# A Piecewise Yield Failure Criterion including the Critical State for Brittle Rock

Authors: Jing Zhang; Fengyu Ren; Zhihua Ouyang; Huan Liu
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

The critical state of rock is an important index for measuring the changes in rock characteristics. However, this state is not unique because of the different researcher assumptions. Based on the theory of the partial differential equation proposed by Vutukuri, according to Mohr’s envelope, a piecewise yield failure criterion (referred to as the Mohr–Wedge criterion), including the critical state for brittle rock, is obtained by introducing the wedge model to solve this equation. The Mohr–Wedge (M–W) criterion consisting of nonlinear and linear components includes the critical state for brittle rock. When the minimum principal stressσ3 is lower than the confining pressure σk, the maximum principal stress σ1 varies nonlinearly with σ3; otherwise, σ1 varies linearly with σ3. This variation conforms to rock deformation features under triaxial compression. In this study, we investigate the rationality of this critical state by an analogy method and illustrate that the critical state mentioned in this criterion is related to the microcracks in the potential failure zone of the rock. Alternatively, the primary object of this study is to reveal the applicability of predicting the yield state for this criterion. The method used in our study is compared to the Mohr–Coulomb (M-C) criterion, the Hoek–Brown (H-B) criterion, and the Exponential (Exp.) criterion by the yield surfaces on the deviatoric plane. Notably, there is a vertex consistent region for the four criteria, but except for this region, the yield state of rock predicted by the four criteria is quite different, depending on the extent of the parameters for the criteria and the magnitude of the slopes of the yield surfaces. The results show that the M-W criterion has certain applicability for predicting the rock yield state by using the multiple data of rock triaxial compression tests in the published literature.


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## Body

## 1. Introduction

The failure criterion, one of the basic theoretical problems in geotechnical mechanics, has been a research hotspot and has important guiding significance for actual geotechnical engineering. The Coulomb criterion proposed that rock failure is mainly a shear failure, and the shear stressτ follows a positive proportion to the normal stress failure σ on the failure shear plane [1]. Through many experimental studies, Mohr [2] proposed that τ and σ should be expressed by a functional relationshipτ=fσ, namely, the Mohr envelope. The formulation of the Mohr envelope provided substantial assistance for numerous scholars to study the failure criterion. Subsequently, Balmer [3] proposed that the nonlinear envelope can be represented by the maximum principal stress σ1 and the minimum principal stress σ3. Using that conclusion, Ucar [4] and Kumar [5] successively obtained the expressions of the shear envelope of the narrow H-B criterion [6] and generalized Hoek–Brown (H-B) criterion [7]. In addition, most scholars [8–14] proposed many available triaxial failure criteria for brittle rock based on experimental observation or theoretical analysis. The applicability of these criteria was mainly discussed in the principal stress plane.Based on the research of Balmer [3], Vutukuri [15] deduced the partial differential equation of stresses by introducing the rock failure angle and envelope slope angle, which proved that the envelope slope angle ∂τΦ/∂σΦ is related to the axial-radial stress curve slope angle ∂σ1/∂σ3. Ouyang and Elsworth [16] utilized a partial differential equation of stresses to propose a phenomenological rock failure criterion, which was a theoretical formula based on the brittle failure of rock in the microshear test, and the wedge model was a rock failure model based on the developmental vertical and horizontal microcracks on the shear potential failure zone [17, 18].The Mohr–Wedge (M-W) yield failure criterion, a theoretical derivation based on microscopic experiments, is a piecewise function including the critical state for brittle rock. The characteristics of rock under different confining pressures are determined by many factors, such as the rock type, earth pressure, and joint. Clearly, the deformation and failure modes of a deep rock mass are different from those of a shallow rock mass [19]. Matthew et al. [20] presented four failure modes for rock slopes, namely, sliding, toppling, slumping, and confined slumping. For the rock critical state, Sankhaneel and Gabriel [21] proposed that the failure of rock is extensile cracking under low confinement conditions and semibrittle shear under higher confinement conditions. In addition, some scholars suggested that the variation in the principal stress is linear under low confining pressure and nonlinear under high confining pressure [22, 23]. However, the M-W criterion reveals a nonlinear relationship under low confining pressure due to the influence of internal microcracks of rock. When the microcracks in the shear potential failure zone are compacted, the variation in the stress is linear under a very high confining pressure due to instantaneous linear elastomer outside the shear potential failure zone.In the following sections, we compare the M-W yield criterion with the Mohr–Coulomb (M-C) criterion [2], the H-B criterion [7], and the Exponential (Exp.) criterion [12] to discuss the rationality of the failure model and the applicability of the criterion. Except for comparing the variation in the principal stress and arranging the parameters of the criterion, we discuss the yield surfaces on the deviatoric plane detailedly. Taking into account the principle of the coordinate invariance of plasticity theory, the description of the yield behavior is not affected by the choice of the coordinate system [24–26]. Consequently, we can obtain the yield surfaces on the deviatoric plane through the coordinate transformation of the deviatoric plane and principal stress space, which transform the three-dimensional yield problem of the principal stress space into the deviatoric plane. Finally, we compared the accuracy of the predicting rock yield state among the three criteria with the triaxial compression test data.


## 2. Criterion Features

The M-W criterion proposed by Ouyang and Elsworth can be summarized as follows [16]:(1)τm=1Aln1+A2σm+W1/2B2+Aσm+W12B+σt2,−W12B≤σm≤1−W12B,−cos2Φ0σm+C,σm>1−W12B,whereτm=σ1−σ3/2, σm=σ1+σ3/2, A=−2K2B/K1, K1=4/π+2, and K2=4/π−2. The variables σ1, σ3, and σt represent the maximum principal stress, minimum principal stress, and uniaxial tensile strength, respectively. C=τm0+cos2Φ01−W1/2B, in which τm0and Φ0are the value of equation (1) and the rock failure angle at σm = (1-W1)/2B, respectively. The constant W1 is the number of cracks per unit volume, and B is the parameter associated with an elliptical crack. The physical significance of the specific parameters is detailed in [16].Obviously, the M-W criterion is a piecewise function consisting of nonlinear and linear components. The critical state between the two components is that the microcracks outside the potential failure zone of the rock sample are compacted. Takingσk as the critical confining pressure, Figure 1 shows the variation in the relationship between σ1 and σ3. When σ3≤σk, σ1 varies nonlinearly with σ3 because of the microcracks, and when σ3>σk, the microcracks are compacted, σ1 varies linearly with σ3, and the rock sample presents as a linear elastic material. This variation conforms to rock deformation features under triaxial compression.Figure 1
Variation in the principal stress for the M-W criterion.In addition, the H-B criterion [7] also can illustrate the change relationships between σ1 and σ3. The expression of the H-B criterion is as follows:(2)σ1=σ3+σcmσ3σc+1,where σc is the uniaxial compression strength and m is the experience parameter.The value ofdσ1/dσ3can be derived from equation (2) as follows:(3)dσ1dσ3=1+m2mσ3/σc+1.The variation in the relationship betweenσ1 and σ3 is nonlinear under low confining pressure. However, when confining pressure σ3 is infinite, the value of dσ1/dσ3is approximately equal to 1, which indicates that the relationship is approximately linear. Therefore, the variation in the relationship between σ1 and σ3 in the M-W criterion is similar to the H-B criterion.


### 2.1. Wedge Model

The wedge model is a rock failure model based on the development of vertical and horizontal microcracks in the shear potential failure zone [16]. There are two features. (1) When the rock bears a load, the force transfer through the potential failure zone must be concentrated on the wedge vertices (similar to the concentrated forces P1 and P2 shown in Figure 2) due to the influence of the microcracks. Therefore, the ratio of the sum of all the concentrated forces P1andP2 to their respective areas is equal to the normal stress σΦ and shear stress τΦ on the macroscopic failure surface of the rock. (2) The rock shear failure can be regarded as the value of the shear stress maximum on the macroaspect. Moreover, the shear failure can also be regarded as the tensile stress (the tensile stress is negative) at the boundary of the wedge particles when the potential failure attains a minimum on the microaspect. The second feature represents the boundary conditions for solving the expression of the rock failure angle Φ and principal stress; the detailed derivation process is described in [16].Figure 2
The microcrack distribution pattern of the potential failure zone in the wedge model [16].


### 2.2. Physical Meaning of the Parameters

Considering that the vast parameters in the criterion may be adverse to practical applications, the parameters are arranged first.(1)
Parameters ofK1 and K2We introduceK instead of K1 and K2 because the dimensions of K1 and K2 are 1 as follows:(4)K=K2K1=π+2π−2.(2)
Parameters ofW1 and σtReference [16] provides the following secondary function:(5)W=Bσ1+σ3+W1,−W1B≤σ1+σ3≤1−W1B,1,σ1+σ3>1−W1B,whereW represents the particle stress concentration factor.The upper limit of equation (5) actually represents the uniaxial tension of the rock. Then, for σ1 = −σt and σ3 = 0, the uniaxial tensile strength σt can be shown as follows:(6)σt=W1B.(3)
Parameters ofA, B, W, and ΦThe equation betweenW and Φ can be found in [16] as follows:(7)tan2Φ=−K2K1W,where Φ represents the rock failure angle.The observed dimensions of the parameters are as follows:A and B are M−1LT2, and W and Φ are 1. Then, referring to the M-C criterion and H-B criterion, the dimensions of both parameters are divided into two types, namely, ML−1T−2 and 1. However, dimension 1 is not an appropriate parameter to integrate the parameters (A, B, W, and Φ). Ultimately, we introduced the parameter σw (the dimension is ML−1T−2) as follows:(8)σw=−1A.Substituting equations (5) and (7) and A=−2K2B/K1into equation (8) and rearranging the equation yield(9)σw=−σ1∗+σ3∗+σt2tan2Φ∗,where σ1∗and σ3∗are the triaxial test data of a group of rock samples and Φ∗ is the rock failure angle when the confining pressure is σ3∗.Substituting equations (4), (6), and (9) into equation (1) gives(10)τm=−σwln1+2σm+σt2σw2−2σm+σt2σw+σt2,−σt2≤σm≤Kσw−σt2,−cos2Φ0σm+C,σm>Kσw−σt2.Although the M-W criterion parameters are integrated, the physical meaning ofσw is unclear. Actually, we can obtain another expression for σw from [16]. The expression of B is as follows:(11)B=ha1−υ2Eb,where h is the microcracks density, a is the major semiaxis of the ellipse microcrack, b is the minor semiaxis of the ellipse microcrack, and E and υ are the elasticity modulus and Poisson’s ratio, respectively.Substituting equation (11) and A=−2K2B/K1into equation (8) and rearranging the equation yield(12)σw=NE1−υ2,whereN=b/2Kha.SubstitutingG=E/21+υ into equation (12) and rearranging the equation yield(13)σw=N4G24G−E,where G is the shear modulus.As for equation (13), we suggest that σw is related to the elasticity modulus, shear modulus, and parameter N, which contain the density and the eccentricity of an elliptical microcrack. It follows that the parameter may be defined as the crack modulus, which is a measure of the rock resistance to cracking. In addition, according to the boundary conditions of equation (10), the parameter K is a constant and the value of uniaxial tensile strength σt is generally not larger. Therefore, the parameter σw determines the magnitude of critical confining pressure, that is, the difficulty of the rock entering the critical state.


## 2.1. Wedge Model

The wedge model is a rock failure model based on the development of vertical and horizontal microcracks in the shear potential failure zone [16]. There are two features. (1) When the rock bears a load, the force transfer through the potential failure zone must be concentrated on the wedge vertices (similar to the concentrated forces P1 and P2 shown in Figure 2) due to the influence of the microcracks. Therefore, the ratio of the sum of all the concentrated forces P1andP2 to their respective areas is equal to the normal stress σΦ and shear stress τΦ on the macroscopic failure surface of the rock. (2) The rock shear failure can be regarded as the value of the shear stress maximum on the macroaspect. Moreover, the shear failure can also be regarded as the tensile stress (the tensile stress is negative) at the boundary of the wedge particles when the potential failure attains a minimum on the microaspect. The second feature represents the boundary conditions for solving the expression of the rock failure angle Φ and principal stress; the detailed derivation process is described in [16].Figure 2
The microcrack distribution pattern of the potential failure zone in the wedge model [16].


## 2.2. Physical Meaning of the Parameters

Considering that the vast parameters in the criterion may be adverse to practical applications, the parameters are arranged first.(1)
Parameters ofK1 and K2We introduceK instead of K1 and K2 because the dimensions of K1 and K2 are 1 as follows:(4)K=K2K1=π+2π−2.(2)
Parameters ofW1 and σtReference [16] provides the following secondary function:(5)W=Bσ1+σ3+W1,−W1B≤σ1+σ3≤1−W1B,1,σ1+σ3>1−W1B,whereW represents the particle stress concentration factor.The upper limit of equation (5) actually represents the uniaxial tension of the rock. Then, for σ1 = −σt and σ3 = 0, the uniaxial tensile strength σt can be shown as follows:(6)σt=W1B.(3)
Parameters ofA, B, W, and ΦThe equation betweenW and Φ can be found in [16] as follows:(7)tan2Φ=−K2K1W,where Φ represents the rock failure angle.The observed dimensions of the parameters are as follows:A and B are M−1LT2, and W and Φ are 1. Then, referring to the M-C criterion and H-B criterion, the dimensions of both parameters are divided into two types, namely, ML−1T−2 and 1. However, dimension 1 is not an appropriate parameter to integrate the parameters (A, B, W, and Φ). Ultimately, we introduced the parameter σw (the dimension is ML−1T−2) as follows:(8)σw=−1A.Substituting equations (5) and (7) and A=−2K2B/K1into equation (8) and rearranging the equation yield(9)σw=−σ1∗+σ3∗+σt2tan2Φ∗,where σ1∗and σ3∗are the triaxial test data of a group of rock samples and Φ∗ is the rock failure angle when the confining pressure is σ3∗.Substituting equations (4), (6), and (9) into equation (1) gives(10)τm=−σwln1+2σm+σt2σw2−2σm+σt2σw+σt2,−σt2≤σm≤Kσw−σt2,−cos2Φ0σm+C,σm>Kσw−σt2.Although the M-W criterion parameters are integrated, the physical meaning ofσw is unclear. Actually, we can obtain another expression for σw from [16]. The expression of B is as follows:(11)B=ha1−υ2Eb,where h is the microcracks density, a is the major semiaxis of the ellipse microcrack, b is the minor semiaxis of the ellipse microcrack, and E and υ are the elasticity modulus and Poisson’s ratio, respectively.Substituting equation (11) and A=−2K2B/K1into equation (8) and rearranging the equation yield(12)σw=NE1−υ2,whereN=b/2Kha.SubstitutingG=E/21+υ into equation (12) and rearranging the equation yield(13)σw=N4G24G−E,where G is the shear modulus.As for equation (13), we suggest that σw is related to the elasticity modulus, shear modulus, and parameter N, which contain the density and the eccentricity of an elliptical microcrack. It follows that the parameter may be defined as the crack modulus, which is a measure of the rock resistance to cracking. In addition, according to the boundary conditions of equation (10), the parameter K is a constant and the value of uniaxial tensile strength σt is generally not larger. Therefore, the parameter σw determines the magnitude of critical confining pressure, that is, the difficulty of the rock entering the critical state.


## 3. Test Data and Parameter Calculations

### 3.1. Triaxial Compression Test Data of Brittle Rock

The triaxial compression test data of brittle rock are derived from the published literature. Table1 presents the test data of Dunham dolomite [27], Mizuho trachyte [8], Jinping sandstone [28], Cedar tonalite [29], Carrara marble [30], and Westerly granite [29], which are digitized from [10]. The values of the confining pressure σ3 in Table 1 are below the critical confining pressure σk. Table 2 presents the strengths of Yamaguchi marble [27], Indiana limestone [31], and Solnhofen limestone [27], which are compressed under various confining pressures and the condition σ3 > σk. The calculation of σk is stated in detail in the next section.Table 1
Triaxial compression test data (MPa): all confining pressuresσ3 below the critical confining pressure σk.
Dunham dolomite (σk = 435.10 MPa)Mizuho trachyte (σk = 198.83 MPa)Jinping sandstone (σk = 154.22 MPa)σ3σ1σ3σ1σ3σ10.0257.00.0100.00.061.625.0400.015.0193.05.0109.545.0488.030.0253.010.0138.665.0568.045.0300.020.0174.685.0624.060.0339.030.0209.0105.0679.075.0365.040.0240.5125.0724.0100.0419.050.0263.060.0288.570.0305.4Cedar tonalite (σk = 89.96 MPa)Carrara marble (σk = 217.05 MPa)Westerly granite (σk = 851.21 MPa)σ3σ1σ3σ1σ3σ10.013.10.0137.00.0201.05.064.425.0234.02.0231.010.088.950.0314.020.0430.020.0112.668.4358.038.0612.030.0148.985.5404.060.0747.065.0228.4161.8558.077.0889.075.0248.7100.01012.0Table 2
Triaxial compression test data (MPa): part of confining pressureσ3 above the critical confining pressure σk.
Yamaguchi marble (σk = 116.44 MPa)Indiana limestone (σk = 40.22 MPa)Solnhofen limestone (σk = 146.86 MPa)σ3σ1σ3σ1σ3σ10.081.00.044.00.00293.06.0113.06.566.06.00335.012.5130.013.785.015.00360.025.0175.020.399.024.00381.040.0210.027.9109.046.00426.055.0246.034.4119.072.00467.070.0272.041.2128.2111.00518.085.0295.048.4135.1195.00595.0100.0324.055.4141.9304.00709.0150.0397.062.3149.1200.0454.068.4156.5


### 3.2. Parameter Calculation

Given that the comparative analyses between the M-W criterion and other criteria are undisposed, we take the M-C criterion, H-B criterion, and Exp. criterion as the comparison to discuss the applicability of M-W criterion.The expression of M-C criterion is as follows:(14)σ1=1+sinϕ1−sinϕσ3+2c⋅cosϕ1−sinϕ,where φ is the angle of internal friction of the material and c is the cohesion of the material.The expression of generalized H-B criterion, introduced by Hoek [32], is expressed as(15)σ1=σ3+σcmσ3σc+sai,where m, s, and ai are the rock mass material constants, given by(16)m=miexpGSI−10028−14D,(17)s=expGSI−1009−3D,(18)ai=12+16e−GSI/15−e−20/3,where GSI is the geological strength index. D is a factor which depends upon the degree of disturbance to which the rock mass has been subjected to blast damage and stress relaxation. For the intact rock, the material constants are denoted by s = 1, ai = 0.5, and m. The expression of H-B criterion for intact rock is shown in equation (2).The H-B criterion is widely used in practical engineering. Many researchers have made an improvement on the Hoek–Brown criterion. The latest major revision of the GSI and its use in equations (16)–(18) was made by Hoek et al. [33]. Hoek and Diederichs [34] used the database of rock mass deformation modulus measurements and proposed that the rock mass deformation modulus can be represented as(19)Erm=Ei0.02+1−D/21+exp60+15D−GSI/11,where Ei is the intact rock deformation modulus.Hoek and Brown [35] summarized the research results of most scholars on the H-B criterion and incorporated the rock mass deformation modulus into the criterion and GSI system, which further demonstrated practical applications of the criterion and the GSI system. However, the parameters of criteria in this work are obtained by using the least square method to fit the criteria and triaxial compression test data. Therefore, the applicability of H-B criterion under the condition of intact rock is discussed to obtain the parameters of H-B criterion more easily.The expression of Exp. criterion is as follows [12]:(20)σ1−σ3=Q∞−Q∞−Q0exp−K0−1σ3Q∞−Q0,where Q0 is the uniaxial compressive strength, Q∞ is the limitation of stress deviator when confining pressure approaches infinity, and K0 is the ratio of strength increasing with confining pressure at confining pressure of zero.The parameter values of the four yield failure criteria are shown in Table3. The parameters of M-C criterion, H-B criterion, and Exp. criterion are determined by using the least square method to fit the criteria and triaxial compression test data, and the determination of σt and σw is similar to these criteria. Here, only the values of Φ0 and C are not obtained. Recall that, in the interpretation of the parameters in equation (1), Φ0 and C are closely related to the critical state where the mean principal stress is σmk and the maximum shear stress is τmk. Then, the expression of σmk is as follows:(21)σmk=Kσw−σt2.Table 3
Parameter values of the three yield failure criteria.
Rock typesM-W criterionH-B criterionM-C criterionExp. criterionσt (MPa)σw (MPa)Φ0 (°)C (MPa)mσc (MPa)φ (°)c (MPa)Q∞ (MPa)Q0 (MPa)K0Dunham dolomite7.05192.7751.26242.168.22286.5838.6166.72683.85257.006.51Mizuho trachyte2.0987.6051.26109.376.63135.4233.3934.75332.19100.006.79Jinping sandstone0.2467.3451.2683.248.2489.0032.2626.13243.8361.607.91Cedar tonalite0.1839.3051.2648.6010.4737.2028.5513.78176.6013.107.51Carrara marble0.9295.0351.26117.825.53153.7925.8152.86467.44137.004.11Westerly granite0.64371.4051.26458.6848.67163.6751.4942.141286.72201.0012.46Yamaguchi marble2.6451.9251.2665.674.4097.6529.3626.81236.5281.004.46Indiana limestone4.1819.3651.2626.442.9750.2421.1617.3189.4344.004.02Solnhofen limestone54.2387.6751.26141.172.77315.8622.55105.48453.90293.003.29Substituting the value ofσmk into equation (1), the value of τmk can be found. Naturally, rearranging equation (1) gives(22)C=τmk+cos2Φ0σmk.The parameter Φ0 is the rock failure angle at the critical state, which is a constant that combines equation (6) with equation (7). Eventually, taking the values of σmk, τmk, and Φ0 into equation (22), the value of C can be obtained.Referring to the calculation formula ofσk (as shown in equation (23)), we can easily solve σk on the premise that σmk and τmk are known.(23)σk=σmk−τmk.


## 3.1. Triaxial Compression Test Data of Brittle Rock

The triaxial compression test data of brittle rock are derived from the published literature. Table1 presents the test data of Dunham dolomite [27], Mizuho trachyte [8], Jinping sandstone [28], Cedar tonalite [29], Carrara marble [30], and Westerly granite [29], which are digitized from [10]. The values of the confining pressure σ3 in Table 1 are below the critical confining pressure σk. Table 2 presents the strengths of Yamaguchi marble [27], Indiana limestone [31], and Solnhofen limestone [27], which are compressed under various confining pressures and the condition σ3 > σk. The calculation of σk is stated in detail in the next section.Table 1
Triaxial compression test data (MPa): all confining pressuresσ3 below the critical confining pressure σk.
Dunham dolomite (σk = 435.10 MPa)Mizuho trachyte (σk = 198.83 MPa)Jinping sandstone (σk = 154.22 MPa)σ3σ1σ3σ1σ3σ10.0257.00.0100.00.061.625.0400.015.0193.05.0109.545.0488.030.0253.010.0138.665.0568.045.0300.020.0174.685.0624.060.0339.030.0209.0105.0679.075.0365.040.0240.5125.0724.0100.0419.050.0263.060.0288.570.0305.4Cedar tonalite (σk = 89.96 MPa)Carrara marble (σk = 217.05 MPa)Westerly granite (σk = 851.21 MPa)σ3σ1σ3σ1σ3σ10.013.10.0137.00.0201.05.064.425.0234.02.0231.010.088.950.0314.020.0430.020.0112.668.4358.038.0612.030.0148.985.5404.060.0747.065.0228.4161.8558.077.0889.075.0248.7100.01012.0Table 2
Triaxial compression test data (MPa): part of confining pressureσ3 above the critical confining pressure σk.
Yamaguchi marble (σk = 116.44 MPa)Indiana limestone (σk = 40.22 MPa)Solnhofen limestone (σk = 146.86 MPa)σ3σ1σ3σ1σ3σ10.081.00.044.00.00293.06.0113.06.566.06.00335.012.5130.013.785.015.00360.025.0175.020.399.024.00381.040.0210.027.9109.046.00426.055.0246.034.4119.072.00467.070.0272.041.2128.2111.00518.085.0295.048.4135.1195.00595.0100.0324.055.4141.9304.00709.0150.0397.062.3149.1200.0454.068.4156.5


## 3.2. Parameter Calculation

Given that the comparative analyses between the M-W criterion and other criteria are undisposed, we take the M-C criterion, H-B criterion, and Exp. criterion as the comparison to discuss the applicability of M-W criterion.The expression of M-C criterion is as follows:(14)σ1=1+sinϕ1−sinϕσ3+2c⋅cosϕ1−sinϕ,where φ is the angle of internal friction of the material and c is the cohesion of the material.The expression of generalized H-B criterion, introduced by Hoek [32], is expressed as(15)σ1=σ3+σcmσ3σc+sai,where m, s, and ai are the rock mass material constants, given by(16)m=miexpGSI−10028−14D,(17)s=expGSI−1009−3D,(18)ai=12+16e−GSI/15−e−20/3,where GSI is the geological strength index. D is a factor which depends upon the degree of disturbance to which the rock mass has been subjected to blast damage and stress relaxation. For the intact rock, the material constants are denoted by s = 1, ai = 0.5, and m. The expression of H-B criterion for intact rock is shown in equation (2).The H-B criterion is widely used in practical engineering. Many researchers have made an improvement on the Hoek–Brown criterion. The latest major revision of the GSI and its use in equations (16)–(18) was made by Hoek et al. [33]. Hoek and Diederichs [34] used the database of rock mass deformation modulus measurements and proposed that the rock mass deformation modulus can be represented as(19)Erm=Ei0.02+1−D/21+exp60+15D−GSI/11,where Ei is the intact rock deformation modulus.Hoek and Brown [35] summarized the research results of most scholars on the H-B criterion and incorporated the rock mass deformation modulus into the criterion and GSI system, which further demonstrated practical applications of the criterion and the GSI system. However, the parameters of criteria in this work are obtained by using the least square method to fit the criteria and triaxial compression test data. Therefore, the applicability of H-B criterion under the condition of intact rock is discussed to obtain the parameters of H-B criterion more easily.The expression of Exp. criterion is as follows [12]:(20)σ1−σ3=Q∞−Q∞−Q0exp−K0−1σ3Q∞−Q0,where Q0 is the uniaxial compressive strength, Q∞ is the limitation of stress deviator when confining pressure approaches infinity, and K0 is the ratio of strength increasing with confining pressure at confining pressure of zero.The parameter values of the four yield failure criteria are shown in Table3. The parameters of M-C criterion, H-B criterion, and Exp. criterion are determined by using the least square method to fit the criteria and triaxial compression test data, and the determination of σt and σw is similar to these criteria. Here, only the values of Φ0 and C are not obtained. Recall that, in the interpretation of the parameters in equation (1), Φ0 and C are closely related to the critical state where the mean principal stress is σmk and the maximum shear stress is τmk. Then, the expression of σmk is as follows:(21)σmk=Kσw−σt2.Table 3
Parameter values of the three yield failure criteria.
Rock typesM-W criterionH-B criterionM-C criterionExp. criterionσt (MPa)σw (MPa)Φ0 (°)C (MPa)mσc (MPa)φ (°)c (MPa)Q∞ (MPa)Q0 (MPa)K0Dunham dolomite7.05192.7751.26242.168.22286.5838.6166.72683.85257.006.51Mizuho trachyte2.0987.6051.26109.376.63135.4233.3934.75332.19100.006.79Jinping sandstone0.2467.3451.2683.248.2489.0032.2626.13243.8361.607.91Cedar tonalite0.1839.3051.2648.6010.4737.2028.5513.78176.6013.107.51Carrara marble0.9295.0351.26117.825.53153.7925.8152.86467.44137.004.11Westerly granite0.64371.4051.26458.6848.67163.6751.4942.141286.72201.0012.46Yamaguchi marble2.6451.9251.2665.674.4097.6529.3626.81236.5281.004.46Indiana limestone4.1819.3651.2626.442.9750.2421.1617.3189.4344.004.02Solnhofen limestone54.2387.6751.26141.172.77315.8622.55105.48453.90293.003.29Substituting the value ofσmk into equation (1), the value of τmk can be found. Naturally, rearranging equation (1) gives(22)C=τmk+cos2Φ0σmk.The parameter Φ0 is the rock failure angle at the critical state, which is a constant that combines equation (6) with equation (7). Eventually, taking the values of σmk, τmk, and Φ0 into equation (22), the value of C can be obtained.Referring to the calculation formula ofσk (as shown in equation (23)), we can easily solve σk on the premise that σmk and τmk are known.(23)σk=σmk−τmk.


## 4. Yield Surface

Assuming that the rock mass is isotropic, the size of three principal stresses is not unique; then the stress state at any point within the rock mass can be described by the movement surface of the stress point in the principal stress space with axes having the dimensions of the stress (σ1, σ2, and σ3). Therefore, the plastic response can be confined to the yield surface, which is the movement surface of the yield stress point. Meanwhile, the yield surface is symmetric to the positive and negative directions of the principal stress axes due to the isotropic mass, which determine the symmetry of the stress offset tensor. Naturally, the space diagonal (as shown in Figure 3) that sets equal angles (θ) with the three principal stress axes is defined in the principal stress space [36], and cosθ=1/3. Whenever any stress point is on the space diagonal, the values of all three principal stresses are equal.Figure 3
The space diagonal and deviatoric plane in the principal stress space.The yield function has always been investigated by the yield surface plotted in the principal stress space. In fact, the deviatoric plane is the foundation of the yield surface. The deviatoric plane is perpendicular to the space diagonal, and theπ-plane is the deviatoric plane through the origin. We define three orthogonal unit vectors denoted by nD, nE, and nF, where nD (as shown in Figure 3) is the unit vector coinciding with the space diagonal and nE and nF are the unit vectors on the deviatoric plane (as shown in Figure 4). Suppose that e1, e2, and e3 are the unit basis vectors of the three principal stress axes (σ1, σ2, and σ3), respectively. With the help of the vector cross product, we know that [36]Figure 4
Orthogonal unit vectors nE and nF on the deviatoric plane: looking down the space diagonal.(24)nD=131,1,1T,(25)nE=e1×nD=120,−1,1T,(26)nF=nD×nE=162,−1,−1T.Recall that the sizes of the principal stresses are not unique, which indicate that the yield surface is symmetric with respect to the line AD, BE, and CF. These symmetries divide the deviatoric plane into six segments possessing similar properties (I, II, III, IV, V, and VI as shown in Figure4). It is hypothesized that σD, σE, and σF are the stresses in the direction of nD, nE, and nF, respectively. Then, the coordinate transformation of the deviatoric plane and principal stress space is as follows [36]:(27)σD=13σ1+σ2+σ3σE=12−σ2+σ3σF=162σ1−σ2−σ3,where σD, σE, and σF are proposed to facilitate the study of yield surfaces. σD is the stress on the space diagonal, which is proportional to the mean stress p. σE and σF are the stresses on the deviatoric plane, which are related to the deviatoric stress q [36]. p and q can be expressed as(28)p=σ1+σ2+σ33,(29)σE2+σF2=23σ12+σ22+σ32−σ1σ2−σ2σ3−σ3σ1=23q=23I12−3I2,where I1 and I2 are, respectively, the first and second principal stress invariants.


### 4.1. M-W Criterion

To create the form of the yield surface on the deviatoric plane, we invert equation (27):(30)σ1=13σD+23σFσ2=13σD−12σE−16σFσ3=13σD+12σE−16σF.It is convenient for the M-W criterion to label the piecewise function. We mark the yield surface of equation (10) as M-W-1and equation (10) as M-W-2.For M-W-1, substituting equation (30) into equation (10) gives the following.For the stress area I (σ1≥σ2≥σ3) of Figure 4,(31)6σF−2σE+4σwln1+S12σw2−S12σw=2σt,where S1=σt+2/3σD+1/2σE+1/6σF.For the stress area VI (σ1≥σ3≥σ2) of Figure 4,(32)6σF+2σE+4σwln1+S22σw2−S22σw=2σt,where S2=σt+2/3σD+1/2σE+1/6σF.For the stress area II (σ2≥σ1≥σ3) of Figure 4,(33)−2σE+2σwln1+S32σw2−S32σw=σt,where S3=σt+2/3σD−2/6σF.SincenD is perpendicular to the deviatoric plane and σD is constant in the deviatoric plane. The expression of σE and σF is shown in equations (31)–(33), which represent the intersection of the yield surface with the stress areas I, IV, and II in Figure 4. To plot the yield surface, we take Solnhofen limestone as an example, and the parameters of the M-W yield criterion are shown in Table 3. Equations (31) and (32) are substituted to obtain the coordinate value of A (as shown in Figure 4), which is the crossing point of the yield surface between stress area I and stress area VI. Naturally, the coordinate value of B (as shown in Figure 4) can be obtained, and then the value range of σE in stress area I can be known. Using Equation (31) and the value range of σE, the yield surface of stress area I can be obtained. Therefore, referring to the yield surface of stress area I, we can graph the M-W-1 yield surface of the other stress areas (II, III, IV, V, and VI). When σD = 0, the yield surface is shown in Figure 5(a).Figure 5
The yield surface of the M-W yield criterion on the deviatoric plane. (a)π plane (σD = 0). (b) Sparse yield surface.
(a)(b)For M-W-2, substituting equation (30) into equation (10) gives the following.For the stress area I (σ1≥σ2≥σ3) of Figure 4,(34)6σF3+cos2Φ0−32σE1−cos2Φ0=12C−43σDcos2Φ0.The graphing steps of the M-W-2 yield surface are the same as those of M-W-1. Recall that the M-W yield criterion is a piecewise expression and the yield surface on the three-dimensional principal stress space exists as an interface because ofσk. When σm=Kσw−σt/2, we mark the value of the interface (as shown in Figure 6) that intersects with the space diagonal is σDmid. When σm=−σt/2, the value of σD approaches the minimum and is exhibited as σDmin. Substituting these boundary conditions into Equation (27), we can obtain the value of σDmin and σDmid, as shown in Table 4.Figure 6
The yield surface of the M-W yield criterion in the principal stress space.Table 4
Critical value of the space diagonal.
Rock typesM-W criterionH-B criterionM-C criterionExp. criterionσDmin(MPa)σDmid(MPa)σDmin(MPa)σDmin(MPa)σDmin(MPa)Dunham dolomite−12.211184.39−60.39−144.71−63.24Mizuho trachyte−3.62569.16−35.38−91.32−24.88Jinping sandstone−0.42439.10−18.71−71.70−13.30Cedar tonalite−0.31256.22−6.15−43.86−3.35Carrara marble−1.59618.99−48.17−189.31−63.83Westerly granite−1.112422.56−5.82−58.08−27.87Yamaguchi marble−4.57335.72−38.44−82.55−32.64Indiana limestone−7.24121.49−29.30−77.46−17.65Solnhofen limestone−93.93509.41−197.50−439.98−126.21As shown in Figure5(b), when σDmin≤σD≤σDmid, the yield surfaces of M-W-1 that are black polygons varied nonlinearly as a function of σD. When σD>σDmid, the yield surfaces of M-W-2 are red irregular hexagons varied linearly as a function of σD. When σD=σDmid, the green irregular hexagon is the interface yield surfaces of M-W-1 and M-W-2. This variation trend is consistent with the change in the principal stress in the principal stress space (as shown in Figure 6).


### 4.2. M-C, H-B, and Exp. Criterion

For stress area I (σ1≥σ2≥σ3) of Figure 4, substituting equation (30) into equations (14), (2), and (20) gives the following:(1)
M-C criterion:(35)−3σE1+sinϕ+σF3−sinϕ=26ccosϕ+22σDsinϕ.(2)
H-B criterion:(36)6σF−2σE=2mσcσD3+σE2−σF6+σc2.(3)
Exp. criterion:(37)6σF−2σE=2Q∞−2Q∞−Q0exp−K0−1Q∞−Q0σD3+σE2−σF6.The graphing steps on the yield surface of the M-C, H-B, and Exp. are the same as those of M-W-1. The parameterσDmin of the three criteria (as shown in Table 4) approaches the minimum when σE and σF are equal to zero in equations (35)–(37). The yield surfaces of M-C criterion, H-B criterion, and Exp. criterion are shown in Figures 7–9, respectively. The results show that the intersection of the M-C surface with a deviatoric plane is irregularly hexagonal, and H-B surface with a deviatoric plane is a nonlinear gradient from triangular towards irregular hexagonal surfaces. The intersection of the Exp. surface with a deviatoric plane is similar to H-B. In the principal stress space, the M-C yield surface varies linearly as a function of σD, and the H-B and Exp. yield surfaces vary nonlinearly as a function of σD. In addition, the yield curve of H-B on the deviatoric plane is a curve of slight curvature, almost equivalent to a straight line [37]. Although the slight curvature is not obvious in Figure 8(a), the nonlinearity of the H-B criterion is obvious along the axes of Oσ1, Oσ2, and Oσ3. Similarly, the Exp. yield curve on the deviatoric plane may be a curve of slight curvature because of the nonlinearity of the Exp. criterion. It is worth noting that all four failure criteria are only used for conventional triaxial compression, and deviatoric plane shape in other stress states will be never used.Figure 7
M-C criterion yield surface. (a) Yield surface on the deviatoric plane. (b) Yield surface on the principal stress space.
(a)(b)Figure 8
H-B criterion yield surface. (a) Yield surface on the deviatoric plane. (b) Yield surface on the principal stress space.
(a)(b)Figure 9
Exp. criterion yield surface. (a) Yield surface on the deviatoric plane. (b) Yield surface on the principal stress space.
(a)(b)


## 4.1. M-W Criterion

To create the form of the yield surface on the deviatoric plane, we invert equation (27):(30)σ1=13σD+23σFσ2=13σD−12σE−16σFσ3=13σD+12σE−16σF.It is convenient for the M-W criterion to label the piecewise function. We mark the yield surface of equation (10) as M-W-1and equation (10) as M-W-2.For M-W-1, substituting equation (30) into equation (10) gives the following.For the stress area I (σ1≥σ2≥σ3) of Figure 4,(31)6σF−2σE+4σwln1+S12σw2−S12σw=2σt,where S1=σt+2/3σD+1/2σE+1/6σF.For the stress area VI (σ1≥σ3≥σ2) of Figure 4,(32)6σF+2σE+4σwln1+S22σw2−S22σw=2σt,where S2=σt+2/3σD+1/2σE+1/6σF.For the stress area II (σ2≥σ1≥σ3) of Figure 4,(33)−2σE+2σwln1+S32σw2−S32σw=σt,where S3=σt+2/3σD−2/6σF.SincenD is perpendicular to the deviatoric plane and σD is constant in the deviatoric plane. The expression of σE and σF is shown in equations (31)–(33), which represent the intersection of the yield surface with the stress areas I, IV, and II in Figure 4. To plot the yield surface, we take Solnhofen limestone as an example, and the parameters of the M-W yield criterion are shown in Table 3. Equations (31) and (32) are substituted to obtain the coordinate value of A (as shown in Figure 4), which is the crossing point of the yield surface between stress area I and stress area VI. Naturally, the coordinate value of B (as shown in Figure 4) can be obtained, and then the value range of σE in stress area I can be known. Using Equation (31) and the value range of σE, the yield surface of stress area I can be obtained. Therefore, referring to the yield surface of stress area I, we can graph the M-W-1 yield surface of the other stress areas (II, III, IV, V, and VI). When σD = 0, the yield surface is shown in Figure 5(a).Figure 5
The yield surface of the M-W yield criterion on the deviatoric plane. (a)π plane (σD = 0). (b) Sparse yield surface.
(a)(b)For M-W-2, substituting equation (30) into equation (10) gives the following.For the stress area I (σ1≥σ2≥σ3) of Figure 4,(34)6σF3+cos2Φ0−32σE1−cos2Φ0=12C−43σDcos2Φ0.The graphing steps of the M-W-2 yield surface are the same as those of M-W-1. Recall that the M-W yield criterion is a piecewise expression and the yield surface on the three-dimensional principal stress space exists as an interface because ofσk. When σm=Kσw−σt/2, we mark the value of the interface (as shown in Figure 6) that intersects with the space diagonal is σDmid. When σm=−σt/2, the value of σD approaches the minimum and is exhibited as σDmin. Substituting these boundary conditions into Equation (27), we can obtain the value of σDmin and σDmid, as shown in Table 4.Figure 6
The yield surface of the M-W yield criterion in the principal stress space.Table 4
Critical value of the space diagonal.
Rock typesM-W criterionH-B criterionM-C criterionExp. criterionσDmin(MPa)σDmid(MPa)σDmin(MPa)σDmin(MPa)σDmin(MPa)Dunham dolomite−12.211184.39−60.39−144.71−63.24Mizuho trachyte−3.62569.16−35.38−91.32−24.88Jinping sandstone−0.42439.10−18.71−71.70−13.30Cedar tonalite−0.31256.22−6.15−43.86−3.35Carrara marble−1.59618.99−48.17−189.31−63.83Westerly granite−1.112422.56−5.82−58.08−27.87Yamaguchi marble−4.57335.72−38.44−82.55−32.64Indiana limestone−7.24121.49−29.30−77.46−17.65Solnhofen limestone−93.93509.41−197.50−439.98−126.21As shown in Figure5(b), when σDmin≤σD≤σDmid, the yield surfaces of M-W-1 that are black polygons varied nonlinearly as a function of σD. When σD>σDmid, the yield surfaces of M-W-2 are red irregular hexagons varied linearly as a function of σD. When σD=σDmid, the green irregular hexagon is the interface yield surfaces of M-W-1 and M-W-2. This variation trend is consistent with the change in the principal stress in the principal stress space (as shown in Figure 6).


## 4.2. M-C, H-B, and Exp. Criterion

For stress area I (σ1≥σ2≥σ3) of Figure 4, substituting equation (30) into equations (14), (2), and (20) gives the following:(1)
M-C criterion:(35)−3σE1+sinϕ+σF3−sinϕ=26ccosϕ+22σDsinϕ.(2)
H-B criterion:(36)6σF−2σE=2mσcσD3+σE2−σF6+σc2.(3)
Exp. criterion:(37)6σF−2σE=2Q∞−2Q∞−Q0exp−K0−1Q∞−Q0σD3+σE2−σF6.The graphing steps on the yield surface of the M-C, H-B, and Exp. are the same as those of M-W-1. The parameterσDmin of the three criteria (as shown in Table 4) approaches the minimum when σE and σF are equal to zero in equations (35)–(37). The yield surfaces of M-C criterion, H-B criterion, and Exp. criterion are shown in Figures 7–9, respectively. The results show that the intersection of the M-C surface with a deviatoric plane is irregularly hexagonal, and H-B surface with a deviatoric plane is a nonlinear gradient from triangular towards irregular hexagonal surfaces. The intersection of the Exp. surface with a deviatoric plane is similar to H-B. In the principal stress space, the M-C yield surface varies linearly as a function of σD, and the H-B and Exp. yield surfaces vary nonlinearly as a function of σD. In addition, the yield curve of H-B on the deviatoric plane is a curve of slight curvature, almost equivalent to a straight line [37]. Although the slight curvature is not obvious in Figure 8(a), the nonlinearity of the H-B criterion is obvious along the axes of Oσ1, Oσ2, and Oσ3. Similarly, the Exp. yield curve on the deviatoric plane may be a curve of slight curvature because of the nonlinearity of the Exp. criterion. It is worth noting that all four failure criteria are only used for conventional triaxial compression, and deviatoric plane shape in other stress states will be never used.Figure 7
M-C criterion yield surface. (a) Yield surface on the deviatoric plane. (b) Yield surface on the principal stress space.
(a)(b)Figure 8
H-B criterion yield surface. (a) Yield surface on the deviatoric plane. (b) Yield surface on the principal stress space.
(a)(b)Figure 9
Exp. criterion yield surface. (a) Yield surface on the deviatoric plane. (b) Yield surface on the principal stress space.
(a)(b)


## 5. Comparison

Taking Solnhofen limestone as an example, four criteria yield surfaces in the principal stress space are shown in Figures6, 7(b), 8(b), and 9(b), respectively. Considering that the nonlinearity of Exp. criterion is similar to H-B criterion, we mainly focus on M-C criterion, H-B criterion, and M-W criterion in the discussion of slope characteristics of yield surface. When σD = 0 MPa, 200 MPa, and 600 MPa, three criteria yield surfaces on the deviatoric plane are shown in Figure 10. All vertices occur on the lines of symmetry where two of the principal stresses are equal. The uppermost vertex corresponds to the condition where σ1 > σ2 = σ3, which represents the conventional triaxial compression tests. Figure 10 shows that with the increase of σD, the vertices of the three yield surfaces undergo three stages of separation, coincidence, and separation again.Figure 10
The position relationship of three criteria yield surfaces when (a)σD = 0 MPa. (b) σD = 200 MPa. (c) σD = 600 MPa.
(a)(b)(c)To make that difference more intuitive and less complicated, we setσE to zero and then only consider the yield surfaces with the σD-σF plane as shown in Figure 11, which is a slice through the three-dimensional yield surface shown in Figures 6, 7(b), and 8(b). When σF > 0, the upper yield surfaces correspond to conventional triaxial compression (σ1 > σ2 = σ3), which also represents the uppermost vertex as shown in Figure 10. When σF < 0, the lower yield surfaces correspond to conventional triaxial tensile stress (σ1 < σ2 = σ3), which represents the lowermost vertex as shown in Figure 10. Figure 11 shows that the yield surfaces corresponding to the three criteria have a vertex consistent region with increasing σF; however, except for the vertex consistent region, the yield state of rock predicted by the three criteria is quite different.Figure 11
The projection of three criteria yield surfaces onto theσD–σF plane.SubstitutingσE = 0 into equations (31) and (34)–(37) gives the following:(1)
M-C yield criterion:(38)σF3−sinϕ=26ccosϕ+22σDsinϕ.(2)
H-B yield criterion:(39)6σF=2mσcσD3−σF6+σc2.(3)
Exp. yield criterion:(40)6σF=2Q∞−2Q∞−Q0exp−K0−1Q∞−Q0σD3−σF6.(4)
M-W yield criterion:(41)6σF+4σwln1+σt+2/3σD+1/6σF2σw2−σt+2/3σD+1/6σF2σw=2σt,(42)6σF3−cos2Φ0=12C−43σDcos2Φ0.That difference depends on the extent of the negative values forσD and the magnitude of the slopes of the yield surfaces. As shown in Figure 11, when σF = 0, the cohesion and the internal friction angle control the value at the starting point for the M-C criterion; the uniaxial compression strength and the experience parameter control the value for H-B criterion; and only the uniaxial tensile strength controls the value for the M-W criterion. Similarly, the parameters Q0, Q∞, and K0 control the value for Exp. criterion according to equation (40). In addition, we can clearly realize the characteristic slopes of the yield surfaces from equations (38)–(42). The slope of the M-C criterion surface is linear. The slopes of the H-B and Exp. criteria surfaces are nonlinear. However, the slope of the M-W yield surface is nonlinear, and when σD > σDmid, the slope becomes linear.Although many factors determine that the four yield surfaces are different, the vertex consistent region may appear due to the increasing value ofσD. In fact, we exploit the upper yield surfaces on the σD-σF plane to fit the triaxial compression test data from Tables 1 and 2, and the fitting results shown in Figures 12 and 13 suggest the existence of the vertex consistent region. Figure 12 shows the fitting results when the confining pressure is below the critical state, and Figure 13 shows the fitting results when the confining pressure is above the critical state. The results show that four yield surfaces have a vertex consistent region, and within this region, the yield state of rock can be better predicted. However, outside this region, the predicted results of the four yield criteria are different. Therefore, we introduce the relative error Er to estimate the dissimilarity degree.(43)Er=∑i=1nσFcalc−σFtest/σFtestn×100%,where Er is the relative error between the test and calculate, σFtest is the deviatoric stress test data, σFcalc represents the calculated data, and n is the number of triaxial test groups for each rock.Figure 12
The yield surfaces on theσD–σF plane: the confining pressure below the critical state. (a) Dunham dolomite. (b) Mizuho trachyte. (c) Jinping sandstone. (d) Cedar tonalite. (e) Carrara marble. (f) Westerly granite.
(a)(b)(c)(d)(e)(f)Figure 13
The yield surfaces on theσD–σF plane: the confining pressure above the critical state. (a) Yamaguchi marble. (b) Indiana limestone. (c) Solnhofen limestone.
(a)(b)(c)Whenσ3 ≤ σk, the relative errors of the fitting results in Figures 12 and 13 are shown in Figure 14. In terms of the relative error, the M-C criterion is higher than the others and the M-W criterion is generally lower than the H-B criterion (except the Dunham dolomite). In addition, the relative error values of M-W criterion and Exp. criterion are lower than the M-C criterion and H-B criterion. It is suggested that the M-W criterion has certain applicability in predicting the rock yield state under a confining pressure below the critical state.Figure 14
Relative error among four yield criteria prediction with test results: all confining pressures below the critical state.


## 6. Discussion

The M-W criterion includes linear and nonlinear sections. In the absence of the derivation process of the yield criterion, the reader can refer to [16]. This study attempted to investigate whether there are differences in predicting the rock yield state for the four yield criteria (M-C, H-B, Exp., and M-W criteria). However, as shown in Figure 13, when σD > σDmid, the M-C criterion and H-B criterion cannot predict the accuracy of the rock yield state, the Exp. criterion can better predict the Indiana limestone plastic state, and the M-W criterion can better predict the Yamaguchi marble plastic state. In addition, none of them can effectively predict the yield state for Solnhofen limestone when the confining pressure is above the critical state. The possible reason may be related to the ductility of the rock.The rock strength increases with the confining pressure. The strength growth rate of rock is generally large under the condition of low confining pressure, but with an increase in the confining pressure, the strength growth rate will gradually decrease [22]. Mogi [38] suggested that the fracture behavior of rock changes from brittle to ductile when the confining pressure continues to increase, where the critical confining pressure is assumed to be σcr. Hoek [31] proposed that the Indiana limestone transition from the brittleness to ductility stage when σcr is equal to σc. Singh et al. [9] analyzed 201 groups of rock triaxial test results to show that the minimum consistent variance was achieved when σcr was approximately equal to σc. However, some scholars put forward a different view to the critical confining pressure. Xu [39] proposed that σcr is approximately equal to 0.85σc∼σc for marble, and Yang [40] suggested that σcr is equal to 0.47σc for marble with medium and coarse grains. In addition, Barton [41] proposed that the critical state is related to the principal stress, namely, σ1 = 3σ3.The different opinions indicate that the critical confining pressure depends on the type of rock, and the value ofσcr may not necessarily be related to σc. As shown in Figure 13, the slope of Exp. yield curve on the σD–σF plane becomes gently when σD > σDmid, which indicates the critical state proposed in this work is reasonable. It is noteworthy from Figure 13(c) that the critical confining pressure σk proposed by the M-W yield criterion seems to be related to the transition from brittleness to ductility. Assuming σcr = σk, the yield surfaces of the M-W yield criterion can be shown in Figure 15 when σD > σDmid (namely, σ3 > σk), which is only a schematic diagram due to the complex characteristics of rocks in the ductile state. If the ductile characteristics of Solnhofen limestone shown in Figure 15 can be confirmed, we can explain the inaccuracy of Figure 13(c).Figure 15
The characteristics of the brittleness and ductility for Solnhofen limestone.Although the M-W yield criterion cannot effectively predict the yield state for Indiana limestone and Solnhofen limestone whenσ3 > σk, the parameter σk can be regarded as the critical confining pressure of the rock during the transition from brittleness into ductility. The indicator of the critical confining pressure of more rocks and the expression of criterion on the condition of σ3 > σk will be studied in future work.


## 7. Conclusions

This study introduced a piecewise rock yield criterion that is a theoretical formula derivation based on microcosmic experiments. Although the actual failure process of the rock may be more complicated than the wedge model, the model is reasonable to a certain extent. When the vertical and horizontal microcracks in the shear potential failure zone of the wedge model are compacted, the failure behavior of the rock enters a critical state. The model accounts for the nonlinear variation in the principal stress when the confining pressure is below the critical state and linear variation in the principal stress when the confining pressure is above the critical state. All of the defining parameters are related to rock damage and can be obtained by a rock axial loading test.Alternatively, we find that the yield surfaces of different criteria can be obtained through the coordinate transformation of the deviatoric plane and the principal stress space. Then, the rock yield state of the three-dimensional space can be transformed into a two-dimensional plane. Increasing the density of the yield surfaces on the deviatoric plane (namely, theσE-σF plane), the M-W yield surface behaves in a nonlinear and linear fashion because of the critical state. Furthermore, we show that different criteria yield surfaces have a vertex consistent region by slicing the three-dimensional yield surface along the σD-σF direction. On the σD-σF plane, the fitting results indicate that the M-W criterion and the Exp. criterion can predict the rock yield state more accurately than the M-C and H-B criteria. This finding suggests that the M-W criterion has certain applicability in predicting the rock yield state.



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*Source: 1612383-2020-12-15.xml*