The Brazilian disc test is a simple and useful technique to determine the tensile strength of rock materials. By using FLAC3D, 63 numerical simulations in total were performed when flattened Brazilian disc coefficient and Poisson’s ratio were different. Based on Griffith theory, the corresponding FISH language was compiled to record the Griffith equivalent stress. Through analysis of numerical simulation results, it is indicated that fracture plane was not the plane going through center of the Brazilian disc, which was in good agreement with the references. In addition, the flattened Brazilian disc coefficients had greater influence on tensile strength than Poisson’s ratio. Based on cusp catastrophe theory, the flattened Brazilian disc coefficient should not exceed 0.035 for the flattened Brazilian disc tests. Consequently, a tensile strength empirical formula considering flattened Brazilian disc coefficient by utilizing the flattened Brazilian disc test was established, which was
The Brazilian disc test is a useful technique to determine the tensile strength of rock materials [
However, threedimension effects of Brazilian test can influence the results [
The flattened Brazilian test has been a popular method in recent years to determine the tensile strength of rock materials [
In this paper, the flattened Brazilian disc coefficient was used to measure the size of the flattened Brazilian disc plane instead of the loading angle. The flattened Brazilian disc with different flattened Brazilian disc coefficient was constructed in FLAC3D to simulate the flattened Brazilian tests. The FISH language was compiled to illustrate the Griffith equivalent stress of the flattened Brazilian discs, and the possible fracture planes of the flattened Brazilian discs were analyzed. Consequently, the empirical formula of tensile strength using flattened Brazilian tests was put forward.
In most references, the loading angle was used to describe the size of loading plane; however, the angle is much complicated to measure in practice. Hence, in this paper, the flattened Brazilian disc coefficient
The flattened Brazilian disc coefficient.
The flattened Brazilian disc coefficient can be expressed as the following equation:
To study the influence of the flattened Brazilian disc coefficient on the flattened Brazilian disc test, the commercial software FLAC3D was utilized. To construct the numerical simulation model, its modeling and meshing were accomplished in ANSYS with Solid185 zone type; subsequently, the constructed numerical simulation models were introduced into FLAC3D, the numerical simulation models with different flattened Brazilian disc coefficient were constructed (Figure
Some flattened Brazilian discs with different flattened Brazilian disc coefficient.
0.01
0.02
0.03
0.04
The uniaxial compression loading was applied on the flattened plane of the flattened Brazilian disc directly, and its loading rate was
Through combination of different flattened Brazilian disc coefficient (9 types) and Poisson’s ratio (7 types), 63 numerical simulations in total were performed.
For most rock materials, they belong to brittle materials, and the fracture mechanism can be explained by Griffith theory [
In order to explore the location of fracture plane of the Brazilian disc test, the corresponding FISH language was compiled to calculate the average Griffith equivalent stress
A selected plane parallel to
To determine the possible fracture plane during loading process, four numerical simulation results were taken as examples to illustrate the possible fracture plane. For convenient comparison of the numerical simulation results, contour of the dimensionless ratio
Contour of the dimensionless ratio
Flattened Brazilian disc coefficient is 0.01
Flattened Brazilian disc coefficient is 0.02
Flattened Brazilian disc coefficient is 0.03
Flattened Brazilian disc coefficient is 0.04
As shown in Figure
Possible fracture plane.
The numerical simulation results indicated that the location of the maximum Griffith equivalent stress is not the center of the Brazilian disc; that is, the cracks would not initiate from the center of the flattened Brazilian discs, which cannot guarantee the accuracy of the tensile strength by utilizing the Brazilian discs test. Through analysis of the numerical simulation results, it is suggested that the accuracy of the tensile strength difference decreases with increasing of flattened Brazilian disc coefficient.
Moreover, these results were also verified by the study by You and Su [
Both numerical simulation results and laboratory tests [
In accordance with the aforementioned numerical simulations results, the fracture plane deviated from the center of Brazilian discs; hence, taking the Griffith equivalent stress of Brazilian discs’ center as the tensile strength of rock materials was not reasonable. However, taking the maximum Griffith equivalent stress
As discussed previously, the stress distribution changed with increasing the flattened Brazilian disc coefficient, and the possible fracture plane was not the plane going through the Brazilian discs center; in other words, the maximum Griffith equivalent stress location changed as well, and the correlation of the tensile strength and the applied load
In order to explore influence of Poisson’s ratio and the flattened Brazilian disc coefficient on the correction coefficient
Influence of Poisson’s ratio and flattened Brazilian disc coefficient on the correction coefficient
As shown in Figure
Through analysis of Figure
Through analysis of Figure
The potential function
Cusp catastrophe function
As shown in Figure
To determine the maximum flattened Brazilian disc coefficient, a series of data sets
Based on the method above, a series of
Different

4  5  6  7  8  9 

Maximum flattened Brazilian disc coefficient  0.015  0.020  0.025  0.030  0.035  0.040 

0.548  141.100  23.990  125.600  48.030  −16.580 

8.834  7.2100  8.922  7.091  8.697  10.230 

>0  >0  >0  >0  >0  <0 
Through analysis of Figure
Because of little influence of Poisson’s ratio on the correction coefficient
Average correction coefficient
Correction coefficient ( 
Flattened Brazilian disc coefficient ( 


0  0.005  0.010  0.015  0.020  0.025  0.030  0.035  
Poisson’s ratio ( 

0.10  0.9398  0.9188  0.9057  0.8288  0.7772  0.7467  0.6393  0.6393 
0.15  0.9404  0.9473  0.9080  0.8398  0.7939  0.7499  0.6537  0.6537 
0.20  0.9402  0.9350  0.9231  0.8391  0.7879  0.7731  0.6461  0.6461 
0.25  0.9428  0.9650  0.9316  0.8607  0.8086  0.7833  0.6723  0.6723 
0.30  0.9586  0.9558  0.9231  0.8750  0.8126  0.7903  0.6596  0.6596 
0.35  0.9712  0.9685  0.9256  0.8977  0.7934  0.7638  0.6823  0.6823 
0.40  1.0002  0.9911  0.9440  0.8813  0.8014  0.7765  0.6874  0.6874 


Average correction coefficient ( 
0.9562  0.9542  0.9230  0.8603  0.7962  0.7691  0.6630  0.6545 
Based on the data in Table
Correlation of the average correction coefficient and flattened Brazilian disc coefficient.
Consequently, correlation of the average correction coefficient and flattened Brazilian correction coefficient can be expressed as follows:
Due to little influence of Poisson’s ratio on the correction coefficient
Based on the numerical simulation results, the tensile strength adopting the flattened Brazilian disc test with variation flattened Brazilian disc coefficient is proposed. Moreover, flattened Brazilian disc coefficient is easily obtained during the Brazilian disc test compared to loading angle; hence, the empirical tensile strength equation is workable, and it could be adopted to estimate the tensile strength in the flattened Brazilian disc test.
The tensile strength is a key mechanical parameter for rock materials, which plays a dominant role in rock engineering. Both direct method and indirect method were used to determine the tensile strength of rock materials; however, the direct method is timeconsuming and hardconducting. Thereafter, the indirect method was more widely used to obtain the tensile strength of rock materials compared to the direct method.
The Brazilian disc test is one of the most widely used indirect methods, whereas some scholars found that the loading location was smashed and the crack did not initiate from the center of the Brazilian disc due to the stress concentration in the loading location, which decrease the accuracy of the tensile strength by using the Brazilian disc test. It is because the most important point of the Brazilian disc test is the crack initiation from the center of the Brazilian disc center. To increase the accuracy of the tensile strength of the Brazilian disc test, some new Brazilian disc tests were proposed, and the flattened Brazilian disc test was one of them.
In order to obtain accurate tensile strength of rock materials by utilizing the flattened Brazilian disc test, the shape parameter of the flattened Brazilian disc was defined as the flattened Brazilian disc coefficient. Then, 63 numerical simulations in total were performed, and the related FISH language was applied to obtain the Griffith equivalent stress based on the Griffith theory. Through analysis of the numerical simulation results, it is found that the crack initiation location (the maximum Griffith equivalent stress location) deviates from the center of the flattened Brazilian disc center. Moreover, the corresponding tensile strength difference becomes larger with increasing of the flattened Brazilian disc coefficient. Thus, it is necessary to determine the appropriate value of the flattened Brazilian disc coefficient to avoid large error between the experimental results and the true value. In this paper, the cusp catastrophe theory was adopted to determine the maximum flattened Brazilian disc coefficient. Based on the cusp catastrophe theory, the flattened Brazilian disc coefficient should be less than 0.035. Additionally, an empirical formula was proposed evaluating the tensile strength of rock materials by using the flattened Brazilian disc test.
The numerical simulation is mainly used in this paper, the numerical simulation result was not verified by the experimental results, and it lacks further experimental study. To validate the numerical simulation results, the experiments study on the flattened Brazilian disc test would be our next tasks.
In this paper, the flattened Brazilian disc tests were conducted, through numerical simulations results analysis, the fracture plane locations of flattened Brazilian disc were determined, and, finally, the tensile strength empirical equation considering flattened Brazilian disc coefficient was obtained; the main conclusions of this paper were as follows:
Through numerical simulations analysis, it was suggested that the fracture plane location was not the plane going through the center of Brazilian discs with increasing flattened Brazilian disc coefficient; it was the plane going through the compression plane ends.
Combining numerical simulation results, an equation for the tensile strength adopting flattened Brazilian disc test was established. Because the maximum Griffith equivalent stress location was not the Brazilian disc center any longer, which was verified by both laboratory tests and numerical simulations, it is necessary to calibrate the empirical formula. The empirical formula for the tensile strength determined by the Brazilian disc test was established, and the empirical equation was
The authors declare that there is no conflict of interests regarding the publication of this paper.
This study was funded by the Open Research Fund Program of Hunan Provincial Key Laboratory of Shale Gas Resource Utilization; Hunan University of Science and Technology (Grant no. E21527); the National Natural Science Foundation of China (Grants nos. 51174088, 51174228); the National Basic Research Program of China (Grant no. 013CB035401); and the Fundamental Research Funds for the Central Universities of Central South University (Grants nos. 2015zzts077, 2014zzts055).