# Relationship between Brittleness Index and Crack Initiation Stress Ratio for Different Rock Types

Authors: Juyu Jiang; Dong Wang; Xinping Han; Shuai Di
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Brittleness and crack initiation stress (σci) are important rock mechanical properties and intrinsically related to rock deformation and failure. We establish the relationship between σci and uniaxial tensile strength (σt) based on the Griffith stress criterion of brittle failure and introduce brittleness indexes B1–B4 based on the ratio of uniaxial compressive strength (σc) to σt. The crack initiation stress ratio (K) is defined as the ratio of σci to crack damage stress. The relationship between brittleness index and K is obtained from laboratory mechanics tests including uniaxial compression and Brazilian splitting tests. The results show that B1 and B2 have an inversely proportional and variant inversely proportional relationship with K, respectively, whereas no apparent relationship is observed between B3 and B4 and K. The fitting of experimental data from igneous, metamorphic, and sedimentary rocks shows that B1 and B2 have a power and linear relationship with K, respectively, whereas no functional relationship is observed between B3 and B4 and K. We collected 70 different types of uniaxial compression test data for igneous, metamorphic, and sedimentary rocks and obtained laws that are consistent within each rock type. The experimental data are used to verify K estimations using a specified constant α based on the experimental data. According to results of the limestone tests, α = 3 for σc < 60 MPa (high porosity), α = 5 for 60 MPa ≤ σc ≤ 90 MPa (moderate porosity), and α = 8 for σc > 90 MPa (low porosity) as well as for igneous and metamorphic rocks. Estimates of K for 127 different rock types using the newly defined brittleness index are in good agreement with the experimental results. This study provides an important new brittleness index calculation method and a simple and reliable method for estimating K.


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## Body

## 1. Introduction

The rock brittleness index is of great significance for mining engineering, tunnel engineering, oil and gas exploration, and rock burst susceptibility and has received extensive attention in the literature. Although brittleness is one of the most important rock properties, there remains no standardized or universally accepted concept of brittleness or a measurement method that exactly defines a rock brittleness index. Morley [1] and Hetenyi [2] defined brittleness as the lack of ductility. Ramsey [3] argued that rocks are brittle when their internal cohesion is broken. The definition of brittleness itself ranges from broad descriptions, such as materials “at or only slightly beyond the yield stress” [4] and “materials that rupture or fracture with little or no plastic flow” (Glossary of Geology and Related Sciences 1960) to specific descriptions such as low elongation values, fracture failure, formation of fines, higher ratio of compressive to tensile strength, higher resilience, higher angle of internal friction, and/or formation of cracks during indentation [5]. Dozens of brittleness indexes have been proposed from different viewpoints. Hucka and Das [5] proposed brittleness indexes B1 and B2 from the viewpoint of the ratio of uniaxial compressive strength (σc) to tensile strength (σt), and Altindag [6–8] proposed brittleness indexes B3 and B4. These indexes are calculated as follows:(1)B1=σcσt,(2)B2=σc−σtσc+σt,(3)B3=σcσt2,(4)B4=σcσt2,where σc is the uniaxial compressive strength and σt is tensile strength.From the viewpoint of stress-strain curves, Tarasov and Potvin [9] substituted the energy ratio with the elastic modulus ratio during loading and proposed brittleness indexes B5 and B6. Bishop [10] gave brittleness index B7 based on the peak strength and residual strength in the stress-strain curve. Hajiabdolmajid and Kaiser [11] proposed index B8 based on the study of Bishop [10] to determine brittleness after considering the peak and residual strain. According to strain and strain energy during loading and recovery, brittleness indexes B9 and B10 are summarized by Hucka and Das [5]. Aubertin et al. [12] and George [13] proposed brittleness indexes B11 and B12 using energy and strain, respectively. Brittleness indexes B5–B12 are defined as follows.(5)B5=dWrdWe=M−EuM,B6=dWαdWe=EuM,where dWr is the postpeak rupture energy, dWe is the withdrawn elastic energy, dWα is the released elastic energy, E is the unloading elastic modulus, and M is the postpeak modulus:(6)B7=τp−τrτp,B8=εp−εrεp,where τp is the peak strength, τr is the residual strength, εp is the peak strain, and εr is the residual strain:(7)B9=εh/ε,B10=Wr/W,where εh is the reversible strain, ε is the total strain, Wr is the reversible strain energy, and W is the total strain energy:(8)B11=A2A1,where A2 represents the energy given by the total area below the stress-stain curve and A1 is the elastic energy stored in the sample, which is obtained here by using the slope of the curve at 50% of the ultimate strength:(9)B12=ε1,where ε1 is the axial strain. The specimen demonstrates brittle properties when ε1<3% and plastic properties when ε1>5%. When ε1 is between 3% and 5%, the specimen exhibits both brittle and plastic properties.From a hardness test viewpoint, Honda and Sanada [14] found that crack development is a direct function of brittleness, and differences between the microindentation and macroindentation hardness values may be taken as a measure of brittleness (B13). The brittleness index B14 was proposed by Lawn and Marshall [15]. J. B. Quinn and G. D. Quinn [16] studied ceramic materials and proposed brittleness index B15:(10)B13=Hμ−Hk,B14=HKIC,B15=HEKIC2,where Hμ is the microindentation hardness, H is the macroindentation hardness, E is Young’s modulus, KIC is the fracture toughness, and k is a constant.The brittleness indexB16 [17] was introduced from punch penetration tests. Copur et al. [18] performed experiments to quantify the force and brittleness index relevant to rock cutting performance. Blocks of eleven different rock samples were collected from different operating mines in Turkey and subjected to indentation tests to calculate brittleness index B17. The results show that B17 is moderately correlated with cutting performance and rock mechanical properties and poorly correlated with coarseness index and cutting size distribution:(11)B16=FmaxP,B17=PdecPinc,where Fmax is the maximum applied force on a rock sample, P is the corresponding penetration at maximum force, Pdec is the average force decrement period, and Pinc is the average force increment period.Rock brittleness indexesB18, B19, and B20 are defined from the viewpoint of Mohr’s circle and impact tests [5, 19, 20] as follows:(12)B18=sinθ,B19=S20,B20=qσc,where θ is the internal friction angle, S20 is the percentage of fines (<11.2 mm), and q is the percentage of fines (28 mesh) formed in the Protodyakonov impact test.Although the results of brittleness indexes are very rich, discussion on the relationship between brittleness index and crack initiation stress ratio (K) for different rock types is rarely reported from the perspective of rock compressive strength and σt. In this study, we establish the relationship between σci and σt using fracture mechanics theory. We theoretically derive the functional relationship between brittleness indexes B1–B4 and K. We select igneous (e.g., granite), metamorphic (e.g., gneiss), and sedimentary (e.g., limestone) rocks for laboratory mechanical tests including uniaxial compression and Brazilian splitting tests. Rock σci is determined using the crack volumetric strain method. Seventy uniaxial compression test data for igneous, metamorphic, and sedimentary rocks were collected to further explore the relationship between B1–B4 and K. We use the newly defined brittleness index to estimate K from 127 different rocks types and compare the results with the experimental results. The feasibility of using K to calculate the brittleness index is verified by the results presented here. This also provides a simple and reliable method for estimating K.


## 2. Crack Initiation Mechanism of Rock under Uniaxial Compression

Lockner [21] argued that deformation and failure is a progressive process characterized by the initiation, propagation, and coalescence of microcracks for most rock samples. Zhou et al. [22–24] and Liu et al. [25] proposed that the complete cracking process can be classified into six levels distinguishable by five characteristic stresses based on the coupled analyses of the acousto-optic-mechanical characteristics. These include (1) crack closure stress σcc, (2) microcrack nucleation stress σcn, (3) crack initiation stress σci, (4) crack damage stress σcd, and (5) peak stress σc. As shown in Figure 1, the stress-strain curve is usually divided into five stages (I–V), where σci and σcd are the focus of rock mechanical studies [26–29]. The determination of σcd has been widely reported, namely, the axial stress corresponding to the volume strain reversal point. Martin [30] proposed a crack volumetric strain method to determine rock σci from uniaxial or triaxial tests. The σci corresponds to stress at the end of the horizontal section with a crack volumetric strain of zero (Figure 1). The crack volume strain εcv is given as follows:(13)εcv=εv−εev,where εv and εev are the total volume strain and elastic volume strain, respectively.Figure 1
Stress-strain diagram of granite showing the stages of crack development (after [30]).The total volume strain is defined as follows:(14)εv=ε1+2ε2,where ε1 and ε2 are the axial and lateral strains, respectively.The elastic volume strain is calculated as follows:(15)εev=1−2νEσ1,where σ1 is the axial stress, E is the elastic modulus, and ν is Poisson’s ratio in the elastic stage.In addition,σci can be determined as the point where volumetric strain starts to deviate from the straight line of stage II, and the two methods can be calibrated to each other.In this study, the definition of the crack initiation stress ratio (K) is introduced [31] and defined as follows:(16)K=σciσcd.Martin [30] and Cai et al. [31] noted that K is an indicator of rock heterogeneity and texture. Low K values indicate high rock heterogeneity. A total of 227 sets of uniaxial compression test data for different rock samples were collected from various literature sources [32], including 138, 53, and 36 sets from igneous, sedimentary, and metamorphic rocks, respectively. The K of igneous rocks ranges from 0.4 to 0.6, that of metamorphic rocks ranges from 0.15 to 0.6, and that of sedimentary rocks ranges from 0.3 to 0.8. In this study, we use the theory of fracture mechanics to study the relationship between brittleness index and K.


## 3. Analysis of the Relationship between Brittleness Index andK Value

### 3.1. Brittle Rock

Under tensile loading, crack initiation often means that tensile fracture is about to occur and thatσci is very close to σt. However, crack initiation is only related to lower stress under compression conditions. After σci is exceeded, cracks begin to stably expand and additional load is required to bring the stress from σci to σc. The Griffith stress criterion is applied to crack initiation when σc = σci, and the relationship between σc and σt is constructed according to the Griffith stress criterion, i.e., RG=σc/σt=8. Hence, Cai et al. [33] proposed using the strength ratio (equation (17) to estimate σt of the brittle rock from σci and σc, as in equation (18):(17)R=RGσcσci=8σcσci,(18)σt=σcR=σci8.Based on the study of Hucka and Das [5] and Altindag [6–8], Wang et al. [34] redefined brittleness indexes B1–B4 after introducing K, but these study only apply to brittle rocks and equations (1)–(4) are written in the following forms:(19)B1=σcσt=8σcσci=8K,B2=σc−σtσc+σt=8σc−σci8σc+σci=8−K8+K,B3=σcσt2=σcσci16,B4=σcσt2=σcσci16.


### 3.2. Weak Rock

Cai et al. [33] argued that the mechanical behavior of sedimentary rocks differs significantly from igneous rocks, e.g., limestone, dolomite, and clay have porosities higher than 20%. Coviello et al. [35] showed that the strength ratio R of Gravina calcarenite is 6.2, whereas Martin and Lanyon [36] reported R from the Opalinus clay of only 3.2, substantially lower than the value suggested by equation (17). This error may be because the fitted Hoek–Brown curves overestimate σt for weak rocks [37]. Another possibility is that the clay and/or the calcite contents in these rocks are relatively high so that the Griffith crack growth fracture mechanism no longer dominates. Other fracture mechanisms (e.g., weak inclusions) may play a role in crack initiation and growth of preexisting defects [33]. Accounting for the difference of crack propagation in compression and tension, Cai et al. [33] noted that R for rocks containing circular pores is as follows:(20)R=3σcσci,and they further proposed R for different rocks as follows:(21)R=ασcσci,where α is a constant depending on the rock texture (e.g., grain size and shape), mineral content (e.g., clay and/or calcite content), and crack initiation mechanism (e.g., Griffith cracks or pore inclusions). Therefore, α = 8 for brittle rocks, α < 8 for porous and weak rocks, α = 3 when the pore flaw is a circular hole, α = 5 when the pore flaw is an elliptical hole, and α = 3 for a 3D spherical hole [38]. However, the determination of α requires further study.Based on the above analysis, theσci of different types of rocks can be estimated from(22)σci=ασt.Hence, equation (22) is introduced into the brittleness index formula. We redefine brittleness indexes B1–B4 for the different types of rock as follows:(23)B1=σcσt=ασcσci=αK,B2=σc−σtσc+σt=ασc−σciασc+σci=α−Kα+K,B3=σcσt2=σcσci2α,B4=σcσt2=σcσci2α.By identifyingσci to correctly evaluate the brittleness index of different rock types from uniaxial compression tests, the key is to identify the right fracture initiation mechanism and hence α. In general, Griffith’s constant of 8 is applicable to most brittle rocks and a large number of tests are required for other rock types. As described above, there is an intrinsic relationship between the brittleness index and K. The correlation is studied from indoor tests and obtained values of α are verified.


## 3.1. Brittle Rock

Under tensile loading, crack initiation often means that tensile fracture is about to occur and thatσci is very close to σt. However, crack initiation is only related to lower stress under compression conditions. After σci is exceeded, cracks begin to stably expand and additional load is required to bring the stress from σci to σc. The Griffith stress criterion is applied to crack initiation when σc = σci, and the relationship between σc and σt is constructed according to the Griffith stress criterion, i.e., RG=σc/σt=8. Hence, Cai et al. [33] proposed using the strength ratio (equation (17) to estimate σt of the brittle rock from σci and σc, as in equation (18):(17)R=RGσcσci=8σcσci,(18)σt=σcR=σci8.Based on the study of Hucka and Das [5] and Altindag [6–8], Wang et al. [34] redefined brittleness indexes B1–B4 after introducing K, but these study only apply to brittle rocks and equations (1)–(4) are written in the following forms:(19)B1=σcσt=8σcσci=8K,B2=σc−σtσc+σt=8σc−σci8σc+σci=8−K8+K,B3=σcσt2=σcσci16,B4=σcσt2=σcσci16.


## 3.2. Weak Rock

Cai et al. [33] argued that the mechanical behavior of sedimentary rocks differs significantly from igneous rocks, e.g., limestone, dolomite, and clay have porosities higher than 20%. Coviello et al. [35] showed that the strength ratio R of Gravina calcarenite is 6.2, whereas Martin and Lanyon [36] reported R from the Opalinus clay of only 3.2, substantially lower than the value suggested by equation (17). This error may be because the fitted Hoek–Brown curves overestimate σt for weak rocks [37]. Another possibility is that the clay and/or the calcite contents in these rocks are relatively high so that the Griffith crack growth fracture mechanism no longer dominates. Other fracture mechanisms (e.g., weak inclusions) may play a role in crack initiation and growth of preexisting defects [33]. Accounting for the difference of crack propagation in compression and tension, Cai et al. [33] noted that R for rocks containing circular pores is as follows:(20)R=3σcσci,and they further proposed R for different rocks as follows:(21)R=ασcσci,where α is a constant depending on the rock texture (e.g., grain size and shape), mineral content (e.g., clay and/or calcite content), and crack initiation mechanism (e.g., Griffith cracks or pore inclusions). Therefore, α = 8 for brittle rocks, α < 8 for porous and weak rocks, α = 3 when the pore flaw is a circular hole, α = 5 when the pore flaw is an elliptical hole, and α = 3 for a 3D spherical hole [38]. However, the determination of α requires further study.Based on the above analysis, theσci of different types of rocks can be estimated from(22)σci=ασt.Hence, equation (22) is introduced into the brittleness index formula. We redefine brittleness indexes B1–B4 for the different types of rock as follows:(23)B1=σcσt=ασcσci=αK,B2=σc−σtσc+σt=ασc−σciασc+σci=α−Kα+K,B3=σcσt2=σcσci2α,B4=σcσt2=σcσci2α.By identifyingσci to correctly evaluate the brittleness index of different rock types from uniaxial compression tests, the key is to identify the right fracture initiation mechanism and hence α. In general, Griffith’s constant of 8 is applicable to most brittle rocks and a large number of tests are required for other rock types. As described above, there is an intrinsic relationship between the brittleness index and K. The correlation is studied from indoor tests and obtained values of α are verified.


## 4. Experimental Materials and Methods

Rock samples used in the tests include granite, gneiss, and limestone collected from a mining site in China. Samples were cut and polished into 34 cylindrical standard test pieces (Φ50 × 100 mm), including 13 granites, 13 gneiss, and 8 limestones. The groupings and numbers of specimens (Φ50 × 25 mm) for the Brazilian tests are the same as those for the uniaxial compression tests. The lithology and uniformity of rock samples are mainly considered in the sampling process. Rock samples are selected to be as representative as possible to achieve realistic results. A WAW-600C universal testing machine at Liaoning Technical University was used to conduct the uniaxial compression and Brazilian tests (Figure2). The upper and lower ends of the cylindrical samples were maintained flat during preparation, and the unevenness of the two ends varied by no more than 0.05 mm. Acoustic testing was performed prior to the start of the uniaxial compression test to eliminate anomalous rock samples. During the loading phase, axial and lateral strain was measured using an extensometer and the loading rate was maintained at 0.03 mm/min until the peak strength was reached. The loading rate was 0.2 MPa/s for the Brazilian tests.Figure 2
WAW-600C mechanical test equipment.To study the relationship between brittleness index andK for different rock types, we use a variety of functional forms (e.g., exponential, logarithmic, power, and linear) to fit the test data. A variety of functional forms are also used to fit the comprehensive data (i.e., synthesis of granite, gneiss, and limestone data) for different rock types. The functional form with the highest correlation coefficient is selected as the final fitting function to best represent the data.


### 4.1. Igneous Rock: Granite

Results of the granite mechanical parameters obtained from the laboratory tests are listed in Table1. Figure 3 shows the fitting analysis of the data and that brittleness index B1 and K have a power function relationship with a correlation coefficient of 0.98. The brittleness index B2 and K demonstrate a linear relationship with a correlation coefficient of 0.97. The fitting results are overall satisfactory with clear data regularity. None of the selected functions adequately represent brittleness indexes B3 and B4 and K. The average value of α obtained from the tests is 7.899, which is in good agreement with the Griffith constant (α = 8).Table 1
Summary of mechanical parameters of granite samples.
No.
σ
ci (MPa)
σ
c (MPa)
σ
t (MPa)
α
σ
ci/σc
Brittleness index
B
1
B
2
B
3
B
4
H01
54.390
118.220
6.873
7.913
0.460
17.200
0.890
406.273
20.156
H02
56.136
109.424
7.017
8.000
0.513
15.594
0.879
383.914
19.594
H03
63.265
116.501
7.775
8.137
0.543
14.983
0.875
452.921
21.282
H04
55.176
116.983
6.754
8.169
0.472
17.320
0.891
395.074
19.876
H05
64.065
108.120
7.876
8.134
0.593
13.727
0.864
425.787
20.635
H06
78.165
117.100
9.371
8.342
0.668
12.497
0.852
548.644
23.423
H07
41.718
118.041
5.306
7.862
0.353
22.245
0.914
313.191
17.697
H08
64.635
135.280
8.770
7.370
0.478
15.425
0.878
593.230
24.356
H09
59.355
104.510
7.653
7.756
0.568
13.656
0.864
399.906
19.998
H10
74.160
101.500
9.564
7.754
0.731
10.613
0.828
485.382
22.031
H11
44.580
108.470
5.899
7.557
0.411
18.389
0.897
319.921
17.886
H12
40.815
124.665
5.161
7.909
0.327
24.157
0.920
321.679
17.935
H13
42.195
111.960
5.420
7.786
0.377
20.658
0.908
303.389
17.418Figure 3
Relationship between crack initiation stress ratioK of granite and different brittleness indexes: (a) B1; (b) B2; (c) B3; (d) B4.
(a)
(b)
(c)
(d)


### 4.2. Metamorphic Rock: Gneiss

Results of the gneiss mechanical parameters obtained from the laboratory test are listed in Table2. Figure 4 shows that brittleness index B1 and K display a power function relationship with a correlation coefficient of 0.91, and B2 and K demonstrate a linear relationship with a correlation coefficient of 0.91. As with granite, no functional relationship is observed between B3 and B4 and K. The average α value obtained from the tests is 7.856, which was also consistent with the Griffith constant (α = 8). The results of these two tests verify that the Griffith crack model is suitable for brittle rock.Table 2
Summary of mechanical parameters of gneiss samples.
No
σ
ci (MPa)
σ
c (MPa)
σ
t (MPa)
α
σ
ci/σc
Brittleness index
B
1
B
2
B
3
B
4
P01
44.890
73.250
5.310
8.454
0.613
13.795
0.865
194.476
13.945
P02
41.260
68.740
6.086
6.780
0.600
11.295
0.837
209.169
14.463
P03
51.075
95.780
6.475
7.888
0.533
14.792
0.873
310.086
17.609
P04
35.250
68.130
4.343
8.117
0.517
15.689
0.880
147.928
12.163
P05
43.815
83.360
5.535
7.916
0.526
15.060
0.875
230.712
15.189
P06
40.230
79.630
5.226
7.699
0.505
15.238
0.877
208.058
14.424
P07
50.655
89.120
6.391
7.926
0.568
13.945
0.866
284.775
16.875
P08
41.625
99.600
5.551
7.499
0.418
17.943
0.894
276.439
16.626
P09
75.450
94.230
9.464
7.972
0.801
9.956
0.817
445.915
21.117
P10
44.010
72.400
5.658
7.779
0.608
12.797
0.855
204.803
14.311
P11
80.475
97.490
10.122
7.951
0.825
9.632
0.812
493.381
22.212
P12
70.725
103.530
8.769
8.066
0.683
11.807
0.844
453.908
21.305
P13
61.620
100.750
7.627
8.079
0.612
13.210
0.859
384.200
19.601Figure 4
Relationship between crack initiation stress ratioK of gneiss and different brittleness indexes: (a) B1; (b) B2; (c) B3; (d) B4.
(a)
(b)
(c)
(d)


### 4.3. Sedimentary Rock: Limestone

Table3 lists the limestone mechanical parameters acquired from the laboratory tests. Figure 5 shows that B1 and K have a power relationship with a correlation coefficient of 0.86, B2 and K have a linear relationship with a correlation coefficient of 0.82, and no functional relationship is observed between B3 and B4 and K. The interior of rocks for this group is well cemented by grains with moderate porosity (5%–12%). The test results show that when the average σc is about 77 MPa, the average α obtained from this test group is 5647.Table 3
Summary of mechanical parameters of limestone samples.
No
σ
ci (MPa)
σ
c (MPa)
σ
t (MPa)
α
σ
ci/σc
Brittleness index
B
1
B
2
B
3
B
4
S01
46.560
75.600
8.700
5.352
0.616
8.690
0.794
328.860
18.134
S02
37.776
80.100
6.870
5.499
0.472
11.659
0.842
275.144
16.587
S03
54.096
89.300
9.620
5.623
0.606
9.283
0.805
429.533
20.725
S04
40.640
64.400
7.500
5.419
0.631
8.587
0.791
241.500
15.540
S05
33.072
84.600
6.540
5.057
0.391
12.936
0.856
276.642
16.633
S06
32.528
60.800
5.330
6.103
0.535
11.407
0.839
162.032
12.729
S07
56.960
78.700
9.030
6.308
0.724
8.715
0.794
355.331
18.850
S08
47.488
86.200
8.160
5.820
0.551
10.564
0.827
351.696
18.754Figure 5
Relationship between crack initiation stress ratioK of limestone and different brittleness indexes: (a) B1; (b) B2; (c) B3; (d) B4.
(a)
(b)
(c)
(d)


## 4.1. Igneous Rock: Granite

Results of the granite mechanical parameters obtained from the laboratory tests are listed in Table1. Figure 3 shows the fitting analysis of the data and that brittleness index B1 and K have a power function relationship with a correlation coefficient of 0.98. The brittleness index B2 and K demonstrate a linear relationship with a correlation coefficient of 0.97. The fitting results are overall satisfactory with clear data regularity. None of the selected functions adequately represent brittleness indexes B3 and B4 and K. The average value of α obtained from the tests is 7.899, which is in good agreement with the Griffith constant (α = 8).Table 1
Summary of mechanical parameters of granite samples.
No.
σ
ci (MPa)
σ
c (MPa)
σ
t (MPa)
α
σ
ci/σc
Brittleness index
B
1
B
2
B
3
B
4
H01
54.390
118.220
6.873
7.913
0.460
17.200
0.890
406.273
20.156
H02
56.136
109.424
7.017
8.000
0.513
15.594
0.879
383.914
19.594
H03
63.265
116.501
7.775
8.137
0.543
14.983
0.875
452.921
21.282
H04
55.176
116.983
6.754
8.169
0.472
17.320
0.891
395.074
19.876
H05
64.065
108.120
7.876
8.134
0.593
13.727
0.864
425.787
20.635
H06
78.165
117.100
9.371
8.342
0.668
12.497
0.852
548.644
23.423
H07
41.718
118.041
5.306
7.862
0.353
22.245
0.914
313.191
17.697
H08
64.635
135.280
8.770
7.370
0.478
15.425
0.878
593.230
24.356
H09
59.355
104.510
7.653
7.756
0.568
13.656
0.864
399.906
19.998
H10
74.160
101.500
9.564
7.754
0.731
10.613
0.828
485.382
22.031
H11
44.580
108.470
5.899
7.557
0.411
18.389
0.897
319.921
17.886
H12
40.815
124.665
5.161
7.909
0.327
24.157
0.920
321.679
17.935
H13
42.195
111.960
5.420
7.786
0.377
20.658
0.908
303.389
17.418Figure 3
Relationship between crack initiation stress ratioK of granite and different brittleness indexes: (a) B1; (b) B2; (c) B3; (d) B4.
(a)
(b)
(c)
(d)


## 4.2. Metamorphic Rock: Gneiss

Results of the gneiss mechanical parameters obtained from the laboratory test are listed in Table2. Figure 4 shows that brittleness index B1 and K display a power function relationship with a correlation coefficient of 0.91, and B2 and K demonstrate a linear relationship with a correlation coefficient of 0.91. As with granite, no functional relationship is observed between B3 and B4 and K. The average α value obtained from the tests is 7.856, which was also consistent with the Griffith constant (α = 8). The results of these two tests verify that the Griffith crack model is suitable for brittle rock.Table 2
Summary of mechanical parameters of gneiss samples.
No
σ
ci (MPa)
σ
c (MPa)
σ
t (MPa)
α
σ
ci/σc
Brittleness index
B
1
B
2
B
3
B
4
P01
44.890
73.250
5.310
8.454
0.613
13.795
0.865
194.476
13.945
P02
41.260
68.740
6.086
6.780
0.600
11.295
0.837
209.169
14.463
P03
51.075
95.780
6.475
7.888
0.533
14.792
0.873
310.086
17.609
P04
35.250
68.130
4.343
8.117
0.517
15.689
0.880
147.928
12.163
P05
43.815
83.360
5.535
7.916
0.526
15.060
0.875
230.712
15.189
P06
40.230
79.630
5.226
7.699
0.505
15.238
0.877
208.058
14.424
P07
50.655
89.120
6.391
7.926
0.568
13.945
0.866
284.775
16.875
P08
41.625
99.600
5.551
7.499
0.418
17.943
0.894
276.439
16.626
P09
75.450
94.230
9.464
7.972
0.801
9.956
0.817
445.915
21.117
P10
44.010
72.400
5.658
7.779
0.608
12.797
0.855
204.803
14.311
P11
80.475
97.490
10.122
7.951
0.825
9.632
0.812
493.381
22.212
P12
70.725
103.530
8.769
8.066
0.683
11.807
0.844
453.908
21.305
P13
61.620
100.750
7.627
8.079
0.612
13.210
0.859
384.200
19.601Figure 4
Relationship between crack initiation stress ratioK of gneiss and different brittleness indexes: (a) B1; (b) B2; (c) B3; (d) B4.
(a)
(b)
(c)
(d)


## 4.3. Sedimentary Rock: Limestone

Table3 lists the limestone mechanical parameters acquired from the laboratory tests. Figure 5 shows that B1 and K have a power relationship with a correlation coefficient of 0.86, B2 and K have a linear relationship with a correlation coefficient of 0.82, and no functional relationship is observed between B3 and B4 and K. The interior of rocks for this group is well cemented by grains with moderate porosity (5%–12%). The test results show that when the average σc is about 77 MPa, the average α obtained from this test group is 5647.Table 3
Summary of mechanical parameters of limestone samples.
No
σ
ci (MPa)
σ
c (MPa)
σ
t (MPa)
α
σ
ci/σc
Brittleness index
B
1
B
2
B
3
B
4
S01
46.560
75.600
8.700
5.352
0.616
8.690
0.794
328.860
18.134
S02
37.776
80.100
6.870
5.499
0.472
11.659
0.842
275.144
16.587
S03
54.096
89.300
9.620
5.623
0.606
9.283
0.805
429.533
20.725
S04
40.640
64.400
7.500
5.419
0.631
8.587
0.791
241.500
15.540
S05
33.072
84.600
6.540
5.057
0.391
12.936
0.856
276.642
16.633
S06
32.528
60.800
5.330
6.103
0.535
11.407
0.839
162.032
12.729
S07
56.960
78.700
9.030
6.308
0.724
8.715
0.794
355.331
18.850
S08
47.488
86.200
8.160
5.820
0.551
10.564
0.827
351.696
18.754Figure 5
Relationship between crack initiation stress ratioK of limestone and different brittleness indexes: (a) B1; (b) B2; (c) B3; (d) B4.
(a)
(b)
(c)
(d)


## 5. Discussion

The experimental results for all rock types display a power relationship betweenB1 and K, a linear relationship between B2 and K, and no determinable relationship between B3 and B4 and K. The relationship between B1 and B2 and K are therefore verified by experiments. We combine data from this study with results from Wang et al. [34] and Cai et al. [33] to further discuss the behavior of B1–B4 and K. Figure 6 shows that the fitted curves of B1 and K also yield a power function relationship with a correlation coefficient of 0.74. B2 and K demonstrate a linear function relationship with a correlation coefficient of 0.70. As in the smaller data set, no functional relationship is detected between B3 and B4 and K. These results are consistent with those achieved from similar lithology, which indicates that this rule can be applied to all rocks, particularly those with higher homogeneity.Figure 6
Relationship between crack initiation stress ratioK and different rock types of rocks with different brittleness indexes: (a) B1; (b) B2; (c) B3; (d) B4.
(a)
(b)
(c)
(d)According to the test results shown in Figure6, B1 and B2 can be defined as(24)B1=a1Kb1,(25)B2=a2+b2K,where a1, a2, b1, and b2 are the fitting constants. In this paper, we have a1 = 7.024, a2 = 0.9991, b1 = −1.054, and b2 = −0.258.We next gathered uniaxial compression test data of 127 different rock types from the literature [8, 39–42], including 58 igneous rocks, 21 metamorphic rocks, and 48 sedimentary rocks. The data were fitted with equation (22) (method 1), equation (24) (method 2), and equation (25) (method 3) to predict K. Our experimental results yield α = 8 for both igneous and metamorphic rocks. However, the porosity of the sedimentary rock used in the experiments is moderate, and the tests show that the limestone σc is 75.05 ± 14.25 MPa and α = 5. The α value for sedimentary rocks is therefore temporarily set to 3 for σc < 60 MPa (high porosity), 5 for 60 MPa ≤ σc ≤ 90 MPa (moderate porosity), and 8 for σc > 90 MPa (low porosity). A large number of additional experiments are required for the accurate determination of α, which can be further explored in subsequent studies.Figure7 and Table 4 show that the K values of igneous, metamorphic, and sedimentary rocks predicted by the three methods are all within reasonable ranges. Values of K for igneous and metamorphic rocks obtained from the three methods are distributed over a relatively large range from 0.2 to 0.8, among which K for igneous rocks concentrate between 0.4 and 0.8 and K for metamorphic rocks concentrate between 0.2 and 0.6. In method 1, K values for sedimentary rocks are concentrated between 0.2 and 0.6, while in methods 2 and 3, they concentrate between 0.4 and 0.8. The three prediction methods present average K values of 0.566, 0.519, and 0.511, respectively, for igneous rocks. Similarly, the average K values for metamorphic are 0.47, 0.461, and 0.454, respectively, and those for sedimentary rocks are 0.37, 0.604, and 0.594. The results are consistent with values reported by Wen Tao of 0.503, 0.5396, and 0.3759 for igneous, metamorphic, and sedimentary rocks, respectively. However, the average K for sedimentary rocks obtained by the fitting methods differ slightly from the statistical experimental results. The primary cause of the differences is that the fitting of equations (24) and (25) can introduce errors owing to a lack of experimental data for sedimentary rocks. Additional experiments are therefore required to verify the fitting equations and further adjust α. The above analysis clearly demonstrates that the presented equations for brittleness index can reliably predict K.Figure 7
Prediction ofK value for igneous, metamorphic, and sedimentary rocks using (a) method 1, (b) method 2, and (c) method 3.
(a)
(b)
(c)Table 4
Percentage ofK value range distribution of different rock types under three prediction methods.
Method
Percentage ofK value range distribution of igneous rock (%)
Percentage ofK value range distribution of metamorphic rock (%)
Percentage ofK value range distribution of sedimentary rock (%)
0.2∼0.4
0.4∼0.6
0.6∼0.8
0.2∼0.4
0.4∼0.6
0.6∼0.8
0.2∼0.4
0.4∼0.6
0.6∼0.8
1
8.6
53.4
31
42.9
28.6
19
50
35.4
2.1
2
13.8
60.3
22.4
42.9
33.3
23.8
6.25
29.2
47.9
3
15.5
58.6
22.4
47.6
33.3
19
6.25
29.2
47.9


## 6. Conclusions

In this study, we determined the relationship between brittleness index and crack initiation stress ratio (K) by fracture mechanics theory and indoor uniaxial compression tests. After theoretical derivation, the brittleness index B1 is found to have an inversely proportional relationship with K and brittleness index B2 has a variant inverse proportional relationship with K, whereas no relationship is observed between brittleness indexes B3 and B4 and K. Analysis of experimental data from igneous, metamorphic, and sedimentary rocks shows that B1 displays a power relationship with K, B2 has a linear relationship with K, and no relationship is observed between B3 and B4 and K. Seventy different types of uniaxial compression test data were collected from igneous, metamorphic, and sedimentary rocks, and consistent behavior is observed within the same rock types.The experimental data verifyK estimations based on Griffith’s theory, and the constant α is temporarily specified on the basis of available experimental data. For example, α = 8 for igneous and metamorphic rocks, whereas sedimentary rocks were defined as (according to results of limestone tests) α = 3 when σc < 60 MPa (high porosity), α = 5 when 60 MPa ≤ σc ≤ 90 MPa (moderate porosity), and α = 8 when σc > 90 MPa (low porosity). Further tests should be conducted to improve the determination of α.K values were estimated on the basis of 127 uniaxial compression test data for different rock types. The results show that K concentrates between 0.4 and 0.8 values for igneous rocks and between 0.2 and 0.6 for metamorphic rocks. In method 1, K values for sedimentary rocks concentrate between 0.2 and 0.6, whereas methods 2 and 3 yield K values between 0.4 and 0.8. Average K values for all rock types obtained by the three prediction methods are consistent with the experimental data. The redefined brittleness index formula can therefore be used for reliable predictions of K.



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