Abstract
The fracture toughness of a rock often varies depending on the specimen shape and the loading type used to measure it. To investigate the mode I fracture toughness using semicircular bend (SCB) specimens, we experimentally studied the fracture toughness using SCB and chevron bend (CB) specimens, the latter being one of the specimens used extensively as an International Society for Rock Mechanics (ISRM) suggested method, for comparison. The mode I fracture toughness measured using SCB specimens is lower than both the level I and level II fracture toughness values measured using CB specimens. A numerical study based on discontinuum mechanics was conducted using a twodimensional distinct element method (DEM) for evaluating crack propagation in the SCB specimen during loading. The numerical results indicate subcritical crack growth as well as sudden crack propagation when the load reaches the maximum. A Kresistance curve is drawn using the crack extension and the load at the point of evaluation. The fracture toughness evaluated by the Kresistance curve is in agreement with the level II fracture toughness measured using CB specimens. Therefore, the SCB specimen yields an improved value for fracture toughness when the increase of Kresistance with stable crack propagation is considered.
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1 Introduction
In rock engineering problems dealing with the stability of structures, controlling crack initiation and propagation is very important. Microcracks and macrocracks affect the rock mass strength and deformation; these factors strongly influence the stability of geological structures such as underground and open pit mines, tunnels, and rock slopes. Rock fracturing also plays a key role in the exploitation of energy resources in that creating new cracks enhances the production of oil, natural gas, and geothermal energy, and also facilitates leakage paths in the sequestration of CO_{2} in geological storage sites.
The fracture toughness is a measure of a material’s resistance to crack propagation. The fracture toughness of rock materials has been determined using various test specimen configurations and methods. The International Society for Rock Mechanics (ISRM) has incorporated chevron bend (CB) (ISRM 1988), short rod (SR) (ISRM 1988), and cracked chevronnotched Brazilian disk (CCNBD) (Fowell 1995) specimens into the standard method for the measurement of the fracture toughness of rock materials. Threepoint bendingtype specimens such as single edge cracked round bar bend (SECRBB) (Ouchterlony 1981), semicircular bend (SCB) (Chong and Kuruppu 1988), and straight notched disk bending (SNDB) (Tutluoglu and Keles 2011) specimens, as well as Brazilian disktype specimens such as cracked straightthrough Brazilian disk (CSTBD) (Fowell and Xu 1994) and flattened Brazilian disk (Wang and Xing 1999) specimens, have also been used for the measurement of the fracture toughness. Among these, the fracture toughness measurement method using the SCB specimen shown in Fig. 1 has been recently approved as an ISRM suggested method (Kuruppu et al. 2014). It is a corebased specimen that possesses inherently favorable characteristics such as simplicity, minimal machining requirements, and easy testability through the application of threepoint compressive loading using a standard test frame.
It is observed that, for the same rock sample, the mode I fracture toughness varies when different specimen types are used for measurement. Chang et al. (2002) measured the fracture toughness of granite and marble using CB, CCNBD, SCB, chevronnotched SCB, and uncracked Brazilian disk test (BDT) (Guo et al. 1993) specimens. The mode I fracture toughness values of granite and marble measured using SCB specimens were 0.68 ± 0.19 MPam^{0.5} with 31 specimens and 0.87 ± 0.15 MPam^{0.5} with 27 specimens, respectively. These values are lower than the fracture toughness values measured using other specimens. Khan and AlShayea (2000) measured the fracture toughness of limestone using SCB, CSTBD, CCNBD, and SECRBB specimens; the average values measured using the SCB, CSTBD, CCNBD, and SECRBB specimens were 0.68, 0.42, 0.61, and 0.55 MPam^{0.5}, respectively. Tutluoglu and Keles (2011) found that the fracture toughness values of andesite measured using CCNBD, SCB, and SNDB specimens were 1.45 ± 0.06 MPam^{0.5} with five specimens, 0.94 ± 0.12 MPam^{0.5} with 21 specimens, and 1.00 ± 0.09 MPam^{0.5} with 20 specimens, respectively. They argued that the variation of the fracture toughness was due to the differences in the size of the fracture process zone (FPZ). They evaluated the size of the FPZ of SCB and SNDB specimens and found that the size of the FPZ of the former was 2.15 times larger than that of the FPZ of the latter. Aliha et al. (2012) reported that the fracture toughness strongly depends on the geometry and loading conditions of the test specimen. They showed that the fracture toughness of Guiting limestone measured using an SCB specimen was higher than that measured using a CSTBD specimen, and they discussed the difference between the values using the maximum tangential stress (MTS) criterion. They found that the higherorder stress term A _{3} was responsible for the variation of the fracture toughness. Iqbal and Mohanty (2007) showed that the fracture toughness values of CB and CCNBD specimens for a brittle rock were comparable.
These studies indicate that the fracture toughness must be dealt with carefully if it is considered representative of the rock material, especially if a nonISRM suggested method is adopted. The size of the FPZ and precritical stable crack growth are the key factors affecting the fracture toughness. Nasseri et al. (2006) measured the fracture toughness and acoustic emission activity in brittle rocks. They found that the variation of fracture toughness is caused by the preexisting microcrack density and its orientation with respect to the fracture propagation direction. The creation of an FPZ surrounding the propagating main crack has been confirmed by acoustic emission techniques. Dai et al. (2007) investigated the effect of crack–microcrack interaction on the anisotropic behavior of fracture toughness. The microstructural investigation of thin sections indicated that the preexisting microcracks caused the variation of the fracture toughness values. The FPZ or crack growth can be estimated in several ways. Optical methods are used to observe moiré fringe patterns during loading and to measure the size of the FPZ. Acoustic emission measurement is also used to estimate the size of the FPZ. The compliance is used to indirectly measure the crack growth during loading. Apart from laboratory studies, numerical modeling is also useful for estimating the crack growth. We investigated the application of the distinct element method (DEM) (Cundall and Strack 1979), which is based on discontinuum mechanics, because crack propagation and microcracking occur in a discontinuous manner. DEM has been used to study crack propagation in rocks or rocklike materials such as concrete. For example, Azevedo and Lemos (2006) presented a DEM/finite element method (FEM) coupling algorithm which enables DEM to be used in the discretization of the fracture zone and for the surrounding areas of a discretization based on the FEM. The hybrid DEM/FEM method was applied for fracture analysis in concrete. They successfully modeled the crack localization process and prepeak load versus displacement curves of both mode I and mixed mode fracture experiments performed using beam specimens. Tan et al. (2009) used the DEM software package PFC2D (Itasca Consulting Group Inc. 2004) for modeling the fracture and damage processes of polycrystalline silicon carbide (SiC) ceramics. They modeled the fracture toughness testing using a specimen subjected to threepoint bend loading and showed that the numerical results agreed with the experimental measurements. D’Addetta et al. (2002) presented a combined particle and lattice model as an improved DEM formulation. It was applied to model the fracture process of cohesive granular materials somewhat similar to sandstone. They were successful in showing the typical microcrack nucleation, growth, and coalescence to form macrocracks under tension, compression, and shear modes of loading. Their simulation results were in agreement with the experimental observations. These studies demonstrate that DEM can be used for simulating the mechanical behavior of rocks and crack propagation behavior, and that it is a useful tool for investigating the FPZ and crack growth during loading.
In this study, we investigated the mode I fracture toughness using an SCB specimen. Furthermore, we used a CB specimen for the purpose of comparison of the fracture toughness. A DEM model of the SCB test specimen was used to investigate the crack growth and FPZ during loading. We evaluated the mode I fracture toughness using the SCB specimen and its Kresistance curve and found that the corrected mode I fracture toughness is comparable to the level II fracture toughness measured using the CB specimen.
2 Methodology
2.1 Test Material
Kimachi sandstone produced in Shimane Prefecture, Japan, was used as the test material. The mechanical properties of the rock are listed in Table 1. An analysis of this material using Xray diffractometry (XRD) revealed that it mainly consists of albite, anorthite, quartz, montmorillonite, and mordenite.
Kimachi sandstone has been found to be slightly anisotropic (Funatsu et al. 2004). The principal directions of anisotropy are known as arrester, divider, and shorttransverse. All tests using SCB and CB specimens reported herein were performed in the arrester orientation.
2.2 Experimental Method
2.2.1 Testing
The tests were carried out using the SCB specimen configuration shown in Fig. 1. Such an SCB specimen can be made from leftover core material after testing CB or SR specimens, so that the variation of the material properties of the rock is kept to a minimum. This specimen has certain inherently favorable properties such as simplicity, minimal machining requirements, and easy testability by means of threepoint compressive loading using a standard test frame (Fig. 2).
Specimens were prepared by sawing or slicing rock cores that were drilled in the direction of bedding planes. Each resulting disk was then cut into two halves, along a plane parallel to the direction of the bedding planes, to form two specimens. Specimens of 50mm radius and 25mm thickness were used. A straight notch was introduced in each specimen using a diamond circular saw, such that the notchlengthtoradius ratio was 0.3, 0.4, or 0.5. The thickness of the saw blade used was 0.3 mm, yielding a notch of similar thickness. The resulting SCB test specimens had their notches in the arrester orientation with respect to the material anisotropy (Chong et al. 1987). The specimens were ovendried at 40 °C for 120 h, and all dimensions were recorded.
The specimens were placed on the loading platform such that the span ratio S/R was 0.8 and then tested to failure under loadline displacement control and at a loading rate of 0.075 mm/min (see Fig. 3). The load, loadpoint displacement (LPD), and crackopening displacement (COD) were recorded as functions of time during each test.
2.2.2 Derivation of Fracture Toughness by the SCB Specimen
The fracture toughness, K _{Ic}, is determined using the peak load, nondimensional stress intensity factor, and specimen dimensions. For the SCB specimens, K _{SCB} is given as (Chong et al. 1987):
where Y is the normalized stress intensity factor, σ _{0} = F _{max}/2Rt, F _{max} is the maximum load, a is the notch length, R is the specimen radius, and t is the thickness. The stress intensity factor Y is a function of the a/R ratio, β, and the halfspantoradius ratio S/R. The bestfit curve for Y is given by Lim et al. (1994) as:
where S is the halfspan length of the support rollers. Equation (2) is valid for 0.1 ≤ β ≤ 0.8. An S/R ratio of 0.8 was chosen for the fracture toughness tests performed with the SCB specimen.
2.2.3 CB Specimen
Fracture toughness measurement by a CB specimen is one of the ISRM suggested methods. The evaluation of fracture toughness is done at two levels. The level I fracture toughness based on fracture load is suitable if the testing material is considered as a linear elastic material. The curve of the normalized stress intensity factor versus notch length for a chevron notch has a minimum value, suggesting that the initial crack growth occurs stably and that the specimen fails upon reaching the minimum value of the stress intensity factor corresponding to the maximum applied load. Therefore, the minimum normalized stress intensity factor is used for evaluating the level I fracture toughness. For nonlinearly behaving materials, the level II fracture toughness corrects the level I fracture toughness by considering the degree of nonlinearity p. Figure 4 shows a typical load versus LPD curve.
The level I fracture toughness, K _{CB}, can be calculated by the following equation (ISRM 1988):
where F _{max} is the maximum load, D is the diameter of the specimen, and:
where S is the halfspan length between support points and a _{0} is the chevron tip distance from the specimen surface.
The level II fracture toughness can be calculated as:
where p is the degree of nonlinearity and F _{c} is the load at the evaluation point. Here, p and F _{c} are determined from the load versus COD curve (ISRM 1988). We used D = 60 mm, a _{0} = 9 mm, and S = 99.9 mm in accordance with the suggested method (ISRM 1988) and the specimens were made with their notches in the arrester orientation with respect to the bedding planes.
2.3 Numerical Model by DEM
Analytical methods based on continuum mechanics are normally used in the design of many geological structures, such as roadway tunnels. However, it is difficult to use continuum mechanics to simulate failures like the separation of materials and shear planes. In this study, a DEMbased twodimensional discontinuum program called PFC2D (Itasca Consulting Group Inc. 2004) is used to simulate crack propagation in rocks. PFC2D simulates the mechanical behavior of a material by representing it as an assemblage of circular particles that can be bonded to one another. The basic mechanical properties such as Young’s modulus and Poisson’s ratio are derived from laboratory tests. In the continuum model, the elastic properties can be used directly. However, in PFC2D, the mechanical behavior of the assemblage is dominated by the microproperties of the particles and the bonds between them. These microproperties cannot be determined from laboratory tests. Thus, the relationship between the microproperties and the macroproperties should be determined by the modeling of rock testing, such as the uniaxial compressive strength (UCS) test and the Brazilian test, prior to the simulation of fracture toughness. Moreover, a clumped particle model (Cho et al. 2007), which combines particles located within a circle, is adopted in this study; each clump behaves as an element having a complicated shape.
2.3.1 Numerical Modeling of the Uniaxial Compressive and Brazilian Tests
Simulations of the uniaxial compressive and Brazilian tests were conducted to calibrate the appropriate input parameters. These simulations were performed according to the works of Potyondy and Cundall (2004). The specimen for the compressive tests is 120 mm in length and 60 mm in width, and the diameter of the specimen for the Brazilian tests is 60 mm. Both the particles and models themselves have thicknesses of one unit of length, which is equal to 1 m. The calibration process is explained in detail by Funatsu et al. (2008). The number of particle elements is about 21,000 for the uniaxial compressive test model. Being a twodimensional code, the PFC2D is unable to simulate the compressive test of a cylindrical specimen. Therefore, we decided to simulate the compressive test of a rectangular specimen having unit thickness.
For the uniaxial compressive tests, the top and bottom wall elements act as the loading platens, and the velocity of the walls is kept constant at 5.0 × 10^{−5} mm/step (i.e., the rate of load application). For the Brazilian tests, the side walls act as platens, and the velocity of the walls is kept constant at 5.0 × 10^{−5} mm/step. The models for the uniaxial compressive tests and Brazilian tests are shown in Fig. 5. The applied stress is measured by dividing the average force acting on opposing walls by the area of the corresponding specimen crosssection (Potyondy and Cundall 2004).
Figures 6 and 7 show the stress–strain curves derived by the physical tests, along with the results of the simulating UCS and Brazilian tests, respectively. With the appropriate selection of DEM parameters, it can be seen that the model simulates the experimental results very well. This is despite the fact that a twodimensional rectangular model was used, which is expected to be weaker in compression than if a cylindrical model was used. The crack distribution indicates that tensile failure occurs along the loading direction. The tensile strength calculated by the numerical model of the Brazilian test is 5.0 MPa. The difference in tensile strength between the numerical simulation and the experiments is only 0.1 MPa. However, there is a limitation of the numerical model in comparison with the test specimen. The former is a twodimensional model and the latter is a threedimensional cylindrical shape. Cho et al. (2007) showed that the twodimensional clumped particle model can reproduce the failure envelope of both hard rock and weak rock. Our results support their findings. A set of input parameters suitable for the modeling of Kimachi sandstone is given in Table 2.
2.3.2 Modeling of the SCB Specimen by DEM
The model of the SCB specimen is shown in Fig. 8b. Figure 8a shows the SCB specimen for comparison. The notch is created by deleting the particles located within the notch. The specimen diameter is 100 mm and initial notch length is 25 mm. The thickness of the test specimen is 25 mm. The numerical model is of unit length thickness (1 m). The support and loading rollers for threepoint bending are created by wall elements. The support rollers are kept separated by a fixed span length of 80 mm, which is same as that used in the physical tests. The element size was defined as shown in Table 2. The particles were randomly packed with a uniform size distribution. The number of particles was 11,663. The loading roller located above the specimen is made to move downward at a constant displacement rate of 5.0 × 10^{−6} mm/step. The loading force used to evaluate the fracture toughness is taken as the force acting between the loading roller and the adjacent material particles. The displacement of the loading roller and the COD is also monitored. In PFC2D, stress cannot be calculated directly; instead, it is calculated as an average value inside the representative area, namely, the measurement circle. The radius of the circle is 3.0 mm, which is three times larger than the average radius of a clump. This diameter was selected to allow estimation of the proper stress state. The measurement circle was located in front of the initial notch tip. Since the stress calculated by the measurement circle is the average value inside the circle, the stress near the notch tip can be underestimated in the case of having a measurement circle with a large radius. In addition, if the crack extends during loading, the calculated stress change can be attenuated by the effect of averaging because the minimum crack extension is 0.2 mm, which is the same value as the minimum ball radius and is smaller than the radius of the measurement circle.
A microcracking in the numerical simulation is defined as a bond breakage between particles. The crack extension is defined as the length from the crack tip to the farther end of the connecting microcracks.
3 Results
3.1 Fracture Toughness of Kimachi Sandstone Measured Experimentally Using SCB and CB Specimens
Figure 9 shows the test results for the fracture toughness using SCB specimens. The dots and bars in the figure respectively indicate the average and standard deviation of the fracture toughness values for the same ratio of notch length to specimen radius. Three tests were performed at each crack length. This figure shows that the fracture toughness measured using SCB specimens is independent of the notch length. The average value of fracture toughness using all data is 0.589 MPam^{0.5}, with a standard deviation of 0.0474 MPam^{0.5}.
The fracture toughness measured using CB specimens is summarized in Table 3. This table shows the level I fracture toughness K _{CB}, the level II fracture toughness K ^{c}_{CB} , and the degree of nonlinearity p.
The fracture toughness of Kimachi sandstone as measured using SR specimens was investigated by Matsuki et al. (1991). The level I fracture toughness K _{SR} had values of 0.86 and 0.85 MPam^{0.5}, with specimen diameters of 80 and 100 mm, respectively. The level II fracture toughness K ^{c}_{SR} has values of 1.01 and 1.02 MPam^{0.5}, with specimen diameters of 80 and 100 mm, respectively. The fracture toughness measured using CB specimens in this study compares well with Matsuki et al.’s results. However, the level I and level II fracture toughness values measured using CB specimens are, respectively, ~35 and 63 % higher than those measured using SCB specimens.
3.2 Numerical Simulation of Fracture Toughness using the SCB Specimen
Figure 10 shows the load–COD curves determined by numerical modeling and through experimental results. The experimental result was from a typical test involving partial unloading cycles as shown in the figure. Note that, in the simulation, the specimen thickness was set to 1 m. Therefore, the load shown in the figure was converted from that corresponding to the thickness of 25 mm used in the experiment. The maximum load and the corresponding COD value were almost the same in the two sets of graphs. The fracture toughness determined by numerical modeling is 0.526 MPam^{0.5}, which is ~10 % lower than the average value of the fracture toughness measured experimentally using SCB specimens.
4 Discussion
4.1 Mode I Fracture Toughness Measured by SCB and CB Tests
There are only a limited number of studies comparing the fracture toughness values measured using SCB specimens and ISRM suggested methods. Chang et al. (2002) compared the fracture toughness values determined using several different types of specimens, such as SCB, CB, CCNBD, and chevronnotched SCB specimens. The fracture toughness values of granite and marble measured using SCB specimens were 0.68 and 0.871 MPam^{0.5}, respectively. The level I fracture toughness values of granite and marble measured using CB specimens were ~1.4 and 1.1 MPam^{0.5}, respectively. These values are almost the same as those measured using CCNBD specimens. Tutluoglu and Keles (2011) reported that the level I fracture toughness values of andesite measured using SCB and CCNBD specimens were 0.94 and 1.45 MPam^{0.5}, respectively. Our results using sandstone showed that the trend is the same, i.e., the fracture toughness measured using SCB specimens is lower than that measured using CB and CCNBD specimens. One of the reasons for the difference in fracture toughness is the different notch type. Both the CB and CCNBD specimens have a chevron notch, whereas the SCB specimen has a straight notch. Furthermore, differences may exist in the development of the process zone, as it is affected by the type of loading, e.g., bending versus tension. Khan and AlShayea (2000) found that the fracture toughness values of limestone measured using CSTBD and CCNBD specimens were, respectively, 0.42 and 0.61 MPam^{0.5}. Chang et al. (2002) reported that the fracture toughness values measured using chevronnotched SCB specimens for granite and marble were 1.39 and 1.11 MPam^{0.5}, respectively. These values are ~2 times and 1.2 times higher than those measured using straight notched SCB specimens. In order to satisfy the requirements of LEFM on which the fracture toughness is based, the size of the FPZ during loading should be small enough so that the material behavior of the rock sample can be considered as linearly elastic. The radius of the FPZ can be derived from the following equation (Schmidt 1980):
For Kimachi sandstone, the value of r _{c} calculated using the level II fracture toughness measured using CB specimens is 6.13 mm. The actual notch length and other dimensions need to be much larger than r _{c}. Furthermore, the fracture toughness may be underestimated unless (1) the crack starts to propagate when the FPZ is fully developed and (2) the fracture toughness is evaluated when the slow stable crack growth reaches its critical limit. The large difference in fracture toughness measured for CB and SCB specimens shows that the level I fracture toughness of the SCB specimen fails to satisfy those conditions.
4.2 Evaluation of Fracture Toughness by the KResistance Curve
When evaluating the fracture toughness, the crack growth that occurs at the critical level of loading cannot be ignored. If the crack growth is measured, the fracture toughness can be corrected as a function of the actual crack length using the Kresistance curve. In this study, we evaluated the crack growth by numerical modeling based on DEM. Figure 11 shows the load versus loadline displacement curve, as obtained by numerical modeling. The number of bond breakages corresponding to microcracking is also shown. The bond breakages suggest that a crack initiates before the load reaches the maximum value and that it propagates rapidly when the load reaches the maximum.
Figure 12 shows the crack growth during loading based on the numerical simulation. The crack growth at each point and the corresponding load can be derived from the numerical modeling. Parts a, b, c, d, and e in Fig. 12 correspond to points a, b, c, d, and e in Fig. 11. The crack growth at the maximum load is ~6 mm, as given in Fig. 12c. Furthermore, Fig. 12d shows the occurrence of sudden, unstable crack growth when the maximum load is reached. Following Eq. (1), the mode I stress intensity factor K _{I} at each point can be calculated as:
where F is the load at each evaluation point and Δa is the crack extension at that point. Y is the normalized stress intensity factor that corresponds to the notch length a + Δa. Similarly, we conducted a numerical simulation of the crack growth of specimens having a/R = 0.3 and 0.4, and constructed the Kresistance curves. Figure 13 shows the DEM models of an SCB specimen having a/R = 0.3 and a/R = 0.4, respectively.
Figure 14 shows the load and the number of bond breakages versus the loadline displacement, and Fig. 15 shows the crack growths for those two cases. Table 4 gives a summary of the values of important parameters in the development of the stress intensity factor for a/R = 0.3, a/R = 0.4, and a/R = 0.5, respectively, and Fig. 16 shows the development of the stress intensity factor with crack growth, which is the Kresistance curve. These figures show that the fracture toughness is about 0.95 MPam^{0.5}, which is comparable to the level II fracture toughness obtained for the CB specimen.
The fundamental process of macrocrack extension in brittle rock is almost always by the opening, growth, and coalescence of microcracks that occurs within the process zone. Barker’s nonlinearity correction incorporates the effect of the finite size of the process zone and yields an improved value known as the level II fracture toughness (Barker 1977). On the other hand, the Kresistance curve indirectly measures the energy release rate with the development of the process zone and, therefore, the two methods are expected to result in the same value for fracture toughness.
5 Conclusions
To investigate the mode I fracture toughness using semicircular bend (SCB) specimens, we experimentally studied the fracture toughness using SCB and chevron bend (CB) specimens, the latter being one of the International Society for Rock Mechanics (ISRM) suggested methods, for comparison. The mode I fracture toughness measured using SCB specimens is lower than the level I and level II fracture toughness values measured using CB specimens. This difference is attributed to differences in notch type, neglecting the stable crack extension, and the differences in the fracture process zone (FPZ) sizes.
A numerical study based on discontinuum mechanics was conducted to evaluate crack propagation in the SCB specimen during loading. The numerical result is validated by comparing the load versus crackopening displacement (COD) curves obtained by the numerical model and by the experiments. The results show that subcritical crack growth, as well as sudden crack propagation, occurs when the load reaches the maximum. For the specimen sizes used for the tests, the crack extension at the maximum load is less than ~7 mm. This is almost the same as the radius of the FPZ as calculated by Schmidt’s formula.
Moreover, the Kresistance curve is determined using the crack extension and the stress intensity factor at the evaluation point. The resistance increases with crack growth and reaches a steady value, which is considered to be the fracture toughness. The fracture toughness evaluated in this matter is in agreement with the level II fracture toughness measured using CB specimens. Therefore, the results show that an improved value for fracture toughness can be determined when the resistance to crack propagation is considered. The suggested numerical method enables the fracture toughness (which is comparable to that given by the ISRM suggested CB specimen method) to be determined by measuring the Kresistance during stable crack propagation in SCB specimens.
Abbreviations
 a :

Notch length
 a _{0} :

Chevron tip distance from the specimen surface
 A _{3} :

Coefficient of the higherorder nonsingular term in the crack tip stress function
 A _{min} :

Minimum normalized stress intensity factor for the CB specimen
 BDT:

Uncracked Brazilian disk test
 CB:

Chevron bend
 CCNBD:

Cracked chevronnotched Brazilian disk
 COD:

Crackopening displacement
 CSTBD:

Cracked straightthrough Brazilian disk
 D :

Diameter of specimen
 DEM:

Distinct element method
 E _{50} :

Tangent Young’s modulus at half the compressive strength
 F :

Load
 F _{c} :

Load at the evaluation point for the CB specimen
 F _{max} :

Maximum load
 FEM:

Finite element method
 FPZ:

Fracture process zone
 K _{Ic} :

Mode I fracture toughness
 K _{CB} :

Level I fracture toughness measured using the CB specimen
 K ^{c}_{CB} :

Level II fracture toughness measured using the CB specimen
 K _{SR} :

Level I fracture toughness measured using the SR specimen
 K ^{c}_{SR} :

Level II fracture toughness measured using the SR specimen
 LEFM:

Linear elastic fracture mechanics
 LPD:

Loadpoint displacement
 MTS:

Maximum tangential stress
 p :

Degree of nonlinearity
 R :

SCB specimen radius
 r _{c} :

Radius of the FPZ
 t :

SCB specimen thickness
 S :

Half span length of support rollers
 SCB:

Semicircular bend
 SECRBB:

Single edge cracked round bar bend
 SNDB:

Straight notched disk bending
 SR:

Short rod
 UCS:

Uniaxial compressive strength
 XRD:

Xray diffractometry
 Y :

Normalized stress intensity factor
 σ _{c} :

Uniaxial compressive strength
 σ _{t} :

Tensile strength
 Δa :

Crack extension
 β :

Ratio of notch length to specimen radius
 ν :

Poisson’s ratio
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We would like to thank Ms. Qian Li for her help with running the simulations.
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Funatsu, T., Shimizu, N., Kuruppu, M. et al. Evaluation of Mode I Fracture Toughness Assisted by the Numerical Determination of KResistance. Rock Mech Rock Eng 48, 143–157 (2015). https://doi.org/10.1007/s0060301405508
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DOI: https://doi.org/10.1007/s0060301405508