Abstract
Conventional triaxial strength criteria are important for the judgment of rock failure. Linear, parabolic, power, logarithmic, hyperbolic, and exponential equations were, respectively, established to fit the conventional triaxial compression test data for 19 types of rock specimens in the Mohr stress space. Then, a method for fitting the failure envelope to all common tangent points of each two adjacent Mohr’s circles (abbreviated as CTPAC) was proposed in the Mohr stress space. The regression accuracy of the linear equation is not as good as those of the nonlinear equations on the whole, and the regression uniaxial compression strength (σc)r, tensile strength (σt)r, cohesion cr, and internal frictional angle φr predicted by the regression linear failure envelopes with the method for fitting the CTPAC in the Mohr stress space are close to those predicted in the principal stress space. Therefore, the method for fitting CTPAC is feasible to determine the failure envelopes in the Mohr stress space. The logarithmic, hyperbolic, and exponential equations are recommended to obtain the failure envelope in the Mohr stress space when the data of tensile strength (σt)t are or are not included in regression owing to their higher R2, less positive x-intercepts, and more accurate regression cohesion cr. Furthermore, based on the shape and development trend of the nonlinear strength envelope, it is considered that when the normal stress is infinite, the total bearing capacity of rock tends to be a constant after gradual increase with decreasing rates. Thus, the hyperbolic equation and the exponential equation are more suitable to fit triaxial compression strength in a higher maximum confining pressure range because they have limit values. The conclusions can provide references for the selection of the triaxial strength criterion in practical geotechnical engineering.
1. Introduction
The Mohr-Coulomb strength criterion has been widely applied in rock engineering. Cohesion c and internal friction angle φ obtained from the conventional triaxial compression tests are commonly used as strength parameters of rock. Rock is a kind of anisotropic material with natural fractures and defects, and inaccuracies, such as loading cell friction, end plate roughness, and membrane stiffness, are inevitable in the triaxial compression tests, so it is hard for Mohr-Coulomb failure envelope to be tangent to all Mohr’s circles [1–4]. Many scholars put forward various methods to obtain more accurate strength parameters of rock. Stafford et al. [4] suggested a method of resorting to a plot of (σ1–σ3)/2 against (σ1 + σ3)/2 for each test and drawing the best fitting straight line through the points and then deriving c and φ from this as shown in Figure 1(a). Another common method is to use a linear equation to fit the relationship between the major principal stress σ1 and the minor principal stress σ3 at failure with the least square method as shown in Figure 1(b), and then 6∼8 points are equidistantly selected on the regression line to determine Mohr’s circles. Finally, the linear regression failure envelope can be drawn based on the common tangent points of all Mohr’s circles, and thus c and φ can be obtained as shown in Figure 1(c). Yang et al. [6] presented a linear equation obtained by fitting the relationship between σ1 and σ3 with the least square method, in which the slope and the y-intercept of the regression line are obtained, and c and φ can be further calculated. Wang firstly calculated the coordinates of common tangent points on each two Mohr’s circles, and then c and φ can be predicted by fitting a straight line to all the common tangent points with the least square method [5]. For example, in Figure 1(d), there are three Mohr circles, three common tangent lines, and six common tangent points, which are used to obtain the linear regression failure envelope.
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With the increasing depth in geotechnical engineering, the rock failure mechanism changes, and the brittle rock under a low confining pressure gradually presents ductile failure feature under a higher confining pressure [7]. Moreover, the differential stress (σ1–σ3) at failure of rock would approach being a constant, and the dip angle of the failure plane is close to π/4 when the maximum confining pressure becomes high enough [8]. In this case, the inaccuracy is quite large using a linear equation to obtain the failure envelope, so that the above-mentioned methods of determining linear strength envelopes are unavailable. Therefore, the nonlinear equations are needed to fit the test data. Specifically, when using Wang’s method to get c and φ, the slope of common tangent line is relatively steep for two adjacent Mohr’s circles under low confining pressures, and then it gradually tends to be gentle with the increase in confining pressure. Theoretically, for two nonadjacent Mohr’s circles, the farther the distance between them, the smaller the slope of common tangent line. Moreover, the common tangent points will significantly differ from those on the adjacent Mohr’s circles, which may result in an unavailable failure envelope. Thus, the common tangent points on only two adjacent Mohr’s circles (CTPAC) are adopted here to obtain the failure envelope. In particular, we use only four common tangent points 1, 3, 4, and 6 generated from the two common tangent lines L1 and L2 and three Mohr’s circles to obtain the rock failure envelope as shown in Figure 1(d).
As mentioned above, the nonlinear strength envelope can obtain a better fitting accuracy and reflects the bearing characteristics of rock more accurately in a larger range of the maximum confining pressure. Many studies have been conducted on the nonlinear strength criteria [8–13], in which You compared 16 strength criteria composed of one, two, and three parameters, respectively. In this paper, some empirical equations for conventional triaxial strength criteria of intact rock are proposed and studied comparatively. From the review authored by Cartrin [14], most empirical formulas of strength criteria are simple elementary function forms, such as power function, exponential function, and logarithmic function, and most of them are in the form of power function. Among them, linear function, parabolic function, and hyperbolic function can be regarded as the variation form of power function. Therefore, this paper mainly focuses on the applicability of linear equation, parabolic equation, power equation, logarithmic equation, hyperbolic equation, and exponential equation in describing the intact rock strength criterion. Finally, more suitable empirical equations are recommended based on the bearing mechanism of rock.
2. Conventional Triaxial Strengths and Empirical Equations
2.1. Test Data of Conventional Triaxial Strengths
Table 1 shows σ1 and σ3 at failure of red sandstone (RS) [15], Tyndall limestone (TL) [16], Lac du Bonnet granite (LG) [16], Dunham dolomite (DD) [17], Inada granite (IG) [17], Mizuho trachyte (MT) [17], Marazuru andesite (MA) [17], Orikable monzonite (OM) [17], Bunt sandstone (BS) [18], Yamaguchi marble (YM) [18], Solnhofen limestone (SL) [18], Carrara marble (CM) [18], Granite (G) [19], Quartzite (Q) [19], and Basalt (B) [19]. Moreover, σ1 and σ3 at failure of powder-grained marble (PGM), fine-grained marble (FGM), middle-grained marble (MGM), and coarse-grained marble (CGM) were obtained using RMT-150B rock mechanics test machine and are also shown in Table 1.
When the test data of uniaxial tensile strength (σt)t and uniaxial compressive strength (σc)t are included in the parameters calculation of the Hoek-Brown strength criterion, a better fitting result than that using only the triaxial data can be obtained [9]. Therefore, whether the tensile strength data (σt)t should be included in regression is discussed here, and the tensile strengths of seven kinds of rock are listed in Table 1.
The marble samples were collected from a quarry in Nanyang City, Henan Province, and the main minerals are calcite, dolomite, and magnesite. According to the requirements of the International Society of Rock Mechanics (ISRM), the standard cylindrical specimens with the diameter of 50 mm and the length of 100 mm were prepared, as shown in Figure 2(a).
After treating wave testing on rock specimens with the ultrasonic detector (Figure 2(b)), a serious test of uniaxial compression, Brazilian splitting and triaxial compression were conducted using the RMT-150B electrohydraulic servo rock mechanics testing system developed by Wuhan Institute of Geotechnical Engineering, Chinese Academy of Sciences (Figure 2(c)). The basic mechanical parameters of rock, such as uniaxial compressive strength, tensile strength, internal friction angle, cohesion, elastic modulus, and Poisson’s ratio, were obtained, as shown in Table 2. The axial and lateral loading capacities of this system are 100 t and 50 t, respectively. One vertical displacement sensor with the stroke of 20 mm was used to monitor the axial deformation of rock specimen, and two horizontal displacement sensors with the stroke of 2.5 mm were used to monitor the transverse deformation of rock specimen.
2.2. Empirical Equations
Six equations (linear, parabola, power, logarithm, hyperbola, and exponent) were used here to fit the failure curves, respectively, in the Mohr stress space. When the conventional strength criteria are fitted to CTPAC in the Mohr stress space, it should be ensured that each two adjacent Mohr’s circles cannot wrap each other, because their common tangent points are not obtained. Besides, the latter Mohr’s circle (the center is located on the right side) should be removed in regression if its diameter is smaller than that of a former one (the center is located on the left side), because it is unreasonable that the common tangent points locate on the right side of the vertex of a Mohr’s circle. The removed data in specimens MA, SL, Q, B, and PGM were marked with as shown in Table 1.
The linear equation iswhere a1 and b1 are the regression.
The parabolic equation iswhere a2 and b2 are the regression constants and a2 > 0, , and σ > b2.
The power equation iswhere a3, b3, and d3 are regression constants and a3 > 0, b3 > 1, and σ > d3.
The logarithmic equation iswhere a4, b4, and d4 are regression constants and a4 > 0.
The hyperbolic equation iswhere a5, b5, and d3 are regression constants and a3 > 0 and σ > d3. The slope of regression failure envelope k, , decreases with increasing σ. When σ = ∞, k = 0, and τ = b5, the radius of the ultimate Mohr’s circle in Mohr stress space b3 does not increase with σ.
The exponential equation iswhere a6, b6, and d6 are regression constants, and a6 > 0 and b6 > 0. In the Mohr stress space, the slope of regression failure envelope k decreases with increasing σ. When σ = ∞, k = 0, τ = a6 + d6, and , the radius of the ultimate Mohr’s circle in the Mohr stress space does not increase with σ3.
3. Regression Results of Linear Strength Criterion
In this paper, the newly added subscript t expresses the test values, the subscript r expresses the regression values, the subscript m expresses the values in the Mohr stress space, and the subscript expresses the values in the principal stress space. Moreover, the subscript u means that (σt)t is not included in regression, while the subscript i means that (σt)t is included in regression. For example, (σt)t means a test tensile strength value of a rock specimen, and (σc)rmu means a regression compressive strength value of a rock specimen obtained in the Mohr stress space with (σt)t not being included in regression.
The linear strength criterion is most commonly used in geotechnical engineering. Thus, it is necessary to analyze the following contents, including the regression results of the linear equation in the Mohr stress space and in the principal stress space, the influences of (σt)t contained in regression on the validity of regression, and the applicability of fitting strength envelopes to CTPAC in the Mohr stress space.
3.1. Regression Squared Correlation Coefficient
Figure 3 shows the regression squared correlation coefficients R2 of linear equation in two different stress spaces. With (σt)t being included in regression in the principal stress space, the minimum (0.632), the average (0.844), and the maximum (0.958) of (R2)rpi of linear equation are all lowest. The regression coefficients (R2)rmi of linear equation in the Mohr stress space with (σt)t included in regression were shown as follows, with the minimum, average, and maximum values of 0.923, 0.948, and 0.983, respectively. However, when (σt)t is not included in regression, the average R2 are all above 0.961 and the maximum R2 are all above 0.991. Thus, R2 of linear equation would generally reduce when (σt)t is included in the regression.
3.2. Regression Strength Parameters
The strength parameters of rock (c, φ) are usually determined by fitting a line to common tangent points on the ultimate Mohr’s circles in the Mohr stress space, and then (σc)r and (σt)r can be calculated from the regression line (linear Mohr-Coulomb criterion) in application. Especially in this paper, the linear strength envelopes were fitted to CTPAC in the Mohr stress space. Besides, (σc)r, (σt)r, cr, and φr also can be predicted by fitting the relationship between σ1 and σ3 at failure in principal stress space using equation (1). If (σc)r, (σt)r, cr, and φr predicted by above two methods have no obvious differences, the linear strength envelope fitted to CTPAC can be considered in the application.
Figure 4 shows (σc)r, Abs. (σt)r, cr, and φr predicted by above two methods. In Figure 4(a), (σc)rpu has a linear relation with (σc)rmu and they are almost equivalent with each other. (σc)rpi and (σc)rmi are also equivalent with each other, and they are both lower than the corresponding (σc)rpu and (σc)rmu. However (σc)rpi and (σc)rmi are closer to (σc)t than (σc)rpu and (σc)rmu. (σc)t is lower than (σc)rpu, and it has an exponential relation with (σc)rpu, which is expressed as follows:in which R2 is 0.894. Therefore, equation (7) can be used to modify (σc)rpu for more accurate estimation of the value of (σc)t.
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In Figure 4(b), Abs. (σt)rpu is about equal to Abs. (σt)rmu, and it has a linear relation with Abs. (σt)rmu, but there is some difference between them in a higher part of Abs. (σt)rmu. Abs. (σt)rpi and Abs. (σt)rmi are close to each other, and they are both lower than the corresponding Abs. (σt)rpu and Abs. (σt)rmu but higher than Abs. (σt)t. The difference between Abs. (σt)rpi, Abs. (σt)rmi, and Abs. (σt)t would also increase with increasing Abs. (σt)rmu on the whole. Thus, it is recommended to get more accurate (σt)r using (σt)t in regression.
Similarly, in Figure 4(c), when (σt)t is not included in regression, crpu is about equal to crmu, and it has a linear relation with crmu. However, there is some difference between them in the higher part of crmu. When (σt)t is included in regression, crpi and crmi are close to each other, and they are both lower than the corresponding crpu and crmu. In Figure 4(d), when (σt)t is not included in regression, φrpu is about equal to φrmu, and it has a linear relation with φrmu. But there is some difference between them in the middle and lower parts of φrmu. When (σt)t is included in regression, φrpi and φrmi are close to each other, and they are mostly higher than the corresponding φrpu and φrmu.
All in all, whether (σt)t is included in regression or not, (σc)r, (σt)r, cr, and φr predicted by linear equation in the principal stress space are similar to those obtained in the Mohr stress space with the method of fitting CTPAC. Consequently, the method of fitting CTPAC is completely suitable to obtain conventional triaxial failure envelopes in the Mohr stress space.
4. Regression Results of Nonlinear Strength Criterion
4.1. Typical Regression Curves of Specimen LG
In this section, CTPAC was fitted with equations (1)–(6), respectively, to obtain the conventional triaxial failure envelopes for 19 kinds of rock specimens when (σt)t is not included in regression and for 7 kinds of rock specimens when (σt)t is included in regression in the Mohr stress space.
Under the maximum normal stress σ of 139 MPa in the Mohr stress space, regression curves of specimen LG were taken as examples to analyze the regression validities of all empirical equations as shown in Figure 5. (σt)t is not included in regression in Figure 5(a), and (σt)t is included in regression in Figure 5(b). Figure 5 shows that the regression curves are close to each other in the middle range of the tangent points, but they are divergent in the two ends, especially when the tangent points are lower. The linear regression curves (equation (1)) are below the tangent points in the middle range of the tangent points, and they are above the tangent points in two ends of the tangent points. Moreover, the parabolic regression curves (equation (2)) also have larger deflections, but they are above the tangent points in the middle of the tangent points and below the tangent points in two ends of the tangent points. Thus, R2 of these two equations should be relatively lower. However, the other nonlinear regression curves show good regression validities, and most of them coincidently go through the tangent points.
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The diverging characteristics in the lower range of the tangent points are described as follows: in Figure 5(a), when (σt)t is not included in regression, the logarithmic regression (equation (4)), hyperbolic regression (equation (5)), and exponential regression curves (equation (6)) are invariably close to tangent points and achieve a good regression result. However, the regression validities are better when (σt)t is included in regression as shown in Figure 5(b); thus the regression cohesions (cr) and the x-intercepts of these three failure envelopes (equations (6), (8), and (10)) are more accurate. Based on the method of uniaxial tension with Mohr’s circles being tangent to the envelopes, the tensile strengths predicted by regression envelopes from low to high are, respectively, as follows: parabolic equation (equation (2)), power equation (equation (3)), logarithmic equation (equation (4)), hyperbolic equation (equation (5)), exponential equation (equation (6)), and linear equation (equation (1)).
4.2. Regression Squared Correlation Coefficients
Figure 6 depicts the relationships between the squared correlation coefficients R2 of the six equations, which are used to fit conventional triaxial failure envelopes in the Mohr stress space and maximum σ3/σc, in which (σt)t is not included in regression as shown in Figure 6(a), while (σt)t is included in regression as shown in Figure 6(b).
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In Figure 6, compared to nonlinear equations, R2 of linear equation (equation (1)) are lowest on the whole and decrease with increasing maximum σ3/σc. When (σt)t is not included in regression, R2 of parabolic equation (equation (2) in Figure 6(a)) are also lower obviously and increase on the whole with increasing maximum σ3/σc. The fit accuracy of other equations is all higher, and their minimums, averages, and maximums of R2 are above 0.985, above 0.996, and close to 1.000, respectively. When (σt)t is included in regression, as for the average value, R2 of nonlinear equations used in the Mohr stress space become higher or remain unchanged but R2 of linear equation (equation (1)) become lower a little in Figure 6(b) compared with those in Figure 6(a). All in all, the power equation (equation (3)), the logarithmic equation (equation (4)), the hyperbolic equation (equation (5)), and the exponential equation (equation (6)) are recommended to fit conventional triaxial failure envelopes for their higher R2.
4.3. The x-Intercepts of the Regression Curves of τ and σ in the Mohr Stress Space
In the Mohr stress space, if the x-intercept of the regression curve is positive, the y-intercept of the regression curves must be negative. In this case, regression cohesion cr, regression uniaxial compression strength (σc)r, and regression tension strength (σt)r of specimens cannot be predicted correctly by the failure envelopes. Therefore, these regression equations are not suitable for fitting the failure envelopes in the Mohr stress space if some x-intercepts of the regression curves are positive. Figure 7 shows the x-intercepts of the regression envelopes. The x-intercepts of the regression nonlinear envelopes are larger than those of the linear envelopes. No matter (σt)t is included or not included in regression, some x-intercepts of the parabolic envelope (equation (2)) and the power envelopes (equation (3)) are positive, which show that cr, (σc)r, and (σt)r may not be obtained when parabolic equation (equation (2)) and power equation (equation (3)) are used in the Mohr stress space. Thus, logarithmic equation (equation (4)), hyperbolic equation (equation (5)), and exponential equation (equation (6)) are optimal to fit the failure envelopes in the Mohr stress space. Moreover, when (σt)t is included in regression (Figure 7(b)), the x-intercepts of the logarithmic regression (equation (4)), hyperbolic regression (equation (5)), and exponential regression curves (equation (6)) are more concentrated than those with (σt)t being not included in regression. All in all, the logarithmic equation (equation (4)), the hyperbolic equation (equation (5)), and the exponential equation (6) are recommended to fit conventional triaxial failure envelopes for they have no positive x-intercepts.
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4.4. cr Predicted by the Relationship between τ and σ in the Mohr Stress Space
We know from Figure 6 that the squared correlation coefficients of logarithmic equation (equation (4)), hyperbolic equation (equation (5)), and exponential equation (equation (6)) are higher, and their failure envelopes go through nearly all tangent points in the Mohr stress space when (σt)t is included in regression; thus the cohesion cr predicted by the above three envelopes should be closer to their true values, and the above three regression envelopes of the same set of tangent points are relatively close to each other. Besides, we know from Figure 7 that some x-intercepts of two regression envelopes of parabolic equation (equation (2)) and power equation (equation (3)) are positive, so some cr cannot get obtained; therefore, cr predicted by power envelopes (equation (3)) and parabolic envelopes (equation (2)) are not shown in Figure 8.
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Figure 8 depicts that cr predicted by nonlinear envelopes are all lower than those predicted by linear envelope (equation (1)). Thus, cr predicted by linear regression envelope (equation (1)) is significantly larger and is far from the true values. In particular, cr predicted by logarithmic (equation (4)), hyperbolic (equation (5)), and exponential (equation (6)) envelopes with (σt)t being included in regression are more focused than those predicted by the above three regression strength envelopes with (σt)t being not included in regression. Thus, using (σt)t in regression could improve the regression validities of logarithmic (equation (4)), hyperbolic (equation (5)), and exponential equations (equation (6)). (σt)t are advised to be included in fitting nonlinear failure envelopes to get more accurate conventional triaxial strength parameters. The logarithmic equation (equation (4)), the hyperbolic equation (equation (5)), and the exponential equation (equation (6)) are all recommended to fit conventional triaxial failure envelopes for they have relatively close cr.
5. Friction and Cohesion Bearing Characteristics of a Rock Specimen
5.1. Negative Friction Bearing Capacity
In order to explain the existence of negative friction bearing capacity, the stress diagram of potential failure plane of a specimen under triaxial compression is shown in Figure 9. When the axial pressure σ1 and the confining pressure σ3 are positive (Figure 9(a)), their components in the potential failure plane are the normal compressive stress component σ3n in the downward vertical direction of the potential failure plane and the shear stress component σ3τ in the upward parallel direction of potential failure plane. The components of axial pressure σ1 in the potential failure plane are the normal compressive stress component σ1n in the downward vertical direction of the potential failure plane and the shear stress component σ1τ in the downward parallel direction of the potential failure plane. Thus, the composition of the total bearing capacity Q can be expressed aswhere S is the failure fracture surface area, both the friction bearing capacity Qf and cohesion bearing capacity Qc are positive, so that the total bearing capacity Q is larger than the cohesion bearing capacity Qc, and the friction bearing capacity Qf > 0 and can be named as a positive one (Figure 9(d)). Besides, the directions of the above two shear stress components are contrary; therefore, when the specimen fails and slides down along the fracture surface, the shear force produced by the confining pressure σ3 in the opposite direction is not conducive to the failure of the potential failure plane and will increase the axial pressure σ1 at failure.
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When the axial pressure σ1 is positive and the confining pressure σ3 is negative (Figure 9(b)), the component directions of the confining pressure σ3 in the potential failure plane will be opposite to the condition that it is positive. That is to say, when the axial pressure σ1 > 0 and the confining tensile stress σ3 < 0, these two shear stress components are both in the downward parallel direction of potential failure plane; therefore, when the specimen fails and slides down along the fracture surface, the shear force produced by the confining pressure σ3 in the same direction is conducive to the failure of the potential failure plane and will decrease the axial pressure σ1 at failure. Under these circumstances, the cohesion bearing capacity Qc is a positive constant, while the friction bearing capacity Qf may be positive or negative according to the value of σ1n + σ3n, so that the total bearing capacity Q may be higher or lower than the cohesion bearing capacity Qc, and the friction bearing capacity Qf can be named as a negative one when Qf < 0 (Figure 9(f)).
When the confining pressure σ3 and the axial pressure σ1 are both negative (Figure 9(c)), the cohesion bearing capacity Qc is still a positive constant, while the friction bearing capacity Qf is negative according to the value of σ1n + σ3n, so that the total bearing capacity Q is lower than the cohesion bearing capacity Qc, and the friction bearing capacity Qf is a negative one (Figure 9(f)).
The above analysis expresses the possibility of the existence of negative friction bearing capacity Qf ; in order to understand the above analysis more clearly in the Mohr stress space as shown in Figure 10, a further explanation is made here.
The tangent point of circle O1 and linear strength envelope is on the vertical axis, and the value of at the vertical coordinate is the cohesion c of a rock. When the ultimate stress circle is on the right side of circle O1, the friction bearing capacity Qf is positive, and the total bearing capacity is higher than the cohesion bearing capacity Qc; when the ultimate stress circle is on the left side of circle O1, the friction bearing capacity Qf is negative, and the total bearing capacity Q is lower than the cohesion bearing capacity Qc.
5.2. Friction and Cohesion Bearing Characteristics of Rock in Linear Strength Criterion
Friction and cohesion will not work for one point at the same time in a rock specimen [8]. That is to say, every point on the potential failure plane cannot provide friction and cohesion at the same time. When the point is not damaged, it provides bearing capacity with cohesion; when the point is damaged, it provides bearing capacity with friction. Evolution schematic diagram of rock bearing capacity composition is shown in Figure 11, in which total bearing capacity Q is the sum of the cohesion bearing capacity Qf and the friction bearing capacity Qc. As shown in Figure 11(a), when a linear equation is used in regression, the friction coefficient of rock k = tanφ is a constant, and the cohesion bearing capacity Qc is also a constant, but the total bearing capacity Q and friction bearing capacity Qf increase linearly as the normal stress σ increases. The contribution proportion of cohesion bearing capacity Qc to increment of total bearing capacity ΔQ is 0, while the increment of friction bearing capacity ΔQf is equal to the increment of total bearing capacity ΔQ with the normal stress increasing, and the ratio of friction bearing capacity Qf to total bearing capacity Q approaches 100% finally. When the normal stress σ is less than 0, there is a negative friction bearing capacity Qf that counteracts a same value of cohesion bearing capacity Qc, thus reducing the total bearing capacity Q.
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5.3. Friction and Cohesion Bearing Characteristics of Rock in Nonlinear Strength Criterion
When nonlinear equations are used in regression, friction coefficients are all reducing variables that decrease with the increase of normal stress. It means that the frictional coefficient is larger when the normal stress is lower, and it is lower when the normal stress is larger. So, the contribution proportion of friction bearing capacity ΔQf to the increment of total bearing capacity ΔQ gradually reduces and even disappears in the end with increasing normal stress. Barton [20] divided shear strength envelope of rock joint under different normal stress into two parts by a critical normal stress. When the normal stress is lower than the critical normal stress, the friction coefficient is large as it is the tangent value of two angles: a friction angle and a climbing angle; when the normal stress exceeds the critical value, the frictional coefficient is small as it is only the tangent value of the friction angle. There is a similar viewpoint here, but the friction coefficient decreases gradually with normal stress increasing. However, the difference is that the friction coefficient in nonlinear strength criteria is always a nonlinear curve but is not only two constants as in Barton’s method [17].
As normal stress increases, total bearing capacity Q increases with a reducing increasing rate as shown in Figure 11(b), the friction coefficient k decreases gradually, the cohesion bearing capacity Qc decreases firstly and then increases before and after the normal stress is 0 MPa, and the friction bearing capacity Qf increases firstly and then decreases. It needs to be explained that the cohesion bearing capacity Qc of the rock specimen under each normal stress is the intercept of the tangent line of each corresponding point on the total bearing capacity curve on the longitudinal axis, the friction coefficient k of the rock specimen under each normal stress is the tangent slope value of each corresponding point on the total bearing capacity curve, and the friction bearing capacity Qf of the rock specimen under each normal stress is the product of the normal stress σ and its corresponding friction coefficient k.
According to the above assumption, when the normal stress is less than 0, there is a negative friction bearing capacity Qf and a larger positive cohesion bearing capacity Qc; when the normal stress is 0, friction bearing capacity Qf is 0 and the cohesion bearing capacity Qc reaches its lowest value; when the normal stress is larger than 0, as the normal stress increases, the friction bearing capacity Qf increases firstly and then decreases to approach 0 gradually, while the cohesion bearing capacity Qc increases to approach the total bearing capacity Q gradually. When the normal stress rises to infinity, the frictional angle will tend to be zero and the increment of friction bearing capacity ΔQf also tends to be zero with increasing normal stress, and the friction bearing capacity Qc is close to the total bearing capacity Q under this condition; thus the total bearing capacity Q and its increment ΔQ will be provided by only the cohesion bearing capacity Qc and its increment ΔQc. Therefore, if the increment of cohesion bearing capacity ΔQc has a limit, the total bearing capacity Q also has a limit; otherwise, the total bearing capacity Q of rock will increase with the increment of cohesion bearing capacity ΔQc in a same quantity. If the former is in existence, the hyperbolic equation and the exponential equation are more suitable for fitting triaxial compression strengths in a higher maximum confining pressure range in the Mohr stress space.
If used to evaluate the strength characteristics of rock in high-stress state at deep or extra deep depts, the applicability of the empirical equations obtained from the test data with a lower maximum confining pressure needs a further study. Only by determining the strength characteristics of rock in the high confining pressure range can a more reasonable rock strength criterion be determined in the lower confining pressure range, and the rock strength criterion can be used without confining pressure range restriction. If a reasonable strength criterion and its parameters are determined only based on the test data in a lower confining pressure range, the applicability of the criterion and parameters under high confining pressure will be more questioned. Therefore, the next step should be to improve the rock mechanical equipment and test methods, obtain the strength test data of higher confining pressure range, determine the characteristics of rock bearing capacity limit, and then determine a more reasonable rock strength criterion.
6. Conclusions
The following conclusions were reached in this research study:(1)Whether (σt)t is included or not included in regression, (σc)r, (σt)r, cr, and φr predicted by linear equation in the principal stress space are similar to those obtained in the Mohr stress space with the method of fitting CTPAC, and thus the method of fitting CTPAC is completely suitable to obtain conventional triaxial failure envelopes in the Mohr stress space.(2)The squared correlation coefficients of linear equation are generally lower than those of nonlinear equations and would further reduce when (σt)t is included in regression. (σc)t are lower than all the corresponding predicted values and have an exponential relation with (σc)rpu when (σt)t is not included in regression. (σc)r and Abs. (σt)r are more closer to their test values, respectively, when (σt)t is included in regression.(3)In the Mohr stress space, logarithmic equation, hyperbolic equation, and exponential equation are recommended to fit conventional triaxial failure envelopes because some x-intercepts of the regression curve of power equation and parabolic equation are positive. Using (σt)t in regression can further improve the regression validities.(4)According to the assumption that the cohesion is constant, the negative friction bearing capacity exists when the ultimate stress circle is on the left side of the critical ultimate stress circle whose tangent point and strength envelope are on the longitudinal axis. Through the analysis of the evolution process of the nonlinear strength envelope, it is considered that when the normal stress is infinite, the total bearing capacity of a rock is about equal to the cohesive bearing capacity, while the friction no longer provides the bearing capacity. It is concluded that the hyperbolic equation and the exponential equation are more suitable to fit triaxial compression strength under higher maximum confining pressures for they have limit values.
Data Availability
The research data used to support the findings of this study are included within the article. Request for more details should be made to the corresponding author.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
The authors acknowledge the financial support of the National Natural Science Foundation of China (Grant no. 51904092), the Key Scientific Research Project of Higher Education Institutions in Henan Province (Grant no. 20B580002), and the Fundamental Research Funds for the Universities of Henan Province (Grant no. NSFRF180336).