## 4 Definition and Estimation of Brittleness Indexes
the yield points for two distinct pre-peak deformation behaviors. BE illustrates a completely brittle failure path, BF represents a perfectly plastic path, and BC, BCO, and BD depict various semi-brittle failure paths. The brittleness of norite is strongly influenced by petrographic factors such as the grain area ratio (GAR), cement and matrix contents, packing density (Pd), packing proximity (Pp), concavo-convex structure, feldspar percentage, and long and tangential contacts [27]. The mechanical properties of rock masses are associated with these petrographic characteristics [27]. Petrographic characteristics also include the average mineral size and their mineralogical composition [28]. Minerals play a crucial role in determining the brittle and ductile behavior of rocks from a structural perspective. It is commonly believed that rocks containing strong or brittle minerals are more likely to exhibit brittle characteristics, while those with a higher proportion of weak or ductile minerals tend to show less brittleness [94]. Brittleness indices B\({}_{48}\) to B\({}_{54}\) are primarily based on mineral compositions, representing the ratios of minerals that contribute to brittle failure. In contrast, indices B\({}_{1}\) to B\({}_{47}\) focus more on mechanical parameters. This suggests that mineralogical compositions alone are insufficient for accurately estimating rock brittleness, and it is essential to consider other rock fabric properties and environmental factors [68]. B\({}_{48}\) to B\({}_{54}\) indices can be determined through mineralogical logging tools or laboratory X-ray diffraction (XRD) analysis [66]. B\({}_{48}\), B\({}_{49}\), and B\({}_{50}\) are expressed as  
\[B_{48}=\frac{W_{qtz}}{W_{qtz}+W_{carb}+W_{clay}} \tag{6}\]  
\[B_{49}=\frac{W_{qtz}+W_{dol}}{W_{qtz}+W_{dol}+W_{lm}+W_{clay}+W_{toc}} \tag{7}\]  
\[B_{50}=\frac{W_{qfm}+W_{cal}+W_{dol}}{W_{tot}} \tag{8}\]  
where \(W_{qtz}\), \(W_{carb}\), \(W_{dol}\), \(W_{lm}\), \(W_{cal}\), \(W_{clay}\), \(W_{toc}\), and \(\mathrm{W_{tot}}\) represent the weights of quartz, carbonate minerals, dolomite, limestone, calcite, clay, total organic content, and total minerals, respectively, while \(W_{qfm}\) refers to the combined weight of quartz, feldspar, and  
Figure 11: A schematic diagram showing the behavior of the material when under conditions of stress and strain [68]. mica. Each mineral equally contributes to the brittleness. However, B48 only considers quartz as the most brittle mineral, while other indices consider other minerals to be more brittle than quartz. For example, B50 considers quartz, feldspar, mica, calcite, and dolomite as brittle minerals [27]. B48 to B54 are commonly used in the choice and design of hydraulic fracturing in shale gas reservoirs based on various aspects of rock composition, particularly the presence of brittle or ductile minerals. The indices suggest that brittleness is solely determined by the content of brittle minerals, but their physico-mechanical properties vary, affecting the overall brittleness of the rock. Based on the insights from the reviewed literature, particularly studies focusing on the mineralogical influence on brittleness indices, a revised brittleness index is proposed for norite that incorporates both its mechanical properties and mineralogical composition. The index incorporates the uniaxial-compressive-strength-to-tensile-strength ratio and the orthopyroxene content, formulated as  
\[BInorite\ =\ a\ (UCS/TS)\ +\ b\ (Opx\%)\ +\ c \tag{9}\]  
where UCS is the uniaxial compressive strength (MPa), TS is the tensile strength (MPa), Opx% is the orthopyroxene volume percentage, and a, b, and c are empirically determined constants. This approach is adapted conceptually from previous studies on rock brittleness, emphasizing the influence of both mechanical properties and mineralogical composition on brittleness behavior. Mineralogical brittleness was initially based only on the weight fraction of quartz ([PERSON] et al.