A nonlinear strength criterion for rock based on the peak value of deviatoric stress under conventional triaxial compression

Abstract

In order to accurately evaluate the rock strength under different confining pressures, a new nonlinear strength criterion with three parameters is proposed based on the peak value of deviatoric stress. A comparison among the proposed criterion, modified Hoek-Brown (MH-B) criterion, modified Mohr-Coulomb (MH-C) criterion and exponential criterion regarded as an outstanding strength criterion in rock engineering is made by using triaxial data of 10 different kinds of rock. It is found that the average coefficient of variation (CV) of the parameters of the proposed criterion is smaller than the other three criteria. It shows that the parameters of the proposed criterion are less sensitive to the range of confining pressure, and the parameters of the proposed criterion, obtained from the first three triaxial tests, are directly used to predict the triaxial strength at higher confining pressure within a small error range. Comparing the predicted values of each strength criterion with rock triaxial testing values, it is found that the mean absolute deviation(MAD) and mean absolute error(MAE) of rock triaxial strength evaluated by the proposed criterion are smaller than the other three criteria. The results show that the proposed criterion has a higher accuracy than the other three criteria and can well predict the triaxial test strength for different types of rock with an universal applicability. In addition, the proposed criterion avoids the defect that MH-B criterion and MM-C criterion holding a constant after the confining pressure reaching uniaxial compressive strength. Finally, the physical meaning of the proposed criterion parameters is discussed. The fitting parameter ${\sigma _{\text{c}}}$ is completely consistent with the experimental value, and can be replaced by the actual uniaxial compressive strength. Therefore, it can be considered that the proposed criterion only contains two criterion parameters ${\sigma _\infty }$ and m. The extreme value of rock deviatoric stress depends on the parameter ${\sigma _\infty }$. The larger the parameter ${\sigma _\infty }$, the greater the extreme value of deviator stress. The increase magnitude of rock triaxial strength with confining pressure at initial stage depends on the parameters value of m. The larger the value of m, the smaller the increase magnitude of triaxial strength with increasing confining pressure at initial stage.

Introduction

In many rock engineerings, such as traffic civil engineering, water conservancy and hydropower projects, resource development, etc., rock strength is the theoretical basis for analyzing rock mass stability, optimizing structural design and evaluating bearing capacity. Rock strength criterion is used to define the strength of rock subjected to given stress field, it reflects the relationship between the stress state and strength parameters of rock under critical failure conditions. Therefore, rock strength criterion is of great significance in rock engineering design and construction^1,2.

A large number of research results show that the failure mechanism of rock is gradually changing from brittle at low confining pressure to ductile at higher confining pressure. The rate of increase in strength is high at low confining pressure. As confining pressure increases, the rate of increase in strength decreases^3,4. Therefore, the rock strength increases nonlinearly with the increase in confining pressure^5,6.

Among many rock strength criteria, Mohr-Coulomb (M-C) criterion and Hoek-Brown (H-B) criterion are the most widely used in geotechnical engineering due to their simple mathematical expressions and clear physical meaning of strength parameters. However, the M-C criterion approximates the Mohr stress circle envelope to a straight line, and expresses the rock strength as a linear function of confining pressure or normal stress, that is, the strength parameter is a fixed value and cannot change with the confining pressure. So the M-C criterion can only be applied to evaluate the strength of rock at low confining pressure. With the increase of confining pressure, the rock strength evaluated by the M-C criterion will deviate greatly from the rock test strength. Compared with M-C criterion, H-B criterion can reflect the nonlinear characteristics of rock strength with the increase of confining pressure. However the prediction results under high confining pressure are significantly larger than the test values, which has been confirmed by a large number of test data^7,8. Therefore, these two traditional strength criteria are difficult to describe the nonlinear characteristics of rock strength under high confining pressure.

In view of the nonlinear strength characteristics of rock under high stress condition, some scholars at home and abroad have proposed or improved several nonlinear rock strength criteria. According to the number of fitting parameters contained in the strength criterion, commonly, the nonlinear strength criteria can be divided into the following categories. The first group is the nonlinear strength criteria with one fitting parameter. In order to overcome the limitation that M-C criterion only can express the rock strength as a linear function of confining pressure or normal stress, Singh et al.⁹ first proposed a quadratic parabola strength criterion based on the critical confining pressure based on the critical state for rock¹⁰. After the analysis of an extensive database of triaxial test data for intact rock and by a trial and error process that explores analogies with the non-linear criterion proposed by Bieniawski¹¹, Shen et al.¹² proposed a nonlinear strength criterion with one parameter. Based on the parabolic Fairhurst¹³ criterion with two parameters, You¹⁴ derived the parabolic criterion with one parameter. However, the nonlinear strength criterion with only one parameter can not be applied to engineering practice because of its small application scope and low fitting accuracy. The second group is the nonlinear strength criteria with two fitting parameters. The parabolic Mohr criterion with two parameter was first proposed by C. Fairhurst¹³ to study the Brazilian splitting strength of disk specimens. Bieniawski¹¹ proposed an empirical strength criterion of power function with two parameters to estimate the strength of rock based on the fact that the strength envelope of rock usually presents a convex shape. However, the physical meaning of the above criteria parameters is not clear. The third group is the nonlinear strength criteria with three fitting parameters. Considering the influence of confining pressure on rock strength, You¹⁵ constructed a exponential strength criterion considering the influence of confining pressure on rock strength. In order to overcome limitation that M-C criterion and H-B criterion cannot describe the nonlinear characteristics of rock strength under high confining pressure. Based on the nonlinear M-C criterion proposed by Singh et al.¹⁶, Li et al.¹⁷ proposed the concept of critical confining pressure coefficient, and proposed nonlinear M-C criterion and H-B criterion with three parameters. However, it is found that it is difficult to accurately determine critical confining pressure coefficient in practical application.

With the continuous upgrading and improvement of rock triaxial test equipment, The confining pressure of triaxial test can be controlled in a relatively large range. Based on the in-depth understanding of rock strength properties, expression of nonlinear strength criterion may be constructed mathematically one after one. Generally speaking, the greater the number of strength parameters, the higher accuracy of strength criterion. However, with the increase of the number of strength parameters, the fitting result of parameters is not unique, which leads to ill condition^18,19.

Therefore, it is necessary to propose a nonlinear strength criterion with a wide range of applicability and high accuracy, and with fewer model parameters and clear physical meanings of the parameters. Based on the changing characteristic of rock strength increasing with confining pressure, a new three parameter nonlinear strength criterion is established based on the viewpoint that the deviatoric stress (${\sigma _1} - {\sigma _3}$) will gradually approach a certain extreme value with the increase of confining pressure. We will use publicly available triaxial test data to validate the proposed nonlinear strength criterion in this study and other typical strength criteria, demonstrating the effectiveness and accuracy of the strength criteria proposed in this study.

Expression of strength criterion

Critical state concept for rock

A large number of conventional triaxial compression tests results of rock show that in the $\tau - \sigma$ coordinate system, the shear strength envelope is nonlinear and concave towards the normal compressive stress axis, where $\tau$ is the shear strength and $\sigma$ is the normal stress on the shear plane, as shown in Fig. 1. The rate of increase in strength is higher at low confining pressure. With the increase of confining pressure, the rate of increase in strength decreases gradually. It has been found that the fiction angle φ of rock during compression is not constant but changes with the confining pressure^10,20. At low confining pressure, the microcracks, which exist in a rock, open up at the onset of the failure, due to which the volume of the rock increases at the time of failure. Rock exhibits dilatant and brittle behaviour.This results in a high friction angle at low confining pressure. Under high confining pressure, rock dilation is suppressed and the failure mechanism shifts from brittle to ductile. So the instantaneous internal friction angle is small. When the confining pressure is sufficiently high, rock become ductile. On further increase in confining pressure the rock enters the critical state.

It can be seen from Fig. 1 that when the envelope passes through the shear stress axis, its tangential gradient is steep, and gradually becomes a horizontal line at sufficiently high confining pressure. Barton¹⁰ termed this phenomenon as the critical state for rock, that is, Mohr envelope of peak shear strength reaches a point of zero gradient. The corresponding peak shear strength also represents the maximum shear strength of the rock. For each rock, there is a critical effective confining pressure above which the shear strength cannot be made to increase.

Fig. 1

Critical state of intact rock.

A new criterion for intact rock strength

Based on the influence characteristics of confining pressure and intermediate principal stress on rock strength, You¹⁸ constructed two exponential formulas to jointly characterize the true triaxial strength criterion for rock.

$${\sigma _1}{\text{=}}{\sigma _{\text{s}}}+H({\sigma _2} - {\sigma _3},{\sigma _3})$$

(1)

$${\sigma _{\text{s}}}={\sigma _3}+{\sigma _{\text{c}}}{\text{+}}({\sigma _\infty } - {\sigma _{\text{c}}})\left\{ {1 - \exp \left[ { - \frac{{({K_0} - 1){\sigma _3}}}{{{\sigma _\infty } - {\sigma _{\text{c}}}}}} \right]} \right\}$$

(2)

$$H({\sigma _2} - {\sigma _3},{\sigma _3})={\sigma _{\text{d}}}\frac{{\gamma ({\sigma _2} - {\sigma _3})}}{{{\sigma _{\text{s}}} - {\sigma _3}}}\exp \left[ {1 - \frac{{\gamma ({\sigma _2} - {\sigma _3})}}{{{\sigma _{\text{s}}} - {\sigma _3}}}} \right]$$

(3)

Where ${\sigma _{\text{s}}}$ is the strength of conventional triaxial compression under confining pressure of ${\sigma _3}$; ${\sigma _{\text{c}}}$, ${\sigma _\infty }$ and ${K_0}$ are parameters, which are material-dependent; $H({\sigma _2} - {\sigma _3},{\sigma _3})$ is a function with parameters $\gamma$ and ${\sigma _{\text{d}}}$ to describe the effect of the intermediate principal stress on strength. ${\sigma _{\text{d}}}$ is also material-dependent, but $\gamma$ may be a constant about 1.7. To better understand the true triaxial index exponential criterion, the function graph is plotted according to Eq. (3), as shown in Fig. 2.

Fig. 2

Effect of the intermediate principal stress on strength from the exponential criterion.

According to the research by You¹⁸, the function $H({\sigma _2} - {\sigma _3},{\sigma _3})$ has the following characteristics: (1) The curve of the function $H({\sigma _2} - {\sigma _3},{\sigma _3})$ is concave downward in the range of 0 < $\frac{{\gamma ({\sigma _2} - {\sigma _3})}}{{{\sigma _{\text{s}}} - {\sigma _3}}}$< 1. It can be observed that the curve has similar characteristics with the Mohr strength envelope of rock under conventional triaxial compression tests in the $\tau - \sigma$ stress space, as shown in Fig. 1; (2) At $\frac{{\gamma ({\sigma _2} - {\sigma _3})}}{{{\sigma _{\text{s}}} - {\sigma _3}}}=1$, the function reaches its maximum value, corresponding Mohr envelope of peak shear strength reaches a point of zero gradient. It is similar to the characteristics of the Mohr strength envelope when rock reach the critical state.

Therefore, based on the true triaxial exponential strength, according to the variation trend of the rock strength envelope shown in Fig. 1, a new nonlinear strength criterion for rock is established based on the viewpoint that the deviatoric stress (${\sigma _1} - {\sigma _3}$) will gradually approach a certain extreme value with the increase of confining pressure. The extreme value of deviatoric stress (${\sigma _1} - {\sigma _3}$) of rock under triaxial test is defined as ${\sigma _\infty }$, and the uniaxial compressive strength of rock is defined as ${\sigma _{\text{c}}}$. The expression of strength criterion is as follows:

When $0 \leqslant {\sigma _3} \leqslant {\sigma _\infty }$,

$${\sigma _1} - {\sigma _3}={\sigma _{\text{c}}}+({\sigma _\infty } - {\sigma _{\text{c}}}){\left( {\frac{{{\sigma _3}}}{{{\sigma _\infty }}}} \right)^m}\exp \left[ {1 - {{\left( {\frac{{{\sigma _3}}}{{{\sigma _\infty }}}} \right)}^m}} \right]$$

(4)

where ${\sigma _1}$ and ${\sigma _3}$ are the maximum principal stress and the minimum principal stress under the conventional triaxial condition respectively; m is the material-dependent parameter.

Combined with the research results^10,21 and Fig. 1, it is considered that the rock deviatoric stress will remain unchanged when it reaches the maximum value, that is:

When${\sigma _3}>{\sigma _\infty },$

$${\sigma _1} - {\sigma _3}{\text{=}}{\sigma _\infty }$$

(5)

From Eq. (4), it can be seen that (a) when ${\sigma _3}=0$, the rock strength is equal to the uniaxial compressive strength, that is, ${\sigma _1}={\sigma _{\text{c}}}$; (b) When ${\sigma _3}={\sigma _\infty }$, the rock reaches the critical state, the deviatoric stress (${\sigma _1} - {\sigma _3}$) is constant. The Mohr envelope of shear strength approaches the horizontal line, and its tangent gradient is approximately 0, that is, ${\sigma _1} - {\sigma _3}={\sigma _\infty }$, $\partial ({\sigma _1} - {\sigma _3})/\partial {\sigma _3}=0.$

Other failure criteria used for comparison

The modified M-C criterion

Based on the concept of rock critical state, Singh et al.¹⁶ proposed a nonlinear M-C strength criterion (SS criterion) by subtracting a quadratic term about confining pressure from the linear M-C criterion. The expression is as follows:

$$\left\{ \begin{gathered} ({\sigma _{\text{1}}} - {\sigma _3}{\text{)=}}{\sigma _{\text{c}}}+\frac{{{\text{2sin}}\varphi }}{{{\text{1}} - {\text{sin}}\varphi }}{\sigma _3} - \frac{{{\text{sin}}\varphi }}{{{\sigma _{\text{c}}}(1 - {\text{sin}}\varphi )}}{\left( {{\sigma _3}} \right)^2}{\text{ 0}} \leqslant {\sigma _3} \leqslant {\sigma _{\text{c}}} \hfill \\ ({\sigma _{\text{1}}} - {\sigma _3}{\text{)}}={\sigma _{\text{c}}}+\frac{{{\text{sin}}\varphi }}{{{\text{(1}} - {\text{sin}}\varphi )}}{\sigma _{\text{c}}}{\text{ }}{\sigma _3}>{\sigma _{\text{c}}} \hfill \\ \end{gathered} \right.{\text{ }}$$

(6)

Data fitting and model verification

Criteria for comparison of performance

There are many methods to determine the parameters of strength criterion and evaluate its prediction accuracy. According to the research by You²³, when using the least square method (that is, taking the least square sum of deviations as the fitting objective) to determine the criterion parameters, the individual data with large error will make the fitting curve deviate from the test data as a whole, which will cause the fitting results not accurate enough. Therefore, in this paper, least absolute deviation is used to determine the strength criterion parameters with the least sum of absolute value of fitting deviations as the fitting objective. The mathematical expression of objective function is as follows:

$$d{\text{=}}\sum\limits_{{i=1}}^{N} {\left| {f({\sigma _3}) - \sigma _{1}^{{\operatorname{test} }}} \right|}$$

(10)

where d is the sum of absolute value of fitting deviations; $f({\sigma _3})$ is the rock strength predicted by the strength criterion under the confining pressure ${\sigma _3}$; $\sigma _{1}^{{\operatorname{test} }}$is the test strength of the rock; N is the number of test groups.

In order to describe the discrepancies between the triaxial strength of rock evaluated by the criterion and the test strength, we use three different error measurements to assess the validity of predictions computed with different criteria: mean absolute deviation (MAD), relative error (RE) and mean absolute error (MAE).Their definitions are:

Mean absolute deviation (MAD)

$$MAD=\frac{{\sum\nolimits_{{i=1}}^{N} {\left| {\sigma _{{1,i}}^{{{\text{pre}}}} - \sigma _{{1,i}}^{{{\text{tes}}}}} \right|} }}{N}$$

(11)

Relative error (RE)

$$RE{\text{=}}\frac{{\left| {\sigma _{{1,i}}^{{{\text{pre}}}} - \sigma _{{1,i}}^{{{\text{tes}}}}} \right|}}{{\sigma _{{1,i}}^{{{\text{tes}}}}}} \times 100{\text{\% }}$$

(12)

Mean absolute error (MAE)

$$MAE{\text{=}}\frac{{\sum\nolimits_{{i=1}}^{N} {\frac{{\left| {\sigma _{{1,i}}^{{{\text{pre}}}} - \sigma _{{1,i}}^{{{\text{tes}}}}} \right|}}{{\sigma _{{1,i}}^{{{\text{tes}}}}}}} }}{N} \times 100{\text{\% =}}\frac{{\sum\nolimits_{{i=1}}^{N} {RE} }}{N}$$

(13)

where $\sigma _{{1,i}}^{{{\text{pre}}}}$ is the predicted value of rock strength criterion; $\sigma _{{1,i}}^{{{\text{tes}}}}$ is the test value of rock strength; N is the number of test groups.Based on the definitions above, it is clear that, the smaller the MAD and MAE, the more reliable the model. RE, that is needed to compute MAE, is the relative error between predicted and testing values for the i-th test.

Sensitivity analysis of proposed criterion parameters

Regression methods are used by rock engineers to determine criterion parameters, which makes the estimation of parameters and its related uncertainty depend on the quantity and quality of test data. Generally speaking, there is a certain degree of difference in the criterion parameters obtained by selecting different groups of triaxial test data.It reflects the dispersion degree of the criterion parameters. The larger the difference is, the higher the dispersion degree of the criterion parameters is. It indicates that the sensitivity of the criterion parameters to the confining pressure is greater. Therefore, when the strength criterion parameters are fitted by different groups, the deviation between the triaxial strength evaluated by the corresponding criterion and the test strength may be greater.

Based on the objective principle, the sensitivity of the proposed criterion parameters is verified by using the triaxial test data from ten different types of rock in the published literature. The ten different types of rock are numbered 1–10, corresponding to Solnhofen limestone²⁴, Yamaguchi marble²⁴, Bunt sandstone²⁵, Jinping sandstone²², Daye marble²⁶, Tyndall limestone²⁷, Georgia marble²⁸, Pottsville sandstone²⁸, Indiana limestone²⁹, Mizuho trachyte³⁰. In this study, the triaxial test data of No.1 Solnhofen limestone is taken as an example to determine the most suitable parameters for the four criteria. Triaxial compression test data of No.1 Indiana limestone are presented in Table 1.

Table 1 Triaxial test data of Solnhofen limestone.

Substituting the corresponding testing values under different confining pressures into the MM-C criterion, the MH-B criterion, the exponential criterion and the proposed criterion. the parameters of the strength criterion are obtained in the following ways:①by considering only the first three data points (including the data point of ${\sigma _3}=0$); ②by considering the first four data points, and so on i.e. by considering the triaxial test data at increasing confining pressures; ③using all nine data points are to determine the criterion parameters. The results of fitting parameters are shown in Table 2.

When it is necessary to compare the degree of dispersion of several groups of data, the influence of measurement scale and dimension of data should be eliminated. The coefficient of variation(CV), that is CV = Standard deviation/Average value, can do this. Therefore, CV is used to describe the variability of criterion parameters with confining stress, that is, the larger CV value is, the more sensitive the criterion parameters are to the confining pressure. Due to the uniaxial compressive strength ${\sigma _{\text{c}}}$ of intact rock is the most common parameter in rock strength criterion, and also a useful parameter for rock mass classification. Therefore, the column of uniaxial compressive strength ${\sigma _{\text{c}}}=2c\cos \varphi /(1 - \sin \varphi )$ is added to the MM-C criterion in Table 2.

Table 2 Best fitting parameters for different criteria.

It can be seen from Table 2 that the uniaxial compressive strength ${\sigma _{\text{c}}}$ predicted by MM-C criterion and MH-B criterion has a large variation, and the corresponding CV values are 0.05 and 0.05, respectively. With the increase of confining pressure, the uniaxial compressive strength values predicted by MM-C criterion and MH-B criterion gradually exceed the uniaxial compressive strength values. An over predication to the tune of 13% has been seen at the maximum confining pressure. The uniaxial compressive strength ${\sigma _{\text{c}}}$ predicted by the exponential criterion and the proposed criterion is basically consistent with the testing strength. The CV values of parameters c, $\varphi$ and n in MM-C criterion are 0.31, 0.37 and 0.74, respectively. The CV values of parameters m and n in MH-B criterion are 0.75 and 0.73, respectively. The CV values of parameters ${\sigma _\infty }$ and ${k_0}$ in exponential criterion are 0.06 and 0.43, respectively. The CV values of parameters ${\sigma _\infty }$ and m in the proposed criterion are 0.04 and 0.08, respectively. It can be known that for the same type of rock, the parameters ${\sigma _\infty }$ and ${k_0}$of the proposed criterion has relatively low sensitivity to the range of confining pressure used in the tests, especially the parameter ${\sigma _{\text{c}}}$ remains unchanged. It shows that the proposed criterion has high accuracy in evaluating the uniaxial compressive strength ${\sigma _{\text{c}}}$ and extreme value of deviatoric stress ${\sigma _\infty }$.

In order to fully compare the differences between the above rock strength criteria, the CV values of each criterion parameter corresponding to 10 different types of rock is obtained, as shown in Table 3.

Table 3 CV values of each strength criterion parameters.

It can be seen from Table 3 that for these 10 kinds of rock, the ${\sigma _{\text{c}}}$ values predicted by the proposed criterion is unchanged, and the corresponding CV values are all 0. It can be seen that the parameter ${\sigma _{\text{c}}}$ of the proposed criterion has the lowest sensitivity to the range of confining pressure, followed by MH-B criterion, exponential criterion and MM-C criterion. In other words, the fitting parameter ${\sigma _{\text{c}}}$ of the proposed criterion with three parameters in this study is completely consistent with the experimental values, and can be replaced by the actual uniaxial compressive strength. Therefore, it can be considered that the proposed criterion contains two strength parameters.

For these ten types of rock, The average CV values of parameters ${\sigma _\infty }$ and m in the proposed criterion are 0.12 and 0.07, which indicates that for different types of rock, the proposed criterion parameters have very low sensitivity to confining pressure in majority of cases. In other words, in order to estimate the strength of rock under high confining pressure, if the proposed criterion is adopted, the strength of rock under high confining pressure can be determined successfully by using the triaxial test results under low confining pressure. It can aviod conducting the relatively difficult and time-consuming triaxial tests with high confining pressure.

Predictive capabilities with triaxial data available

When applying strength criterion to specific rock engineering, the parameters of strength criterion are usually determined by triaxial tests of rock, which are usually carried out under low confining pressure. In view of this, in order to be consistent with the actual application situation, due to the limited space of this study, taking No. 1 Solnhofen limestone as an example, it is assumed that the test data with confining pressure of about 20 MPa can be obtained. The criterion parameters are obtained by using these test data only. These parameters are used to determine the triaxial strength for other confining pressures. The predicted results have been compared with experimental values, as shown in Fig. 3.

MM-C criterion

The parameters of the MM-C criterion are c = 60.83 MPa, $\varphi$= 44.9°, n = 0.09. Then, the following equation can be obtained:

$${\sigma _{\text{1}}}{\text{=5}}{\text{.8}}{\sigma _3}+293 - 0.09\sigma _{3}^{2}$$

(14)

For the confining pressure range of ${\sigma _3}$> 26.4 MPa, then the following equation can be obtained:

$${\sigma _{\text{1}}}{\text{=}}356.27{\text{+}}{\sigma _3}$$

(15)

MH-B criterion

The parameters of the MH-B criterion are m = 9.86, ${\sigma _{\text{c}}}$= 293 MPa, n = 0.09. Then, the following equation can be obtained:

$${\sigma _{\text{1}}}{\text{=}}{\sigma _3}+{(2889{\sigma _3}+85849)^{0.5}} - 0.068\sigma _{3}^{2}$$

(16)

For the confining pressure range of ${\sigma _3}$> 26.4 MPa, then the following equation can be obtained:

$${\sigma _{\text{1}}}{\text{=}}355.22{\text{+}}{\sigma _3}$$

(17)

Exponential criterion

The parameters of the exponential criterion are ${\sigma _\infty }$= 360.1MPa, ${\sigma _{\text{c}}}$= 293 MPa, ${k_0}$= 9.6. Then, the following equation can be obtained:

$${\sigma _1}{\text{=}}{\sigma _3}{\text{+}}360.1 - 67.1\exp ( - 0.13{\sigma _3})$$

(18)

Proposed criterion

The parameters of the proposed criterion are ${\sigma _\infty }$= 418.2 MPa, ${\sigma _{\text{c}}}$= 293 MPa, m = 0.5. Then, the following equation can be obtained:

$${\sigma _1}{\text{=}}{\sigma _3}{\text{+}}293{\text{+}}125.2{(\frac{{{\sigma _3}}}{{418.2}})^{0.5}}\exp \left\{ {1 - {{(\frac{{{\sigma _3}}}{{418.2}})}^{0.5}}} \right\}$$

(19)

It can be clearly observed from Fig. 3 that under low confining pressure, the triaxial strength predicted by the four criteria is basically consistent with the test strength. With the increase of confining pressure, the predicted values of MM-C criterion, MH-B criterion and exponential criterion are lower than the triaxial strength test values of rock, while the relative error in prediction by the proposed criterion in this study is significantly lower than the other three criteria. When the triaxial test data under low confining pressure are used to determine the criterion parameters, the reason for the deviation in prediction by MM-C criterion and MH-B criterion may be that the critical confining pressure coefficient n fitted by MM-C criterion and MH-B criterion is not accurate enough. For the exponential criterion, the reason may be that the extreme value of deviatoric stress ${\sigma _\infty }$ is larger than the actual value.

Fig. 3

Comparison of experimental and predicted strength (Solnhofen limestone).

Predictive capabilities with all the triaxial data

Based on the objective principle, this section continues to use triaxial test data of the above ten types of rock to check the applicability of the proposed criterion. The specific process is as follows: ①Substituting all the triaxial test data of rock into equations of the four strength criteria, the corresponding criterion parameters are fitted by using least absolute deviation. The specific mathematical expressions of each strength criterion are obtained; ②The obtained mathematical expressions are used to predicted the triaxial strength of the rock under different confining pressures, and the MAD and MAE are calculated according to Eq. (11) and Eq. (13), respectively, as shown in Table 4; Fig. 4.

Fig. 4

Comparison of four criteria. (a) MAD, (b) MAE.

Table 4 Fitting results and MAD and MAE using four nonlinear criterions for conventional triaxial strength.

It can be seen from Table 4 that the MAE and MAD in prediction by the proposed criterion is minimum for seven rock types (No. 2, 3, 4, 6, 7, 8 and 10 in Table 4) out of total ten considered in this study. Exponential criterion show least MAE and MAD for two rock types (No. 1and 9 in Table 4), and MH-B criterion was found to give minimum MAE and MAD for one rock type(No. 5 in Table 4). For these 10 kinds of rock, the R² of the proposed criterion in this study is more than 0.99. The average MAE and MAD in prediction by the proposed criterion are 1.37% and 3.06, which are lower than the corresponding values of the other three criteria. It also shows that the predicting accuracy of the proposed criterion is more higher than the other three criteria in majority of cases. It is concluded that the proposed criterion has good applicability and can evaluate the triaxial strength of different types of rock with high accuracy. Applicability of predictions computed with different criteria is not only depend on predicting accuracy. In order to compare the applicability of different criteria, the author will make a more in-depth comparison of strength criteria in the follow-up work.

Based on the concept of critical state, the above 10 types of rock are divided into two categories. The first group is the rock reaching the critical state. Taking No.1 Solnhofen limestone and No.9 Indiana limestone as examples, the two kinds of rock enter the critical state when the confining pressure is equal to 0.34${\sigma _{\text{c}}}$and 0.91${\sigma _{\text{c}}}$, respectively. Meantime, the corresponding Mohr envelope of peak shear strength is approximately a horizontal line. The predicted results of each strength criterion and experimental results are shown in Fig. 5.

It can be seen from Fig. 5 that the predicted uniaxial compressive strength of No. 1 Solnhofen limestone and No. 9 Indiana limestone by MM-C criterion and MH-B criterion is greater than the testig strength if the parameters of the strength criterion are obtained by using all triaxial test data (from low confining pressure to high confining pressure). The predicted values of MM-C criterion and MH-B criterion for No. 1 Solnhofen limestone are larger than the testing values at high confining pressure. It can be seen from Table 3 that the average CV values of the critical confining pressure coefficient n in MM-C criterion and MH-B criterion is larger, which indicates that the critical confining pressure coefficient n is more sensitive to the range of confining pressure. In other words, the critical confining pressure coefficient n fitted by MM-C criterion and MH-B criterion is not accurate enough, which leads to large deviation in prediction by MM-C criterion and MH-B criterion. The predicted values of the proposed criterion and exponential criterion are closer to the testing values than those of MM-C criterion and MH-B criterion, indicating that the proposed criterion and exponential criterion have a higher accuracy.

Fig. 5

Fitting results of criteria for the first kind of rock. (a) Solnhofen limestone, (b) Indiana limestone.

The second group refers to rock that did not reach the critical state under traditional triaxial test condition. Taking No.4 Jinping sandstone and No.6 Tyndall limestone as examples, the deviatoric stress increases gradually with the increase of confining pressure, and its growth rate has not reached 0. It can be seen from Fig. 6 that the predicted values of the four strength criteria is basically consistent with the test values at the high confining pressure. However, when the confining pressure is low, the proposed criterion in this paper is closer to the testing values than the other three strength criteria. It shows that the proposed criterion has a higher accuracy in the whole range of confining pressure.

Fig. 6

Fitting results of criteria for the second kind of rock. (a) Jinping sandstone, (b) Tyndall limestone.

To sum up, the new strength criterion in this study is established based on the viewpoint that the deviatoric stress will gradually approach a certain extreme value with the increase of confining pressure. It not only has a high accuracy for the strength of the above two kinds of rock, but also avoids the defect that it is difficult to accurately determine the critical confining pressure coefficient n in MM-C criterion and MH-B criterion based on the critical state.

Probability of predictions

The performance of the proposed criterion can easily be compared with the other criteria by checking the probability of predicting the triaxial strength within certain permissible error. For this purpose the parameters of the above four strength criteria are obtained by using all available triaxial test data(i.e., for the full range of ${\sigma _3}$ values) of above 10 kinds of rock. Then RE in prediction is computed for all data points. Now we use the cumulative distribution functions (CDF) of RE values to assess the quality of the fits provided by different criteria. the CDF indicates the ‘probability’ that the prediction error is less than such threshold (probabilities are obtained by dividing the number of cases where the RE is smaller than the threshold by the total number of cases considered). The probability so obtained is shown in Fig. 7. With this definition, curves with a ‘higher’ position in the plot indicate criteria that provide a better fit to the available data. The probability of predicting the strength successfully within a given permissible error is higher if the proposed criterion is used. For example, if the permissible relative error is 10%, then the probability of predicted triaxial strength within this error by the proposed criterion is 100% whereas if the MM-C criterion is used, the probability is about 92%. Further, if the proposed criterion is used, it is probable that the maximum error will be within about 10%, whereas the maximum error in case of exponential criterion, MH-B criterion and MM-C criterion may be much higher.

Fig. 7

Probability of predicted strength to be within permissible error using four criteria.

Effect of parameters in the proposed criterion

The proposed criterion (Eq. (4)) contains three parameters, i.e. ${\sigma _\infty }$, ${\sigma _{\text{c}}}$ and m. It can be seen from the above the fitting parameter ${\sigma _{\text{c}}}$ of the proposed criterion is completely consistent with the experimental values, and can be replaced by the actual uniaxial compressive strength. Therefore, this paper only analyzes the influence of parameters ${\sigma _\infty }$ and m ($0<m<1$) on the predicted results. No.3 Bunt sandstone is selected as examples. When the parameter ${\sigma _\infty }$ is fixed, it can be seen from Fig. 8(a) that under the same confining pressure, the larger the parameter m, the smaller the predicted strength of rock is. As m increases, the rate of increase in strength decreases. When the parameter m is fixed, it can be seen from Fig. 8(b) that under the same confining pressure, the larger the parameter ${\sigma _\infty }$, the larger the predicted values of rock strength is. The predicted values of deviatoric stress increases to the extreme value ${\sigma _\infty }$ with confining pressure.

Fig. 8

Effect of strength parameters for predicted results of the strength criterion in this paper: (a) effect of strength parameter m (${\sigma _\infty }$= 344.67 MPa); (b) effect of strength parameter ${\sigma _\infty }$ (m = 0.76 MPa).

Equivalent M-C criterion parameters

At present, most geotechnical engineering software are written based on the M-C strength criterion. The key to the application of the proposed criterion is to calculate the equivalent cohesion c and internal friction angle $\varphi$ of the M-C criterion through the proposed criterion parameters ${\sigma _\infty }$, ${\sigma _{\text{c}}}$and m. According to Balmer³¹, a failure envelope expressed in terms of ${\sigma _1}$ and ${\sigma _3}$ can be transformed into an equivalent convex Mohr envelope that is tangent to the Mohr circles at failure. Under the condition of triaxial compression, for a given pair of ${\sigma _1}$ and ${\sigma _3}$ at failure, the Mohr circle shown in Fig. 9 is defined as:

Fig. 9

Failure criterion in $({\sigma _1} - {\sigma _3})$ plane and its corresponding Mohr-envelope.

$${\left( {\frac{{{\sigma _1} - {\sigma _3}}}{2}} \right)^2}{\text{=}}{\left[ {\left( {\frac{{{\sigma _1}+{\sigma _3}}}{2}} \right) - {\sigma _\alpha }} \right]^2}+\tau _{\alpha }^{2}$$

(20)

where ${\sigma _\alpha }$ and ${\tau _\alpha }$ are the normal and shear stresses acting on the failure plane with angle $\alpha$.

The derivative of ${\sigma _1}$ in Eq. (20) with respect to ${\sigma _3}$ gives the normal stress on the failure plane as:

$${\sigma _\alpha }{\text{=}}{\sigma _3}{\text{+}}\frac{{{\sigma _1} - {\sigma _3}}}{{\frac{{\partial {\sigma _1}}}{{\partial {\sigma _3}}}+1}}$$

(21)

Substituting Eq. (21) into Eq. (20), an expression for the shear stress on the failure plane as:

$${\tau _\alpha }{\text{=}}\frac{{{\sigma _1} - {\sigma _3}}}{{\frac{{\partial {\sigma _1}}}{{\partial {\sigma _3}}}+1}}{\left( {\frac{{\partial {\sigma _1}}}{{\partial {\sigma _3}}}} \right)^{1/2}}$$

(22)

The slope angle ${\varphi _i}$ of the tangent line drawn at point A is the instantaneous friction angle at the normal stress ${\sigma _\alpha }$. Taking Eqs. (20)-(21)into account, the following equation can be obtained:

$$\tan {\varphi _i}{\text{=}}\frac{{\frac{{\partial {\sigma _1}}}{{\partial {\sigma _3}}} - 1}}{{2{{\left( {\frac{{\partial {\sigma _1}}}{{\partial {\sigma _3}}}} \right)}^{1/2}}}}$$

(23)

The instantaneous internal friction angle ${\varphi _i}$ and the instantaneous cohesion ${c_i}$ at point A can be computed from Eqs. (21)-(23):

$${\varphi _i}{\text{=arc}}\tan \frac{{\frac{{\partial {\sigma _1}}}{{\partial {\sigma _3}}} - 1}}{{2{{\left( {\frac{{\partial {\sigma _1}}}{{\partial {\sigma _3}}}} \right)}^{1/2}}}}$$

(24)

$${c_i}=\frac{{{\sigma _1} - {\sigma _3}\frac{{\partial {\sigma _1}}}{{\partial {\sigma _3}}}}}{{2{{\left( {\frac{{\partial {\sigma _1}}}{{\partial {\sigma _3}}}} \right)}^{1/2}}}}$$

(25)

When ${\sigma _3} \leqslant {\sigma _\infty }$,

$$\frac{{\partial {\sigma _1}}}{{\partial {\sigma _3}}}=1+({\sigma _\infty } - {\sigma _{\text{c}}})\frac{m}{{{\sigma _\infty }}}{(\frac{{{\sigma _3}}}{{{\sigma _\infty }}})^{m - 1}}\exp \left[ {1 - {{(\frac{{{\sigma _3}}}{{{\sigma _\infty }}})}^m}} \right]\left[ {1 - {{(\frac{{{\sigma _3}}}{{{\sigma _\infty }}})}^m}} \right]$$

(26)

When ${\sigma _3}>{\sigma _\infty }$, $\frac{{\partial {\sigma _1}}}{{\partial {\sigma _3}}}=1$.

Taking No.9 Indiana limestone as examples, the instantaneous cohesion ${c_i}$ and internal friction angle ${\varphi _i}$ associated with the proposed criterion can be obtained by using Eqs. (24)-(25), as shown in Fig. 10. It can be seen that the ${\varphi _i}$ is not a constant value, but decreases with the increase of confining pressure. Its decay rate gradually decreases to 0. The ${c_i}$ increases with the increase of confining pressure, and finally increases to 0.5${\sigma _\infty }$. The rate of increase gradually decreases with the increase of confining pressure. When ${\sigma _3}>{\sigma _\infty }$, the ${c_i}$ and ${\varphi _i}$ of rock no longer change with confining pressure.

Fig. 10

The instantaneous cohesion and internal friction angel for Indiana limestone changing with confining pressure.

Conclusion

Based on the viewpoint that the deviatoric stress will gradually approach a certain extreme value with the increase of confining pressure, a nonlinear three-parameter strength criterion with three-parameter has been proposed. Additionally, using triaxial test data from different types of rock, a precise comparison was conducted between the proposed criterion and the other three typical strength criteria, leading to the following conclusions:

• (1) The parameters of the proposed nonlinear strength criterion exhibit the lowest sensitivity to the range of confining pressure based on the triaxial test results of ten rock types. Compared to the MM-C criterion, MH-B criterion, and exponential criterion, the proposed nonlinear criterion maintains consistency and accuracy in evaluating strength under low confining pressure. This avoids the need to perform difficult and expensive high confining pressure triaxial tests to obtain the strength of rock under high confining pressure. The average MAE and MAD in prediction by the proposed nonlinear criterion are lower than the corresponding values of the other three criteria. It indicates that the proposed nonlinear criterion has a high degree of consistency between the evaluated strength and experimental strength under high confining pressure and has broad applicability.

• (2) The proposed criterion provides the best predictive capabilities with an cumulative distribution functions(CDF) curve clearly above the others. Therefore, compared to the other three strength criteria, the maximum relative error of the proposed nonlinear criterion in this study is within 10%, while the maximum relative error of the other three criteria may be much higher. This indicates that the prediction accuracy of this article is relatively high.

• (3) The effects of nonlinear strength criterion parameters on strength curves are investigated. The fitting parameter (uniaxial compressive strength ${\sigma _{\text{c}}}$) exhibits perfect agreement with experimental values and can be replaced by the actual uniaxial compressive strength. Therefore, it can be considered that proposed strength criterion contains two criterion parameters. The parameter ${\sigma _\infty }$ determines the extreme value of the triaxial deviatoric stress of rock, and the larger the parameter ${\sigma _\infty }$, the greater the extreme value of the deviatoric stress; the parameter m determines the increase magnitude of rock triaxial strength with increasing confining pressure at initial stage. The larger the value of m, the smaller increase magnitude of rock triaxial strength with increasing confining pressure at the initial stage.