Variation of the intrinsic rock properties on Hoek-Brown failure criterion parameters

The Hoek-Brown (H-B) criterion is one of the most commonly used rock failure criteria in recent years. This criterion includes a constant parameter called m i which is a fundamental parameter for estimating rock strength. Due to the im portance of the m i parameter in the H-B criterion, it is necessary to conduct comprehensive studies on various aspects of the effect of this parameter on the behavior of rocks. Therefore, in this study, using numerical simulation of the Triaxial Compressive Strength (TCS) tests in PFC-2D code, the effects of microscopic properties of different rocks on the H-B parameter m i have been studied. Based on the results of this study, it was found that the effects of micro-parameters on the H-B parameter m i can be different depending on the type of rock, however this parameter has an inverse relationship to the micro-parameters of bond tensile strength and bond fraction of the rocks. Also, the m i parameter increases with an increase in the micro-parameters of the friction coefficient, the friction angle, the particle contact modulus, and the contact stiffness ratio of rocks.


Introduction
Between all of the failure criteria presented, the Hoek-Brown (H-B) (1980) empirical criterion is one of the most well-known. This criterion has become an indispensable tool for rock engineers due to its simplicity, proper compatibility to laboratory data, as well as its applicability to rock masses, and it has been used successfully in various aspects of rock engineering in recent years (Merifield et  Depending on the application, different types of H-B criterion have been proposed. The mathematical equation for an intact rock is expressed as follows (Hoek & Brown, 1980a): (1) Where: σ 1 : the major principal stress, σ 3 : the minor principal stress or confining pressure, m i : H-B material constant, σ ci : the uniaxial compressive strength of the intact rock. According to the H-B criterion, one of the parameters that has a significant effect on the failure of rocks is m i , which is a dimensionless parameter. The strength parameter m i which is generally assumed to be a curve-fitting parameter to achieve the H-B failure envelope, can be determined by statistical evaluation of the results of experimental studies, linear or non-linear approaches (Hoek and Brown, 1980a;Hoek and Brown, 1980b;Hoek, 1983;Shah and Hoek, 1992;Colak and unlu, 2004), and its values are distributed from 7 to 35 depending on the rock material characteristics (Hoek, 2007). Due to its significance in H-B failure criterion, many researchers have dedicated their studies to parameter m i (Colak and Unlu, 2004

Material and methods
In the current study, the mechanical properties of three rock types namely andesite, limestone, and sandstone have been used to conduct investigations. Numerical simulations have been performed using Particle Flow Code 2D (PFC-2D), which is one of the most popular software based on the Discrete Element Method (DEM). Due to its high capability to simulate the mechanical behavior of different rocks, PFC-2D has been considered by many researchers and has been used in a wide range of numerical studies in recent years. (Potyondy and Cundall, 2004;Calvetti, 2008 In these studies, the numerical samples follow the ISRM recommendations, and the diameter and height of the samples were considered to be 54 mm and 115 mm, respectively Figure 1. Before starting the analysis, it is essential to define the base values of the micro-parameters. The base values of the micro-parameters from each rock are given in Table 1, which is determined with a calibrating operation on the experimental data. For the calibration process, the mechanical behavior of the synthetic samples under the compression test was reproduced and compared with the experimental tests. In this paper, the TCS test has been used for calibrating the micro-properties of synthetic samples. Also, to ensure the accuracy of the numerical results, the failure envelopes of one of the rock samples has compared with laboratory tests, which is shown in Figure 2. Moreover, in Table 2, the values of rock strength parameters in both numerical and laboratory modes are compared with each other. According to Figure 2 and Table 2, the values obtained from numerical models are very close to the laboratory values in all three rock samples. Therefore, it can be concluded that the created models are sufficiently accurate to perform more investigations.

Results and discussion
After validation and ensuring the accuracy of the answer provided by numerical simulations, to study the effect of micro-parameters on the m i parameter, seven fac-    tors including bond tensile strength, bond cohesion, the friction coefficient, the friction angle, bond fraction, the particle contact modulus (Ec) and the contact stiffness ratio (Kn/Ks) have been selected. Since in current research the effects of micro-parameters on three different rocks have been evaluated, the results have been presented in three sections, which are discussed as follows.

Andesite
The results of the studies about the behavior of m i parameter in andesite rock are shown in Figure 3. According to Figure 3, it can be seen that the value of m i decreases with an increase in the amount of bond tensile strength, which changes from 0 to 15 MPa. By increasing the amount of particles bond cohesion, which has varied from 10 to 40 MPa, the m i parameter decreased firstly (C=20 MPa), then increased slightly and the changes became small. With an increase in the amount of the friction coefficient, which changed from 0.2 to 0.8, m i also increased. The rate of these changes decreased between the values of 0.4 and 0.6 and then increased again as before. Changes in the values of the friction angle led to an increase in the m i parameter. The rate of these changes was smaller at lower angles. The m i parameter also decreased with an increase in the bond fraction which varied from 0.5 to 1. Moreover, as the stiffness ratio increased, the m i parameter increased linearly. Also, with an increase in the value of the modulus of elasticity, the m i parameter increased slightly.
The effect of micro-parameters on the failure envelope of andesite rock are shown in Figure 4. According to Figure 4, it can be seen that the greatest impact of the tensile strength was on the minor stress. This micro-parameter also had an effect on the major stress of the rock, but its amount is small. The micro-parameters of cohesion, the friction coefficient and the friction angle caused the strength of the rock samples to increase. These micro-parameters only affected the major stress of the rock. Bond fraction lead to an increase in the value of the major and minor stress. The stiffness ratio and particle contact modulus also affected both stresses, but the greatest effect had occurred on the major stress.

Sandstone
The results of studies on sandstone samples are shown in Figures 5 and 6. Based on the results of Figure 5, it   can be seen that increasing the amount of the micro-parameter of bond tensile strength leads to a decrease in the value of the m i parameter, so that its value has changed from 16 to 7, which reached about half of its value. By increasing the micro-parameter of bond cohesion, m i changed slightly at first (40 MPa < C < 60 MPa), particle contact modulus, the m i parameter first increased slightly (E c <14) and then it decreased again (E c >14). According to Figure 6, it can be seen that, same as andesite rock, the greatest effect of the micro-parameter of bond tensile strength was on the minor stress and its effect on the major stress was relatively smaller. The micro-parameters of bond cohesion, the friction coefficient and the friction angle only affected the major stress Rudarsko-geološko-naftni zbornik i autori (The Mining-Geology-Petroleum Engineering Bulletin and the authors) ©, 2021, pp. 73-84, DOI: 10.17794/rgn.2021.4.7 of the rock. The bond fraction and the particle contact modulus increased both the maximum and minimum stresses. Also, the stiffness ratio micro-parameter affected both major and minor stresses, but the greatest effect occurred on the minor one.

Limestone
The results of studies on limestone samples are shown in Figures 7 and 8. According to Figure 7, it can be seen that with an increase in bond tensile strength, the m i parameter decreased so that its value reached half of the original value (it changed from 32 to 15). With an increase in bond cohesion, the m i parameter first decreased (20 MPa < C < 40 MPa), then it increased (40 MPa < C < 65 MPa) and finally decreased again. Increasing the friction coefficient, which varied from 0.4 to 1, led to an increase in the m i parameter so that the value of m i changed from 12 to 18. The rate of changes in m i parameter were small at first (0.4 to 0.6), then these changes increased (0.6 to 0.8) and finally decreased again. As the friction angle increased, m i also increased. However, the rate of this changes was relatively low. By increasing the amount of bond fraction micro-parameter, which varied from 0.7 to 1, the m i parameter decreased. This reduction was linear and the m i changes ranged from 22 to 17. Increasing the stiffness ratio also led to an increase in the value of the m i parameter. The changes of the m i parameter were small at first (1 <K N /K S < 1.5), then increased (1.5 <K N /K S < 2) and finally became small again. Moreover, by increasing the amount of particle contact modulus, m i first increased slightly (E c <15), and then the m i value decreased again.
According to Figure 8, it can be seen that similar to the earlier samples, the greatest effect of the bond tensile strength micro-parameter was on the minor stress and its effect on the major stress was relatively smaller. Increasing the micro-parameter of bond cohesion and bond fraction increased the major and minor stress of the limestone. The micro-parameters of friction coefficient and friction angle affected the major stress of the rock. Also, the effect of these micro-parameters on the minor one were very minimal. Increasing the micro-parameter stiffness ratio increased the major stress. This parameter also led to a reduction in minor stress, which is lower in higher values. Also, increasing the micro-parameter of the particle contact modulus increased the major and minor stresses, and these changes were greater than in the major stresses.

Conclusion
In this paper, using numerical simulations of triaxial compressive strength tests in PFC-2D code, the effects of micro-parameters in rocks on the H-B parameter m i were researched. To perform the analyses, the mechanical behaviour of three different rock types: andesite, sandstone and limestone, have been simulated and the effects of micro-parameters of bond tensile strength, bond cohesion, the friction coefficient, the friction angle, bond fraction, the particle contact modulus (Ec) and the contact stiffness ratio (Kn/Ks) have been evaluated. According to the performed analyses, the results of this research can be summarized as follows: • The m i parameter of the H-B criterion has an inverse relationship to the micro-parameters of bond tensile strength and bond fraction of the rocks, so that by increasing these micro-parameters, the m i parameter decreases by about half of its original value. • The effects of the bond cohesion micro-parameter on m i is varied in different rocks. • The m i parameter increases with an increase in the micro-parameters of the friction coefficient, the friction angle, the particle contact modulus, and the contact stiffness ratio of rocks. However, these effects vary depending on the type of rock. • The greatest effect of the bond tensile strength micro-parameter was on the minor stress and its effect on the major stress is relatively smaller. • The greatest effects of micro-parameters of the friction coefficient and the friction angle was on the major stress and their effects on a minor one are very small. • Increasing the amount of the contact stiffness ratio micro-parameter reduces the minor stress in different rocks.