## 2 Methodology  
### Basic Theories  
The basic theories of all Brazilian tests can be simplified into a plane stress model pertaining to the elasticity theory. In this model, the tensile stress field is developedby the compression acting at geometric boundaries of discs. The compression is loaded with two parallel plates which are flattened and rigid. Generally, the contact is assumed to be rigid, where the disc and plates in contact do not deform, or their deformation is negligible. However, significant deformation occurs at the loading point of concrete discs. The distribution of stress at the boundaries can be more accurately solved by Hertz's solution. As a result, the theoretical basis of the Brazilian test can be described as a plane stress model where diametrical concentrated compressive forces act at diametrically opposite points of a cylindrical disk with \(D\) and \(L\) represented as its diameter and thickness, respectively. However, the results of the Brazilian test have shown evidence that the crack begins near to the loading points due to the stress singularity at the boundaries. To improve the stress singularity, the distributed loads can be applied at the boundaries of discs by increasing the contact area between the rigid plate and disc. One common method is to use a pair of steel-loading jaws with a radius 1.5 times that of the disc sample to exert a load on the Brazilian disc. Under the assumption that cracks initiate at the centre of discs, the stress state along the central line in the BTC with rigid plates can be described by  
\[\left\{ \begin{aligned} \sigma_{x}&=\frac{-2P}{ \pi DL}\\ \sigma_{y}&=\frac{2P}{\pi DL}\Big{(}\frac{4D^{2}}{D^{2 }-4y^{2}}-1\Big{)}\\ \tau_{xy}&=0\end{aligned} \right. \tag{1}\]  
where \(\sigma_{x}\) describes the normal stress which is perpendicular to the loading direction, \(\sigma_{y}\) is the normal stress along the loading direction, \(\tau_{xy}=\) is the shear stress, \(P\) is the force loaded at the boundary, \(D\) is the diameter of the disc specimen, and \(L\) is the thickness of the disc specimen. Accordingly, the theoretical tension strength of rigid plates loaded at the centre of the disc according to the Griffith criterion can be described by  
\[\sigma_{T}=\frac{-2P}{\pi DL} \tag{2}\]  
The Brazilian test with rigid jaws also assumes that a crack initiates at the centre of the disc. The stress state along the central line for the Brazilian test with rigid jaws can be described by  
\[\left\{ \begin{aligned} \sigma_{x}&=\frac{-2P}{\pi DL }\Big{[}\frac{\left(1-\rho^{2}\right)\sin 2\theta}{\rho^{4}-2\rho^{2}\cos 2 \theta+1}-\tan^{-1}\Big{(}\frac{1+\rho^{2}}{1-\rho^{2}}\tan\theta\Big{)}\Big{]} \\ \sigma_{y}&=\frac{2P}{\pi DL}\Big{[}\frac{-\left(1- \rho^{2}\right)\sin 2\theta}{\rho^{4}-2\rho^{2}\cos 2\theta+1}-\tan^{-1} \Big{(}\frac{1+\rho^{2}}{1-\rho^{2}}\tan\theta\Big{)}\Big{]}\end{aligned} \right. \tag{3}\]  
where \(\theta\) is the contact angle. The ISRM suggests two steel-loading jaws designed to contact a disc-shaped sample at diametrically opposed surfaces over an arc of contact of approx \(10^{\circ}\) at failure; \(\rho=R_{0}/R_{1}\), where \(R_{0}\) is the radius of the disc specimen, and \(R_{1}\) is the radius of the jaws. According to the \"Griffith criterion\", the tension strength at the centre of disc can be described by  
\[\sigma_{T}=\frac{-2P}{\pi DL}\frac{\sin\theta\cos^{2}\theta}{\theta} \tag{4}\]  
Table 1 summarises the stress distribution along the central line of the disc and the corresponding tensile strength at the disc centre according to the Griffith cracking criterion for Brazilian tests with concentrated boundary conditions (BTC) and distributed boundary conditions (BTD). Figure 1 shows the distribution of tensile and compressive stress along the central line of discs with different boundary conditions. Obviously, the stress distribu  
tion remains the same at the centre of the disc but different at the periphery for the two boundary conditions.