Experimental verification of parameter m in Hoek-Brown failure criterion considering the effects of natural fractures

Xu Wei, Jinping Zuo,b,*, Yue Shi, Hiyn Liu, Yunqin Jing, Chng Liu

a School of Mechanics and Civil Engineering, China University of Mining and Technology, Beijing,100083, China

b State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Beijing,100083, China

Keywords:Hoek-Brown parameters Critical crack parameter Naturally fractured limestone Ultrasonic test

ABSTRACT Hoek-Brown failure criterion is one of the widely used rock strength criteria in rock mechanics and mining engineering.Based on the theoretical expression of Hoek-Brown parameter m of an intact rock,the parameter m has been modified by crack parameters for fractured rocks.In this paper,the theoretical value range and theoretical expression form of the parameter m in Hoek-Brown failure criterion were discussed. A critical crack parameter B was defined to describe the influence of the critical crack when the stress was at the peak, while a parameter b was introduced to represent the distribution of the average initial fractures.The parameter m of a fractured rock contained the influences of critical crack(B),confining pressure (σ3) and initial fractures (b). Then the triaxial test on naturally fractured limestones was conducted to verify the modification of the parameter m. From the ultrasonic test and loading test results of limestones,the parameter m can be obtained,which indicated that the confining pressure at a high level reduced the differences of m among all the specimens. The confining pressure σ3 had an exponential impact on m, while the critical crack parameter B had a negative correlation with m. Then the expression of m for a naturally fractured limestone was also proposed.

1. Introduction

Natural fractures have significant influence on the mechanical properties and engineering stability of rock mass (Elsworth and Day, 1999; Rutqvist et al., 2013). On one hand, the essence of deformation and failure in rock mass is the process of fracture initiation, propagation, interaction and penetration under engineering disturbance. On the other hand, the discontinuity and anisotropy caused by the complex fractures in the rock mass will change the stress conditions of rock mass,which makes it difficult to predict the actual mechanical characteristics of the rock mass,affecting the failure form and instability of the engineering rock mass (Katsuki et al., 2019). Therefore, it is of great significance to investigate the effects of fractures on rock mass.

Through analyzing a large amount of triaxial test and field test data of rock mass, the Hoek-Brown (HB) failure criterion was developed (Hoek and Brown, 1980a). The HB failure criterion is applicable to many types of rocks, and associated parameters can be obtained from conventional laboratory tests, mineral composition and discontinuity surface (Barton, 2013; Shen et al., 2018). In the classical HB failure criterion, the values and significance of parameters m and s are particularly concerned.Over the years,the two parameter values have still been discussed and improved.

The parameters m and s can reflect the features of rock materials and the structural impact caused by fractures (Hoek and Brown,1980a,b). Hoek and Brown (1980b) and Hoek (1994) empirically summarized various types of rocks, and later several scholars applied the rock mass rating (RMR) and geological strength index(GSI) to formulate the parameters in detail (Marinos and Hoek,2000; Hoek and Carranza-Torres, 2002; Cai et al., 2004;Eberhardt, 2012; Marinos, 2017; Bewick et al., 2019). Sonmez and Ulusay (1999) introduced initial damage to the parameters m and s for easier quantization of their values on site.Cai(2010)proposed a method to use initiation stress σciduring compression to represent the tensile strength σt, and then the uniaxial compressive strength σcand initiation stress σciwere applied to the estimate of HB strength parameter miof the brittle rocks. Shi et al. (2016)proposed an anisotropic index (αβ) to develop a modified HB failure criterion for describing the triaxial strength behavior of the rocks.Aladejare and Wang(2019)employed the Bayesian approach to probabilistically characterize the HB strength parameter mibased on Hoek’s guideline chart, regression model and uniaxial compression test.Zuo et al.(2008,2016)proved the validity of the classical HB failure criterion by fracture mechanics,and pointed out that the parameter m gained from the intact rock materials was associated with crack angle, strength and permeability of rock materials.

In the above studies, many scholars discussed the parameter m in HB failure criterion by field tests and theoretical analyses, but few researches have been reported on fractured rock mass (Hoek and Martin, 2014). Many scholars have carried out a series of laboratory compression tests on fractured rock mass and given the explanation from the perspectives of joint factor (Singh et al.,2002), joint dip angle (Nasseri et al.,1997; Medhurst and Brown,1998; Singh et al., 2015), nonlinear friction (Barton, 2013), and dominant distribution direction (Zhong et al., 2014). However,these experimental results and conclusions are somewhat contradictory, which are not easy to be illustrated with a universal rock strength theory. In this paper, according to the theory of the HB parameter m presented by Zuo et al. (2008, 2015, 2016, 2020), the relationships of parameter m of brittle rocks with initial filling cracks at different overburden depths were studied. Taking some naturally fractured limestones for examples,the specific expression of the parameter m of samples was determined using the triaxial test results, which was used to verify the previous assumption.

2. Discussion on Hoek-Brown parameter m of rock

2.1. Theoretical expression of Hoek-Brown parameter m for intact rock

Through analysis of a large number of rock triaxial test and field test data,Hoek and Brown(1980a)proposed the classic HB strength criterion as follows:

where σ1and σ3represent the maximum and minimum principal stresses, respectively; σcis the compressive strength of rock; constant s describes the rock integrity ranging from 0 to 1; and constant m indicates the hardness of the rock ranging from 0.0000001 to 25 in the classic HB strength criterion(Hoek and Brown,1980a).As important parameters to describe the characteristics of rock,the values of s and m were obtained empirically.The accuracy of s and m would not only cause the errors in the application of HB strength criterion, but also affect the judgments on engineering safety.

Zuo et al.(2008)used the Griffith hypothesis,which assumed a distribution of microcracks in rocks, to theoretically derive a strength formula consistent with HB strength criterion,which was highly appraised by Hoek and Martin (2014) and Hoek and Brown(2019). Based on the relevant criteria of composite cracks in linear elastic fracture mechanics (Zuo et al., 2008, 2015), the relationship between σ1and σ3can be written as

For an intact rock(s=1),the theoretical expression of m in Eq.(1) is

where f is the friction coefficient;and Ω is a fracture factor derived from different criteria of fracture mechanics,including the criterion of σθmax(the maximum circumferential tensile stress, which is applied to judging when cracking starts), Gθmax (the maximum strain energy release rate,which is focus on the maximum energy release caused by the compound cracking),and Smin(the minimum strain energy density under plane stress, which can describe the direction of crack initiation). The expression of Ω is presented as

where μ is the Poisson’s ratio.

Brook (1993) suggested that the strength ratio (σc／|σt|) is an important factor which is approximated equal to 10(for mudstones and limestones), 15 (for sandstones) and 20 (for granites). Rock types may influence σc／|σt|, but it is not the dominant factor.Sheorey (1997) indicated that the relationship between σcand σtgenerally satisfies σc／|σt| = 10 when tensile strength test data are not available. Cai (2010) also introduced that σc／|σt| values are about 8 (for the Griffith maximum tensile stress criterion under two directions of stress) and 12 (for the Griffith maximum tensile stress criterion under three directions of stress). Moreover, he found that σc／|σt| is always gradually close to 10 when the crack initiation stress σciis approached to the uniaxial compressive strength σc,irrespective of rock types.Based on above,we assumed σc／|σt|=10.According to Eqs.(3)and(4),the range of f is from 0.2 to 0.8, while the range of Ω is from 0.8 to 1.1. The range of m is shown in Fig.1.

In Fig.1,it can be seen that the range of m is mainly influenced by the range of f when σc／|σt| = 10. When f/Ω is constant, the ranges of σcand |σt| should be revisited. According to the above description, σc／|σt| is a ratio of uniaxial compressive strength to uniaxial tensile strength,which is determined by the rock type.The f/Ω is considered to be in the range of 0.2-0.8, σcranges from 10 MPa to 300 MPa,while|σt|ranges from 0.1 to 20 MPa.The range of m is shown in Fig. 2.

Fig.1. Relationship between m, f and Ω at σc／|σt| = 10.

Fig. 2. Relationship between m, σc and |σt| at different f/Ω values.

When s = 1, m is nearly up to 32 for some extremely hard rock masses(Hoek and Brown,2019).However,the parameter m is 0-25 in the classical HB criterion, which is more suitable for most rock masses (Hoek and Brown,1980a,b). Therefore, we will discuss the general condition of rock masses with m value not greater than 25.From Fig.2,the general range of m with different values of f/Ω can be easily seen as several surfaces. When f/Ω is a constant, both σcand |σt| can affect the parameter m.

According to laboratory experimental results of basalt samples from Mentougou in Beijing, China, the m values of rock specimens collected at different depths are obtained in light of above theories(Fig. 3).

The tensile and compressive strengths of rock increase continuously with increasing overburden depth. It is found that the m value calculated by the σθmaxcriterion is remarkably higher than those calculated by the other two criteria and the Smincriterion gains the minimum value. All the m values range from 7 to 10, in accordance with the range given by HB criterion.The m value is not fixed in horizontal: with the increase of depth, m declines in a global sense except that calculated by the Smincriterion and m value drops violently after the depth reaches 800 m.It is illustrated that as the hardness of rock decreases,the brittleness of rock changes to ductility with increasing depth.

In Eq.(3),the parameter m can represent the hardness of a rock directly since it can be considered to be intact(s=1).However,for the fractured rock masses (0 ＜s ＜1), originally filled cracks may exist in its interior meso-structures, suggesting that the influence of overburden depth cannot be ignored. In addition, large errors will occur when employing Eq.(3)to indicate the features of rocks.Therefore, it is necessary to modify the parameter m.

2.2. Modification of Hoek-Brown parameter m for fractured rock

On the macroscopic scale, the fractures are not uniformly distributed in the rock. In order to study the rock damage, a schematic diagram of the whole failure process of rock specimen under uniaxial compression is shown in Fig. 4.

In Fig. 4, φndenotes the angle of a crack and the surface of specimen, and σ1nis the axial stress imposed on the specimen.During the process of loading, each σ1nwill produce a new crack with a crack angle φn, and a penetrating crack (with critical crack angle of φc)finally appears when the stress σ1creaches its peak.In other words, the whole process of loading is associated with the changes of crack angle and axial stress.

Zuo et al. (2015) proposed a critical crack angle to describe the strength characteristics of the intact rock specimen (s = 1), which satisfies a relationship as follows:

where φ defines the angle of crack distribution zone on a tested specimen(Zuo et al.,2015,2016),a is a fracture parameter of a rock specimen, and b is a material coefficient related to rock lithology.Eq.(5)indicates that the rock damage is caused by accumulation of cracks, i.e. as the axial load increases, the cracks gradually accumulate until the failure occurs globally. Therefore, the material coefficient B of this rock is equal to the sum of the material coefficient corresponding to each crack under different stress states:

Fig. 3. Relationships between depth, uniaxial compressive strength, tensile strength and parameter m calculated by different criteria.

Fig.4. Schematic diagram of the whole failure process of a rock specimen:(a)Specimen is initially loaded; (b)Cracks initiate and propagate during the loading process;and(c)A penetrating crack is produced when specimen failure occurs and a layout of specimen surface is also plotted.

Assuming that φnand σ1nare two independent parameters,hence we just need to discuss their respective value ranges.When 0°＜ φn＜ 90°, the cosine function is positive. But when 90°＜φn＜180°,the cosine function is negative.Moreover,σ1nand σtare in opposite directions. Then Eq. (7) can be obtained as follows:

Fig. 5. The specimens of naturally fractured limestone.

Thus, we have

Dimensional analysis is applied to Eqs. (3), (5) and (8), and a critical crack parameter B0can be defined to describe the effect of the crack. Generally, the rock performed “softer” in deformation when the cracks are generated and propagated, thus B0would inversely correlate with HB parameter m. It can be shown as

For naturally fractured rock, the critical crack parameter B is defined to be concerned with the initial defects. According to Eq.(9),since the initial natural fractures may affect the strength of the rock,a linear assumption is made accordingly.The rock at the same overburden depth satisfies following relation:?

where σ1maxis the peak stress obtained from the triaxial compression test in laboratory; material constant a changes with confining pressure under different conditions; and coefficient b indicates the influence of all the initial natural fractures on the rock,which can be written as

where φ0represents the average angle of natural fractures, which can be obtained by statistical analysis of the initial fractures on all untested specimens.

For the fractured rock (0 ＜s ＜1), the parameter m can be expressed as

where a0is a coefficient related to the overburden depth, which decreases with increasing overburden depth, demonstrating the drop in hardness with the increasing depth.

Compared with Eq.(3),the effects of critical crack,initial defects and overburden depth are considered in Eq. (12). In order to validate Eqs. (10)-(12), a group of naturally fractured limestone specimens are selected for the experiments.

3. Calculated value of parameter m of naturally fractured limestone from triaxial test results

3.1. Sample preparation

The limestone used in this experiment was sampled from Lvliang,Shanxi Province,China.The rock sample was taken in a coal mine floor passed through by a fault, which is mainly gray, brittle,and calcareous. There are irregular and oblique natural fractures distributed in the specimens. According to the geological investigation, the limestones are mainly composed of calcite (92%-97%),dolomite (2%-6%), and pyrite (0%-1.2%). The size of the tested specimens is 50 mm in diameter by 100 mm in height,as shown in Fig. 5.

All the specimens were measured by ultrasonic test,and several intact specimens were selected for preliminary testing and calculation, as listed in Table 1.

X-ray computed tomography(CT)scan test and circumferential P-wave velocity anisotropy test have been carried out to understand the naturally structural characteristics of these limestone specimens. The anisotropy test results and model reconstruction obtained from X-ray scanning results of three typical limestone specimens are shown in Fig. 6.

From above test results, we can know that the complex structural characteristics of limestone specimens can be reflected by the difference of P-wave velocity, which can provide information for later calculation.

3.2. Test results of limestone

Some limestone specimens were adopted for ultrasonic and triaxial tests,and the test results are listed in Table 2.Owing to the uniqueness of each specimen, the parameters s and m cannot be obtained directly by fitting under confining pressure. In order to determine m value, the P-wave velocity measured by the ultrasonic test was used to characterize the parameter s of the limestone, so that m can be calculated by the HB formula.

Table 1Information of limestone specimens.

3.3. Parameter s of limestone

For the fractured rock, the specimen was assumed to be complete,thus the fractures determined the integrity of the specimen.In the HB failure criterion,the integrity of the rock was described by parameter s, which ranged from 0 to 1. The parameter s can be written by Eq. (13) under uniaxial compression:

On the other hand, some researchers used the methodology of ultrasonic detection to study the effect of initial fracture (Guo et al., 2011; Chawre, 2018; Marinos and Carter, 2018; Rose et al.,2018). There is a relationship among the initial damage factor D,vPand VP:

where vPis the minimum initial P-wave velocity of each specimen,and VPis the P-wave velocity of relevant rock material(see Table 1).

A certain connection between parameters s and D is assumed as the initial damage affects the integrity of the rock (Sonmez and Ulusay, 1999; Suorineni et al., 2009; Guo et al., 2011). Hoek and Carranza-Torres (2002) and Hoek and Brown (2019) proposed that the initial damage factor D has an exponential relationship with parameter s.To understand the relationship between s and D,several groups of limestone were chosen for ultrasonic P-wave test and uniaxial compression test. Then the experimental results are fitted in Fig. 7.

In Fig. 7, the values of the initial damage factor D for most specimens range between 0.3 and 0.8,suggesting that the integrity of rocks declines rapidly with increasing D. It indicates that the natural fissures have a significant impact on the rock integrity.The fitting relationship is expressed as

From Eq. (14), we can see that the integrity of each limestone specimen can be calculated by P-wave velocity before loading,which can roughly describe the reduction of the compressive strength by internal fractures. As for Eq. (15), although the applicability of the fitting relationship for these limestone specimens will be reduced in the two cases where D approaches 0 or 1, the fitting results are consistent with those of most test specimens.Moreover, it is also difficult to find completely broken rock or completely intact rock in practice. Thus, the exponential relationship of s and D is valid for fractured rock in natural states such as those tested limestones.

The parameter m under confining pressure can be obtained to analyze the mechanical properties of the rock according to Eq.(12).

3.4. Parameter m of limestone

In the classical HB failure criterion,both the parameters s and m affect the rock strength.Combining the ultrasonic test results with the relationship of s and D as discussed in Section 3.3, the parameter m can be calculated under triaxial compression,as depicted in Fig. 8.

Fig.6. Typical description of limestone specimens with(a)bedding,(b)bedding and fractures,and(c)naturally infilled fractures.Specimen photo,three-dimensional reconstructed model, and anisotropic results of P-wave velocity (m/s) are shown in sequence from left to right. Numbers around the circles are in degree.

The peak differential stresses of referred group range from 100 MPa to 260 MPa, while the parameter m varies from 0.81 to 21.94, showing a remarkable difference. Without considering the influence of accidental errors on the test, when the differential stress of the limestone is higher,the parameter m will be assigned with a higher value and the rock strength will increase under the same confining pressure. With the increase in confining pressure,the parameter m generally declines in the same stress level,and the difference between the maximum and minimum values of m also decreases in the whole stress range.This could be explained by the fact that increase in confining pressure can reduce the parameter m and to make the specimen “soften”. Natural fractures in the specimens may play a dominant role in the stress state under low confining pressure, which would show a significant difference.Meanwhile, the rising confining pressure may transform the specimens from hard to soft under high confining pressure,making the difference between the maximum and minimum ranges of m decrease.

Table 2Test results of limestone specimens.

4. Experimental verification of parameter m of naturally fractured limestone

In order to simplify the calculation, Eq. (10) is employed to discuss and compare the calculated and fitting values.At the same level of confining pressure,the parameter B is concerned with the material constant and the coefficient of initial fractures. According to Section 2.1,|σt|is usually about 10%of the compressive strength of an intact rock.In this context,the value of|σt|is set as 16.79 MPa.The coefficient of initial fractures b is associated with the average initial state of the rock specimen. The average angle of initial fractures of all naturally fractured limestones is calculated,and the parameter b0is obtained (Table 3).

The parameter B is obtained from the statistics of the critical crack angle and the peak stress when the specimen breaks. The broken specimen is depicted in Fig. 9b, from which φcis obtained.Besides, the peak stress of σ1max(Fig. 9a) is determined from the test results.

For the same group of rock specimens,the parameters a and b in Eq. (10) would remain unchanged under various confining pressures. With the parameter m calculated from the test results, the relationships between B and m under different confining pressures are fitted by Eq. (10), as shown in Fig.10.

Fig. 8. Relationship between differential strength and parameter m.

Table 3Initial parameters of naturally fractured limestones.

Fig.9. Calculation method of the critical crack angle φc(specimen No.20): (a) Critical crack produced under triaxial condition; and (b) Photograph of the specimen laid out to measure φc.

Except several test results,most data points agree well with the fitting relations. The parameters a and b in the fitting curve are tabulated in Table 4.

Taking into account the confining pressure,the relationships of fitting parameters a and b with confining pressure are plotted in Fig.11.It can be seen from Table 4 and Fig.11 that the parameter a decreases with the rise of confining pressure, which satisfies following exponential relationship:

From Section 2.2, the parameter a can be assumed to be

where f/Ω is equal to 0.83 by analysis of test results.The parameter a is considered as an intrinsic value(0.8)when σ3=0 on the basis of Eq. (16), which is close to the calculated value of the test results,indicating that the assumption of parameter a is reasonable. It demonstrates a gradual decrease in hardness of specimen under confining pressure.

The parameter b fluctuates around 0.04. The average value of parameter b fitted in Table 4 is 0.0407, which is close to the calculated value of b0in Table 3, illustrating that the assumption of parameter b is also reasonable.Consequently,the coefficient of initial fractures of limestone in this study is approximately 0.04.Therefore, the relationship between the parameters m and B of these fractured limestone specimens is defined as

Fig.10. Fitting relationships between parameters B and m under different confining pressures of (a) 10 MPa, (b) 20 MPa, (c) 30 MPa, and (d) 40 MPa.

Table 4Parameters of the fitting curve.

Fig.11. Relationships of fitting parameters with confining pressure.

As discussed in Section 3.4,the parameter m is a factor reflecting the effects of the confining pressure and the fracture in the specimen.In Eq.(18),the numerator reflects the influence of confining pressure. The property of the specimen is changed from hard to soft. Moreover, the denominator shows the differences in specimens:when the parameter B appears to be smaller under the same confining pressure, the strength of specimen would improve obviously, i.e. as the crack angle and strength increase, the parameter m would become larger.

According to the basic derivation in the literature (Zuo et al.,2008, 2015), the triaxial and initially fractured conditions are considered in the expression of m. Integrating with the laboratory test results,Eq.(12)is verified and the expression of m in this type of the limestone is obtained, indicating the effectiveness of the modification of parameter m.

5. Conclusions

(1) The parameter m in HB failure criterion was theoretically derived by Zuo et al. (2008, 2015), and is herein discussed and calculated by test results. The results verify that the parameter m decreases and the rock becomes softer with increase in overburden depth. The theoretical expression illustrates the properties of the intact rock material.

(2) The parameter m is modified by the critical crack, initial fractures and confining pressure. Based on the linear assumption about the compressive process of brittle rocks,a parameter B is defined to represent the relationship between the critical crack and strength of brittle rocks.The expression of m for brittle rock with fractures is derived.The parameter m in the expression is determined by the confining pressure and initial fractures of the rock material.

(3) The modification is verified by experimental results of naturally fractured limestones. The HB parameter m is calculated by the integrity prediction of rock with different fractures. The natural fractures play a dominant role in the specimen under low confining pressure,and the differences of m among all the specimens decrease due to the high confining pressures. According to the experimental results,the fitted and calculated values are compared and analyzed,and then the expression of parameter m for the naturally fractured limestones is obtained (see Eq. (12)).

Declaration of competing interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

The authors would like to acknowledge the financial support from Beijing Outstanding Young Scientist Program, China (Grant No. BJJWZYJH01201911413037), the National Natural Science Foundation of China(Grant No.41877257)and Shaanxi Coal Group Key Project, China (Grant No.2018SMHKJ-A-J-03).