Journal of Engineering Geology

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./files/site1/files/1.pdfExtended Abstract
(Paper pages157-174)
Introduction
Considering the fact that the estimation of mode  fracture toughness by testing is time-consuming and expensive. It might be associated with certain practical difficulties. Therefore, many researchers have attempted to propose experimental relationships in order to capture these problems. Gunsallus et al. (1984) and Bhagat (1985) experimentally found that mode  fracture toughness is related to tensile strength. Whittaker et al. (1992) have also proposed a number of relationships between mode I fracture toughness, tensile strength, point load index, uniaxial compressive strength and the velocity of sound waves. Bearman (1999) obtained an experimental relationship between mode I fracture toughness and point load index, while Brown et al. (1997) presented an experimental relationship between this parameter and density. Up to now no significant research effort has been made in this field in Iran, only Ayatollahi and Fatehi addressed rock fracture toughness. Although, Ayatollahi has not presented any experimental relationships. In the present research the three-point bending test was used on a cylindrical specimen containing a straight crack in order to determine the mode  fracture toughness, and the Brazilian test was employed to determine tensile strength.
Materials and Methods
The tests were carried out on six types of rocks, namely gray sandstone,
tuff, lithic tuff, travertine, andesite, and limestone. Sandstone, travertine, and limestone are sedimentary rocks, while andesite is an extrusive igneous rock, and tuff and lithic tuff are pyroclastic rocks (pyroclastic rocks resulting from volcanic eruptions that harden by sedimentation). Therefore, the studied rocks have different origins. In order to carry out the Brazilian and the three-point bending test, cores were prepared from these blocks. In order to perform the three-point bending test, specimens with diameter of 73 mm with a thickness of 30 mm were used. The samples were cut in two semicircular by a cutting machine, and a notch with length of 15 mm is created by a diamond saw.  Notch is vertical in the center of the semicircular samples.
The Brazilian test was performed on disc shaped specimens. In order to perform the Brazilian test, specimens with diameter of 51 mm and thick of 25 mm were used. The specimens are carefully placed under the curved jaws of the machine and then loaded until fracture.
Results and Discussion
A summary of the Brazilian and the three-point bending test results are presented in Table 1. The average value of test result pertaining to each rock is reported in Table 1.
Table 1. Summary of the Brazilian and the three-point bending test results

Specimen	Tensile Strength (MPa)	Fracture Toughness (MPa√m)
Limestone	3.74	1.23
Sandstone	7.14	1.63
Tuff	16.36	2.17
Lithic Tuff	4.34	1.01
Andesite	13.25	1.86
Travertine	8.27	1.14

In this study, it was attempted to propose an experimental relationship between mode I fracture toughness and the tensile strength of the rock.
In order to determine the relationship between the tensile strength and the fracture toughness, the tensile strength vs. fracture toughness diagram was plotted in Excel to obtain Eq. 1 and the coefficient of determination (R²) (Figure 1).

The coefficient of determination (R²) in Eq. 1 shows that almost 80 percent of the mode I fracture toughness variations can be estimated using the linear relationship (Eq. 1). The relationship is applicable for determining the mode I fracture toughness resulting from the three-point bending test on semicircular specimens containing a straight crack.

In the following, the results of this study are compared to those reported by Whittacker (1992) and Zhang (2002).
In order to examine the accuracy of the presented relationships, the Root Mean Square Error (RMSE) measure was used which is computed from Eq. 2. In the best case, RMSE is zero. 

In the relationships,   represents the fracture toughness obtained from testing while  is the fracture toughness estimated using the relationships.
Comparison of the obtained results indicate that the proposed relationship has the capability of precise estimation of the mode I fracture toughness of rocks.
Conclusion
Given the many difficulties associated with the direct estimation of fracture toughness, indirect estimation methods have been proposed. One of such methods is the estimation of mode I fracture toughness using tensile strength. A linear relationship with a coefficient of determination of 0.7977 was proposed. The accuracy of this relationship has been verified by comparing its results to those from previous studies.

 

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In cases such as explosion, fire, deep drilling and geothermal energy extraction, rocks are exposed to high temperatures influencing the rock toughness. Thus, the aim of this study is to investigate the effect of temperature on the fracture toughness of the rocks. In this study, the effect of temperature on the mode I fracture toughness is investigated. To this end, three-point bending tests were performed on semicircular specimens of four types of natural rocks including sandstone, limestone, tuff, andesite, and a series of concrete specimens to determine the fracture toughness. The specimens were first heated to 100, 200, 300, 500 and 700 °C. After reaching the desired temperatures, the specimens were cooled. A series of tests was performed on the specimens at ambient temperature (25 °C). The heating rate in the electric furnace was 15 °C/min in accordance with the temperature rise in fires. Petrographic studies and X-ray diffraction analysis (XRD) were performed to identify the composition of the rocks. Furthermore, the effective porosity and the weight loss of heated specimens were determined to study the behavior of rocks. Comparison of the test results indicated the higher impact of temperature on the fracture toughness of fine-grained rocks. In addition, the fracture toughness decreased by increasing the effective porosity and decreasing the weight loss. According to the results, the mode I fracture toughness of sandstone, tuff, limestone, andesite and concrete specimens underwent a heating-cooling cycle up to 700 ^°C respectively decreased 45, 17, 44 and 9.5 and 37 percent compared with that of unheated specimens.
 

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Introduction
Rock masses have an enormous geometrical discontinuities such as void, notch, crack and flaw. These geometrical discontinuities which play as stress concentrator, cause to reduce the load bearing capacity of rock masses. In rock masses, the crack is the most important geometrical discontinuity assessed frequently by civil, mechanical and mining engineers and researcher. The fracture mechanics which is a branch of mechanical engineering science, has been often used for investigating the cracked rock samples. The fracture toughness is one of the important parameters in the fracture mechanics which describes the resistance of materials against the crack growth. On the other hand, since orientation of cracks relative to the loading directions can be arbitrary, brittle fracture in rocks may happen due to a combination of two major fracture modes, i.e. crack opening mode (mode I) and crack sliding mode without any opening or closing the crack flanks (mode II). In order to obtain the fracture toughness of rocks, several test configurations under pure mode I have been proposed. One of the parameters that has the influence on the fracture toughness of rocks and other materials is the thickness of test sample. Previous experimental results showed that the fracture toughness of rocks increases by increasing the specimen thickness until a specific thickness. After that, the fracture toughness decreases for thicker samples until plane strain condition occurs. Then, the fracture toughness becomes a fixed value when the thickness of sample varies.
The all preceding studies have been dealt with considering the effect of specimen thickness on fracture toughness focusing only the mode I fracture toughness and there is few research concerning the thickness effect on the mode II fracture toughness of rocks. Therefore, the aim of this paper is to investigate experimentally the effect of specimen thickness on the mode II fracture toughness.
Material and methods
To investigate the thickness effect on the mode II fracture toughness of rocks, several fracture tests were conducted on the semi-circular bend (SCB) specimens. The SCB specimen is a semi-disk of radius R and thickness t including an edge crack of length a loaded under three-point bending. When the crack is along the applied load and the bottom supports are symmetric relative to vertical crack, the SCB sample is under pure mode I loading. One of the methods for achieving the mixed mode loading in SCB sample is the asymmetry distances of bottom supports from the vertical crack located at the middle of bottom edge (see Figure 1). The pure mode II in this type of SCB sample is attained at a specific distances, i.e. at specific values of S₁ and S₂. These values of supporting distance can be obtained from finite element analysis.

Figure 1. The schematic of SCB sample.
The fracture tests were done both on pure mode I and pure mode II, for the sake of comprehensiveness. Therefore, 32 SCB samples with 4 different thicknesses and 4 repetition for each specimen size were tested for both pure mode I and pure mode II. The specimens were cut from Ghorveh marble sheets with different thicknesses by water jet machine. Then, the specimens were cracked artificially by a high speed rotary diamond saw blade. The specimen dimensions and loading conditions are presented in Table 1. Finally, the cracked SCB samples were tested by using a 300 kN ball-screw universal test machine. Table 1 also gives the average of four fracture loads (P_f) obtained for each thickness of specimen.
Table 1. The specimen dimensions and loading conditions.

	S.D.  (N)	Pf  (N)	S2 (mm)	S1 (mm)	a (mm)	t (mm)	R (mm)
Pure mode I	150	3220	57	57	28.5	15	95
Pure mode II	350	4726	11	57
Pure mode I	360	6711	57	57	28.5	25	95
Pure mode II	882	9445	11	57
Pure mode I	1450	20285	57	57	28.5	50	95
Pure mode II	4179	25441	11	57
Pure mode I	4672	31810	57	57	28.5	80	95
Pure mode II	4686	36848	11	57

Results and discussion
The mode I and mode II fracture toughness (K_Ic and K_IIc) can be calculated for SCB samples from following equations:
(1)
(2)
where P_f is fracture load, R and t are the radius and thickness of SCB sample, respectively K_I^* and K_II^* are geometry factors which depend on geometrical ratios a/R, S₁/R and S₂/R and independent of specimen dimensions and magnitude of applied load. These dimensionless parameters are often obtained from finite element analysis. For tested SCB samples, the values of K_I^* and K_II^* were extracted from previous studies as shown in Table 2. Substituting the fracture loads and specimen dimensions from Table 1 and the values of K_I^* and K_II^* given in Table 2 into Eqs. (1) and (2), the mode I and mode II fracture toughness were calculated as listed in Table 2. Figure 2 also shows the variations of mode I and mode II fracture toughness with respect to specimen thickness. As seen from this figure, the fracture toughness for both pure modes increases for thicker samples until a specific thickness. After that, the values of K_Ic and K_IIc decrease by increasing the specimen thickness. For plane strain condition in which the thickness of specimen is relatively large, the values of K_Ic and K_IIc are nearly constant.
 
 
Table 2. The dimensionless parameters K_I^* and K_II^* for tested SCB samples and their corresponding fracture toughness.

	KIIc (MPa.√m)	KIc (MPa.√m)	KII*	KI*	t	R
Pure mode I	0.0	1.125	0.0	0.644	15	95
Pure mode II	0.897	0.0	0.35	0.0
Pure mode I	0.0	1.411	0.0	0.644	25	95
Pure mode II	1.075	0.0	0.35	0.0
Pure mode I	0.0	2.126	0.0	0.644	50	95
Pure mode II	1.448	0.0	0.35	0.0
Pure mode I	0.0	2.083	0.0	0.644	80	95
Pure mode II	1.311	0.0	0.35	0.0

The other point assessed in the present study is the dependency of fracture path on specimen thickness in mode II loading. It was shown that the fracture trajectory becomes more curvilinearly when the thickness of specimen increases.

Figure 2. The variations of K_Ic and K_IIc versus the specimen thickness.
Conclusion
The effect of specimen thickness on the mode I and mode II fracture toughness of rock was investigated experimentally using the SCB specimens. The experimental results showed that the fracture toughness for both pure modes increases when the thickness of specimen increases until a specific thickness. After that, the values of K_Ic and K_IIc decrease by increasing the specimen thickness. For plane strain condition in which the thickness of specimen is relatively large, the values of K_Ic and K_IIc are nearly constant. Also, it is shown the crack grows more curvilinearly for thicker SCB samples../files/site1/files/142/1.pdf