## 2 Existing Brittleness Indices
### Test-Based Brittleness Indices (TBIs)  
Laboratory loading tests are a natural source of information regarding mechanical behavior of rocks, including their failure processes. As a result, the results of these studies can be used to evaluate brittleness of the rock samples. Once again, lack of a universal definition of rock brittleness led to existence of different ways to introduce brittleness indices calculation procedures. [PERSON] et al., 2016 [3] provided a comprehensive overview of brittleness indices proposed for laboratory tests analysis: we will highlight some of them below. The majority of TBIs used in practice are related to analysis of stress-strain curve obtained from triaxial loading tests. Figure 3 represents the typical result of single-stage loading test: a cylindrical sample is subjected to axial effective stress \(\sigma_{a}\) and radial effective stress \(\sigma_{r}\). Radial stress is maintained at a certain level, while axial stress is gradually increased until the failure of the sample. Differential stress \(\Delta\sigma=\sigma_{a}-\sigma_{r}\) and axial and radial strains of the sample \(\varepsilon_{a}\) and \(\varepsilon_{r}\) are the parameters of stress-strain state of the sample obtained at any point of time during the test. Experimentally obtained axial strain dependency on differential stress plotted as a solid black line in Figure 3 is generally used to evaluate brittleness. This line has certain specific zones: linear elastic sector \(OA\), inelastic strain accumulation \(AB\), failure point \(B\), post-peak rupture \(BQ\), and relaxation zone \(QK\). Specifics of behavior of the rock in these regions are used in many studies to estimate test-based brittleness indices (TBIs). The first definitions of brittleness [6; 7; 8; 9; 10; 11; 12] have been summarized by [PERSON] and [PERSON], 1974 [2] in a form related to loading test. While many definitions of brittleness were provided in a qualitative way (e.g., brittleness is lack of ductility), the following definitions were given in a quantitative form [2]  
\[TBI_{1}=\frac{\textit{reversible}\textit{strain}}{total\textit{ strain}}=\frac{OC}{OL},\quad TBI_{2}=\frac{\textit{reversible}\textit{energy}}{total\textit{ energy}}=\frac{\textit{S}_{CBL}}{\textit{S}_{OABL}}\ . \tag{11}\]  
The first interpretation of the loading curve has been used here: the stress-strain curve was complemented by theoretical unloading path \(BC\)--potential linear unloading with a slope equal to the one observed at linear elastic loading region \(OA\). As a result, the failure point can be characterized with total strain, equal to the length of segment \(OC\), and its reversible part, equal to the length of segment \(OL\). Simultaneously, the energetic concept can be used in a similar way: the area of curvilinear trapezoid \(OABL\) represents the total amount of energy stored in the sample at failure point, while the area of triangle \(CBL\) stands for reversible part of stored energy. The second definition should be discussed here from energetic concept of failure process. It has already been mentioned that failure of the rock is associated with energy release: recall that critical energy release rate \(G_{c}\) has been used for LBI calculation. In fact, within the failure mechanics concepts [28], this energy is associated with the area of free surface emerging during failure process. This free surface is, in turn, closely related to brittleness. Consider two media: very brittle and very ductile samples. A brittle sample subjected to external loading will fail with many micro fractures or one big fracture. In both cases, the cumulative area of free surfaces of all parts of the sample after failure will be large. At the same time, failure of the ductile sample will be associated with plastic flow and dislocations emergence and movement. The cumulative area of free surface of the sample after failure will be less, compared to the brittle sample.