# Study of the Relationship between Mode I Fracture Toughness and Rock Brittleness Indices
## 3 Results  
Regression results between KIC and brittleness indices B1 to B10 were shown in Figures 1-10, respectively. The results showed weak correlations, i.e., coefficient of correlation (R\({}^{2}\)) < 0.5, between KIC and indices B1 (Figure 1), B2 (Figure 2), B3 (Figure 3), B5 (Figure 5), B6 (Figure 6), B7 (Figure 7) and B8 (Figure 8). Moreover, the correlations between KIC and each of the indices B6 (Figure 6), B7 (Figure 7) and B8 (Figure 8) showed an inverse trend in contrast to the other indices. KIC--index B6 relationships were investigated only for exponential and linear models since index B6 had negative values that were not valid mathematically for power and logarithmic models. On the other hand, the KIC--index B4 relationship (Figure 4) was good, i.e., R\({}^{2}\) > 0.5, for the exponential model, while their correlation was weak for linear, logarithmic and power models. KIC showed a good correlation with index B9 (Figure 9) for exponential and linear models and with index B10 (Figure 10) for all models. A summary of the relationships with R\({}^{2}\) > 0.5 is presented in Table 3. Additionally, the Root-Mean-Square Error (RMSE) is presented in Table 3 for the relationships with (R\({}^{2}\) > 0.5). The RMSE was lowest for the KIC--index B10 power model with a value of 0.1307, which is lower by more than 50% of the nearest RMSE values for the remaining relationships listed. In general, power and exponential models had the lowest RMSE compared to linear and logarithmic models. Figure 3: \(\rm K_{IC}-rock\) brittleness (B3) relationships. Figure 2: \(\rm K_{IC}-rock\) brittleness (B2) relationships. Figure 4: \(\rm K_{IC}-\)rock brittleness (B4) relationships. Figure 5: \(\rm K_{IC}-\)rock brittleness (B5) relationships. Figure 6: K\({}_{\rm IC}-\)rock brittleness (B6) relationships. Figure 7: K\({}_{\rm IC}-\)rock brittleness (B7) relationships. Figure 8: K\({}_{\rm IC}-\)rock brittleness (B8) relationships. Figure 9: K\({}_{\rm IC}-\)rock brittleness (B9) relationships. The \(p\)-value can be used to determine the significance of the proposed relationships. When the variables have a normal distribution, then the \(p\)-value, obtained from [PERSON]'s correlation, can be used to judge the relationships' significance; otherwise, the \(p\)-value obtained from [PERSON]'s correlation should be used [63; 64; 65]. The \(p\)-value (obtained using R) indicates the probability that the null hypothesis, i.e., the correlation is not significant, is true. For a 95% confidence level (i.e., \(\alpha\) =0.05), a \(p\)-value lower than 0.05 indicates a significant correlation [64]. The Shapiro-Wilk test [66; 67; 68] was used to test the normality of brittleness indices (B4, B9 and B10) and \(\mathrm{K_{IC}}\) datasets. The results of the Shapiro-Wilk test, obtained using R, are shown in Table 4. The dataset of brittleness index (B10) showed evidence of non-normality; hence, the significance of \(\mathrm{K_{IC}}\) -index B10 relationships was assessed using the \(p\)-value from [PERSON]'s correlation. A summary of the significant relationships proposed in this paper is presented in Figure 3:0  
Figure 3:1 \(\mathrm{K_{IC}}-\mathrm{rock}\) brittleness (B10) relationships.