Characterization of Hoek–Brown constant mi of quasi-isotropic intact rock using rigidity index approach

An accurate determination of Hoek–Brown constant mi is of great significance in the estimation of the failure criteria of brittle rock materials. So far, different approaches such as rigidity index method (R-index), uniaxial compressive strength-based method, and tensile strength-based method, and the combination of these two methods (combination based method) have been proposed to calculate the value of mi. This paper aims to thoroughly review the previously existing methods to calculate the value of mi and make comparison between the obtain results to propose the new material constants that provide the best fit with the experimental data. In order to fulfill this goal, a large number of data for different quasi-isotropic intact rock types from the literature were collected and statistically analyzed. Additionally, based on rock types, new material constants are introduced for igneous, sedimentary, and metamorphic rocks. The obtained results proves that for different rock groups (igneous, sedimentary, and metamorphic rocks), R-index method provides the best fit with the experimental data among the others, and it is also independent of rock type. Interestingly enough, there is significant differences in the predicted mi values using different methods, which is more probably due to the quantity and quality of data used in the statistical analysis.

Hoek-Brown material constant m b Hoek material constant for rock mass GSI Geological Strength Index TS Tensile strength UCS Uniaxial compressive strength R Rigidity index b Intermediate fracture mechanics parameter l The coefficient of friction for pre-existing sliding crack surfaces r 1 ; r 3 Major and minor principal stress r c Uniaxial compressive strength (UCS) r ci Crack initiation stress r t Tensile strength (TS)

Introduction
The Hoek-Brown failure criterion is widely used in rock mechanics and rock engineering practice for determining the strength of brittle intact rock and rock masses. This nonlinear semiempirical failure criterion was introduced by Hoek and Brown [33], and the following form was suggested for intact rock (see also [17]) (Eq. 1): Although several practical, empirical, and probabilistic approaches have been presented in the literature to address uniaxial compressive strength (UCS) [12,34,36,73,80], the accurate determination of m i is still challenging task and is influenced by many factors such as mineral composition, grain size, and cementation of rock. Generally, m i presents curve-fitting parameter for Hoek-Brown failure envelope [37]. However, researches by Zuo et al. [84] showed that m i is not a curve-fitting parameter but has physical meanings and can be derived from micro-mechanical principles (Hoek and Martin [38]).
Hoek and Brown [33][34][35] suggested that these values should be determined by numerous triaxial tests, applying different confining pressures (r 3 ) between zero and 0.5 r c . These laboratory tests are time-consuming, expensive, and in many cases, there are not enough (or suitable) samples. Singh et al. [70], Peng et al. [55], and Shen and Karakus [65] demonstrated that the reliability of m i values measured from triaxial test analysis depends on the quality and quantity of test data used in the analysis. They concluded that the range of r 3 could have a significant influence on the calculation of m i . It is the reason why several methods were developed for determining the Hoek-Brown constant (m i ) [72].
There are several techniques available to calculate m ivalues in the absence of triaxial test. These approaches are entitled, Guidelines [32,33], R-index [10,33,52,59,60,62], UCS-based model [65,74], tensile strength (TS)-based approach [77], and crack initiation stress-based model [10]. These methods are based on the rock lithological classifications and rock properties that can be easily obtained at an early stage of a project, which can be used in preliminary designs of engineering projects when triaxial test data are not available.
Aladejare and Wang [1] published a paper and introduced a new method for the determination of m i value. They developed a Bayesian approach for probabilistic characterization of Hoek-Brown constant m i through Bayesian integration of information from Hoek's guideline chart, regression model, and site-specific UCS data. The UCS data used in this paper were obtained from testing granite samples collected from the Forsmark site, Sweden. In addition, several sets of simulated data are used to explore the evolution of m i as the number of site-specific data increases.
Wei et al. [78] experimentally investigated the effect of confining pressure r 3 and critical crack parameter on determination of mi value. They applied ultrasonic test and load test results of limestone and verified the exponential impact of r 3 on limestone, while the negative correlation was observed between critical crack parameter and m i .
Recently, Wen et al. [79] developed an empirical relation for estimation m i from r 3 and b, where r 3 is minor principal stress and b is an angle between the major principal stress and weak plane based on substantial uniaxial and triaxial compression test data. They considered the effect of anisotropy on the value of Hoek constant (m i ).
In the present study, however, we focused on the quasiisotropic intact rocks and utilized the existing methods in the literature for predicting m i values. Furthermore, the detailed comparison is drawn between the obtained values of each approach to calculate which method provides the best estimation for the investigated rocks based on the calculated error function. Accordingly, new material constants are proposed for different igneous, sedimentary, and metamorphic rocks.

Determination of Hoek-Brown constant m i
In the absence of triaxial tests, there are several different methods available to estimate the Hoek-Brown constant mi, which are referred to as the R-index method [33], UCSbased model [65,74], TS-based model [77], and combination method [3,66] below. By using these methods, the H-B constant m i is obtained for the collected data, the achieved results are compared, and then, the error function was calculated and accordingly considering the lowest error value, the best data fit was presented and finally new material constants introduced.

Guideline method
Hoek and Brown [33] and Hoek [32] provided values of m i constants for different types of rocks. The values of m i are between 7 and 35; however, several factors that influence these values, such as mineral composition, foliation, grain size (texture), and cementation are among others [10].
The updated values of constant m i for intact rock were collected by [10], using the published values of Hoek [32] ( Table 1). Possible data ranges are shown by a variation range value immediately following the suggested m i value. For example, for sandstones, the m i values can vary between 13 and 21, and for slates, between 3 and 11.

Calculation the rigidity of the rock (R-index)
This method was introduced by [33] and also developed by [48,[59][60][61]. Cai [10] published the paper and showed that Hoek-Brown's strength parameter (m i ) could be determined from the ratio of the uniaxial compressive strength (r c ) and the tensile strength (r t ), and the suggested relationship is: As can be seen from Fig. 1, when R [ 8, the error for approximating m i (Eq. 2) by R (Eq. 4) is less than 1.6%. It has been observed that when the R is higher than 8, the value of R is nearly equal to Hoek-Brown material constant (m i ). Note, that according to [38] m i is not a curve fitting parameter, but has a physical meaning and can be derived from micro-mechanics principles. The referred m i model is express in Eq. (3): where r c and r t are the uniaxial compressive strength (UCS) and tensile strength (TS) of intact rock, respectively. While l is the coefficient of friction for pre-existing sliding crack surfaces, b is an intermediate fracture mechanics parameter that can be obtained from experimental data. It means when R [ 8: In recent years, Hoek and Brown [34] analyzed several published data and proposed the following approximate relationship between the compressive to tensile strength ratio, r c jr t j , and the Hoek-Brown parameter m i (see Fig. 2): It means that the Hoek-Brown constant (m i ) can be calculated using the following relationship, using the uniaxial compressive strength (r c ) and the tensile strength (r t ) data of the intact rock.

UCS-based model
Firstly, Shen and Karakus [65] and later Vásárhelyi et al. [74] also emphasized the difficulties in determining the m i values of rocks. They suggested normalizing the Hoek- where a and b are constants that depend on rock types.

TS-based model
The tensile strength-based calculation method was suggested by Wang and Shen [77]. In this method, the m i value is determined from the tensile strength (r t ) of the intact rock.
where A and B are material constants that depend on rock types. In general, the constants A and B are curvefitting parameters.
However, according to the theory of Zuo et al. [84] the m i value can be estimated from the coefficient of friction for the pre-existing sliding crack surfaces, and fracture mechanics. From this point of view, the constants A and B for the TS-based model seem to have its physical meaning, which have possible relations with the micro-mechanics principles. However, investigating the potential relationships between these constants and other rock properties and explaining the possible physical meanings need to carry out further research to get appropriate amount of reliable data.

Combination method (UCS and TS)
Analyzing the published data of Sheorey [66], Arshadnejad and Nick [3] suggested a new equation to calculate the Hoek-Brown material constant (m i ) based on uniaxial compressive strength (r c ) and the tensile strength (r t ) parameters of the intact rock: where a, b, and c are material constants. The suggested constants are summarized in Table 2, according to Arshadnejad and Nick [3].

Crack initiation stress-based model
According to [10], the Hoek-Brown constant (m i ) depends on the confining pressure. For practical estimate of m i , it was found that for strong, brittle rocks, applicable to high confining zone the following equation could be used: For low confinement stress to tension zone, especially for the tension zone the suggested equation is: In these equations, r c and r ci are the uniaxial compressive strength and the crack initiation stress, respectively.

Bayesian approach
The proposed approach [1] derives the probability density function (PDF) of m i based on the integration of Hoek's guideline chart, regression model, and site-specific UCS data, under a Bayesian framework. A large number of equivalent samples of m i are generated from the PDF using Markov Chain Monte Carlo (MCMC) simulation.  3 Experimental data A wide range of data from literature by [72] were selected and analyzed. The published data of the investigated rocks are illustrated in Appendices 1, 2, and 3 for igneous, sedimentary, and metamorphic rocks, respectively. In these tables, calculated data are also given. Statistical analyses of the measured m i values for different rock types are summarized in Table 3. In this table, parameters such as number of data sets, maximum (Max.), minimum (min.), average (Ave.), and standard deviation (std.) of measured m i for three different groups are presented (see the appendix for more details). for andesite and rhyolite, respectively. For agglomerate, the value of m i is between 16 and 22, but the published value is 7.92, which does not fit the guideline range.

Analyzing the data
For sedimentary rocks, the minimum value of m i for shale is 3.76 which is close to the estimated range of m i which is between 4 and 8 (see Appendix 2). Also, the maximum published value of m i for sandstone is 35.11 which is much higher than the estimated range of m i value and based on guideline which varies between 13 and 21. Moreover, for dolomite, the value of m i based on guidelines is between 6 and 12, whereas the value of m i based on published data is between 7.8 and 17.5. For limestone, the value of m i based on guideline is between 7 and 15; however, based on published data, the value of m i is between 5.3 and 14.6.
For metamorphic rocks, it is interesting to know that both maximum and minimum published values are for slate with the amounts of 1.42 and 29.54, respectively (see Appendix 3). However, based on guideline the values of m i for slate changes between 3 and 11. It means that the range of published value for slate does not fit the estimated range given by the guidelines. For gneiss, the value of m i based on guidelines is between 23 and 33; however, based on published data, the value ranges between 5.3 and 27. For schist, the published value of m i is 20.42, but the values based on guideline vary between 9 and 15. For quartzite, the published value of m i is between 7.36 and 22.7; however, based on guideline it is between 17 and 23. Also, the values of m i were calculated according to [10].
The differences between the published values of m i and the guideline values are displayed in Table 4 and Fig. 3. As shown, the published values of m i do not fit well with the guideline. As it is clear from the graph that there is a good consistency between published experimental data for m i and R-index method, whereas other methods have significant errors in estimation of m i for all studied rock samples. The obtained results based on different proposed calculation method are presented in Appendices 4, 5, and 6 for igneous, sedimentary, and metamorphic rocks, respectively. It should be noted that the constant parameters were derived from the existing Eqs. 2, 6, 7, 8, and 9, respectively.

Evaluation of R-index model
According to our analyses for determination of m i in respect to Eq. 2 (R-index method) and Eq. 5, it was observed that m i value is independent of rock type. The correlations between m i and the ratio between uniaxial compressive strength (UCS) and tensile strength (TS), which is called R-index method, for different rocks (sandstone, shale, slate, and gneiss) are presented in Fig. 4. As shown for different rock types, correlation value R 2 ¼ 1 and it is not influenced by rock type.

Evaluation of UCS-based model
The relationship between m i and uniaxial compressive strength (UCS) for the investigated rock samples (UCSbased method) shows an opposite result, compared to R-index method. It is evident that based on this approach, no correlation was found for the studied rock samples (Fig. 5).

Evaluation of TS-based model
Based on our analyses for determination of m i value by applying TS-based method, the established correlations for the investigated rock types were weak except for slate where the value of R 2 ¼ 0:66 ( Fig. 6).
In order to examine the relationship between m i and rigidity of rock (R-index method) for different rock groups, analyses were also carried out based on three different rock types (igneous, sedimentary, and metamorphic) and the results are illustrated in Fig. 7. Additionally, the error function was calculated for each rock type. The calculated  [4,5,6,8,9,11,15,19,20,22,31,39,44,47,54,63,69] (see data in Appendix 3)   Table 5 (according to rock group) and Table 6 (according to different investigated rock samples), and the statistical results are illustrated in Table 7.
In the proposed method by Arshadnejad and Nick [3], (combination method), (i), (j), and (k) are presented as material constants. Specifically explaining, based on their analyses for different rock groups (igneous, sedimentary,  Table 2 for their obtained results.

Evaluation of combination method (UCS and TS methods)
We analyzed the data set based on Eq. 9 as proposed by [3]. The results are depicted in Fig. 8. The graphs clearly illustrate that the Hoek constant (m i ) is independent of rock type. Accordingly, it is worth to mention the point that the intact rocks are known to contain a wide-ranging uncertainty owing to inherent heterogeneities. For rock mass classification in respect to uncertainties, GSI (Geological Strength Index) provides good classification of rock mass category as it relies on structure as well as surface conditions of existing discontinuities which are highly associated with high uncertainties in rock mass. Ván and Vásárhelyi [72] investigated the sensitivity of Hoek material constant for rock mass (m b ) in relation to GSI. Based on their measurement, they calculated that 10% deviation in the GSI value; the relative sensitivity of m b is at least double the uncertainties of the GSI values and may be 7 times higher in case of large disturbance parameters and low and high GSI values.
In the present research, to ensure the quantity and quality of data, the statistical analysis with the help of SPSS software was performed on quasi-isotropic rocks. The results are presented in Fig. 9. As shown, normal distribution function and standard deviation were calculated for each rock group. According to Fig. 9, for igneous rock, the calculated standard deviation is 7.21 with the mean value of 15.36, for sedimentary rock is 6.32 with the mean value of 12, and for metamorphic rocks is 10.07 with the mean value of 13.37. Interestingly enough, metamorphic rocks exhibit higher standard deviation, which can be can associated with the foliated texture of the investigated metamorphic rocks.
Similarly, Aladejare and Wang [1] investigated the values of m i for 30,000 equivalent granite samples based on Bayesian approach and realized that the histogram peaks at Based on the empirical model by Wen et al. [79], most of the estimated values of m i are within the upper/lower limit of 90%, which accounts for 96.6% of all the data, and that all the estimated values are within the upper/lower limit of 80%. This result indicates that the developed relation estimates of m i agree well with the experimental observations, demonstrating that the developed relation has an excellent ability to estimate m i accurately. The comparison graph between our results and recently published paper by Wen et al. [79] is presented in Fig. 10.
Moreover, Wen et al. [80][81][82] thoroughly investigated the impact of minor principal stress (r 3 ) on rock ductilebrittle behavior. Based on their measurement, under a low minor principal stress, rocks experience dilatancy and brittleness that causes the exciting microcrack inside the rock to coalesce, which result in an increase in volume; In contrast, applying the high minor principal stress, the rock undergoes ductility which leads to microcrack opening.
In the recently published paper by He et al. [27], they suggested that the constant m i of the H-B criterion can be continuously estimated during drilling. Therefore, a method to estimate the constant m i from drilling data was proposed. Based on the proposed method for m i determination, for the granite, slate, the obtained values of m i in their work were lower than the suggested values for granite Table 5 Calculated values of m i by different methods (Eqs. 2,5,7,8,9,10) Rock type R-index Hoek and Brown  and sandstone from [33], and they were higher than the suggested value for slate and limestone.
To establish the correlations for specific rock types, Shen and Karakus [65], Wang and Shen [77] considered five of the most common rock types (coal, granite, limestone, marble, and sandstone) from the database in the Rocscience [61], in which there are at least 12 groups of data with 115 triaxial tests available. Results illustrate that four rock types have trends of decreasing m i with increasing r ci ; however, such a correlation for limestone was not observed. It is worth noting that this finding is also in agreement with our obtained results for limestone. This may be due to the fact that limestone has a wide array of test data with different compositions and cementations; for example, there are different guidelines-based m i values for three types of limestone, crystalline (m i = 12 ± 3), sparitic (m i = 10 ± 2), and micritic (m i = 9 ± 2), in the Hoek guidelines [32]. Therefore, the data sets for limestone are quite widely scattered compared with those for the other four rock types.
The observed value of m i for limestone is between 5.3 and 14.6; In contrast, according to guideline, the m i value of limestone is between 7 and 15, according to R-index method m i is between 5.32 and 14.67, according to UCSbased method m i is between 19.8 and 22.1, based on TSbased method m i is between 6.18 and 13.05, and based on combination method m i is between 4.9 and 13. Thus, R-index method gives the best estimate of m i value among the others. However, based on [65], UCS-based method provides a better prediction of m i value than the guidelinesbased and R-index methods and according to [77], TSbased model exhibits the best performance for granites, limestone, and marble.
Additionally, the observed value of m i for slate is between 1.42 and 30.97; however, based on guideline, the m i value of slate is between 3 and 11, based on R-index method m i is between 1.59 and 31.23, based on UCS-based method m i is between 7.2 and 22.3, based on TS-based method m i is between 3.82 and 48.42, and based on combination method m i is between 1.98 and 32.44. So, R-index method gives the best estimate of m i value among the others. The achieved value of m i is inconsistent with the obtained value of m i by [16], which they used the internal friction angel method for estimating the value of m i for dolomite, slate, dike, and granodiorite. The obtained values of m i were 6.28, 5.1, 12.8, and 13.7, respectively.
Moreover, the observed value of m i for shale is between 3.76 and 25.31, but, according to guideline, the m i value of shale is between 4 and 8, according to R-index method m i is between 3.7 and 24.9, according to UCS-based method m i is between 17.82 and 22.1, based on TS-based method m i is between 6.52 and 29.87, and based on combination method m i is between 3.65 and 24.3. Therefore, R-index Furthermore, the observed value of m i for sandstone is between 3.97 and 35.1; nevertheless, according to guideline, the m i value of sandstone is between 13 and 21, according to R-index method m i is between 3.9 and 35, according to UCS-based method m i is between 17.82 and 22.23, based on TS-based method m i is between 5.48 and 29.87, and based on combination method m i is between 3.83 and 38.5. As a result, R-index method gives the best estimate of m i value among the others. Nevertheless, based on [77], UCS-based model gives the best prediction for sandstone.
As well as that, the observed value of m i for quartzite is between 7.36 and 30.1, while, according to guideline, the m i value of quartzite is between 17 and 23, according to R-index method m i is between 7.36 and 30.14, according to UCS-based method m i is between 8.45 and 21.1, based on TS-based method m i is between 5.66 and 10.43, and based on combination method m i is between 6.55 and 31.10. Hence, R-index method gives the best estimate of m i value among the others.
In addition, the observed value of m i for gneiss is between 5.34 and 32.33, whereas, according to guideline, the m i value of gneiss is between 23 and 33, according to R-index method m i is between 5.3 and 32.3, according to UCS-based method m i is between 8.07 and 22.46, based on TS-based method m i is between 4.68 and 11.44, and based on combination method m i is between 4.92 and 34.28. Therefore, R-index method gives the best estimate of m i value among the others.

Conclusion
This paper comprehensively reviewed the proposed methods for determination of m i value using the triaxial data set published by [66]. New linear and nonlinear correlations were found. New material constants were suggested for igneous, sedimentary, and metamorphic rocks such as granite, quartzdiorite, sandstone, shale, limestone, gneiss, quartzite, and slate. Furthermore, the error function was calculated for each rock type. According to our analyses, R-index approach with new material constants provides the best fit for all the studied lithotypes including igneous, sedimentary, and metamorphic rocks. Moreover, it is worth mentioning that comparison graph between out achieved results for quasi-isotropic rocks and anisotropic rocks was developed. The data scattering of quasi-isotropic rocks differs from the anisotropic rocks in that quasi-isotropic rocks exhibit normal distribution in narrower range between 1.43 and 35.11, whereas anisotropic rocks display wider range of data distribution between 1 and 80.
The combination methods (UCS) and (TS) provide the best fit for investigated rock data, when the determination coefficient and error function are considered. For various rock types such as igneous, sedimentary, and metamorphic rocks, the different approaches resulted in non-uniform correlations. For instance, TS-based approach works well for the granite and gives estimation fit with R 2 ¼ 0:89. For quartzites, UCS-based approach displays the best approximation with R 2 ¼ 0:91, and for slate, TS-based model exhibits the best correlation with R 2 ¼ 0:67.
R-index method, the value of m i is not influenced by rock type. In other words, this method gives the best estimation for all the investigated rock types and is independent of rock type. However, in all the other approaches, the rock type plays a significant role in correlation values and results.  Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/.