## 3 Numerical Modeling
### Effectiveness of the Proposed Numerical Model  
Assessment of the predictions of the built material numerical model is carried out through comparison with available experimental results and analytical solutions of the indirect tensile strength. Consequently, the numerical results were compared with the experimental one obtained by [PERSON] et al. [6], [PERSON] et al. (2017) [8], and the numerical results obtained by [PERSON] et al. [7]. As indicated in Table 1, the experimental indirect tensile strength of Brisbane tuff disk with a diameter of 52 mm (R = 26 mm) with different contact angles of 15\({}^{\circ}\), 20\({}^{\circ}\), and 30\({}^{\circ}\) has been obtained by [PERSON] et al. [6]. [PERSON] et al. [7], in their work, also compared their numerical results with [PERSON] et al. (2012) [6]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Loading arch Angle (\({}^{\circ}\))**} & \multirow{2}{*}{**[PERSON] et al. [6]**} & \multirow{2}{*}{**[PERSON] et al. [7]**} & \multicolumn{2}{c}{**Present Analysis**} \\ \cline{3-5}  & & & **Numerical** & **Analytical** \\ \hline
15 & 8.04 & 8.20 & 8.175 & 8.366 \\
20 & 8.29 & 8.45 & 8.217 & 8.562 \\
30 & 8.70 & 8.64 & 8.737 & 9.602 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Predicted indirect tensile strength \(\sigma_{t}\) (MPa) compared to [6; 7]. Figure 1: The Brazilian test configuration (loading arch). Table 1 compares the predicted tensile strength by FLAC\({}^{\text{2D}}\) code with those suggested by [6; 7] and the analytical solution obtained using Equation (3). Based on the stress state in the center, the tensile strength is evaluated (Equation (3)) [5; 23]. The elastic modulus and the Poisson ratio of the Brisbane tuff tested by [6] were 26 GPa and 0.25, respectively. According to the results inTable 1, we can conclude that the present analysis provides an acceptable estimation of the indirect tensile strength with a maximum absolute relative difference of 1.65% comparatively with the experimental results of [PERSON] et al. [6]. Likewise, the maximum absolute relative difference between the results of [PERSON] et al. [6] and those obtained by [PERSON] et al. [7] is 1.95%. As shown in Table 1, the analytical indirect tensile strength obtained by Equation (3) is overestimated with an absolute relative difference of 9.39% if compared with the experimental results of [PERSON] et al. [6]. To confirm the good agreement of our numerical model, a comparison was made between stresses estimated numerically and those obtained by analytical equations (Equations (1) and (2)). Thus, before few steps prior to initiation failure, the numerical values for \(\sigma_{\text{r}}\) and \(\sigma_{\text{\theta}}\) through the specimen of Brisbane tuff tested by [PERSON] et al. [6] in the case of arch loading angle \(2\alpha=15^{\circ}\), obtained numerically and analytically [25] (Equations (1) and (2)), are shown in Figure 2 for comparison purposes. From Figure 2, it is clear that the proposed numerical and analytical methods lead to the same values of the stresses around the loading diameter. For the normal stress \(\sigma_{\text{\theta}}\), a maximum relative difference between analytical and numerical solutions is 6.37%. However, a maximum deviation equal to 3.63% is marked between analytical and numerical stress along the loading diameter, \(\sigma_{\text{r}}\). As indicated in [30], the Splitting Tension Test (or Brazilian test) makes it possible to impose a practically homogeneous tensile stress field in the central zone of the disk in 0.8H and a compression in 0.1H on both sides (Figure 3). Figure 2: Distribution of vertical and horizontal stresses through the loading diameter in the disk tested by [6] in the case \(2\alpha=15^{\circ}\) (compression\(-\), tension\(+\)). The results of Figure 2 show clearly that the analytical and numerical stresses along the loading diameter provide an acceptable estimation with analytical solutions. For the case of disks tested by [PERSON] et al.