## 2 Existing Brittleness Indices
### Test-Based Brittleness Indices (TBIs)
As a result, total energy stored in a ductile sample at the failure point will be higher than energy accumulated in brittle sample, as a considerable part of it was released to form free surfaces. That means that definition of brittleness index TBI\({}_{2}\) is, in fact, backed up by failure mechanics and remains in agreement with energetic concept of failure process. Various modifications of energy-based TBI\({}_{2}\) have been developed to take post-peak behavior into account [29; 30; 31; 32; 33]: it was shown that post-peak rupture of the sample has an important effect on rock brittleness. There are still other methods to evaluate brittleness index from results of special tests [34; 35; 36; 37; 38; 39], their comprehensive review was provided by [PERSON] et al., 2016 [14], but their consideration is not within the scope of the current study. There are several established ways to calculate brittleness from post-peak rupture of the sample. Following the energy concept, certain operations provide the following definitions of brittleness indices [30]  
\[TBI_{3}=\frac{M-E}{M},\quad TBI_{4}=\frac{E}{M}\ \, \tag{12}\]  
Figure 3: Typical single-stage loading test result for brittleness index determination. Modified after [PERSON] et al., 2021 [27]. where \(E\) is tangential static Young's modulus obtained from the linear zone of stress-strain curve, and \(M\) is unloading modulus--the slope of the sector \(BQ\) in Figure 3 corresponding to post-peak rupture associated with increase in strain and decrease in differential stress. Existence of different methods to evaluate brittleness from loading curve leads to the same problem as for log-based brittleness indices: usage of different TBIs for the same data can result into contradicting estimations. The problem of choosing the proper brittleness index determination suitable for the specific rock is still relevant. Nevertheless, contrary to the case of log-based indices, there are insights regarding the way to verify the suitability of the chosen brittleness index. A number of studies [3; 33] highlight the role of confining pressure in rock brittleness. Simultaneous analysis of stress-strain loading curve and corresponding acoustic emission emerging from microfracture emergence and propagation was performed by [PERSON] et al., 2016 [3] to study brittle to ductile transition: it was shown that increase in confining pressure for sets of triaxial loading tests leads to a corresponding decrease in rock brittleness. [PERSON] et al., 2020 [33] reported that this decrease is typical for rocks of different mineral compositions. As a result, there is an opportunity to check whether a suggested TBI is valid to evaluate brittleness in a specific rock by checking its dependence on confining pressure. The other actual problem is related to normalization of brittleness. Equation (11) are physically consistent, but their direct usage to compare brittleness indices of completely different media can be complicated, as there is no strict definition of what do minimum and maximum values of TBI mean. [PERSON] et al., 2021 [27] proposed an additional normalization of brittleness: any rock sample was considered with respect to its ideally ductile and ideally brittle analogs (lines \(OABG\) and \(OADF\) in Figure 3 respectively). These hypothetical rock analogs were proposed in a way to represent the medium with same elastic moduli (and plastic flow potential for ideally ductile rock) as the studied sample, but with rock strength considerably higher than the studied sample. Ideally ductile analogue of the rock was considered as the potential maximum of total energy storable by the sample without failure. Plastic flow consideration made it possible to evaluate the potential strength of the sample giving an opportunity to introduce ideally brittle analog of the rock--a potential rock that deforms linearly up until its potential failure point \(D\). Brittleness index of ideally ductile rock was taken as zero for TBI evaluation, and ideally brittle rock was characterized by BI of one.