# Investigating the Influences of Indentation Hardness and Brittleness of Rock-Like Material on Its Mechanical Crushing Behaviors

Authors: Jiuqun Zou; Jihuan Han; Weihao Yang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Indentation hardness and brittleness are the important factors to be considered in the study of rock-like materials’ mechanical crushing behaviors. The brittleness of rock-like materials is defined as the ratio of uniaxial compressive strength to tensile strength in this paper. In order to investigate the influences of hardness and brittleness on rock-like materials’ crushing behaviors, quartz sand and high strengthα-hemi-hydrated gypsum were used to prepare rock-like materials with different hardness and brittleness through different mass ratios. The artificial rock-like materials can eliminate the effects of natural rock’s weak structure plane on experimental results. The indentation test, Uniaxial compressive test, and Brazilian tensile test were conducted for characterizing the indentation hardness and brittleness of this artificial rock-like materials. The experimental results showed that brittleness increased with the increase of indentation hardness with high correlation coefficient. The confining stress presented a positive impact on the indentation hardness of the rock-like materials. Based on those mechanical properties, the numerical rock models were calculated to study rock crushing behaviors using discrete element method (DEM). A series of rock crushing tests were conducted to investigate the influences of hardness and brittleness of rock-like materials on rock crushing behaviors using a conical pick cutter. The numerical results showed that the normalized specific energy was negatively correlated with indentation hardness index (IHI). The normalized specific energy decreased with the increase of brittleness index (BI) with a high correlation coefficient. This study is beneficial in utilizing IHI and BI to evaluate the mechanical properties, failure patterns, and mechanical crushing efficiency of rock-like materials.


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## Body

## 1. Introduction

Understanding of rock hardness and brittleness has a crucial importance for practical underground construction engineering. The correlation between hardness and brittleness of rock materials and their mechanical properties and failure patterns has been the focus of research hotspot in related fields. Szwedzicki [1] employed the indentation test for measuring rock hardness and stated that this test could be used for characterization of rocks’ mechanical properties. There exists a relationship between the value of the indentation hardness and the uniaxial compressive strength (UCS). Yagiz [2] stated that the indentation hardness could be directly used for measuring rock toughness and brittleness, and the measured index had a close correlation with UCS and density of rocks. Shalabi et al. [3] suggested that there is a good relationship between the engineering properties (such as strength, modulus of elasticity, and Poisson’s ratio) of the intact rock and its hardness (Schmidt, shore scleroscope, abrasion, and total hardness). Xuefeng et al. [4] investigated the influences of rock brittleness on rock crushing behaviors and suggested that rock brittleness had a dominant effect on rock failure patterns. Fan et al. [5] investigated the mechanical characterization of shale through instrumented indentation test and stated that the indentation hardness could be used to obtain the elastic modulus, hardness, and brittleness. Therefore, hardness and brittleness can be used as the important indexes to characterize the mechanical properties and failure patterns of rock materials.In addition, many numerical studies had been conducted on the failure of brittle rock materials utilizing discrete element method. Li et al. [6] studied the brittle rock indentation process and rock-tool interactions by particle flow code (PFC). The results revealed that the tensile cracks are resulted from the chipping process and the shear cracks are induced by the crushing force. Li et al. [7] investigated the crack initiation and propagation mechanism in brittle rocks using grain-based finite-discrete element method. The compression testing under different end friction, slenderness, loading rate, and confining stress are systematically performed to investigate the external influences on rock fracturing. It can be seen that the fragmentation patterns of brittle rocks are related to many mechanical properties of rocks and external factors. The DEM has advantages in the numerical investigation of brittle rocks’ mechanics fragmentation.In practical underground constructions, mechanical rock crushing is a continuous, efficient and safe technique exhibiting great superiority in geotechnical engineering [8]. Many researchers [8–10] have investigated rock crushing behaviors through different indentation tests. Specific energy (SE) is always used to evaluate the rock crushing efficiency, which is defined as the amount of work required to break per unit volume of rocks. Yilmaz et al. [11] stated that the hybrid dynamic hardness had the potential to be used as an index rock material property that can be employed in preliminary estimations of specific energy (SE) and cutting force for a wide range of rocks under different operational conditions. Ji et al. [12] investigated the effects of ductility-brittleness change on rock fragmentation process and its failure mode. As the brittleness of the model increased, the normalized specific energy decreased and the efficiency of the rock crushing increased. Consequently, this paper investigates the correlation between mechanical properties and failure patterns of rock materials and their brittleness as well as indentation hardness based on rock crushing behaviors.Although using only one mechanical parameter always cannot accurately predict the rock crushing behaviors, a large number of studies have shown that there exists a close correlation between those rock mechanical parameters (such as strength, modulus of elasticity, toughness, Poisson’s ratio, and density) and hardness as well as brittleness. Balci et al. [13] tested 23 different rock specimens and investigated the predictability of SE from physical and mechanical properties. The results showed good correlations between SE and rock properties, such as UCS, Brazilian tensile strength (BTS), static and dynamic elastic moduli, and the Schmidt hammer value. Singh [14] observed that the in situ SE strongly depended upon the brittleness, UCS, Protodyakonov strength index, coefficient of internal friction, and the initial critical penetration stress. Therefore, it is worth investigating the influence of rock indentation hardness and brittleness on rock crushing behaviors. Because rock failure process is not visible, the mechanism of influence of hardness and brittleness on rock crushing behaviors remains unclear. Very few studies have investigated the hardness and brittleness of rock materials on their crushing behaviors utilizing pick cutters. Moreover, the IHI and BI indexes can be used to quantitatively characterize rocks’ mechanical crushing behaviors.Indentation test is the most common method to obtain IHI of rock materials. In generally, the IHI is calculated by equation (1) [1, 2]. Here, Fmax is the maximum thrusting force of indenter at the specific penetration, P is the penetration depth. Rock brittleness is defined as the ability of rocks to deform continuously without permanent deformations along with the application of stress surpassing the stress limit corresponding to rock microcracking [4, 15]. Liu [16] studied the relations between rock brittleness and rock fragmentation efficiency by disc cutter and summarized the existing 35 different brittleness indexes, which can be classified into seven categories with respect to strength, strain, strain energy, hardness, Mohr envelope, etc. Although the rock brittleness has not yet been standardized [16], most previous studies stated that BI could be calculated by the equations in Table 1 and stated that it exhibited a good correlation with rock crushing performance [16, 17].(1)IHI=FmaxP.Table 1
Brittleness indexes based on UCS and BTS [16].
AuthorsEquationsAltindagBI1=UCS×BTS/2Hucka and DasBI2=UCS/BTSHucka and DasBI3=UCS−BTS/UCS+BTSAltindagBI4=UCS×BTS/2As shown in Figure1, natural rocks contain many joints and fissures [18], which are the important factors affecting rock mechanical properties. Gypsum has proven to be a good rock-like material for rock mechanics experiments and rock fragmentation behaviors [19, 20], while the artificial rock-like materials may be able to eliminate the testing errors caused by the inhomogeneity and discontinues of natural rocks. Therefore, quartz sand and high-strength α-hemi-hydrated gypsum were used to prepared rock-like materials with different hardness and brittleness through different mass ratios. These rock-like materials can eliminate test errors caused by the inhomogeneity and discontinues of natural rocks. A series of indentation tests were conducted using a new indentation apparatus under confining stress conditions. In addition, the rock crushing models were calculated using a discrete element method code, i.e., Particle Flow Code in two dimensions (PFC2D). PFC2D is widely used in the simulation of rock crushing behaviors [4, 6, 21]. In this paper, the crushing models are calculated based on the UCS and BTS of the rock-like specimens. The cutting force and SE during fragmentation and failure patterns were investigated. The contribution of this study is that for the first time, the influence mechanism of rock-like materials’ hardness and brittleness on their crushing behaviors and failure patterns is investigated by artificial rock-like materials and DEM. The artificial rock-like materials can eliminate the effects of natural rock’s weak structure plane on study results. This investigation is of great significance for understanding the correlation between hardness and brittleness of rock materials and their fragmentation as well as failure patterns in rock crushing behaviors of underground engineering.Figure 1
The fissures in natural rocks (CT images).


## 2. Indentation Test with Different Confining Stress Levels

### 2.1. Testing Apparatus and Specimen Preparation

In this paper, the indentation hardness testing is performed on a newly designed apparatus, which enables the indentation test of cylindrical specimen can be conducted under confining stress conditions. A schematic diagram of the newly designed apparatus is shown in Figure2. The movement of indenter is controlled by a stepping motor. The penetration displacement is monitored by a linear variable differential transformer (LVDT), and the loading force is monitored by a 20 kN load sensor. A volume-pressure-controller (VPC) was used to load confining pressure. The conical indenter has a 90° cone and 0.25 mm radius spherical tip. The bearing platform is a hollow cylinder, and the indenter can pass through the bearing platform. A free rock surface is prepared for indentation-testing on the top of rock specimen.Figure 2
Schematic diagram of the newly designed triaxial testing system.The detail of the mass ratios of rock-like materials is shown in Table2. The particle size of quartz sand is 0.18 mm (80 mesh sieve). As shown in Figure 3, the preparation process of specimens is as follows:(1)
According to the designed mass ratio, the ingredients were weighed, mixed, and stirred at a high speed for 3 minutes.(2)
The mixture was loaded into moulds and placed on the vibrostand to shake for 5 minutes.(3)
Specimens were removed from moulds after 2 hours and maintained at room temperature for 28 days. All the end faces of specimens were corrected according to ISRM recommendations. The ratio of retarder to gypsum is equal to 0.1%; the ratio of defoamer to water is equal to 0.5%.Table 2
The mass ratio of ingredients of materials.
No.GypsumQuartz sandWaterA1014B101.54C1024D1044Figure 3
Preparation of the rock-like specimens.As shown in Figure3, the dimensions of specimens for uniaxial compressive testing were designed as 50 mm and 100 mm in diameter and height. Moreover, the dimensions of specimens for Brazilian splitting testing were designed as 50 mm and 25 mm in diameter and height. In the indentation tests, the size of the specimens affects the indentation test results and, as such, guidelines as observed from the following studies may be used for eliminating the size effects. According to the cavity expansion model (CEM) theory [9, 22, 23], the ratio of the depth of the plastic zone induced by an axisymmetric indenter to the diameter of the rock specimen should be less than 0.167. Based on CEM theory, the ratio of the plastic zone depth-to-penetration depth could be calculated with the equations (2)–(7). In the equations, r∗ is the critical radius of the plastic zone, n is a number characterizing the indenter geometry (for spherical and conical indenters, n = 2.0), Kp is the passive coefficient, Kd is the dilatancy coefficient (here, dilatancy angle ψ is equal to friction angle φ, ψ=φ = 20°), E1 is the Young’s modulus of rocks (E1 = 3.63 GPa), E2 is Young’s modulus of indenter (E2 = 450 GPa). v1 is Poisson’s ratio of rocks (v1 = 0.16), v2 is Poisson’s ratio of indenter (v2 = 0.27), β is the angle between the indenter and the free surface of the rock (for this indenter, β = 45°), σc is the UCS of this rock-like material (σc = 18.53 MPa), and α is the tip contact radius (α = 0.25 mm). By substituting the parameters to equations (2)–(7), the plastic zone radius-to-sample diameter ratio is equal to 0.045, which is less than 0.167. Therefore, the dimension of rock sample with 36 mm in diameter is reasonable.(2)1+μξn+Kd/Kd−μξKp−1/Kp=γ,(3)γ=23−2nKp+nE1tanβn+1π2−nσc1+v1,(4)Kd=1+sinψ1−sinψ;Kp=1+sinφ1−sinφ,(5)μ=nλKpnKp+Kd+1−nKpKd,(6)λ=Kp−1Kd−1+1−2v1Kp+1Kd+1−n−1KpKd3−n1+v1n−1Kp,(7)ξ=r∗α.


### 2.2. Uniaxial Compressive Strength and Brazilian Tensile Strength Tests

In order to investigate the effects of hardness and brittleness on rocks fragmentation mechanism, the rock-like specimens with four different mass ratios were prepared in this study. A series of uniaxial compressive tests and Brazilian splitting tests were conducted to calculate the indexes of BI. The stress-strain curves of uniaxial compressive tests are shown in Figure4, the displacement-stress curves of Brazilian tensile tests are shown in Figure 5. It can be seen that the failure patterns all exhibited obvious plunge after peak stress, which suggested that these rock-like materials could characterize the brittle breaking characteristics of rocks. As shown in Table 3, the BI2 values range from 4.08 to 6.3, and the BI3 values range from 0.62 to 0.73. Figure 6 shows that the specimens in the uniaxial compression test mainly exhibited splitting failure induced by the propagation of vertical cracks. It also can be seen in the Brazilian tensile strength tests, as shown in Figure 7. Consequently, the rock numerical models used for investigating rock crushing behaviors are calculated based on the UCS, BTS, Young’s modulus, and the failure patterns.Figure 4
Stress-strain curves of rock-like materials of uniaxial compressive tests.Figure 5
Displacement-stress curves of rock-like materials of Brazilian tensile tests.Table 3
The experimental results of rock-like materials.
No.UCS (MPa)BTS (MPa)BI1 (MPa2)BI2BI3BI4 (MPa)IHI (kN·mm−1) unconfinedIHI (kN·mm−1) 5 MPa confining stressA16.984.0434.304.080.625.860.87691.1545B20.293.2232.676.300.735.721.15451.4264C16.703.0925.805.400.695.080.93181.7456D20.163.2632.866.180.725.731.11191.7558Figure 6
Failure patterns of specimens in uniaxial compressive tests.Figure 7
Failure patterns of specimens in Brazilian tensile tests.


### 2.3. Indentation Tests under the Unconfined Condition and Confining Stress Condition

A series of indentation tests were conducted under unconfined stress and 5 MPa confining stress conditions. The experimental results are shown in Figures8 and 9. The velocity of the pick cutter is set to 1.0 mm/min according to Wang’s studies [8]. The confining stress restrained the propagation of vertical cracks. Since the slope of the curve is unstable at the initial stage of the indentation test, the slope of the fitted line near the maximum load is used as the indentation hardness, as shown in Figure 8. Moreover, the slope of the fitted line at the penetration displacement of 6 mm is used as the IHI under 5 MPa confining stress conditions. The IHIs under 5 MPa confining stress conditions are always less than that under unconfined stress conditions. According to the results in Table 3 and failure patterns in Figure 10, it can be seen that propagating crack length has negative correlation with IHI. As shown in Figures 10 and 11, there existed a compaction stage at the beginning of indentation test, and then the expansion of the crushed rocks induced the initiation and propagation of tension cracks. It can be seen in Figure 12 that the index BI2 [16] increased with the increase of IHI with a high correlation coefficient of 0.999. As shown in Figure 13, BI3 exhibited the same change law with IHI while BI1 and BI4 did not show a prominent correlation with IHI. The Figure 14 shows that the IHI has evident positive correlation with UCS. Therefore, the index BI2 is more suitable for the brittleness study in this paper.Figure 8
Displacement-force curves of indentation tests under unconfined stress conditions.Figure 9
Displacement-force curves of indentation tests under 5 MPa confining stress conditions.Figure 10
Failure patterns of specimens in indentation tests under unconfined stress conditions.Figure 11
Failure patterns of specimens in indentation tests under 5 MPa confining stress conditions.Figure 12
Relationship between IHI and BI2 under unconfined stress conditions.Figure 13
Relationship between IHI and BI3 under unconfined stress conditions.Figure 14
The correlation between IHI and UCS.


## 2.1. Testing Apparatus and Specimen Preparation

In this paper, the indentation hardness testing is performed on a newly designed apparatus, which enables the indentation test of cylindrical specimen can be conducted under confining stress conditions. A schematic diagram of the newly designed apparatus is shown in Figure2. The movement of indenter is controlled by a stepping motor. The penetration displacement is monitored by a linear variable differential transformer (LVDT), and the loading force is monitored by a 20 kN load sensor. A volume-pressure-controller (VPC) was used to load confining pressure. The conical indenter has a 90° cone and 0.25 mm radius spherical tip. The bearing platform is a hollow cylinder, and the indenter can pass through the bearing platform. A free rock surface is prepared for indentation-testing on the top of rock specimen.Figure 2
Schematic diagram of the newly designed triaxial testing system.The detail of the mass ratios of rock-like materials is shown in Table2. The particle size of quartz sand is 0.18 mm (80 mesh sieve). As shown in Figure 3, the preparation process of specimens is as follows:(1)
According to the designed mass ratio, the ingredients were weighed, mixed, and stirred at a high speed for 3 minutes.(2)
The mixture was loaded into moulds and placed on the vibrostand to shake for 5 minutes.(3)
Specimens were removed from moulds after 2 hours and maintained at room temperature for 28 days. All the end faces of specimens were corrected according to ISRM recommendations. The ratio of retarder to gypsum is equal to 0.1%; the ratio of defoamer to water is equal to 0.5%.Table 2
The mass ratio of ingredients of materials.
No.GypsumQuartz sandWaterA1014B101.54C1024D1044Figure 3
Preparation of the rock-like specimens.As shown in Figure3, the dimensions of specimens for uniaxial compressive testing were designed as 50 mm and 100 mm in diameter and height. Moreover, the dimensions of specimens for Brazilian splitting testing were designed as 50 mm and 25 mm in diameter and height. In the indentation tests, the size of the specimens affects the indentation test results and, as such, guidelines as observed from the following studies may be used for eliminating the size effects. According to the cavity expansion model (CEM) theory [9, 22, 23], the ratio of the depth of the plastic zone induced by an axisymmetric indenter to the diameter of the rock specimen should be less than 0.167. Based on CEM theory, the ratio of the plastic zone depth-to-penetration depth could be calculated with the equations (2)–(7). In the equations, r∗ is the critical radius of the plastic zone, n is a number characterizing the indenter geometry (for spherical and conical indenters, n = 2.0), Kp is the passive coefficient, Kd is the dilatancy coefficient (here, dilatancy angle ψ is equal to friction angle φ, ψ=φ = 20°), E1 is the Young’s modulus of rocks (E1 = 3.63 GPa), E2 is Young’s modulus of indenter (E2 = 450 GPa). v1 is Poisson’s ratio of rocks (v1 = 0.16), v2 is Poisson’s ratio of indenter (v2 = 0.27), β is the angle between the indenter and the free surface of the rock (for this indenter, β = 45°), σc is the UCS of this rock-like material (σc = 18.53 MPa), and α is the tip contact radius (α = 0.25 mm). By substituting the parameters to equations (2)–(7), the plastic zone radius-to-sample diameter ratio is equal to 0.045, which is less than 0.167. Therefore, the dimension of rock sample with 36 mm in diameter is reasonable.(2)1+μξn+Kd/Kd−μξKp−1/Kp=γ,(3)γ=23−2nKp+nE1tanβn+1π2−nσc1+v1,(4)Kd=1+sinψ1−sinψ;Kp=1+sinφ1−sinφ,(5)μ=nλKpnKp+Kd+1−nKpKd,(6)λ=Kp−1Kd−1+1−2v1Kp+1Kd+1−n−1KpKd3−n1+v1n−1Kp,(7)ξ=r∗α.


## 2.2. Uniaxial Compressive Strength and Brazilian Tensile Strength Tests

In order to investigate the effects of hardness and brittleness on rocks fragmentation mechanism, the rock-like specimens with four different mass ratios were prepared in this study. A series of uniaxial compressive tests and Brazilian splitting tests were conducted to calculate the indexes of BI. The stress-strain curves of uniaxial compressive tests are shown in Figure4, the displacement-stress curves of Brazilian tensile tests are shown in Figure 5. It can be seen that the failure patterns all exhibited obvious plunge after peak stress, which suggested that these rock-like materials could characterize the brittle breaking characteristics of rocks. As shown in Table 3, the BI2 values range from 4.08 to 6.3, and the BI3 values range from 0.62 to 0.73. Figure 6 shows that the specimens in the uniaxial compression test mainly exhibited splitting failure induced by the propagation of vertical cracks. It also can be seen in the Brazilian tensile strength tests, as shown in Figure 7. Consequently, the rock numerical models used for investigating rock crushing behaviors are calculated based on the UCS, BTS, Young’s modulus, and the failure patterns.Figure 4
Stress-strain curves of rock-like materials of uniaxial compressive tests.Figure 5
Displacement-stress curves of rock-like materials of Brazilian tensile tests.Table 3
The experimental results of rock-like materials.
No.UCS (MPa)BTS (MPa)BI1 (MPa2)BI2BI3BI4 (MPa)IHI (kN·mm−1) unconfinedIHI (kN·mm−1) 5 MPa confining stressA16.984.0434.304.080.625.860.87691.1545B20.293.2232.676.300.735.721.15451.4264C16.703.0925.805.400.695.080.93181.7456D20.163.2632.866.180.725.731.11191.7558Figure 6
Failure patterns of specimens in uniaxial compressive tests.Figure 7
Failure patterns of specimens in Brazilian tensile tests.


## 2.3. Indentation Tests under the Unconfined Condition and Confining Stress Condition

A series of indentation tests were conducted under unconfined stress and 5 MPa confining stress conditions. The experimental results are shown in Figures8 and 9. The velocity of the pick cutter is set to 1.0 mm/min according to Wang’s studies [8]. The confining stress restrained the propagation of vertical cracks. Since the slope of the curve is unstable at the initial stage of the indentation test, the slope of the fitted line near the maximum load is used as the indentation hardness, as shown in Figure 8. Moreover, the slope of the fitted line at the penetration displacement of 6 mm is used as the IHI under 5 MPa confining stress conditions. The IHIs under 5 MPa confining stress conditions are always less than that under unconfined stress conditions. According to the results in Table 3 and failure patterns in Figure 10, it can be seen that propagating crack length has negative correlation with IHI. As shown in Figures 10 and 11, there existed a compaction stage at the beginning of indentation test, and then the expansion of the crushed rocks induced the initiation and propagation of tension cracks. It can be seen in Figure 12 that the index BI2 [16] increased with the increase of IHI with a high correlation coefficient of 0.999. As shown in Figure 13, BI3 exhibited the same change law with IHI while BI1 and BI4 did not show a prominent correlation with IHI. The Figure 14 shows that the IHI has evident positive correlation with UCS. Therefore, the index BI2 is more suitable for the brittleness study in this paper.Figure 8
Displacement-force curves of indentation tests under unconfined stress conditions.Figure 9
Displacement-force curves of indentation tests under 5 MPa confining stress conditions.Figure 10
Failure patterns of specimens in indentation tests under unconfined stress conditions.Figure 11
Failure patterns of specimens in indentation tests under 5 MPa confining stress conditions.Figure 12
Relationship between IHI and BI2 under unconfined stress conditions.Figure 13
Relationship between IHI and BI3 under unconfined stress conditions.Figure 14
The correlation between IHI and UCS.


## 3. Discrete Element Analysis of Crushing

### 3.1. Microparameters Calculating Based on UCS and BTS

In order to investigate the relation between hardness, brittleness, and rock crushing behaviors, the numerical models of rock-like materials were built through the flat-joint contact model (FJM) [24] in PFC2D [25]. As shown in Figure 15, the flat-joint contact simulates the behavior of an interface between two notional surfaces, each of which is connected rigidly to a piece of body. The particles of a flat-jointed material are considered as faced grains, each of which is depicted as a core and a number of skirted faces. The interface is evenly discretized into several elements. According to previous researchers’ work, the numerical models of FJM is found to closely match real rocks’ macroproperties [26]. Therefore, the numerical models of rock-like materials were all calculated utilizing FJM in this study.Figure 15
Mechanism of the interaction between particles in FJM [25].According to the mechanical parameters in Table3 and the calibration procedure in Figure 16, the calculating results are shown in Figure 17. The microparameters can be calculated as shown in Table 4. The ratio of maximum to minimum diameter is set to 1.66 [27]. Table 5 shows the comparison of results obtained from numerical modeling and experimental tests. It suggested that the modeling results agree well with the results obtained from physical experiments. It can be seen that the failure patterns of numerical results exhibited the same characteristics with that of experimental results as shown in Figure 6. The different errors of all the parameters can be controlled within ±3.5%. In Figure 17, the blue cracks represent the shear cracks and the green cracks represent the tension cracks. Figures 17(a)–17(d) are the stress-strain curves and failure patterns of UCS tests, and Figure 17(e)–17(h) are the stress-strain curves and failure patterns of BTS tests.Figure 16
The microparameters calibration procedure of numerical models.Figure 17
The calculating results of numerical modeling (from left to right are materialsA, B, C, and D).
(a)(b)(c)(d)(e)(f)(g)(h)Table 4
The microparameters used in this PFC2D model.
Micro-parametersABCDMinimum disk diameterdmin (mm)0.50.50.50.5Maximum to minimum disk diameter ratiodmax/dmin1.661.661.661.66Young’s modulus of particle and bondEc=E¯c (GPa)3.113.653.373.87Ratio of normal to shear stiffness of particle and bondkn/ks = k¯n/k¯s1.61.61.61.6Element number,N4444Particle friction coefficientμ0.70.70.70.7Diameter of particled (mm)0.360.360.360.36Parallel-bond tensile strengthσ¯c (MPa)17.056.6213.86.94Parallel-bond cohesion strengthc¯ (MPa)8.212.557.5511.16Parallel-bond friction angleφ¯ (°)13131313Table 5
Comparison of results obtained from numerical modeling and experimental tests.
MaterialABCDItemExp.Num.Diff.Exp.Num.Diff.Exp.Num.Diff.Exp.Num.Diff.UCS (MPa)16.9817.00.12%20.2919.88−2.02%16.7016.40−1.80%20.1619.76−1.98%BTS (MPa)4.043.90−3.47%3.223.250.93%3.093.141.62%3.263.302.76%E1 (GPa)3.223.220.0%3.803.800.0%3.493.490.0%3.994.000.25%


### 3.2. Effects of Hardness and Brittleness on Rock Crushing Behaviors

In the numerical crushing tests, the numerical models of rock-like materials were established according to the experimental results (i.e., UCS, BTS, and Young’s modulus). More microscopic crushing characteristics can be observed in numerical analysis. The SE can be calculated by equation (8), where Fi is the recorded crushing force data and Vchippings is the total volume of crushing chippings. Since the SE is positively related to UCS, the normalized SE¯ is proposed by equation (9). The failure patterns of crushing chippings of rock-like materials are shown in Figures 18 and 19.(8)SE=∫0dFidxVchippings,(9)SE¯=SEUCS.Figure 18
The failure patterns of crushing chippings of rock-like materials under unconfined conditions.
(a)(b)(c)(d)Figure 19
The failure patterns of crushing chippings of rock-like materials under 5 MPa confining stress conditions.
(a)(b)(c)(d)Figure20 shows that SE¯ is negatively correlated with IHI. The normalized specific energy SE¯ decreased exponentially with the increase of BI2 with a high correlation coefficient of 0.9217, as shown in Figure 21. Figure 18 shows that the main failure patterns of crushing behaviors keep in line with the failure patterns of rock tests in Figure 17. According to the numerical results of crushing force in Table 6, it can be seen that the mean of crushing forces under unconfined stress conditions is larger than that under 5 MPa confining stress conditions in the crushing tests of material A and C, while the standard deviation values of crushing forces are opposite. Comparing Figures 18 and 19, it can be seen that the propagation depth of cracks was restrained, but the index SE¯ increased. Under 5 MPa confining stress conditions, SE¯ is negatively correlated with IHI; the normalized specific energy SE¯ decreased with the increase of BII∗ with a high correlation coefficient of 0.98256, as shown in Figures 22 and 23.Figure 20
The relationship betweenSE¯ and IHI.Figure 21
The relationship betweenSE¯ and brittleness.Table 6
The mean and standard deviation values of crushing force data.
Crushing force (kN)Crushing forces ofACrushing forces ofBCrushing forces ofCCrushing forces ofDForce directionX-directionY-directionX-directionY-directionX-directionY-directionX-directionY-directionUnconfined stress conditionsMean159.0592.96123.0172.03161.2093.04116.6868.20Standard deviation81.5248.6088.4551.2486.6850.1881.8348.005 MPa confining stress conditionsMean154.4392.1289.3751.67152.9888.56122.4472.73Standard deviation89.5856.2776.5346.9995.0855.5973.0844.72Figure 22
The relationship betweenSE¯ and IHI.Figure 23
The relationship betweenSE¯ and brittleness.The numerical results show that the average size of chippings of materialsB and D are larger than that of materials A and C. Moreover, the propagation length of tensile cracks is larger than that of shear cracks. As shown in Tables 7 and 8, the number of shear cracks of materials A and C is greater than that of materials B and D, but the number of tension cracks of materials A and C is less than that of materials B and D. Therefore, the failure patterns of rock-like materials with higher brittleness are more prone to tensile failure in the crushing testing.(10)Re=2×cd¯/221−ρ.Table 7
The orientation and quantity of microcracks in rock crushing tests (unconfined conditions).
TypeABCDShear cracksTension cracksTable 8
The orientation and quantity of microcracks in rock crushing tests (5 MPa confining stress conditions).
TypeABCDShear cracksTension cracksIn equation (10), Re is the equivalent diameter of chipping, c is the number of particles of chipping, and d¯ is the mean diameter of particle. The pore ratio ρ of the numerical model is 0.12. Comparing A and C, it is found that the larger the indentation hardness, the larger the proportion of the particles (equivalent diameter at 0–0.008 m range) in the cumulative gradation curve, as shown in Figures 24 and 25. The same is true between B and D.Figure 24
The distribution of equivalent diameter of rock chippings under unconfined stress conditions.Figure 25
The distribution of equivalent diameter of rock chippings under 5 MPa confining stress conditions.


## 3.1. Microparameters Calculating Based on UCS and BTS

In order to investigate the relation between hardness, brittleness, and rock crushing behaviors, the numerical models of rock-like materials were built through the flat-joint contact model (FJM) [24] in PFC2D [25]. As shown in Figure 15, the flat-joint contact simulates the behavior of an interface between two notional surfaces, each of which is connected rigidly to a piece of body. The particles of a flat-jointed material are considered as faced grains, each of which is depicted as a core and a number of skirted faces. The interface is evenly discretized into several elements. According to previous researchers’ work, the numerical models of FJM is found to closely match real rocks’ macroproperties [26]. Therefore, the numerical models of rock-like materials were all calculated utilizing FJM in this study.Figure 15
Mechanism of the interaction between particles in FJM [25].According to the mechanical parameters in Table3 and the calibration procedure in Figure 16, the calculating results are shown in Figure 17. The microparameters can be calculated as shown in Table 4. The ratio of maximum to minimum diameter is set to 1.66 [27]. Table 5 shows the comparison of results obtained from numerical modeling and experimental tests. It suggested that the modeling results agree well with the results obtained from physical experiments. It can be seen that the failure patterns of numerical results exhibited the same characteristics with that of experimental results as shown in Figure 6. The different errors of all the parameters can be controlled within ±3.5%. In Figure 17, the blue cracks represent the shear cracks and the green cracks represent the tension cracks. Figures 17(a)–17(d) are the stress-strain curves and failure patterns of UCS tests, and Figure 17(e)–17(h) are the stress-strain curves and failure patterns of BTS tests.Figure 16
The microparameters calibration procedure of numerical models.Figure 17
The calculating results of numerical modeling (from left to right are materialsA, B, C, and D).
(a)(b)(c)(d)(e)(f)(g)(h)Table 4
The microparameters used in this PFC2D model.
Micro-parametersABCDMinimum disk diameterdmin (mm)0.50.50.50.5Maximum to minimum disk diameter ratiodmax/dmin1.661.661.661.66Young’s modulus of particle and bondEc=E¯c (GPa)3.113.653.373.87Ratio of normal to shear stiffness of particle and bondkn/ks = k¯n/k¯s1.61.61.61.6Element number,N4444Particle friction coefficientμ0.70.70.70.7Diameter of particled (mm)0.360.360.360.36Parallel-bond tensile strengthσ¯c (MPa)17.056.6213.86.94Parallel-bond cohesion strengthc¯ (MPa)8.212.557.5511.16Parallel-bond friction angleφ¯ (°)13131313Table 5
Comparison of results obtained from numerical modeling and experimental tests.
MaterialABCDItemExp.Num.Diff.Exp.Num.Diff.Exp.Num.Diff.Exp.Num.Diff.UCS (MPa)16.9817.00.12%20.2919.88−2.02%16.7016.40−1.80%20.1619.76−1.98%BTS (MPa)4.043.90−3.47%3.223.250.93%3.093.141.62%3.263.302.76%E1 (GPa)3.223.220.0%3.803.800.0%3.493.490.0%3.994.000.25%


## 3.2. Effects of Hardness and Brittleness on Rock Crushing Behaviors

In the numerical crushing tests, the numerical models of rock-like materials were established according to the experimental results (i.e., UCS, BTS, and Young’s modulus). More microscopic crushing characteristics can be observed in numerical analysis. The SE can be calculated by equation (8), where Fi is the recorded crushing force data and Vchippings is the total volume of crushing chippings. Since the SE is positively related to UCS, the normalized SE¯ is proposed by equation (9). The failure patterns of crushing chippings of rock-like materials are shown in Figures 18 and 19.(8)SE=∫0dFidxVchippings,(9)SE¯=SEUCS.Figure 18
The failure patterns of crushing chippings of rock-like materials under unconfined conditions.
(a)(b)(c)(d)Figure 19
The failure patterns of crushing chippings of rock-like materials under 5 MPa confining stress conditions.
(a)(b)(c)(d)Figure20 shows that SE¯ is negatively correlated with IHI. The normalized specific energy SE¯ decreased exponentially with the increase of BI2 with a high correlation coefficient of 0.9217, as shown in Figure 21. Figure 18 shows that the main failure patterns of crushing behaviors keep in line with the failure patterns of rock tests in Figure 17. According to the numerical results of crushing force in Table 6, it can be seen that the mean of crushing forces under unconfined stress conditions is larger than that under 5 MPa confining stress conditions in the crushing tests of material A and C, while the standard deviation values of crushing forces are opposite. Comparing Figures 18 and 19, it can be seen that the propagation depth of cracks was restrained, but the index SE¯ increased. Under 5 MPa confining stress conditions, SE¯ is negatively correlated with IHI; the normalized specific energy SE¯ decreased with the increase of BII∗ with a high correlation coefficient of 0.98256, as shown in Figures 22 and 23.Figure 20
The relationship betweenSE¯ and IHI.Figure 21
The relationship betweenSE¯ and brittleness.Table 6
The mean and standard deviation values of crushing force data.
Crushing force (kN)Crushing forces ofACrushing forces ofBCrushing forces ofCCrushing forces ofDForce directionX-directionY-directionX-directionY-directionX-directionY-directionX-directionY-directionUnconfined stress conditionsMean159.0592.96123.0172.03161.2093.04116.6868.20Standard deviation81.5248.6088.4551.2486.6850.1881.8348.005 MPa confining stress conditionsMean154.4392.1289.3751.67152.9888.56122.4472.73Standard deviation89.5856.2776.5346.9995.0855.5973.0844.72Figure 22
The relationship betweenSE¯ and IHI.Figure 23
The relationship betweenSE¯ and brittleness.The numerical results show that the average size of chippings of materialsB and D are larger than that of materials A and C. Moreover, the propagation length of tensile cracks is larger than that of shear cracks. As shown in Tables 7 and 8, the number of shear cracks of materials A and C is greater than that of materials B and D, but the number of tension cracks of materials A and C is less than that of materials B and D. Therefore, the failure patterns of rock-like materials with higher brittleness are more prone to tensile failure in the crushing testing.(10)Re=2×cd¯/221−ρ.Table 7
The orientation and quantity of microcracks in rock crushing tests (unconfined conditions).
TypeABCDShear cracksTension cracksTable 8
The orientation and quantity of microcracks in rock crushing tests (5 MPa confining stress conditions).
TypeABCDShear cracksTension cracksIn equation (10), Re is the equivalent diameter of chipping, c is the number of particles of chipping, and d¯ is the mean diameter of particle. The pore ratio ρ of the numerical model is 0.12. Comparing A and C, it is found that the larger the indentation hardness, the larger the proportion of the particles (equivalent diameter at 0–0.008 m range) in the cumulative gradation curve, as shown in Figures 24 and 25. The same is true between B and D.Figure 24
The distribution of equivalent diameter of rock chippings under unconfined stress conditions.Figure 25
The distribution of equivalent diameter of rock chippings under 5 MPa confining stress conditions.


## 4. Conclusions

In this paper, quartz sand and high-strengthα-hemi-hydrated gypsum were prepared to make rock-like materials with different hardness and brittleness through different mass ratios. Gypsum has proven to be a good rock-like material for rock mechanics experiments and rock fragmentation behaviors. The numerical models were calculated based on the mechanical parameters of those rock-like materials. The influence mechanism of indentation hardness and brittleness of rock-like materials on their crushing behaviors was investigated using the DEM, i.e., PFC2D. The main conclusions can be drawn as follows.The indexBI2 increased exponentially with the increase of IHI with high-correlation coefficient. The index IHI has evident positive correlation with UCS. Hardness and brittleness can be used as important indexes to characterize the mechanical properties and failure patterns of rock materials. According to the numerical results of the crushing tests, the normalized specific energy SE¯ is negatively correlated with IHI. SE¯ decreased exponentially with the increase of BI2. The failure patterns of rock-like materials with higher brittleness are more prone to tensile failure in the crushing testing.This investigation is of great significance for understanding the correlation between hardness and brittleness of rock materials and their fragmentation as well as failure patterns in underground constructions. The FJM in PFC is more suitable for the numerical study of rock-like materials than finite element method, and it can obtain more coincident mechanical parameters and more microscopic failure characteristics.



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