To analyze the effect of the width of a prefabricated crack on the dimensionless stress intensity factor of notched semicircular bend (NSCB) specimens, ABAQUS software was employed to perform numerical calibration of the crack tip stress intensity factor for the width of prefabricated cracks in the range of 0.0∼2.0 mm. The relative errors of the dimensionless stress intensity factor for different widths of prefabricated cracks were analyzed. The results indicate that the dimensionless stress intensity factor shows an approximate linear increase as the width of the prefabricated crack increases. The longer is the length of the prefabricated crack, the “faster” is the increase in speed. The effect of the dimensionless support spacing on the increase in the speed of the dimensionless stress intensity factor due to the increase in crack width is minimal. When the prefabricated crack width is 2.0 mm, the maximum relative error of the dimensionless stress intensity factor is 4.325%. The new formula for the dimensionless stress intensity factor that eliminates the influence of the width of a prefabricated crack is given, which provides a theoretical basis for the more accurate fracture toughness value measured using an NSCB specimen.
Rock mass is a typical noncontinuous natural medium, whose interior presents many defects, such as joints and cracks [
Currently, the influence of the width of a prefabricated crack on the dimensionless stress intensity factor of the NSCB specimen has not been reported. Analysis of the extent to which the width of the prefabricated crack influences the dimensionless stress intensity factor is necessary to improve the method that measures the fracture toughness of NSCB specimens.
For this reason, the dimensionless stress intensity factor of the crack tip was calibrated by ABAQUS software for different widths of the prefabricated cracks of the NSCB specimens in this paper. The influence of the width of a prefabricated crack on the dimensionless stress intensity factor is examined; the relative error of the dimensionless stress intensity factor for different widths of the prefabricated cracks is analyzed; and the corrected formula of the NSCB dimensionless stress intensity factor that eliminates the effect of crack width is obtained. It is worth pointing out that it is not easy to prefabricate a crack on rock, and the crack tip of the manually prepared specimen is not an ideal sharp type, but a Ushaped crack with a certain radian. Therefore, the next numerical research work in this paper is to treat the prefabricated crack into a Ushaped crack, which is consistent with the actual crack form in the specimen used by researchers to test rock fracture toughness.
Figure
Loading diagram of the NSCB specimen: (a) plane diagram and (b) crosssectional diagram.
To verify the accuracy of the equivalent regional integration method with respect to the stress intensity factor calibration in this paper, the geometric parameters that were employed in the method suggested by the ISRM were selected. In the case of the dimensionless support spacing
Comparison between calibration result of this paper and that obtained by formula (
Figure
According to the range of support spacing and crack length given by the internationally recommended method, the dimensionless stress intensity factor with the common width of a prefabricated crack (0.0∼2.0 mm) is calibrated for the NSCB specimens. The geometric and mechanical parameters of the specimens are shown in Table
NSCB specimen parameters.
Elastic modulus, 
Poisson’s ratio, 
Dimensionless crack length, 
Specimen thickness to diameter ratio, 
Crack width, 2 
Dimensionless support spacing, 

63.94  0.27  0.4, 0.5, and 0.6  0.4  0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0  0.5, 0.6, 0.7, and 0.8 
The NSCB specimens with prefabricated crack widths of 0.0∼2.0 mm, dimensionless lengths of the prefabricated cracks of 0.4∼0.6, and a dimensionless support spacing of 0.5∼0.8 were analyzed using ABAQUS finite element software. The NSCB specimen loading is shown in Figure
NSCB models with different widths of the prefabricated cracks.
Since the degree of the density of the mesh near the tip of the prefabricated crack will have a substantial influence on the calibration result of the dimensionless stress intensity factor, the prefabricated crack tip for different widths in Figure
(a) NSCB specimen meshing. (b) Zoomed view of the mesh of the crack tip.
Based on the equivalent region integration method, the Jintegral at the crack tip is obtained by ABAQUS software, and the stress intensity factor of the mode I crack is calculated by the following formula:
In the case of a fixed support spacing and crack length, when the influence of crack width is considered (i.e., 2b > 0 mm), the calibration value of the dimensionless stress intensity factor is larger than that of the ideal crack (i.e., 2b = 0 mm). With an increase in the width of the prefabricated crack, the dimensionless stress intensity factor continuously increases.
When the support spacing is fixed, the length of the prefabricated crack is within the range
When the length of the prefabricated crack is fixed, the dimensionless stress intensity factor of the NSCB specimen increases with an increase in the width of the prefabricated crack within the range of the support spacing
Dimensionless stress intensity factor variation diagram of NSCB specimens. (a)
The relative error
Errors of dimensionless stress intensity factors of the NSCB specimens due to different widths of prefabricated cracks.
Crack width, 2 



 

0.4  0.5  0.6  0.4  0.5  0.6  0.4  0.5  0.6  0.4  0.5  0.6  
0.0  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000 
0.2  0.274  0.277  0.351  0.293  0.240  0.412  0.283  0.288  0.453  0.313  0.367  0.376 
0.4  0.513  0.609  0.784  0.559  0.633  0.858  0.566  0.648  0.879  0.589  0.704  0.823 
0.6  0.787  0.941  1.176  0.825  0.960  1.286  0.827  0.990  1.304  0.883  1.056  1.281 
0.8  1.060  1.274  1.588  1.064  1.331  1.682  1.088  1.331  1.716  0.939  1.393  1.728 
1.0  1.334  1.661  1.980  1.330  1.636  2.095  1.371  1.673  2.155  1.436  1.729  2.139 
1.2  1.573  1.855  2.351  1.623  1.920  2.474  1.654  1.997  2.567  1.712  2.112  2.551 
1.4  1.813  2.132  2.702  1.862  2.247  2.952  1.959  2.321  2.979  1.988  2.418  2.962 
1.6  2.018  2.409  3.053  2.128  2.531  3.266  2.199  2.609  3.418  2.264  2.770  3.362 
1.8  2.223  2.658  3.403  2.394  2.814  3.629  2.438  2.915  3.830  2.558  3.107  3.797 
2.0  2.428  2.907  3.733  2.634  3.120  3.991  2.721  3.239  4.325  2.835  3.444  4.197 
Figure
Maximum relative error of the dimensionless stress intensity factor for NSCB specimens.
As shown in Figure
To obtain a more general situation, the crack width is nondimensional to obtain a more general relationship between the dimensionless stress intensity factor and the dimensionless crack width. The threedimensional scatter plot and the fitted surface are shown in Figure
Fitting curves of the dimensionless crack width and the dimensionless stress intensity factor. (a)
As shown in Figure
Value of parameters in formula (
Support spacing  Parameter value  









5.9  −2.776  −18.806  −7.413  28.427  11.109 

7.355  −3.683  −22.66  −6.03  34.191  14.464 

8.814  −5.088  −26.525  −1.74  39.977  18.582 

10.104  −5.278  −29.693  −1.507  45.064  20.545 
When the support spacing ranges from 0.5 to 0.8, the correlation between the fitted surface by formula (
In theory, accurate rock fracture toughness test values can be obtained by the correction calculation of Formula (
Fracture toughness is one of the most important parameters in fracture mechanics. Accurate measurement of fracture toughness is of great significance for safety protection of engineering structures. The test results of fracture toughness are affected not only by the precision of test equipment, sample preparation, and human factors but also by the dimensionless stress intensity factor. Therefore, accurate calibration of dimensionless stress intensity factors of corresponding samples is a prerequisite for accurate measurement of fracture toughness.
Generally speaking, the research on dimensionless stress intensity factors mainly focuses on the geometry of the specimen, the length of the crack, and the angle between the crack plane and the load. However, when processing the specimen to test the fracture toughness, the prefabricated crack will inevitably have a certain width, which will make the stress distribution of blunt notched crack different from that of ideal zero width. Ayatollahi et al. [
The influence of the width of a prefabricated crack on the dimensionless stress intensity factor of NSCB specimens was analyzed using ABAQUS numerical software. The following main conclusions are obtained:
As the width of the prefabricated crack increases, the dimensionless stress intensity factor of the NSCB specimen continuously increases. The longer is the length of the prefabricated crack, the “faster” is the increase in speed. The dimensionless support spacing has a minimal effect on the speed of change in the dimensionless stress intensity factor as the width of the prefabricated crack increases.
Based on the dimensionless stress intensity factor obtained by calibration using the ideal crack, the errors in the dimensionless stress intensity factor caused by different widths of the prefabricated cracks are obtained when the widths are less than or equal to 2.0 mm. When the support spacing is
The corrected formula that considers the influence of the width of a prefabricated crack on the dimensionless strength factor is given for any dimensionless support spacing in the range of 0.5∼0.8. The degree of fit using the formula exceeds 0.99. Using this formula and the specimen fracture load and geometric parameters, the fracture toughness test value of an NSCB specimen considering the influence of the width of a prefabricated crack can be obtained.
Crack length
Half of crack width
Thickness of specimen
Elastic modulus
Jintegral
Mode I stress intensity factor
Fitting constants related to formula (
Concentrated load
Crack initiation load
Radius of specimen
Support spacing
Poisson’s ratio
Dimensionless stress intensity factor
Dimensionless stress intensity factor of the 0 mmwidth crack
Dimensionless stress intensity factor of the 2
Dimensionless crack length
Dimensionless crack width
Dimensionless specimen thickness
Dimensionless support spacing
Relative error
Chevron notched threepoint bend round bar specimen
Cracked chevron notched Brazilian disc specimen
6node triangular plane stress element
8node quadrilateral element
Cracked straightthrough flattened Brazilian disk
Diametrally compressed ring
Edge cracked triangular
Edgenotched disc bend
Holedcracked flattened Brazilian disc specimen
International Society for Rock Mechanics
Chevron notched short rod specimen
Theory of critical distances.
The data supporting this research article are available from the corresponding author via email.
The authors declare that they have no conflicts of interest.
This study was financially supported by the National Key Research and Development Program of China (Grant no. 2016YFC0600701), the National Natural Science Foundation of China (Grant nos. 51674101, 51674170, and 51822403), and the Key Laboratory Open Project Fund in Henan Province (Grant no. S201605). The authors express their gratitude for the supportive help and instructions of the managers.