## 4 Definition and Estimation of Brittleness Indexes
Strength brittleness indices quantify a material's fracture behavior by evaluating strength, stress-strain characteristics, and crack initiation stress, as presented in Table 2 with their corresponding formulas and parameters. Brittleness indices \(\mathrm{B_{3}}\), \(\mathrm{B_{4}}\), and \(\mathrm{B_{5}}\) have been used to explore the relationship between rock brittleness and various factors such as drillability, sawability, and drilling rate [91]. Additionally, \(\mathrm{B_{4}}\) has been applied to different rock types, including norite, to predict fracture toughness values wherein higher \(\mathrm{B_{4}}\) values are associated with lower brittleness in rocks [63]. [PERSON] et al. [92] (2016) introduced \(\mathrm{B_{13}}\), but both \(\mathrm{B_{12}}\) and \(\mathrm{B_{13}}\) fail to capture the full stress-strain path, limiting their ability to represent brittle failure. [PERSON] et al. [93] (2016) improved upon this with \(\mathrm{B_{15}}\), which considers both stress and strain, but it still neglects post-failure processes. [PERSON] et al. [89] (2015) developed \(\mathrm{B_{16}}\), which accounts for both rupture strength loss and inelastic strain but ignores variations in rock behavior within the elastic region. The study by [PERSON] et al. [68] (2018), as shown in Figure 11, outlines the brittleness indices based on stress or strain characteristics. Figure 11 comprise various class I stress-strain curves, showing different pre-peak and post-peak brittle characteristics. O represents the origin, O' is the point of elastic strain recovery, and B is the failure point. A and A' are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Measurement Methods** & **Formulation** & **Authors** & **Measurement Parameter** \\ \hline \multirow{4}{*}{\begin{tabular}{c} Based on strength \\ parameter \\ \end{tabular} } & \(B_{1}=\frac{\sigma_{c}}{\sigma_{t}}\) & [76] & \\ \cline{2-3}  & \(B_{2}=\frac{(\sigma_{c}-\sigma_{t})}{(\sigma_{c}+\sigma_{t})}\) & [76] & \\ \cline{2-3}  & \(B_{3}=\frac{\sigma_{c}\sigma_{t}}{2}\) & [64] & \\ \cline{2-3}  & \(B_{4}=\sqrt{\frac{\sigma_{c}\sigma_{t}}{2}}\) & [64] & \\ \hline \multirow{2}{*}{\begin{tabular}{c} Based on strength and \\ crack initiation \\ \end{tabular} } & \(B_{5}=\frac{\sigma_{c}}{\sigma_{t}}=8\frac{\sigma_{c}}{\sigma_{t}}=\frac{8}{ \sigma_{c}}\) & [82] & \\ \cline{2-3}  & \(B_{6}=\frac{(\sigma_{c}-\sigma_{t})}{(\sigma_{c}+\sigma_{t})}=\frac{(8-K)}{(8+K)}\) & [82] & \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} Based on stress-strain curve \\ curve \\ \end{tabular} } & \(B_{7}=\frac{(\gamma_{p}-\tau_{r})}{\tau_{p}}\) & [83] & \\ \cline{2-3}  & \(B_{8}=\frac{(\epsilon_{p}-\epsilon_{r})}{\epsilon_{r}}\) & [83] & \\ \cline{2-3}  & \(B_{9}=\frac{\epsilon_{r}}{\epsilon}\) & [84] & \\ \cline{2-3}  & \(B_{10}=\frac{\left(\frac{\sigma_{r}-\epsilon_{r}^{\prime}}{\epsilon_{r}^{ \prime}}\right)}{\epsilon_{r}^{\prime}}\) & [85] & \\ \cline{2-3}  & \(B_{11}=\frac{W_{r}}{W}\) & [86] & \\ \cline{2-3}  & \(B_{12}=\frac{dW_{r}}{dW_{r}}=\frac{(M-E)}{M}\) & [87] & \\ \cline{2-3}  & \(B_{13}=\frac{dW_{r}}{dW_{r}}=\frac{E}{M}\) & [88] & \\ \cline{2-3}  & \(B_{14}=\frac{(\gamma_{p}-\tau_{r})}{\tau_{p}}\) & \(\frac{1}{2}[k_{a}]\) & [89] & \\ \cline{2-3}  & \(B_{15}=\frac{\left(\frac{\sigma_{r}-\sigma_{r}}{\epsilon_{r}}\right)}{(\epsilon_{ r}-\epsilon_{r})}+\frac{\left(\left(\sigma_{r}-\sigma_{r}\right)\left(\epsilon_{r}- \epsilon_{r}\right)\right]}{(\epsilon_{r}^{\prime}\sigma_{r})}\) & [67] & \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Based on stress-strain curve \\ curve \\ \end{tabular} } & \(B_{16}\) & \\ \cline{2-3}  & \(=\frac{\left(\epsilon_{gRT}-\epsilon_{a}\right)}{(\epsilon_{a}-\epsilon_{a})}+ \infty\) & [90] & \\ \cline{2-3}  & \(+\infty\) & \(SC_{BRT}+\eta\), \(SC_{BRT}\) & \\ \cline{2-3}  & \(=\frac{\left[\epsilon_{r}-\epsilon_{r}\right)}{\left[\sigma_{r}\left(\sigma_{r}- \epsilon_{r}\right)\right]}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of brittleness indices based on strength, stress-strain curve, and crack initiation stress.