Rock fracture toughness study using cracked chevron notched Brazilian disc specimen under pure modes I and II loading – A statistical approach

Theoretical and Applied Fracture Mechanics 69 (2014) 17–25

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Rock fracture toughness study using cracked chevron notched Brazilian disc specimen under pure modes I and II loading – A statistical approach M.R.M. Aliha a, M.R. Ayatollahi b,⇑ a

Welding and Joining Research Center, School of Industrial Engineering, Iran University of Science and Technology, Narmak, 16846-13114 Tehran, Iran Fatigue and Fracture Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, 16846-13114 Tehran, Iran b

a r t i c l e

i n f o

Article history: Available online 11 December 2013 Keywords: CCNBD specimen Pure mode I Pure mode II Rock GMTS criterion Statistical analysis

a b s t r a c t Fracture toughness of a white marble is studied experimentally using several cracked chevron notched Brazilian disc (CCNBD) specimens under pure mode I and pure mode II loading. Even in the presence of natural scatters in the test data, it was observed that the average mode II fracture toughness KIIc was considerably larger than that of mode I fracture toughness KIc such that the mean fracture toughness ratio (KIIc/KIc) was about 2. Using the generalized maximum tangential stress theory, the obtained mode II test results were estimated in terms of mode I fracture toughness data. The enhanced KIIc value in the CCNBD specimen could be related to the influence of very large negative T-stress value that exists in the mode II CCNBD specimens. The statistical analyses of test data were performed successfully to predict the Weibull parameters of mode II results in terms of mode I Weibull parameters. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction During the past decades extensive experimental, theoretical and numerical research studies have been performed on fracture behavior of different types of rock materials. Due to brittle or quasi-brittle behavior of rocks, especial test configurations are usually required for obtaining the strength properties and fracture toughness of such materials. For example, most of the test specimens for rock experiments have cylindrical or disc shape configurations. This is mainly because of the convenience in the specimen preparation from cylindrical rock cores. Accordingly, the international society for rock mechanics (ISRM) has suggested some standard test specimens and methods for obtaining the mode I fracture toughness (KIc) of rocks using disc type specimens. The chevron notch short rod (CNSR) specimen, the chevron notch cylindrical specimen subjected to symmetric three-point bend loading, the cracked chevron notched Brazilian disc (CCNBD) specimen subjected to diametral compression and the edge cracked semi circular bend specimen under three-point bend loading are four ISRM suggested test methods for experimental determination of a versatile and reliable value for mode I fracture toughness of rocks [1–3]. Since introducing fatigue pre-cracks in brittle materials such as ceramics, graphite and rocks is very difficult, it is usually preferred to create a chevron notch instead of straight crack in the test samples made from these materials [1,2,4–6]. According to ISRM, the ⇑ Corresponding author. Tel.: +98 21 77240201; fax: +98 21 77240488. E-mail address: [email protected] (M.R. Ayatollahi). 0167-8442/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tafmec.2013.11.008

chevron notches allow a stable crack growth from the notch tips prior to the final fracture load and the chevron notched specimens can produce reasonable, versatile and reproducible data for fracture toughness of rocks. Hence, many of rock fracture researchers have employed one of these samples in their fracture toughness experiments [7–13]. However, among the mentioned test methods, the CCNBD specimen has received more attention by researchers for conducting rock fracture toughness tests. Simple geometry, ease of diametral loading by two flat platens and ability of CCNBD testing with the conventional test machines are some of the advantages of this specimen. Meanwhile, cracked rock masses are often subjected to complex mixed mode I/II (tensile–shear loads) conditions. The CCNBD specimen is also able to introduce different mode mixities ranging from pure mode I to pure mode II easily by changing the crack orientation relative to the direction of applied diametral compressive load. Hence, a large number of papers have employed this circular shape configuration for fracture toughness studies [11–24]. The un-cracked Brazilian disc specimen has been also widely used as an ISRM suggested specimen for obtaining the indirect tensile strength of rocks or geo-materials [25–27]. However, a review of literature shows that most of the papers have focused mainly on determining KIc using the CCNBD specimen and only a few papers deal with pure mode II fracture toughness experiments using this specimen. On the other hand, due to the presence of inherent heterogeneity, porosity, bedding planes, etc., a large scatter in the fracture toughness data is often expected to exist for rock materials. For such cases, the average of test data obtained from a small number

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of specimens fails to provide a versatile and reliable value for fracture toughness, and hence it is very useful to investigate the fracture toughness of rocks statistically using a larger number of test samples. In this paper, pure mode I and pure mode II fracture toughness of a marble rock is obtained experimentally using several CCNBD specimens and the scatter in the obtained results are analyzed using a probabilistic model. It is shown that the mean mode II fracture toughness and also the probabilistic curve of pure mode II data can be predicted very well from pure mode I results when a generalized maximum tangential stress criterion is used. 2. Cracked chevron notched Brazilian disc specimen Fig. 1 shows the geometry and loading configuration of the cracked chevron notched Brazilian disc (CCNBD) specimen. In this specimen, two chevron notches are cut at the center of disc from each side by using a high speed sawing machine with a very narrow diamond blade. The following dimensionless parameters are often used for characterizing the geometry of chevron notch in the CCNBD specimen:

a0 ¼ a0 =R a1 ¼ a1 =R aB ¼ B=R aS ¼ DS =R

ð1Þ

where R is the radius of disc, B is the thickness of disc and Ds is the diameter of circular cutting blade, respectively. When the crack orientation angle (a) relative to applied load is zero, the CCNBD specimen is subjected to pure mode I condition. For a > 0, the CCNBD specimen experiences mixed mode I/II loading. According to the ISRM [2], in order to obtain a valid plane strain fracture toughness value using the CCNBD specimen, some geometrical requirements should be provided for the overall dimensions of circular disc and also for the created notch. The important requirements are mentioned below and also schematically are shown in Fig. 2.

a1 P 0:4 line 1 a1 P aB =2 line 2 ab 6 1:04 line 3 a1 6 0:8 line 4 aB P 1:1729a1:6666 line 5 1 aB P 0:44 line 6

ð2Þ

The initial crack length ratio a0/R should be chosen between 0.2 and 0.3 [2] and by selecting three main dimensions of CCNBD specimen (i.e. a0, a1 and aB), the other parameters can be determined from the following equations:

B

Fig. 1. Geometry and loading condition of CCNBD specimen.

Fig. 2. Geometrical requirements of CCNBD specimen for obtaining valid fracture toughness values [2].

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a20 þ ða21  a20 þ a2B =4Þ=a2B   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hc ¼ aS  a2S  a21 R ¼ aS  a2S  a20 R þ B=2

aS ¼ DS =R ¼

ð3Þ

In the next section the modes I and II fracture toughness testing of a rock material using the CCNBD specimen is described.

3. Fracture toughness experiments For conducting rock fracture toughness tests using the CCNBD specimen, a white marble called coarse grain Harsin marble excavated from the west of Iran was selected. For manufacturing the test specimens, three rods of length 900 mm and diameter approximately 100 mm were prepared from Harsin marble. In order to reduce the possible effects of rock anisotropy, the marble cylinders were marked before cutting with a narrow line along the longitudinal direction of the cylinders. This line was used for loading all the CCNBD specimens along a fixed diametral direction. Then, the rods were sliced with a diamond saw blade to obtain disc specimens of approximately 30 mm thickness. Two chevron notches were cut in the center of discs from each side using a thin rotary diamond saw with a diameter of Ds = 80 mm and a thickness of t = 1 mm. The indentation depth (hc) of the rotary saw at each side of the disc was about 17 mm to introduce two V shape chevron notches in the circular disc. The diamond saw was cooled by water during the chevron notch cutting process. The procedure for cutting the chevron notches in the circular disc specimen has been shown schematically in Fig. 3. A total number of 44 CCNBD specimens were manufactured from the Harsin marble rock. The prepared test specimens were then located in appropriate positions inside the two flat plates of test machine and loaded by a diametral compressive force P. While for pure mode I loading, the crack direction should be exactly along the applied diametral force (i.e. a = 0°), pure mode II condition in the CCNBD specimen is achieved when the crack inclination angle is set up in a specific angle depending on the crack length ratio (a/R). Some researchers have determined this angle for the Brazilian disc specimen using numerical and theoretical methods. For example, Ayatollahi and Aliha [28] analyzed the Brazilian disc specimen by the finite element method and obtained the corresponding pure mode II crack inclination angle (aII) for different a/R ratios. Fig. 4 shows the variations of aII versus a/R according to the data given in [28]. Using the measured values of a0 and a1, the average crack length a is

M.R.M. Aliha, M.R. Ayatollahi / Theoretical and Applied Fracture Mechanics 69 (2014) 17–25

19

Fig. 3. Introducing chevron notches at the center of disc specimen.

found to be about 22.5 mm. Thus by dividing this value to the average radius of manufactured discs (R = 52 mm), the crack length ratio a/R = 0.43 is considered for the mode II experiments in the present study. Therefore, the corresponding value of aII is found from Fig. 4to be about 24°. From the total 44 manufactured CCNBD specimens, 22 samples were tested under pure mode I (with a = 0°) and the rest of specimens were tested under pure mode II conditions (i.e. with aII = 24°). Fracture toughness tests were conducted by a servo-hydraulic tensile–compressive test machine and the CCNBD specimens were loaded monotonically up to the final fracture load. The load–displacement curves were nearly linear and all the samples were fractured suddenly at a critical fracture load showing the predominantly linear elastic fracture behavior of tested CCNBD specimens. While crack growth for mode I samples was self-similar and along the chevron notch plane, the crack in mode II samples extended out of the plane of chevron notches. Fig. 5 shows the photos of typical CCNBD marble specimens fractured under pure mode I and pure mode II loading. For determining pure mode I fracture toughness (KIc), ISRM [2] suggests that only the fracture load (Pmax) of CCNBD specimen is required from the test process, while the total time for each fracture test should be less than 20 s. Therefore, the loading rate was chosen equal to 0.2 kN/s to satisfy the ISRM requirements. Mode I fracture toughness for the CCNBD specimen is determined from [2]:

Pmax K Ic ¼ pffiffiffiffi Y min B D

ð4Þ

where the critical non-dimensional stress intensity factor Y min is obtained in terms of a0, a1, a2 from the following equation:

Y min ¼ uev a1

ð5Þ

Pure mode II crack inclination angle (αII) - degrees

32 30 28

where u and v are the constant parameters given by [2] in terms of

a0 and aB. Table 1 presents the geometrical parameters, the critical fracture loads and the corresponding values of KIc for the tested mode I CCNBD specimens. Pure mode II fracture toughness KIIc value was determined from the following equation:

Pmax K IIC ¼ pffiffiffiffiffiffiffi pRB

rffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a1  a0 Y II R a  a0

ð6Þ

where YII is pure mode II geometry factor which depends on the crack length ratio (a/R) in the Brazilian disc specimen. This geometry factor can be determined either from the theoretical equation given in [29]

Y II ¼ 2 sin 2u

 2i2 n X l Si Bi ðuÞ a i¼1

ð7Þ

or numerically using a finite element model [28] in which the parameters given in Eq. (7) have been defined by Atkinson et al. [29]. Variations of YII with a/R for the Brazilan disc specimen, extracted from Ayatollahi and Aliha [28], have been presented in Fig. 6. It is worth mentioning that the real test specimens have three-dimensional shape and the effect of thickness may influence the fracture parameters and the experimental results. Hence, some researchers have studied three dimensional fracture behavior of cracked specimens [22,30–34]. Although these studies show a slight difference between the 2-D and 3-D fracture parameters, but using simple and commonly used 2-D fracture parameters (obtained from Refs. [28,29]) are still valid for investigating the crack growth resistance of real 3-D test configurations. For example, Abd-Elhady [22], analyzed the effect of specimen thickness on the mixed mode stress intensity factors of three-dimensional Brazilian disc specimen. Based on his numerical results, the influence of thickness on mode II stress intensity factor is negligible and on mode I stress intensity factor is typically less than 8% for different thickness-to-diameter ratios. The results obtained from the KIIc experiments are presented in Table 2 for 22 CCNBD specimens tested under pure mode II loading conditions. 4. Results and discussion

26 24 22 20 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

a/R Fig. 4. Variations of pure mode II inclination angle (aII) in terms of crack length ratios in the CCNBD specimen [28].

The values of modes I and II fracture toughness obtained from the experiments are shown in Fig. 7. The average values of KIc and KIIc for the tested Harsin marble are about 1.12 MPa m0.5 and 2.25 MPa m0.5, respectively. The mode I fracture toughness value is in the range reported for similar marble rocks. For example, Chang et al. [7], Xeidakis et al. [35] and Whittaker et al. [36], determined experimentally the values of KIc for coarse and medium grain marbles using the Brazilian disc and three-point bend rectangular beam specimens. The range of their reported fracture toughness values is

20

M.R.M. Aliha, M.R. Ayatollahi / Theoretical and Applied Fracture Mechanics 69 (2014) 17–25

Fig. 5. Two typical CCNBD specimens fractured under (a) pure mode I and (b) pure mode II loading.

Table 1 Geometrical parameters, fracture loads and fracture toughness of CCNBD specimens tested under pure mode I loading. Specimen no.

D (mm)

B (mm)

a0

a1

aB

aS

u [2]

v [2]

Pmax (kN)

KIc (MPa m0.5)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

100.65 105.4 105.4 105.4 106.5 106.5 105.4 100.65 100.65 105.4 106.5 106.5 106.5 106.5 106.5 100.65 100.65 105.4 105.4 100.65 100.65 106.5

30 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.0 29.2 29 29.1 30 30 29.2 28.8 30 30 29.3

0.282 0.265 0.248 0.291 0.279 0.211 0.261 0.279 0.265 0.267 0.259 0.314 0.245 0.286 0.280 0.271 0.262 0.275 0.273 0.258 0.233 0.280

0.659 0.622 0.618 0.629 0.620 0.604 0.621 0.620 0.622 0.623 0.615 0.630 0.612 0.621 0.620 0.655 0.653 0.625 0.622 0.653 0.647 0.620

0.596 0.554 0.554 0.554 0.548 0.548 0.554 0.548 0.554 0.554 0.548 0.545 0.548 0.545 0.546 0.596 0.596 0.554 0.546 0.596 0.596 0.546

0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8

0.277 0.278 0.278 0.278 0.279 0.280 0.279 0.279 0.278 0.277 0.277 0.275 0.283 0.279 0.276 0.276 0.275 0.276 0.278 0.275 0.273 0.276

1.783 1.786 1.785 1.785 1.787 1.788 1.786 1.787 1.786 1.784 1.782 1.777 1.789 1.787 1.782 1.782 1.781 1.781 1.784 1.781 1.778 1.782

10.658 14.548 10.559 13.946 12.571 11.259 12.361 12.108 14.554 14.369 14.080 12.686 14.841 12.292 14.125 10.172 11.200 13.749 10.867 11.166 11.867 14.125

1.003 1.305 0.939 1.244 1.120 1.012 1.120 1.150 1.278 1.275 1.216 1.069 1.367 1.091 1.221 0.893 1.036 1.197 0.949 1.030 1.075 1.210

between 1.065 and 1.19 MPa m0.5 which is in good agreement with the results obtained in this research for coarse grain Harsin marble. The mode II fracture toughness is about twice the mode I fracture toughness. But the conventional mixed mode fracture criteria such as the maximum tangential stress [37], the minimum strain energy density [38], the maximum energy release rate [39] and the cohesive zone model [40] suggest that the ratio of mode II over mode I fracture toughness (KIIc/KIc) is a figure less than one. For example, the maximum tangential stress (MTS) criterion proposes the follow-

ing relations for the fracture initiation angle (h0) and the KIIc/KIc ratio:

K II sin h0 ¼ K I 1  3 cos h0

ð8Þ

K IIc 1 ¼ K Ic 1:5 sin h0 cos h20

ð9Þ

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M.R.M. Aliha, M.R. Ayatollahi / Theoretical and Applied Fracture Mechanics 69 (2014) 17–25 2.6

KIc (Test results) Average value of KIc

KIc , KIIc (MPa. m0.5)

2.4

Geometry factor YII

KIIc (Test results)

3.0

2.2

2.0

1.8

Average value of KIIc

2.5

2.0

1.5

1.0

1.6 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

a/R

Specimen Number

Fig. 6. Variations of mode II geometry factor with a/R in the CCNBD specimen [28].

Fig. 7. Fracture toughness data (KIc and KIIc) obtained from CCNBD specimens for the Harsin marble.

For pure mode II condition, the mode II fracture angle h0II and KIIc/KIc are obtained from Eqs. (8) and (9) as 70.5° and 0.87, respectively. However, based on more accurate fracture criteria such as the generalized MTS (GMTS) criterion [41] (or modified versions of other conventional fracture criteria like minimum strain energy density), considering the effects of non-singular stress terms including the T-stress provides a more precise description for the tangential stress around the crack tip and improves theoretical predictions for mode II and mixed mode fracture toughness of brittle materials [42–48]. Using the GMTS criterion the direction of mode II crack growth and the onset of pure mode II fracture can be obtained from:

rffiffiffiffiffiffiffi @ rhh 16 b 2r c ho sin cos ho ¼ 0 T II ¼ 0 ) 3 cos ho  1  3 @h a 2

ð10Þ

K IIc 1 ¼ qffiffiffiffiffi 2 K Ic 2rc b T II a sin ho  32 sin ho cos h2o

ð11Þ

b II is significantly negative in the CCNBD specimen. that the value of T According to the GMTS criterion outlined in Eqs. (10) and (11) the enhanced fracture toughness ratio KIIc/KIc in the tested CCNBD specimen can be attributed to this negative T-stress. The critical distance rc in rock materials can be estimated from Schmidt’s maximum tensile strength theory [49] as:

rc ¼ 0:269

K 2Ic

ð12Þ

r2t

The average tensile strength rt was determined for the tested Harsin marble experimentally as 4.7 MPa. Therefore, the critical distance rc for this rock material is estimated to be 14.66 mm, which is in the typical range reported for large grain rock materials [9,50,51]. Recall that rc for rock materials is often of a relatively significant length, particularly for large grain rocks.From the practical consideration it is important to examine whether the fracture toughness data obtained from the ISRM-suggested mode I specimen can be used for predicting the experimental results of CCNBD specimen tested under pure mode II loading conditions. Using the GMTS criterion given by Eq. (11), the mode II fracture toughness b II . Fig 9 shows the mode can be predicted in terms of KIc, rc and T II mean fracture toughness value predicted from KIc for the tested

b II is the nonwhere rc is a critical distance from the crack tip and T dimensional form of T-stress under pure mode II loading (TII) b II ¼ T II pffiffiffiffiffiffi defined as T pa=K II . Fig. 8 shows the variations of Tb II for the cracked Brazilian disc specimen under pure mode II loading extracted from Ayatollahi and Aliha [28]. It is seen from this Figure

Table 2 Geometrical parameters, fracture loads and fracture toughness of CCNBD specimens tested under pure mode II loading. Specimen no.

D (mm)

B (mm)

a0

a1

aB

aS

Pmax (kN)

KIIc (MPa m0.5)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

100.65 100.65 100.65 105.4 100.65 100.65 105.4 105.4 105.4 105.4 105.4 106.5 106.5 106.5 106.5 106.5 106. 5 106.5 100.65 105.4 105.4 106.5

29.6 30 29.7 29.2 27.7 29.6 29.1 28.7 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.0 29.2 29.1 30.0 29.0 29.2 31.5

0.261 0.268 0.261 0.258 0.312 0.269 0.233 0.268 0.248 0.240 0.267 0.274 0.246 0.238 0.241 0.263 0.263 0.255 0.232 0.240 0.275 0.220

0.651 0.655 0.651 0.621 0.651 0.652 0.614 0.620 0.618 0.616 0.623 0.619 0.612 0.610 0.611 0.615 0.616 0.614 0.647 0.615 0.625 0.621

0.588 0.596 0.590 0.554 0.550 0.588 0.552 0.554 0.554 0.554 0.554 0.548 0.548 0.548 0.548 0.544 0.548 0.546 0.596 0.550 0.554 0.591

0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8

15.017 10.740 9.451 16.244 9.939 12.688 16.864 12.703 15.611 12.940 15.294 16.962 14.209 19.703 15.988 14.297 17.546 14.841 12.368 15.195 16.782 14.179

2.430 1.737 1.594 2.500 1.807 2.075 2.658 1.935 2.361 2.012 2.360 2.576 2.117 2.892 2.543 1.983 2.634 2.266 1.927 2.383 2.529 2.107

M.R.M. Aliha, M.R. Ayatollahi / Theoretical and Applied Fracture Mechanics 69 (2014) 17–25

Pure mode II fracture initiation angle (θ0II)

22

TˆII

a/R

0 Test data Average curve fitted to test data Predicted curve using the GMTS criterion

-10 -20 -30 -40 -50 -60 1

2

b II with a/R in the CCNBD specimen subjected to pure mode II Fig. 8. Variations of T loading condition.

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22

Specimen Number Fig. 10. Prediction of mode II fracture initiation angle (h0II) for the tested CCNBD specimen using the GMTS criterion.

CCNBD specimens. It is seen from this Figure that a good agreement exists between the predicted KIIc value (i.e. KIIc = 2.128 MPa m0.5) and the mean value of 2.25 MPa m0.5 experimentally obtained for mode II fracture toughness of Harsin marble tested with the CCNBD specimen. Similarly, the direction of mode II fracture initiation angle (h0II) measured from pure mode II broken samples can be predicted theoretically using the GMTS criterion (i.e. Eq. (10)). Fig. 10 compares the experimentally measured fracture angles and the theoretical predictions for the 22 mode II CCNBD specimens. Again it is seen that the average value of h0II = 41.1° measured from the broken mode II samples is in very good agreement with the angle of h0II = 38° predicted by the GMTS criterion. In addition to determining the average fracture toughness values for modes I and II, the experimental results can also be used to study the statistical distribution of toughness where the T-stress influences onset of fracture. This is examined in the following section.

Here N is equal to 22 for both modes I and II cases. Based on the weakest link theory, the largest flaw existing in the material (i.e. the weakest link) provides a favorite site for initiation and propagation of fracture. Wallin [52] proposed a probability function for brittle fracture using a general statistical model suggested by Weibull [53]. This probability function for mode I loading can be written in terms of either two or three Weibull parameters as:

  m  K Ic two PF ðK Ic Þ ¼ 1  EXP  K 0;I  parameterðm; K 0;I Þmodel   m  K Ic  K min;I three PF ðK Ic Þ ¼ 1  EXP  K 0;I  K min;I  parameterðm; K 0;I ; K min;I Þmodel

4.1. Statistical analysis As shown in Fig. 9, similar to other brittle materials, there is a natural scatter in fracture load for both modes I and II tests. The scatter in the test results obtained for mode II fracture toughness is more than that for mode I fracture toughness. Thus, the scatters in modes I and II rock fracture toughness data are analyzed using a probabilistic failure analysis. The failure probability PF of the tested CCNBD samples made of Harsin coarse grain marble can be written as:

PF ¼

n Nþ1

n ¼ 1; 2; . . . ; N

ð13Þ

where n is number of the test, sorted in order of increasing fracture load, and N is the total number of tests for each loading condition.

KIc (Test results) Average value of KIc KIIc (Test results) Average value of KIc IIc Predicted mean value using GMTS criterion

KIc, KIIc (MPa. m0.5)

3.0 2.5 KIIc = 2.25 MPa. m0.5

2.0 KIIc = 2.128 MPa. m0.5

1.5 KIc = 1.12 MPa. m0.5

1.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Specimen Number Fig. 9. Prediction of mode II fracture toughness (KIIc) in terms of KIc for the tested CCNBD specimen using the GMTS criterion.

ð14Þ

ð15Þ

where m is a fitting parameter, Kmin,I is the mode I threshold fracture toughness below which the probability of fracture is zero and K0,I is a reference fracture toughness corresponding to a fracture probability of 0.623. The Weibull parameters are found by fitting a curve to the obtained statistical fracture toughness data. For example, for mode I test data the following equation was minimized using the least square method to obtain three Weibull distribution parameters (m, K0,I, Kmin,I):

f ðm; K 0 ; K min Þ ¼

   m 2 N  X K Ic ðiÞ  K min PF ðiÞ  1  EXP  K 0  K min i¼1 ð16Þ

Thus, using a MATLAB code, the Weibull distribution parameters were obtained for pure mode I data as: m ¼ 1:81; K 0;I ¼ pffiffiffiffiffi pffiffiffiffiffi 1:16 MPa m and K min;I ¼ 0:843 MPa m. Similarly the fitting parameters for the two-parameter Weibull model were found as: pffiffiffiffiffi m ¼ 7:613; K 0;I ¼ 1:17 MPa m. Fig. 11 shows the predicted statistical curves using two and three Weibull parameter models for mode I fracture toughness of the tested CCNBD samples made of Harsin marble. In general, the micro-mechanism of brittle fracture in rocks for mode I loading is expected to be similar to that for mode II loading. For both modes, fracture initiates along the direction where the tangential stress is maximum, but depending on the loading mode this direction is in two different initiation angles. Therefore, a model similar to pure mode I is used here to explore a statistical description of the mode II fracture toughness data. The failure probability for mode II loading condition was determined using an extension of the Wallin model developed by Fowler et al. [54]

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M.R.M. Aliha, M.R. Ayatollahi / Theoretical and Applied Fracture Mechanics 69 (2014) 17–25 1.0 Experimetal data Weibull distribution - 2 parameter

Probability of fracture

Probability of fracture

1.0 0.8 0.6 0.4 0.2 0.0

Experimetal data Weibull distribution - 3 parameter

0.8 0.6 0.4 0.2 0.0

0.8

1.0

1.2

1.4

0.8

1.0

1.2

1.4

Mode | fracture toughness (MPa.m0.5)

Mode | fracture toughness (MPa.m0.5)

m = 7.613, K 0, I = 1.17MPa m

m = 1.81, K 0, I = 1.16 MPa m and K min,I = 0.843 MPa m

Fig. 11. Probability of mode I fracture for CCNBD specimen using two and three parameter Weibull models.

Mode I experimetal data Mode I Weibull distribution - 3 parameter Mode II experimetal data Mode II Weibull distribution - 3 parameter (Predicted from mode I)

1.0

1.0

0.8

0.8

Probability of fracture

Probability of fracture

Mode I experimetal data Weibull distribution - 2 parameter Mode II experimetal data Weibull distribution - 2 parameter (Predicted from mode I data)

0.6

0.4

0.2

0.0

0.6

0.4

0.2

0.0 1.0

1.5

2.0

2.5

1.0

3.0

1.5

2.0

2.5

Fracture toughness (MPa.m0.5)

Fracture toughness (MPa.m0.5)

(a)

(b)

3.0

Fig. 12. Prediction of probability curves for mode II fracture toughness of CCNBD specimen using mode I fracture data; (a) two-parameter Weibull distribution curves and (b) three-parameter Weibull distribution curves.

  m  K Ic  K min;II three PF ðK IIc Þ ¼ 1  EXP  K 0;II  K min;II

Experimetal data Weibull distribution - 2 parameter Weibull distribution - 3 parameter (Predicted from mode I data) Weibull distribution - 2 parameter (Predicted from mode I data)

 parameterðm; K 0;II ; K min;II Þmodel

ð18Þ

1.0

Using the KIIc/KIc ratio, the mode I fitted parameters Kmin,I and K0,I were modified to directly estimate the mode II parameters Kmin,II and K0,II from

Probability of fracture

0.8

0.6

K IIc K min;II K 0;II ¼ ¼ K Ic K min;I K 0;I

0.4

Now the mode II Weibull parameters can be found from the mode I ones as:

0.2

for 3  parameter : mII ¼ mI ¼ 1:81

0.0

K 0;II ¼ 1.5

2.0

2.5

3.0

  pffiffiffiffiffi K IIc  K 0;I ¼ 2:013  1:16 ¼ 2:336 MPa m K Ic

Mode II fracture toughness (MPa.m0.5) Fig. 13. Comparison of two and three parameter Weibull probability curves for mode II fracture toughness.

and Hadidimoud et al. [55]. The failure probability for mode II is given by

  m  K Ic two PF ðK IIc Þ ¼ 1  EXP  K 0;II

 parameterðm; K 0;II Þmodel

K min;II ¼

  pffiffiffiffiffi K IIc  K min;I ¼ 2:013  0:843 ¼ 1:696 MPa m K Ic

ð20Þ

for 2  parameter : mII ¼ mI ¼ 7:613 K 0;II ¼

ð17Þ

ð19Þ

  pffiffiffiffiffi K IIc  K 0;I ¼ 2:013  1:17 ¼ 2:3361 MPa m K Ic

K min;II ¼ 0

ð21Þ

24

M.R.M. Aliha, M.R. Ayatollahi / Theoretical and Applied Fracture Mechanics 69 (2014) 17–25

where KIIc/KIc in Eqs. (20) and (21) (already determined from Eq. (11)) depends on the T-stress. The values of Kmin,II and K0,II calculated using Eqs. (20) and (21) were replaced in Eqs. (17) and (18). The curves predicted for the mode II results using mode I statistical parameters are shown in Fig. 12. It is seen from this figure that the proposed statistical model based on the mode I data provides good predictions for the mode II fracture toughness data. Fig. 13 compares the statistical curves of two and three-parameter Weibull models for mode II data that shows reasonable accuracy of both models for predicting the KIIc data in terms of KIc results in the tested CCNBD specimens. It is seen from this Figure that the mode II statistical parameters predicted theoretically using Eq. (19) and those obtained from fitting the Weibull model to mode II experimental data are in very good agreement. For example based on Fig. 13, the values of K0,II predicted from the two and three-parameter Weibull models are 2.35 and 2.33 MPa m0.5, respectively, which both are very close to the K0,II value of 2.39 MPa m0.5 obtained from fitting the curve to the experimental data. Therefore, according to the findings of this research the fracture behavior of mode II results for the CCNBD specimen (i.e. the mean fracture toughness, the direction of fracture and the statistical parameters) can be estimated well by knowing only the pure mode I fracture toughness results of CCNBD specimens and employing the GMTS criterion. Consequently, there is no need to perform rather complicated pure mode II tests to obtain KIIc from the CCNBD experiments. 5. Conclusions 1- Several modes I and II fracture toughness experiments were conducted on an Iranian white marble using CCNBD specimen. 2- The enhancement in mode II fracture toughness KIIc, relative to KIc in the tested specimens was mainly due to the negative T-stress that exists in the CCNBD specimen when it is subjected to pure mode II loading conditions. 3- The GMTS criterion was able to predict very well both the fracture initiation direction (h0II) and KIIc in terms of mode I data for the tested CCNBD samples. 4- Using two and three parameter Weibull models, the statistical distribution of modes I and II fracture toughness results were obtained by fitting the Weibull parameters to the experimental data. 5- The mode II fracture probability curves were predicted successfully in terms of mode I statistical Weibull probability distribution curves.

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