# Study of the Relationship between Mode I Fracture Toughness and Rock Brittleness Indices
## 4 Discussion  
Based on coefficient of correlation (R\({}^{2}\)) and \(p\)-value, the proposed relationships (Table 3) were significant. These relationships were compared with K\({}_{\rm IC}\)--rock brittleness relationships from the literature (Table 5). In this paper, there were weak correlations between K\({}_{\rm IC}\) and each of the indices B1 and B2. Similar weak correlations were reported in [23; 24] between K\({}_{\rm IC}\) and index B1 and in [22; 23; 24] between K\({}_{\rm IC}\) and index B2. This indicated that indices B1 and B2 may not be good predictors of K\({}_{\rm IC}\). For strength-based brittleness index B3, there was no significant correlation with K\({}_{\rm IC}\) in this study, despite the significant K\({}_{\rm IC}\)--index B3 relationship reported in [22], i.e., Equation (20) in Table 5.  
The relationship between K\({}_{\rm IC}\) and index B4 proposed in this study, i.e., Equation (11) in Table 3, was significant but it had an R\({}^{2}\) lower than the relationships reported in [23], i.e., Equations (21) and (22) in Table 5, that were proposed using data from [18] and [69], respectively. A comparison between the K\({}_{\rm IC}\)--index B4 relationships is shown in Figure 11. There were significant differences between Equations (21) and (22), particularly at B4 values higher than 20, while Equation (11) showed an overall average trend between Equations (21) and (22). The proposed K\({}_{\rm IC}\)--index B9 linear relationship, i.e., Equation (13) in Table 3, was compared to Equation (18) in Table 5. There were significant differences between the two relationships (Figure 12), particularly at B9 values higher than nearly 65. These dissimilarities between the relationships proposed in this paper and the proposed ones in the literature, for indices B3, B4 and B9, may be owed to two main reasons. The first reason may be the differences in the values of rock properties, namely \(\sigma_{\rm t}\) and \(\sigma_{\rm c}\). Indices B3 and B4 were criticized for reflecting rock strength rather than brittleness [42; 43]. Based on the formula of indices B3, B4 and B9, they would be sensitive for variations in \(\sigma_{\rm t}\) and \(\sigma_{\rm c}\). So, considering other types of rocks (i.e., weaker or stronger rocks) would affect the value of these indices and in turn affect their correlation significance with K\({}_{\rm IC}\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **K\({}_{\rm IC}\)** & \(\sigma_{\rm t}\)** & \(\sigma_{\rm c}\) & **E** & \(\upsilon\) & **B4** & **B9** & **B10** \\ \hline Statistic & 0.9789 & 0.97599 & 0.9542 & 0.762 & 0.9497 & 0.9755 & 0.9544 & 0.9348 \\ \hline \(p\)-value & 0.5351 & 0.4201 & 0.0589 & 1.98 \(\times\) 10\({}^{-7}\) & 0.0389 & 0.4071 & 0.06 & 0.0103 \\ \hline  & & & & & & & \\ \end{tabular} \(p\)-value \(>\) 0.05, the null hypothesis is accepted where there is no difference between the sample distribution and a normal distribution, i.e., there is not enough evidence of non-normality. Otherwise, the alternative hypothesis is accepted.  
\end{table} 
Table 4: Results of Shapiro–Wilk normality test. \begin{table}
\begin{tabular}{c c c c} \hline
**Equation** & & **No.** & **Reference** \\ \hline \(K_{IC}=(0.01)\ B9+(0.39)\) &, \(R^{2}=0.61\) & (18) & [24] \\ \hline \(K_{IC}=(0.09)\ B10+(0.16)\) &, \(R^{2}=0.68\) & (19) & [24] \\ \hline \(K_{IC}=(0.11)\ B3^{0.43}\) &, \(R=0.96\) & (20) & [22] \\ \hline \(K_{IC}=(0.107)\ B4^{0.8663}\) &, \(R^{2}=0.9262\) & (21) & [23] \\ \hline \(K_{IC}=(0.3952)\ B4^{0.4315}\) &, \(R^{2}=0.5857\) & (22) & \\ \hline \end{tabular}
\end{table} 
Table 5: K\({}_{\rm IC}\)—rock brittleness indices from the literature.