Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens

Engineering Geology 66 (2002) 79 – 97 www.elsevier.com/locate/enggeo

Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens Soo-Ho Chang *, Chung-In Lee, Seokwon Jeon School of Civil, Urban and Geosystem Engineering, Seoul National University, Seoul, South Korea Accepted 17 December 2001

Abstract Rock fracture mechanics has been widely applied to blasting, hydraulic fracturing, mechanical fragmentation, rock slope analysis, geophysics, earthquake mechanics, hot dry rock geothermal energy extraction and many other practical problems. But a standard method to accurately determine fracture toughness of rocks, one of the most important parameters in fracture mechanics as an intrinsic property of rock, has not been yet well established. To obtain rock fracture toughness, disc-type specimens were used in this study. Rock fracture toughness under mixed-mode conditions was measured by using the straightthrough crack assumption (STCA) applied to the cracked chevron-notched Brazilian disc (CCNBD) specimen and the semicircular bend (SCB) specimen. Size effects, in terms of specimen thickness, diameter and notch length on fracture toughness, were investigated. From the mixed-mode test results, fracture envelopes were obtained by applying various regression curves. The mixed-mode test results were also compared with the three well-known mixed-mode failure criteria. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Fracture toughness; Disc-type specimen; Mixed mode; Fracture envelope

1. Introduction Linear elastic fracture mechanics has been developed to describe crack growth and fracture within a material under essentially linear elastic conditions. It is based on the assumption that the influence of applied loads upon crack extension can be represented in terms of certain parameters that characterize the stress –strain intensity near the crack tip. The introduction of the fracture mechanics approach to engineering geology and rock engineering has led to

*

Corresponding author. Fax: +82-2-877-0925. E-mail address: [email protected] (S.-H. Chang).

the development of rock fracture mechanics, which mainly refers to the discrete initiation and propagation of an individual crack or cracks in geological materials subjected to a particular stress field (Whittaker et al., 1992). The explosion in rock fracture mechanics research has touched many diverse areas including blasting, hydraulic fracturing and in situ stress determination, mechanical fragmentation, rock slope analysis, earthquake mechanics, earthquake prediction, plate tectonics, magmatic intrusions, hot dry rock geothermal energy extraction, fluid transport properties of fracturing rock masses, propagating oceanic rifts, crevasse penetration and other glaciological problems, the development of steeply dipping extension fractures that are nearly ubiquitous at the earth’s surface and are

0013-7952/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 3 - 7 9 5 2 ( 0 2 ) 0 0 0 3 3 - 9

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normal to the plane of the crack. Mode II is the inplane sliding or shear mode, in which the crack faces are mutually sheared in the direction normal to the crack front. Mode III is the tearing or out of plane mode, in which the crack faces are sheared parallel to the crack front (Fig. 1). Among many different testing methods for rock fracture toughness, the International Society for Rock Mechanics (ISRM) suggested the chevron bend (CB) and the short-rod (SR) specimens in 1988 and cracked chevron-notched Brazilian disc (CCNBD) specimen in 1995. But it should be noted that most studies are relevant to mode I (opening mode) with some studies on mode II (in-plane shear mode) or the mixed-mode. Considering specimen geometries, tensile (mode I) cracks are induced during CB and SR tests. In addition, it has been reported that they are not appropriate for testing the fracture toughness of rock under mode II or mixed-mode cases (Fowell and Chen, 1990; Fowell and Xu, 1993; Lim et al., 1994a,b). Therefore, in this study, rock fracture toughness was measured using disc-type specimens such as CCNBD, semicircular bend (SCB), uncracked Brazilian Disc Test (BDT) and chevron-notched SCB specimens. Disc-type specimens are simple in geometry and have many advantages in terms of specimen preparation, testing and analysis (Table 1).

Fig. 1. Three basic modes of crack propagation.

formed through folding, upwarping and rifting and the modeling of time-dependent rock failure (Atkinson, 1987; Whittaker et al., 1992). A fundamental feature of rock fracture mechanics lies in its ability to establish the relationship between rock fracture strength to the geometry of a crack or cracks and the fracture toughness, the most fundamental parameter in fracture mechanics describing resistance of a material to crack propagation. It follows that for quasi-brittle geological materials, crack propagation is the major cause of material failure in many cases. Thus, assessment of fracture toughness is important to the understanding of behaviour of structures involving geological materials. In addition, rock fracture toughness has been applied as a parameter for classification of rock materials, an index for rock fragmentation process and a material property in the interpretation of geological features and in stability analysis of rock structures, as well as in modeling of fracturing in rock (Ouchterlony, 1988). According to the applied stress condition, a crack propagates under the three basic failure modes or the mixed-mode condition. Mode I is the tensile opening mode, in which the crack faces separate in a direction

2. Disc-type specimens for the fracture toughness test Four different kinds of disc-type specimens were used for measuring rock fracture toughness under

Table 1 Comparison between the various rock fracture toughness testing methodsa Item of comparison

CCNBD

SCB

CB

SR

Method of obtaining mixed-mode conditions Size of specimen Preparation apparatus Set-up of equipment Loading machines Loading method Reproducible data Requirement of testing machine

Rotate specimen

Vary notch angle

None

None

Small Simple Simple Compressive Compressive loading Excellent Ordinary

Small Simple Simple Compressive Three-point bending Excellent Ordinary

Long Simple Complex Compressive Three-point bending Reasonable High

Small Complex Complex Tensile Tensile loading Reasonable High

a

From Fowell and Chen (1990) and Lim et al. (1994b).

S.-H. Chang et al. / Engineering Geology 66 (2002) 79–97

modes I and II and mixed-mode conditions. Mode I fracture toughness was obtained by using disc-type specimens and this was compared with those obtained with the CB specimen, which was suggested by the ISRM in 1988. The theories and methods for disc-type specimens used in this study are as follows. 2.1. CCNBD specimen The cracked chevron-notched Brazilian disc (CCNBD) specimen (Fig. 2) has the same geometry and shape as the conventional Brazilian disc used for measuring the indirect tensile strength of rock, except that the CCNBD specimen has a chevron notch. Shetty et al. (1985) first used it for measuring the fracture toughness of ceramics, and applied the stress intensity factor solutions of a cracked straight-through Brazilian disc (CSTBD) with a through notch to the

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Table 2 Standard geometrical dimensions of the CCNBD specimen Description

Values

D (mm) B (mm) a0 (mm) a1 (mm) Ds (mm) hc (mm) Y* min (dimensionless) Notch width (mm) am (mm)

75.0 30.0 9.89 24.37 52.0 16.95 0.84 V 1.5 19.31

Dimensionless expression aB = B/R = 0.80 a0 = a0/R = 0.2637 a1 = a1/R = 0.65 as = Ds/D = 0.6933

am = am/R = 0.5149

CCNBD by means of the straight-through crack assumption (STCA) method. Afterwards, ISRM presented the suggested method for determining mode I fracture toughness using a CCNBD specimen because it has many advantages over other methods (Fowell, 1995). But this yields a solution only for mode I fracture toughness. The standard geometrical dimensions of a CCNBD specimen suggested by ISRM are presented in Table 2 and Fig. 2. All the geometrical dimensions are converted into dimensionless parameters with respect to the specimen radius R as follows. a0 ¼ a0 =R a1 ¼ a1 =R aB ¼ B=R as ¼ Ds =ð2RÞ

Fig. 2. The geometry of a CCNBD specimen.

ð1Þ

The chevron notch is made with two cuts from both sides of the disc along the disc-rotating axis on the same diametrical cutting plane, which is to be the designed crack orientation direction. First, the gap is set between the disc surface and the rotating wheel to zero. The first cut is made by moving the disc toward the rotating wheel up to the designed cutting depth hc. After this cut, the specimen together with the fixture are removed from the fixing vice and turned 180j. The specimen is cut to the same depth hc as the first cut (Fig. 3). The cutting depth is determined according to the specimen radius and the designed dimensionless geometric parameters such as a0, a1 or aB.

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where u and v are constants determined by a0 and aB only (Fowell, 1995). Rock fracture toughness under the mixed-mode condition, however, can not be calculated by the ISRM suggested method. Therefore, the STCA method was used to obtain fracture toughness under mode II and mixed-mode conditions in this study. Firstly, the numerical solutions of the stress intensity factors for the CSTBD specimen with a through notch length of 2a can be expressed in the following forms (Shetty et al., 1985). P pffiffiffiffiffiffi P pffiffiffi paNI ¼ pffiffiffiffiffiffi aN I pRB pRB P pffiffiffiffiffiffi P pffiffiffi paNII ¼ pffiffiffiffiffiffi aNII KII ¼ pRB pRB

KI ¼

ð4Þ ð5Þ

where P is the load applied in compression, a is half the notch length and NI and NII are dimensionless stress intensity factors depending on the dimensionless notch length a (a/R) and the notch inclination angle with respect to loading direction, h (Fig. 2), respectively. In this study, NI and NII solutions for the CSTBD specimen provided by Atkinson et al. (1982), Shetty et al. (1985) and Fowell and Xu (1993) were applied to calculate fracture toughness values by using the STCA method. Atkinson et al. (1982) developed NI and NII solutions for the CSTBD specimen given by five-term approximation (for 0.1 V a V 0.6) and small crack approximation (for a V 0.3). NI ¼ Fig. 3. Cutting procedure of the CCNBD specimen.

n X

Ti ðaÞ2i2 Ai ðhÞ

i¼1

The mode I fracture toughness is calculated by the ISRM suggested method as follows.

ðby five  term approximationÞ n X NII ¼ 2sin2h Si ðaÞ2i2 Bi ðhÞ

ð6aÞ

i¼1

KIC

Pmax ¼ pffiffiffiffi Ymin * B D

ð2Þ

ðby five  term approximationÞ NI ¼ 1  4sin2 h þ 4sin2 hð1  4cos2 hÞa2

where Ymin * is the critical dimensionless stress intensity value for a specimen which is determined by the specimen geometry a0, a1 and aB only, and is calculated using the following formula.

ðby small crack approximationÞ h i NII 2 þ ð8cos2 h  5Þa2 sin2h

Ymin * ¼ ueva1

ðby small crack approximationÞ

ð3Þ

ð6bÞ

ð7aÞ

Z NII ¼ t 2þ ð8cos2 h  5Þa2 b sin2h ð7bÞ

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where the first five values of Ti and Si and the corresponding Ai(h) and Bi(h) are given in Tables 3 and 4, respectively. Unless the dimensionless notch length a matched the values listed in the Table 3, linear interpolation or extrapolation was used to calculate them. Shetty et al. (1985) used the following simple third-order polynomial (for 0.1 V a V 0.6) to fit the results of Atkinson et al. (1982) with an error of less than 0.1%. NI ¼ 0:991 þ 0:141a þ 0:863a2 þ 0:886a3

ð8Þ

Fowell and Xu (1993) calculated the NI solution by the following polynomial (for 0.05 V a V 0.95). pffiffiffiffiffiffiffiffi NI ¼ p=að0:0354 þ 2:0394a  7:0356a2 3

4

þ 12:1854a þ 8:4111a  30:7418a

5

 29:4959a6 þ 62:9739a7 þ 66:5439a8  82:1339a9  73:6742a10 þ 73:8466a11 Þ

ð9Þ

For comparison, dimensionless mode I stress intensity factors (h = 0) calculated by Eqs. (6a), (7a), (8) and (9) are shown in Fig. 4. By applying the STCA method to p a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CCNBD ffi specimen, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii.e., replacing B with B  ða  a0 Þ= p ða1  a0 Þ in Eqs. (4) and (5), we can obtain the following relation for the CCNBD specimen. It is assumed that the crack front width increases linearly

Table 3 The first five values of Ti and Si a

0.1 0.2 0.3 0.4 0.5 0.6

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Table 4 The first five angular constants of Ai(h) and Bi(h) A1 A2 A3 A4 A5 B1 B2 B3 B4 B5

1  4sin2h 8sin2h(1  4cos2h)  4sin2h(3  36cos2h + 48cos4h)  16sin2h(  1 + 24cos2h  80cos4h + 64cos6h)  20sin2h(1  40cos2h + 240cos4h  448cos6h + 256cos8h) 1  5 + 8cos2h  3 + 8(1  2cos2h)(2  3cos2h) 3 + 16(1  2cos2h)  12(1  2cos2h)2  32(1  2cos2h)3 5  16(1  2cos2h)  60(1  2cos2h)2 + 32(1  2cos2h)3 + 80(1  2cos2h)4

from zero to B as the dimensionless crack length increases from a0 to a1. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P pffiffiffi a1  a0 a KI ¼ pffiffiffiffiffiffi NI a  a0 pRB

ð10Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P pffiffiffi a1  a0 a KII ¼ pffiffiffiffiffiffi NII a  a0 pRB

ð11Þ

Fig. 5 shows an example of the STCA method. The stress intensity factor for a chevron notch decreases with increasing crack length for a constant applied load and then a sharp crack initiated at the tip of the chevron notch requires a continually increasing load to grow stably at KI = KIC to the base of the notch, i.e., a = a1. At this crack length, the crack should become unstable since further growth would require a decreasing load. Thus fracture toughness is calculated from the instability crack length (a = a1), the corresponding values of NI and NII, and the maximum load. 2.2. SCB specimen

T1

T2

T3

T4

T5

S1

S2

S3

S4

S5

1.014998 1.009987 1.060049 1.039864 1.135551 1.089702 1.243134 1.160796 1.387239 1.257488 1.578258 1.390654

0.503597 0.502341 0.514907 0.509959 0.533477 0.522272 0.559734 0.539824 0.594892 0.563966 0.642124 0.597985

0.376991 0.376363 0.382430 0.379956 0.391640 0.386086 0.404603 0.394822 0.421949 0.406869 0.445387 0.424037

0.376991 0.376363 0.383392 0.380584 0.393835 0.387518 0.408597 0.397403 0.428353 0.410966 0.454861 0.430072

0.314159 0.314159 0.318086 0.316245 0.325033 0.320834 0.334831 0.327411 0.347941 0.336447 0.365559 0.349219

The application of a tensile load may cause rupture from an existing defect situated away from the crack tip. However, the application of compressive loads tends to restrict regions of tensile stresses around the crack tips. Therefore, the possibility of rupture from existing defects located some distance from the crack tips is significantly reduced. In addition, it is very difficult to glue any specimen to the holders usually required for tensile loading. Consequently, tests should preferably be done with compressive loadings where tensile (mode I) fractures are induced (Lim et al., 1994b). In order to satisfy this requirement, Chong

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Fig. 4. Dimensionless mode I stress intensity factors for the CSTBD specimen.

and Kuruppu (1984) proposed using a semicircular core specimen with a single edge notch, subjected to a three point loading (Fig. 6).

Mixed-mode conditions can be obtained by changing the angle between loading direction and the notch in the same way as for the CCNBD specimen.

Fig. 5. Dimensionless stress intensity factors for straight-through and chevron cracks in a CCNBD specimen.

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where dimensionless stress intensity factors, YI and YII, are determined by the dimensionless notch length (a/D), the notch inclination angle with respect to the loading direction (a), and the loading span (Fig. 6). In this study, fracture toughness under mode I and mixed-mode conditions was calculated from the stress intensity factor solutions provided by Chong et al. (1987) and Lim et al. (1993). In addition, the loading span normalized with specimen diameter (s/D) was set to 0.4, and dimensionless notch lengths (a/D) had a range from 0.066 to 0.3971. 2.3. Chevron-notched SCB specimen

Fig. 6. Semicircular bend (SCB) test specimen.

Chong et al. (1987) developed a formula for KI by using both the strain energy release rate method and the elliptical displacement approach. pffiffiffiffiffiffi P pa Yk KI ¼ ð12Þ DB where Yk is the dimensionless stress intensity factor as a function of the dimensionless crack length, a/D, with D being the disk diameter. Yk can be approximated by a third-order polynomial as follows. Yk ¼ 4:47 þ 7:40

 a 2  a 3 a  106:0 þ433:3 D D D

Kuruppu (1997) performed a 3D finite element analysis to obtain the crack tip stress intensity factors of a chevron-notched SCB specimen as a function of the crack length. Fig. 7 shows the shape of the chevron-notched SCB specimen. The initial crack length a0 is 6 mm and the thickness t is 25 mm. All specimens must have a span to radius ratio (S/R) of 0.8 and a chevron notch angle h of 90j. The average value of the stress intensity factor determined along the crack front is normalized as follows. Knd ¼

KI pffiffiffi t R P

where Knd is the normalized stress intensity factor, P is the applied load and t and R are the specimen thickness and radius, respectively. Fig. 8 shows the

ð13Þ for 0.25 V a/D V 0.35 and s/D = 0.4 with s being the loading span. Lim et al. (1993) also developed KI and KII solutions of the SCB specimen (for 0.05 V a/D V 0.4 and s/D = 0.25, 0.305, 0.335 or 0.4). pffiffiffiffiffiffi P pa YI KI ¼ DB KII ¼

pffiffiffiffiffiffi P pa YII DB

ð14Þ

ð15Þ

Fig. 7. The chevron-notched SCB test specimen.

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Fig. 8. Normalized mode I stress intensity factor for the chevron-notched SCB specimen.

normalized stress intensity factor vs. the normalized crack length. The stress intensity factor of the chevron-notched SCB specimen has a minimum value in the same way as other chevron-notched specimens. Initial crack

growth in a chevron notch occurs in a stable manner during which the load increases and the specimen fails immediately beyond its maximum load bearing capacity. Therefore, the maximum load was used as the critical fracture load. This value along with the

Fig. 9. Typical failure curve for a BDT specimen.

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minimum value of the stress intensity factor given in Fig. 8 was used to determine the mode I fracture toughness. 2.4. BDT specimen The uncracked Brazilian Disc Test (BDT) specimen has the same geometry as a conventional Brazilian disc without any notch or crack (Guo et al., 1993). The nonrequirement of a notch or crack is the advantage of this method. One of the features of the BDT specimen is that it uses the local minimum load as the critical load instead of the maximum load. The local minimum load is defined as Pmin in Fig. 9. The starting point of the bc region where unstable crack propagation is finished and crushing or secondary cracks are developed is defined as the local minimum load. Most tests were stopped at the beginning of the bc stage, when the local minimum load can be clearly identified from the recorded graph. 2.5. CB specimen The chevron bend (CB) specimen with the short rod (SR) specimen was suggested by the ISRM (Ouchterlony, 1988). Only mode I fracture toughness can be measured by using a CB specimen. There are Levels I and II tests that consider nonlinearity of rocks, but only the Level I test was carried out in this study.

3. Experiments Keochang Granite and Yeosan Marble produced in Korea were the main rock types that were used for testing. CCNBD specimens 75 and 54 mm in diameter and 15 – 35 mm in thickness were used to investigate effects of specimen size on fracture toughness values. Specimens 100 and 75 mm in diameter and 15 – 40 mm in thickness were used for SCB specimens. BDT specimens were prepared with the same shape as the conventional Brazilian disc used to measure indirect tensile strength of rock. The diameter and thickness of the chevron-notched SCB specimens were 75 and 23 –32 mm, respectively. The cutting machine for preparing a chevron notch for the CCNBD specimen was manufactured to satisfy

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the ISRM suggested geometrical conditions for the CCNBD specimen. The diameter and thickness of the diamond saw were 50 and 0.8 mm, respectively. A dial gauge with 1/100 mm precision was attached to the cutting machine to precisely control the cutting depth. Servo-controlled hydraulic testing machines, MTS 815 and 810, were used for compressive and threepoint bending testing, respectively. The test was displacement controlled at the rate of 0.01 mm/s.

4. Mode I fracture toughness results 4.1. CCNBD tests Fig. 10 shows the results of mode I fracture toughness test performed on the CCNBD specimens with a 75-mm diameter for granite and marble. In the case of marble, the specimens with thickness of less than 15 mm had larger fracture toughness than others. This coincided with the fact that fracture toughness becomes larger as thickness gets smaller. It is thought that reasonable fracture toughness values may be obtained if a specimen with thickness of larger than 15 mm is used. Four indices except the ISRM suggested method in Fig. 10 represent fracture toughness values obtained by applying the STCA method to the CCNBD specimen. Comparing the results from the STCA method, those by five-term approximation, third-term approximation, and Fowell and Xu’s solutions were very close to each other. But Atkinson’s small crack assumption showed very large difference from the others. The small crack assumption has simpler formulas and is applied to a case of a less than 0.3 (Eqs. (7a) and (7b)). Considering that a normalized notch length (a1) used in this study was 0.5 f 0.6, it is thought that Atkinson’s small crack assumption could not be applied to the sample geometry and shape used in this study. In addition, the results from the STCA method except for the small crack assumption were approximately 15% lower than those by the ISRM method. This discrepancy may rise from the errors by approximation of CSTBD stress intensity factor solutions to CCNBD. Fig. 11 shows the effect of notch length and shape on fracture toughness. Notch length was normalized by specimen radius, and a1 – a0 represents a normal-

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Fig. 10. Effect of thickness on mode I fracture toughness (CCNBD, D = 75 mm).

ized length of a chevron notch. There was little effect of notch length, as shown in Fig. 11. a1 and a0 also showed little effect on fracture toughness.

Fig. 12 was obtained from the results of CCNBD specimens of 54-mm diameter. There was little effect of size, thickness and notch length on mode I fracture

S.-H. Chang et al. / Engineering Geology 66 (2002) 79–97

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Fig. 11. Effect of normalized chevron notch length on mode I fracture toughness (CCNBD, D = 75 mm).

toughness, as it was in the case of the 75-mm diameter specimens. But results from the STCA, except those of the small crack assumption, were close to one another,

as well as to the results from the ISRM suggested method. Using the 50-mm diameter diamond saw manufactured for this study, the dimensionless notch

Fig. 12. Effect of thickness on mode I fracture toughness (CCNBD, D = 54 mm).

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S.-H. Chang et al. / Engineering Geology 66 (2002) 79–97

Fig. 13. Mode I fracture toughness values versus normalized notch length curve from SCB tests.

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length a1, was set to about 0.8. In case of using such a large notch dimension in comparison to the specimen diameter, it is known that errors in the solutions used in the STCA become larger (Whittaker et al., 1992). Consequently, it is thought that due to such errors, results from the STCA were close to those from the ISRM method. Therefore, if the STCA is applied and a 50-mm diameter saw should be used, specimens with a diameter larger than 54 mm must be used in order to reduce such errors. Because a1 was greater than 0.6, Atkinson’s five-term approximation was not applied here. 4.2. SCB tests Mode I fracture toughness values obtained from SCB specimens were not constant with the notch length. The values were very scattered and smaller than those from CCNBD tests, as shown in Fig. 13. The reason for such a result may be that precracking was not performed. Since crack tips obtained from precracking are in crystal grains, in which case it usually resists crack propagation, more energy is required to cause cross-grain (transgranular) crack propagation (Whittaker et al., 1992). Due to this fact, a lower failure load was obtained in this study in which precracking was not carried out and consequently lower fracture toughness values. Although the mode I fracture toughness values for marble had a large scatter, they were somewhat close to those from CCNBD tests. The reason may be because mineral composition of marble is weaker than granite, and consequently less energy for crack propagation through grains is required. In the case of Lim et al.’s (1994a,b) study, reasonable mode I fracture toughness values were obtained without precracking for soft rocks such as mudstone and oil shale. Therefore, it is thought that precracking is indispensable for SCB tests for hard rocks. 4.3. Chevron-notched SCB tests The chevron-notched SCB specimen has the same specimen geometry as the SCB specimen with a through-notch. The only difference is that the chevron-notched SCB specimen uses a chevron notch instead of a through-notch. In contrast to results from SCB tests, mean fracture toughness values of the

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chevron-notched SCB tests were very close, although slightly greater, to those from CCNBD tests. Consequently, a chevron notch is considered to be very useful for obtaining reasonable rock fracture toughness values when compared to a through-notch of SCB specimens. When a V-shaped chevron notch is used, the length of the crack front is gradually increased as the crack propagates and any further crack propagation requires a further increase in the applied load. This leads to a stable crack propagation which can be readily controlled and self-precracking is essentially achieved during testing. In addition, the tests with chevron-notched specimens require neither crack length nor displacement measurement, nor complicated fracture toughness calculation techniques, but just require the recording of the maximum applied load. As a result, considering difficulties in precracking as described in Section 4.2, it may be reasonable to use a chevron notch. 4.4. BDT tests Although a standard deviation of mode I fracture toughness values obtained from BDT specimens is greater than those from other specimens, the BDT specimen has reasonable mode I fracture toughness values. Therefore, considering the fact that uncertainty of a notch or a crack may be avoided, the BDT specimen may be useful in measuring mode I fracture toughness. However, it has a limitation in that only mode I fracture toughness can be measured. Table 5 Summary of mode I fracture toughness values Granite a

Mean

Marble b

S.D.

Number Meana S.D.b

CCNBD 1.3509 0.0634 19 (D = 75 mm) CCNBD 1.3376 0.0815 8 (D = 54 mm) SCB 0.6836 0.1874 31 Chevron1.3932 0.0274 5 notched SCB BDT 1.2894 0.1545 9 pffiffiffiffi a (MPa m). b Standard deviation.

Number

1.0605 0.0785 19 1.1815 0.1134

9

0.8711 0.1536 27 1.1133 0.0367 5

0.9865 0.1586 10

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Fig. 14. Relationships between physico-mechanical properties and fracture toughness of rock.

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4.5. Summary of mode I fracture toughness results Mode I fracture toughness values obtained in this study are summarized in Table 5. The results obtained by four different methods were very close to one another, except for those of the SCB specimens. The 75-mm diameter CCNBD specimen has a smaller standard deviation than the other methods except for the chevron-notched SCB specimen. Although the chevron-notched SCB specimen has the smallest standard deviation, it was used only five times. Therefore, further research may be necessary to certify the repeatability of the results obtained with the chevronnotched SCB specimen. 4.6. Comparison between CB tests and CCNBD tests Mode I fracture toughness values obtained with CCNBD specimens were compared with those of CB specimens for five different rock types. Values of the CCNBD specimens were slightly higher than those of the CB specimens, but there is a close relationship between them (R = 0.97). 4.7. Relationship between physico-mechanical properties and mode I fracture toughness of rock From a practical viewpoint, it is necessary and important to predict fracture toughness of rock based on its relationships with physico-mechanical properties of the rock. This work is of special significance because the fracture behaviour of rocks can be more readily evaluated and predicted through these relationships. The relationships between physico-mechanical properties of rock and mode I fracture toughness were investigated for eight different rock types (Fig. 14). The fracture toughness values were determined from the CCNBD specimens suggested by the ISRM (Fowell, 1995). Among some of physico-mechanical properties, the relationship between fracture toughness and acoustic wave velocity, especially P-wave velocity, was the closest (R = 0.80). This result agreed with those of Huang and Wang (1985). It is generally known that seismic velocity varies with mineral composition, density, porosity, and especially the degree of fracturing and fissures. Therefore, considering the fact that fracture toughness represents the resistance of a material to fracturing, it is thought that acoustic

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wave velocity can represent the resistance of rock to fracturing or failure better than any other physicomechanical property.

5. Mixed-mode and mode II fracture toughness results Mixed-mode and mode II fracture toughness values were measured using the STCA and SCB specimens, but the results of the SCB specimens showed a great difference from those of the STCA as in the case of mode I results (Section 4.2). Therefore, when mixed-mode results were analyzed and interpreted, those from the SCB specimens were omitted. In this study, NI and NII solutions by five-term approximation (Eqs. (6a) and (6b)) were applied to calculate mixedmode and mode II fracture toughness values from the STCA method. A notch angle (h) relative to a loading direction for mode II condition (mode I stress intensity factor equal to zero, NI = 0) was numerically determined by the secant method used to estimate a root (namely, h at NI = 0), and mode II fracture toughness values were measured by precisely maintaining the determined notch angle. The notch angles for mode II condition were slightly over 20j for CCNBD specimens used in this study. Mode II fracture toughness values were measured three times for both granite and marble. The average mode II fracture toughness values for granite and marblepffiffiffiffi were 1.7924 F 0.0423 and 1.3481 F 0.1085 MPa m , respectively, and their normalized values with mode I fracture toughness values are summarized in Table 6. Mixed-mode fracture toughness values were measured by maintaining the angle (h) between 0j and the determined notch angle for mode II condition.

Table 6 Comparison of KIIC/KIC ratios by experiment, three mixed mode criteria and an empirical regression curve Rock type KIIC/KIC Experimental results Granite Marble

Gmax rmax Smin

Regression curve (C in Eq. (16))

1.3268 F 0.0313 0.627 0.866 1.043 1.2569 1.3481 F 0.1085 0.627 0.866 0.935 1.5584

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Fig. 15. Comparison between experimental results and three mixed mode criteria.

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Fig. 16. KI versus KII empirical regression curves (granite).

Experimental results were compared with three mixed-mode fracture criteria (Whittaker et al., 1992) which include the maximum energy release rate criterion, Gmax, the maximum stress criterion, rmax,

and the minimum strain energy density criterion, Smin. The three mixed-mode fracture criteria have provided a sound theoretical basis for predicting the location of fracture initiation and direction of propagation in rock

Fig. 17. KI/KIC versus KII/KIIC empirical regression curves (granite).

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subjected to a mixed-mode load, and have been applied to the numerical modeling of fracture initiation and propagation in rock. This study was intended to find which criterion was the closest to experimental results by comparing mixed-mode fracture toughness values with fracture envelopes obtained from the three mixed-mode fracture criteria. Experimental results were higher than the values predicted by the three criteria (Fig. 15). Among the three criteria, the minimum strain energy density criterion, Smin, which considers material properties, was the closest to the experimental results. This result agrees with those by Ingraffea (1981). Therefore, the minimum strain energy density criterion is most preferable to get the mixed-mode fracture criterion, based on this study. Some researchers suggested various empirical regression curves for the fracture envelope (Whittaker et al., 1992; Awaji and Sato, 1978). The x and y axes can be represented by KI vs. KII (Fig. 16) or KI/KIC vs. KII/KIC (Fig. 17) according to the used regression curves. The correlation coefficient of the regression curves for the fracture envelope showed little difference from one another. Therefore, it is reasonable to select an appropriate empirical regression curve for the tested materials or testing objectives. The KIIC/KIC ratios obtained from the experiment, the three mixed-mode criteria, and the empirical regression curve (Palaniswamy and Knauss, 1978) given by Eq. (16) are compared in Table 6. The ratio obtained from the experiment showed some difference from the ratio predicted by the three mixed-mode criteria, but the ratio predicted by the empirical regression curve was close to the experimental result. So, when it is difficult to measure mode II fracture toughness, it can be reasonably predicted from the regression result. KI þ KIC



KII CKIC

2 ¼1

ð16Þ

where C is KII/KIC ratio, i.e., an empirical constant.

6. Conclusions The results, analysis and discussions of this study allowed the following conclusions to be drawn.

(1) Among the mode I fracture toughness testing methods used in this study, the CCNBD specimen had the smallest standard deviation in mode I fracture toughness, and the effects of the diameter, notch length and thickness on fracture toughness were negligible for the geometry and shape of the CCNBD specimen used in this study. In addition, the CCNBD specimen can be used to measure mixed-mode and mode II fracture toughness values by the STCA method. It is also unnecessary to perform precracking for the CCNBD specimen because it uses a chevron notch which induces self-precracking during testing and leads to a stable crack propagation. Consequently, it is concluded that the CCNBD specimen is the most preferable and versatile among disc-type specimens used in this study. (2) Results from SCB tests showed large scatters and inconsistent patterns with a notch length. The reason could be found in the fact that precracking was not carried out. So it is recommended that precracking should be carried out, especially for hard rocks. However, considering difficulties of precracking, the use of a chevron notch is preferred. (3) Mode I fracture toughness values from chevronnotched SCB and BDT tests were very close to those by CCNBD tests. Results from BDT tests had a relatively large scatter. However, considering that uncertainty by an artificial notch can be avoided, BDT is thought to be a good method for mode I fracture toughness measurements. (4) Mode I fracture toughness values from CB and CCNBD tests showed very close relationships with one another. Among various physico-mechanical properties of rocks, P-wave velocity and mode I fracture toughness showed the best relationship. Therefore, it is deduced that P-wave velocity can represent the resistance of rock to fracturing or failure better than any other physico-mechanical property. (5) Mixed-mode fracture toughness values showed some differences from the fracture envelopes predicted by the three mixed-mode fracture criteria. Among the three mixed-mode failure criteria, the minimum strain energy density criterion was the closest to the experimental results. (6) The ratio of modes II to I fracture toughness predicted by an empirical regression curve was very similar to the experimental result. Therefore, when it is difficult to measure mode II fracture toughness, it

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can be reasonably predicted from the regression result. Acknowledgements The authors would like to thank Dr. Mahinda Kuruppu in Western Australian School of Mines for his recommendations and help in SCB and chevronnotched SCB tests. References Atkinson, B.K., 1987. Fracture Mechanics of Rock. Academic Press, London. Atkinson, C., Smelser, R.E., Sanchez, J., 1982. Combined mode fracture via the cracked Brazilian disk test. Int. J. Fract. 18 (4), 279 – 291. Awaji, H., Sato, S., 1978. Combined mode fracture toughness measurement by the disk test. J. Eng. Mater. Technol. 100, 175 – 182. Chong, K.P., Kuruppu, M.D., 1984. New specimen for fracture toughness determination for rock and other materials. Int. J. Fract. 26, R59 – R62. Chong, K.P., Kuruppu, M.D., Kuszmaul, J.S., 1987. Fracture toughness determination of layered materials. Eng. Fract. Mech. 28 (1), 43 – 54. Fowell, R.J., 1995. Suggested method for determining mode I fracture toughness using cracked chevron notched Brazilian disk (CCNBD) specimen. Int. J. Rock Mech. Mineral Sci. Geomech. Abstr. 32 (1), 57 – 64. Fowell, R.J., Chen, J.F., 1990. The third chevron-notch rock fracture specimen—the cracked chevron-notched Brazilian disk. Proc. 31st U.S. Symp. Rock. Balkema, Rotterdam, 295 – 302. Fowell, R.J., Xu, C., 1993. The cracked chevron notched Brazilian disc test—geometrical considerations for practical rock fracture

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