# Shape Effect and Accuracy Analysis of Rock Tensile Strength Test
## 4 Accuracy Analysis of Diameter Correction Factor and Test Method  
### Diameter Correction Factor  
From the above analysis, for the same rock material but differently shaped specimens, the tensile strength initially decreases and then gradually stabilizes as the specimen diameter increases, reflecting the size effect related to diameter. In engineering practice, the diameter of the cored rock depends on the drill bit size; thus, not all rock cores measure 50 mm [35]. Consequently, if non-standard-sized specimens are used for tensile strength testing, the results must be corrected. Here, the ratio of the measured strength from the splitting numerical simulation test to the preset material strength is adopted as the correction factor. After curve fitting, the formula for the correction factor k in the disc-splitting test is derived, as shown in Equation (2):  
\[\text{k}_{\text{Disc}}=0.899\;7D^{0.052.7} \tag{2}\]  
The formula is used to modify the data of the field splitting test, and the results are shown in Figure 11.  
Figure 10: The relationship between the direct tensile strength and diameter of dumbbell specimens. Figure 9: Stress distribution diagram of straight dumbbell specimen. Panels (**a–d**) show the stress distribution contour changes at different time points, while (**e**) shows the final fracture result. It can be observed that the slope of the fitted curve describing the relationship between the corrected splitting tensile strength and diameter decreases and becomes smoother. This indicates that the correction reduces the influence of diameter on the splitting tensile strength. Similarly, the correction factors for the diameters of the other two types of specimens can be obtained through curve fitting, as shown in Equations (3) and (4). \[\text{k}_{\text{Curved dumbbell}} = 0.537\;2D^{0.227}\;\text{\text{\  
However, it shows a significant deviation from the actual tensile strength, with an error reaching 49.9%, as shown in Table 5.  
This is primarily because, during the loading of a dumbbell-shaped specimen with a linear transition, a significant stress concentration can form at the transition region near the specimen's root. This stress concentration may lead to early failure at that location, resulting in a lower measured tensile strength. This phenomenon is especially pronounced in brittle rocks, where a linear-transition shape facilitates crack propagation at stress concentration sites, eventually causing fracture. By contrast, in a dumbbell-shaped specimen with a curved transition, the smooth contour helps distribute the applied tensile stress more evenly, reducing the likelihood of stress concentration. Consequently, cracks typically initiate in the central region of the specimen and spread uniformly, resulting in a more stable fracture pattern. This design thus provides test values closer to the material's true tensile strength. Due to stress concentration, dumbbell-shaped specimens with a linear transition may fail earlier, typically yielding lower tensile strength values. In contrast, dumbbell-shaped specimens with a curved transition generally exhibit slightly higher tensile strength values, thanks to improved stress distribution, and their test results are more consistent and reliable. A linear-transition dumbbell-shaped specimen can be considered a simplified version of the curved-transition specimen, replacing the rounded corners with oblique straight lines. Consequently, in terms of testing accuracy, the curved-transition design outperforms the linear-transition design [36; 37; 38]. Further analysis shows that under the condition of isotropic rock, compared to direct tension, the disc specimen, with its regular shape, easily controlled force boundaries, and good alignment, can more easily achieve a concentrated and stable tensile failure zone under the same testing conditions, with a more uniform stress distribution.