.25
13.0
0.48
0.47
—It can be observed from Table2 that the rock strength parameter s is greater than 1.0, which is contradicted with the defined range of s (from 0 to 1.0). The reason is that the cohesion c and the friction angle φ are obtained based on the Mohr–Coulomb strength criterion. There is a maximum limit of confining pressure σ3max for linear fitting between the Mohr–Coulomb criterion and Hoek–Brown criterion [29], as shown in Figure 6.Figure 6
Relationships between the minimum and maximum principal stresses for equivalent Hoek–Brown and Mohr–Coulomb criteria [29].In terms of the upper limit of the maximum confining pressureσ3max obtained in rock engineering, it is mainly influenced by the engineering experience and classification. It is suggested in this paper that the value of the maximum confining pressure σ3max can be reached when the strength parameter s value is equal to 1.0. In this elastic state, the rock specimen is in peak strength under the confining pressure. Furthermore, the internal structure of the specimen is unfractured; therefore, Hoek–Brown and Mohr–Coulomb criteria could be used to fit its stress-strain curve. On the contrary, once this upper limit of the maximum confining pressure is exceeded, it is unmeaningful to use the Hoek–Brown criterion or Mohr–Coulomb criterion since the calculated results display deviation from the real ones. In other words, it is no necessary to define a value to s.As indicated in Tables1 and 2, the cohesion and friction angle at the peak and the starting point of the residual stage change with the confining pressure. It may be concluded that cohesion c and friction angle φ are a function of the confining pressure σ3. By conducting secondary fitting of c, ϕ, and σ3 in Table 1, the fitting result is shown in Figures 7 and 8, and the fitting functions are(23)cpσ3=−0.01σ32+0.77σ3+3.67,R2=0.95,(24)φpσ3=0.04σ32−2.23σ3+36.75,R2=0.98,(25)crσ3=−0.02σ32+0.89σ3−1.39,R2=0.95,(26)φrσ3=−0.02σ32−1.88σ3+36.90,R2=0.96.Figure 7
Relationship ofc or φ changing with confining pressure at the peak.Figure 8
Relationship ofc or φ changing with confining pressure at the starting point of the residual stage.Substituting (23) and (24) into (19), the maximum confining pressure σ3max is calculated to be 12.1 MPa when s=1 in (19).It can be found from Table2 that when the specimen reaches the peak strength, mp2MPa>mp5MPa>mp8MPa>mp12MPa and sp2MPa<sp5MPa<sp8MPa<sp12MPa.Within a certain range of confining pressure and increasing confining pressure at the peak,mp becomes smaller while sp becomes greater. It can be concluded that the increase of confining pressure results in smaller hardness at the peak and better rock quality. With the increase of confining pressure, rock develops gradually from brittle to ductile, which results in increased plasticity, reduced brittleness, and reduced hardness. It is interesting to note that as a result of the increase of confining pressure, the internal extent of the damage becomes smaller and the fragmentation influences less when the rock reaches the peak.By substituting the corresponding values ofm⁡and⁡p for the confining pressures of 2, 5, 8, and 12 MPa into (5) and (6), the analytical results of such two equations can be obtained and substituted into (1). Thus, the post-peak strain softening model of the Hawkesbury sandstone can be established. Figure 9 displays the comparison between the whole stress-strain curves of numerical simulation and triaxial compression tests for the Hawkesbury sandstone.Figure 9
Comparison between Hawkesbury sandstone triaxial compression test data and numerical simulation curves.


## 5.2. Discussion of Results

In order to verify the analytical results, the same analytical method was carried out with laboratory results of the representative rock types: Indiana limestone [26], Tennessee marble [7], and Hegang granite [7]. Table 3 presents the axial strains and the rock parameters m,s of different rock types corresponding to the peak strength and residual strength. The values of Sp/Sr are related to different confining pressures. Figures 9–11 illustrate the comparison chart between the numerical simulation and experimental data for the Hawkesbury sandstone, Tennessee marble, and Hegang granite, respectively.Table 3
Rock strength parametersm,s of different rock types at the peak position and at the starting point of the residual stage.
Rock type
Confining pressure (MPa)
ε
p (10−3)
m
p
s
p
ε
r (10−3)
m
r
s
r
S
p
/
S
r
Indiana limestone
2
3.4
7.67
0.47
5.5
1.17
0.03
16.94
4
4.2
3.81
0.71
6.3
1.25
0.05
16.60
8
5.0
2.86
0.86
8.1
2.15
0.19
4.79
12
5.9
2.70
0.98
8.0
2.25
0.40
2.52
Tennessee marble
3
2.4
4.1
0.84
3.2
2.54
0.12
7.00
7
2.8
4.0
0.85
3.8
2.50
0.15
5.67
14
3.2
3.9
0.92
4.9
2.42
0.22
4.18
21
3.5
3.2
1.00
6.0
1.80
0.31
3.23
Hegang granite
2
4.2
18.83
0.37
5.6
2.36
0.02
24.18
5
5.5
14.44
0.44
8.8
2.45
0.02
19.82
10
6.7
13.50
0.64
12.6
2.53
0.09
7.29
15
7.5
11.99
0.98
20.0
2.95
0.15
6.63Figure 10
Comparison between Indiana limestone triaxial compression test data and numerical simulation curves.Figure 11
Comparison between Tennessee marble triaxial compression test data and numerical simulation curves.As can be seen from Figures9–12, numerical simulation is consistent with experimental data trends within a certain range of confining pressure, and the numerical simulation value agrees well with actual experimental values, especially for peak strength and residual strength. It indicates that the post-peak strain softening model established in the paper is reasonable, and it can accurately describe the mechanical post-peak behaviour of rock.Figure 12
Comparison between Hegang granite triaxial compression test data and numerical simulation curves.For better understanding of the post-peak softening characteristics and the degree of fragmentation and the degree of brittleness of rock, the brittleness index is proposed forSp/Sr. Figure 13 illustrates the relationship among Sp/Sr, confining pressure, and rock type.Figure 13
Relationships betweenSp/Sr and confining pressure for different types of the rock.As the confining pressure decreases,Sp/Sr gradually decreases and tends to become stable finally. In other words, as the confining pressure increases, the rock gradually develops brittleness to ductility. Influenced by the confining pressure, the performance of brittleness of different types of rock is varying. The results also show that the reaction of the Hawkesbury sandstone is the most violent, followed by the Hegang granite and Tennessee marble.


## 6. Conclusion

A reasonable post-peak strain softening model plays an important role in the analysis of rock behaviour. Based on the nonlinear Hoek–Brown criterion, a post-peak strain softening model was proposed in this paper, and the conclusions are listed below:(1)
Based on the Hoek–Brown criterion, the maximum principal strainε1 was selected as the softening parameter. The piecewise linear function was used to evaluate the rock strength parameters m,s, and the post-peak strain softening model was obtained accordingly.(2)
By theoretical derivation, the analytical relation of parametersm,s, cohesion c, and friction angle φ is determined. The cohesion c and friction angle φ under different confining pressures can be obtained regarding any points of the post-peak stage. This developed relationship helps to establish a nonlinear post-peak strain softening model.(3)
The laboratory test results of the Hawkesbury sandstone show that with the increase of confining pressure, the rock hardness in peak strength becomes smaller. However, the degree of rock fragmentation in peak strength becomes lower within the range of confining pressure from 0 toσ3min. When the rock strength parameter s reaches 1, the upper limit of the confining pressure σ3max for best fitting between the Hoek–Brown criterion and Mohr–Coulomb criterion can be obtained.(4)
The analytical results of the developed strain softening model have been verified with laboratory results of the Hawkesbury sandstone, Indiana limestone, Tennessee marble, and Hegang granite. The results of the analytical method and laboratory tests are in good agreement, and it is concluded that the proposed model can predict the post-peak mechanical properties of the rock accurately.(5)
Different types of rock may have different values ofSp/Sr. The value of Sp/Sr is related to the influence of the confining pressure on the brittleness of rock. As the confining pressure increases, the value of Sp/Sr obtained from the same rock type becomes small, which results in the weakened brittleness.



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