Evaluation of Rock Brittleness Based on Complete Stress–Strain Curve

Abstract

As a basic mechanical property of rocks, brittleness is closely related to the drillability, wellbore stability, and rockburst characteristics of reservoir rocks. Accurate evaluation of rock brittleness is of great significance for guiding oil and gas production and reservoir reconstruction. This paper systematically introduced the commonly used brittleness evaluation methods based on the stress–strain curve and analyzed their theoretical background and mathematical models. Combined with practical engineering application, the characterization effect of commonly used brittleness indexes in various rock samples is verified and optimized, and it is obtained that the brittleness index (B₁₇ in this paper), based on the stress–strain curve and considering energy conversion, has the best characterization result for rock brittleness, which has good differentiation for different rock samples. At the same time, considering that the stress–strain curve under high confining pressure may result in a significant yield plateau phenomenon before and after the peak strength, the endpoint of the plastic yield plateau is used to replace the peak point as the starting point for the drop of bearing capacity. The revised brittleness index is consistent with the changing trend of the original curve, which verifies the reliability of the model. Finally, the method for characterizing the brittleness of Class II curves is supplemented, and the combined brittleness index of rock is established, which verifies the rationality and correctness of the index and provide a more general evaluation method for rock brittleness in engineering.

1. Introduction

The brittleness of rock refers to the property that the bearing capacity of rock will be rapidly lost when it produces minimal plastic deformation in the process of loading. As an important basic mechanical property of rocks, brittleness is closely related to reservoir rock drillability [1,2,3], well wall stability [4,5], rockburst characteristics [6,7,8], etc. Brittleness assessment has an important reference value for the safety of some tunnel support projects and destabilization aspects of block collapse [9,10]. Effective evaluation of rock brittleness is of great significance for the guidance of oil and gas production and reservoir reconstruction.

At present, researchers have established various brittleness evaluation indicators in terms of rock mineral composition and porosity [11,12,13,14,15,16], rock strength parameter [17,18,19,20,21], Young’s modulus and Poisson’s ratio [22,23,24], hardness and fracture toughness [25,26,27], impact penetration test [28], acoustic emission characteristics [29,30,31,32], etc. An effective and direct method to evaluate rock brittleness is by using characteristic parameters from full stress–strain curves obtained from laboratory mechanical experiments. Therefore, most of the new brittleness indexes in recent years are based on the stress–strain curve of rock [33]. Whether from the perspective of energy accumulation and dissipation or the perspective of the characteristics of the stress–strain curve, such indicators have a clear physical significance, and the parameters are relatively simple to obtain.

Due to the lack of corresponding theoretical background, the conclusions of some brittleness indexes used to evaluate rock brittleness are contradictory. The reliable and applicable brittleness index should not only have a sufficient scientific basis, but it should also cover the entire rock fracture process. In this paper, from the point of view of the complete stress–strain curve of rock, the commonly used and the latest brittle evaluation indexes are analyzed. Combined with the typical stress–strain curves obtained in practical engineering, the applicability of various brittleness indexes is verified and optimized, and the brittleness calculation method of rock class II curve is supplemented in order to provide a more general evaluation method for brittleness evaluation in rock engineering.

2. Research Progress of Rock Brittleness Evaluation Based on Stress–Strain Curve

Obtaining mechanical parameters such as rock strain and strength by triaxial compression test is the most common method to study the characteristics of rock failure, and it is also one of the more intuitive means to study rock brittleness. Due to the difference in lithology and loading conditions, different rocks will show different stress–strain curve characteristics, but after simplification, they can all be described by the four curves in Figure 1. In fact, the pre-peak stage of the stress–strain curve of different rocks is similar and can be roughly divided into four stages, the difference is that the proportions of each stage are different. The first stage is the initial compaction stage (AB segment). The primary cracks in the rock are compacted under small external forces, and the curve is concave. The endpoint of this stage is the crack closure stress. Second is the elastic deformation stage (BC segment). The stress–strain curve at this stage is approximately a straight line, and the elastic modulus of rock can be determined by the slope of the stress–strain curve at this stage. The endpoint of this stage is the crack initiation stress. Third is the stable crack growth stage (CD segment). With the increase of the load, new cracks are initiated, and the evolution speed of lateral strain starts to increase. In this stage, the crack damage stress (also known as long-term strength of rock [34,35,36]) is the endpoint. Finally, there is the unstable crack growth stage (DE segment). The internal microcracks of the rock are interconnected and form a macro tension or shear band, and the rock is subject to macro failure. In this stage, the peak stress is the endpoint.

At the post peak stage, due to the difference between the test method and the mechanical properties of the rock itself, the stress–strain curve may have significant differences in morphology. If limited by the response rate of the testing machine, it may not be possible to obtain a complete post peak curve, which is represented by curve ①. It is also possible to obtain a complete post peak curve when the rock reaches the balance of the next stage, which is represented by curve ②, and the tortuous part represents the characteristics of unsteady failure. Of course, for most rocks, in the triaxial compression test, the post peak stage will show strain softening, that is, with the increase of strain, the stress will gradually drop, which is represented by curve ③. In addition, when the confining pressure is large enough, a plastic yield platform appears in some rocks after the stress reaches the peak value, showing the brittle plastic transformation characteristics, which is represented by curve ④.

A large number of triaxial compression tests show that the brittle failure of rock is closely related to the whole process of its internal microcrack initiation, propagation, penetration and failure. As the embodiment of rock mechanical properties, the stress–strain curve reflects the whole process of rock from deformation to ultimate loss of bearing capacity under external load [37]. Use of the full stress–strain curve is an effective means to obtain characteristic parameters for the evaluation of rock brittleness. Therefore, many scholars have proposed a variety of methods to characterize rock brittleness based on the full stress–strain curve.

2.1. Brittleness Index Based on Stress–Strain Curve and Characteristic Parameters

In recent years, many scholars have conducted in-depth and detailed discussions on the evaluation of rock brittleness based on the stress–strain curve. For instance, the rock brittleness index (B₁) established by Bishop (1967) [21] is related to the post-peak stress drop. The greater the stress drop, the stronger the rock brittleness. Hucka and Das (1974) [17] defined the brittleness index as the ratio of the elastic strain to the total strain (B₂) when the rock fails. Andreev (1995) [38] defined brittleness as a lack of ability to withstand plastic deformation, used the absolute plastic strain before rock failure as an index (B₃) to characterize rock brittleness, and used 0.3 and 0.5 as the characteristic values for brittle and ductile failure. The brittleness index reflects the influence of confining pressure on plastic deformation, but ignores the post-peak damage state, which may be inconsistent with the actual situation in an engineering application.

Li et al., (2012) [39] believed that the pre-peak and post-peak stages should be considered to characterize the brittleness of rocks and proposed the brittleness index (B₄) based on the full stress–strain curve. Zhou et al., (2014) [40] put forward the brittleness evaluation index (B₅), which considers both the relative value and the absolute rate of the post-peak stress drop of the rock mass, but the improved brittleness index only considers the post-peak stress–strain state and does not consider the influence of the pre-peak mechanical behavior changes on rock brittleness. Since then, Chen et al., (2018) [41] have established a new brittleness index (B₆) by comprehensively considering the pre-peak stress growth rate and post-peak stress drop rate. Liu et al., (2019) [42] proposed a brittleness evaluation method (B₇) based on post-peak stress drop rate and pre-peak brittleness damage failure difficulty, explored the influence of the variation of different rock meso-structure parameters on the degree of rock brittleness based on PFC2D software, and analyzed the relationship between different meso-structure parameters and the degree of brittleness, indicating that the established brittleness index can better reflect the brittleness differences of different rock samples. Cao et al., (2020) [43] concluded that post-peak brittle failure of granite should satisfy both the conditions of fast stress drop rate and fast strain growth rate, and proposed a new brittleness evaluation index (B₈) that can reflect the whole process of granite deformation and damage, and combined this with a test to verify that the proposed brittleness index can accurately reflect the brittleness characteristics of granite with different moisture content. Zhang et al., (2021) [44] proposed a new method for evaluating the brittleness characteristics of rocks (B₉) by considering the pre-peak stress growth rate, growth magnitude and post-peak stress–strain relationship. It was verified through experimental data that this brittleness index can not only accurately evaluate the brittleness of different rocks (marble, tuff, sandstone and granite), but also better reflect the brittle characteristics of rocks under complex stress states. Kuang et al., (2022) [45] proposed a new brittleness evaluation index (B₁₀) that simultaneously considers the pre-peak stress growth rate, post-peak stress drop rate and the combined strain relationship at the peak point. The index has a clear physical meaning and can reflect the change in brittleness of various types of rocks under confining pressure. Several common brittleness indexes based on the shape of stress–strain curve and characteristic parameters are shown in

Table 1. Brittleness indexes based on stress–strain curve and characteristic parameters.

Brittleness Index	Description	Researchers
$B_1=\frac{{\sigma }_p-{\sigma }_r}{{\sigma }_p}$	${\sigma }_p$ and ${\sigma }_r$ are the peak stress and residual stress, respectively	Bishop, 1967 [21]
$B_2=\frac{{\epsilon }_{\mathrm{e}}}{{\epsilon }_{\mathrm{p}}}=\frac{{\sigma }_{\mathrm{P}}}{E{\epsilon }_{\mathrm{P}}}$	${\epsilon }_{\mathrm{e}}$ is the recoverable (or elastic) strain	Hucka and Das, 1974 [17]
$B_3={\epsilon }_{1i}\times 100\%=\left({\epsilon }_{\mathrm{p}}-\frac{{\sigma }_{\mathrm{P}}}{E}\right)\times 100\%$	${\epsilon }_{1i}$ is the unrecoverable axial strain	Andreev, 1995 [38]
$B_4=B_i+B_j$, $B_i=({\epsilon }_p-{\epsilon }_n)/\left({\epsilon }_m-{\epsilon }_n\right)$  $B_j=\alpha CS+\beta CS+\eta$,$CS=\left({\epsilon }_P/{\sigma }_p\right)\left({\sigma }_p-{\sigma }_r\right)/\left({\epsilon }_r-{\epsilon }_P\right)$	$B_i$ and $B_j$ are the pre-peak index and post-peak index, respectively; ${\epsilon }_m$ and ${\epsilon }_n$ are the maximum and minimum peak strains, respectively; ${\epsilon }_P$ is the peak strain; $\alpha$, $\beta$ and $\eta$ are the standardized coefficients	Li et al., 2012 [39]
$B_5=B_{i1}{B}_{i2}$, $B_{i1}=\frac{{\sigma }_p-{\sigma }_r}{{\sigma }_p}$,$B_{i2}=\frac{\lg \left|k_{ac\left(AC\right)}\right|}{10}$	$B_{i1}$ and $B_{i2}$ are the relative magnitude and absolute rate of post-peak stress drop, respectively; $k_{ac\left(AC\right)}$ is the absolute value of slope from yield starting point to residual starting point	Zhou et al., 2014 [40]
$B_6=B_{i1}+B_{i2}=\frac{\left({\sigma }_p-{\sigma }_{ci}\right){\epsilon }_p}{\left({\epsilon }_p-{\epsilon }_{ci}\right){\sigma }_p}+\frac{\left({\sigma }_p-{\sigma }_r\right){\epsilon }_p}{\left({\epsilon }_r-{\epsilon }_p\right){\sigma }_p}$	${\sigma }_{\mathrm{ci}}$ is the crack initiation stress; ${\epsilon }_{\mathrm{ci}}$ is the crack initiation strain	Chen et al., 2018 [41]
$B_7=B_{\mathrm{i}}{B}_{\mathrm{j}}=\frac{1}{{\epsilon }_p}\cdot \frac{\left({\epsilon }_p/{\sigma }_p\right)\left({\sigma }_p-{\sigma }_r\right)}{\left({\epsilon }_p-{\epsilon }_r\right)}$	${\sigma }_p$ and ${\sigma }_r$ are the peak stress and residual stress, respectively; ${\epsilon }_p$ and ${\epsilon }_r$ are the peak strain and residual strain, respectively	Liu et al., 2019 [42]
$B_8=\lg \left(\frac{1}{{\epsilon }_{\mathrm{p}}}\right)/5+\frac{{\sigma }_p-{\sigma }_r}{{\sigma }_p\cdot t}+\frac{{\epsilon }_r-{\epsilon }_p}{{\epsilon }_p\cdot t}$	$t$ is the time interval between the peak strength point on the stress–strain curve and the starting point of residual strength (s)	Cao et al., 2020 [43]
$B_9=B_i+B_j=\frac{2}{\pi {\epsilon }_{\mathrm{p}}}\arctan \frac{{\sigma }_{\mathrm{p}}}{{\epsilon }_{\mathrm{p}}}+\frac{\left({\sigma }_{\mathrm{p}}-{\sigma }_{\mathrm{r}}\right)/{\sigma }_{\mathrm{p}}}{\left({\epsilon }_{\mathrm{r}}-{\epsilon }_{\mathrm{p}}\right)/{\epsilon }_{\mathrm{r}}}$	${\sigma }_p$ and ${\sigma }_r$ are the peak stress and residual stress, respectively; ${\epsilon }_p$ and ${\epsilon }_r$ are the peak strain and residual strain, respectively	Zhang et al., 2021 [44]
$B_{10}=B_i{B}_j{B}_k$, $B_i=\frac{{\sigma }_{\mathrm{p}}-{\sigma }_{\mathrm{ci}}}{{\epsilon }_{\mathrm{p}}-{\epsilon }_{\mathrm{ci}}}$,$B_j={\mathrm{e}}^{\frac{10\left({\epsilon }_{\mathrm{p}}-{\epsilon }_{\mathrm{r}}\right)}{{\sigma }_{\mathrm{p}}-{\sigma }_{\mathrm{r}}}}$, $B_k=1/{\epsilon }_{\mathrm{p}}$	$B_k$ is the coefficient of strain control; ${\sigma }_{\mathrm{ci}}$ is the crack initiation stress; ${\epsilon }_{\mathrm{ci}}$ is the crack initiation strain	Kuang et al., 2022 [45]

.

It can be seen from Table 1 that the pre-peak curve reflects the ability of the rock to resist inelastic deformation before damage, while the post-peak curve reflects the magnitude of the bearing capacity of the rock after damage, and a sole consideration of the pre-peak or post-peak curve form cannot fully reflect the rock brittleness. The study shows that the smaller the total strain of the rock before stress damage, the steeper the straight-line section before the peak, the larger the post-peak stress drop, and the faster the stress drop is under that same post-peak stress drop, the stronger the brittleness is.

2.2. Brittleness Index Based on Stress–Strain Curve and Energy Evolution

In essence, rock deformation and failure are the comprehensive results of energy dissipation and energy release. Energy dissipation is mainly used to induce rock damage, leading to deterioration of rock properties and loss of strength. Energy release is the internal cause of sudden rock failure [46]. Therefore, the establishment of brittleness index from the perspective of energy can better reflect the nature of rock failure.

Hucka and Das (1974) [17] believed that the energy source of rock fracture was the elastic energy stored in the rock and defined the brittleness index (B11) as the ratio of elastic strain energy to the total energy at the peak point. Although the index considered the energy before the peak, it did not describe the energy release after the peak. Based on the post-peak energy balance theory, Tarasov and Potvin (2013) [47] proposed the use of the ratio of post-peak fracture energy or post-peak additional (released) energy to the elastic energy consumed in the whole process as the brittleness index (B₁₂ and B₁₂’). The brittleness index can describe the Class I and Class II brittle failure behavior of rocks and can reflect the whole process of brittle ductile transformation of rocks. It has good monotonicity and continuity, but there are also some problems. For example, the stress–strain curves ABG and ABE in Figure 2 have the same pre-peak behavior and post-peak modulus, and the results are the same when characterized by the brittleness indexes B₁₂ and B₁₂’, though obviously the two curves have different brittle failure characteristics.

Xia et al., (2016) [48] proposed a rock brittleness index (B₁₃) characterized by the ratio of the post-peak stress drop rate of the rock and the elastic energy released during rock destruction to the total energy stored before the peak. The greater the index, the higher the brittleness of rock. Ai et al., (2016) [49] believed that more reasonable rock brittleness evaluation indicators should include: (1) the ability of materials to resist inelastic deformation before failure; (2) the decline of rock bearing capacity after brittle failure; (3) cohesion and friction of rock materials; and (4) the whole process from brittleness to plasticity. In addition, the evaluation results should be continuous and monotonous. On this basis, a new brittleness index (B₁₄, B₁₅) is defined based on the fracture energy, post-peak release energy and pre-peak dissipation energy. It is found that the brittleness of different rocks (red sandstone, shale and granite) changes differently with the increase of confining pressure. Zhang et al., (2017) [50] believed that the dissipated energy before the peak and the fracture energy after the peak are the essential factors determining whether brittle fracture occurs in rocks. Considering the energy evolution in the whole process of rock failure, a new brittleness index (B₁₆) is established based on the linear elastic modulus, yield modulus and weakening modulus of the curve. It is found that the pre-peak dissipation energy and post-peak fracture energy of different rock materials increase with the increase of confining pressure, and that the brittle plastic transformation occurs in the rocks, but that the granite maintained strong brittleness. Kivi et al., (2018) [51] considered that the smaller the plastic deformation in the yield stage of the pre-peak region, the stronger the fracture process in the post-peak stage, and the more brittle the material character. Based on energy conversion, a new brittleness index (B₁₇) is established, which describes the degree of plastic yield deformation, and the degree and rate of strength degradation during rock failure. However, this index only considers Class I failure behavior of rock, and whether it is applicable to Class II failure remains to be discussed. Chen et al., (2020) [52] concluded that the higher the pre-peak accumulation rate and post-peak dissipation rate of rocks, the stronger their brittleness. Therefore, they established the brittleness index (B18) based on the whole process of elastic energy evolution, analyzed the influence of confining pressure and water pressure on the brittleness of different rocks, and verified the rationality of the brittleness index through laboratory tests. Wen et al., (2021) [53] believed that the ideal brittle rock has no energy dissipation process before the stress reaches the peak strength. Considering the energy evolution characteristics of the rock before and after the peak, they established the brittleness index evaluation method (B₁₉) to comprehensively describe the brittleness of the rock. This index only considers the energy evolution characteristics of the rock’s Class I curve and does not involve the rock’s Class II curve. Gong et al., (2022) [54] established a new brittleness index (B₂₀) based on the dissipation ratio of peak elastic strain energy to failure energy and residual elastic strain energy. Combined with triaxial compression tests of different rocks, it verified the universality of the brittleness index to evaluate rocks. Some common brittleness indexes based on the stress–strain curve and energy evolution are shown in

Table 2. Brittleness indexes based on stress–strain curves and energy evolution.

Brittleness Index	Description	Researchers
$B_{11}=\frac{W_{te}}{W_{\mathrm{te}}+W_{\mathrm{d}}}=\frac{\frac{{\sigma }_p^2}{2E}}{{{\displaystyle \int }}_0^{{\epsilon }_p}{\sigma }_i\mathrm{d}{\epsilon }_i}$	$W_{te}$ is the total elastic energy; $W_{\mathrm{d}}$ is the dissipated plastic energy	Hucka and Das, 1974 [17]
$B_{12}=\frac{W_{\mathrm{r}}}{W_{\mathrm{ue}}}=\frac{M-E}{M}$, $B_{12}\prime =\frac{W_{\mathrm{a}}}{W_{\mathrm{ue}}}=\frac{E}{M}$	$W_{\mathrm{r}}$ is the rupture energy; $W_{\mathrm{a}}$ is the additional (or released) energy; $W_{\mathrm{ue}}$ is the consumed elastic energy; $E$ and $M$ are the Young’s modulus and drop modulus, respectively	Tarasov and Potvin, 2013 [47]
$B_{13}=B_i+B_j$, $B_i=\frac{{\sigma }_p-{\sigma }_r}{{\epsilon }_r-{\epsilon }_p}$$B_j=\frac{\left({\sigma }_p-{\sigma }_r\right)({\epsilon }_r-{\epsilon }_p)}{{\sigma }_p{\epsilon }_p}$	$B_i$ and $B_j$ are the pre-peak brittleness index and post-peak brittleness index, respectively	Xia et al., 2016 [48]
$B_{14}=\frac{W_{\mathrm{r}}+W_{\mathrm{d}}}{W_{\mathrm{ue}}+W_{\mathrm{d}}}=\frac{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)\left(M-E\right)}{2EM}+\left({{\displaystyle \int }}_0^{{\epsilon }_p}{\sigma }_i\mathrm{d}{\epsilon }_i-\frac{{\sigma }_p^2}{2E}\right)}{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)}{2E}+\left({{\displaystyle \int }}_0^{{\epsilon }_p}{\sigma }_i\mathrm{d}{\epsilon }_i-\frac{{\sigma }_p^2}{2E}\right)}$	$W_{\mathrm{ue}}$ is the consumed elastic energy	Ai et al., 2016 [49]
$B_{15}=\frac{W_{\mathrm{a}}}{W_{\mathrm{ue}}+W_{\mathrm{d}}}=\frac{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)}{2M}}{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)}{2E}+\left({{\displaystyle \int }}_0^{{\epsilon }_p}{\sigma }_i\mathrm{d}{\epsilon }_i-\frac{{\sigma }_p^2}{2E}\right)}$	$W_{\mathrm{a}}$ is the additional (or released) energy	Ai et al., 2016 [49]
$B_{16}=\frac{E}{D}\frac{M-E}{M}$, $B_{16}\prime =\frac{E-D}{D}\frac{E}{M}$	$E$ and $M$ are the Young’s modulus and drop modulus, respectively; $D$ is the pre-peak yield modulus	Zhang et al., 2017 [50]
$B_{17}=\frac{1}{2}\left(\frac{W_{\mathrm{ue}}}{W_{\mathrm{r}}}+\frac{W_{\mathrm{ue}}}{W_{\mathrm{te}}+W_{\mathrm{d}}}\right)=\frac{1}{2}\left(\frac{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)}{2E}}{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)}{2E}+{{\displaystyle \int }}_{{\epsilon }_p}^{{\epsilon }_r}{\sigma }_i\mathrm{d}{\epsilon }_i}+\frac{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)}{2E}}{{{\displaystyle \int }}_0^{{\epsilon }_p}{\sigma }_i\mathrm{d}{\epsilon }_i}\right)$	$W_{\mathrm{r}}$ is the rupture energy; $W_{\mathrm{ue}}$ is the consumed elastic energy	Kivi et al., 2018 [51]
$B_{18}=\frac{1}{2}\left(\frac{W_{\mathrm{te}}}{W_{\mathrm{te}}+W_{\mathrm{d}}}+\frac{W_{\mathrm{ue}}}{W_{\mathrm{r}}}\right)=\frac{1}{2}\left(\frac{\frac{{\sigma }_p^2}{2E}}{{{\displaystyle \int }}_0^{{\epsilon }_p}{\sigma }_i\mathrm{d}{\epsilon }_i}+\frac{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)}{2E}}{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)}{2E}+{{\displaystyle \int }}_{{\epsilon }_p}^{{\epsilon }_r}{\sigma }_i\mathrm{d}{\epsilon }_i}\right)$	$W_{te}$ is the total elastic energy; $W_{\mathrm{d}}$ is the dissipated plastic energy; $W_{\mathrm{r}}$ is the rupture energy; $W_{\mathrm{ue}}$ is the consumed elastic energy	Chen et al., 2020 [52]
$B_{19}=\frac{W_{pre}-W_c-W_{te}^{\prime }}{W_{te}-W_{te}^{\prime }}\cdot \frac{W_{ue}}{W_{ue}-W_{post}}$ $=\frac{{{\displaystyle \int }}_0^{{\epsilon }_p}{\sigma }_i\mathrm{d}{\epsilon }_i-{{\displaystyle \int }}_0^{{\epsilon }_{cc}}{\sigma }_i\mathrm{d}{\epsilon }_i+\frac{{\sigma }_{cc}^2}{2E}-\frac{{\sigma }_{cd}^2}{2E}}{\frac{{\sigma }_p^2}{2E}-\frac{{\sigma }_{cd}^2}{2E}}\cdot \frac{\frac{{\sigma }_p^2}{2E}-\frac{{\sigma }_r^2}{2E}}{\frac{{\sigma }_p^2}{2E}-\frac{{\sigma }_r^2}{2E}+{{\displaystyle \int }}_{{\epsilon }_p}^{{\epsilon }_r}{\sigma }_{i}d{\epsilon }_i}$	$W_{te}$ is the total elastic energy; ${\mathrm{W}}_{\mathrm{te}}^{\prime }$ is the elastic energy in elastic stage; ${\mathrm{w}}_{\mathrm{c}}$ is the compaction stage energy; ${\mathrm{W}}_{\mathrm{pre}}$ is the total energy before peak; ${\epsilon }_{cc}$ and ${\sigma }_{cc}$ are the crack closure strain and stress, respectively	Wen et al., 2021 [53]
$B_{20}=\frac{W_{\mathrm{ue}}-W_{\mathrm{r}}^{\prime }}{W_{\mathrm{te}}}=1-\frac{2E{W}_{\mathrm{r}}^{\prime }+{\sigma }_r^2}{{\sigma }_p^2}$$W_{\mathrm{r}}^{\prime }=\left\{\begin{array}{c}{{\displaystyle \int }}_{{\epsilon }_p}^{{\epsilon }_r}{\sigma }_{1}d{\epsilon }_1-\frac{{\sigma }_r^2}{2E},{\epsilon }_p<{\epsilon }_r-\frac{{\sigma }_r}{E}\\ {{\displaystyle \int }}_{{\epsilon }_p}^{{\epsilon }_r}{\sigma }_{1}d{\epsilon }_1-\frac{{\sigma }_r^2}{2E}+{\left({\epsilon }_p-{\epsilon }_r+\frac{{\sigma }_r}{E}\right)}^{2}E/2,{\epsilon }_p>{\epsilon }_r-\frac{{\sigma }_r}{E}\end{array}\right.$	$W_{\mathrm{r}}^{\prime }$ is the failure energy density; $W_{\mathrm{ue}}$ is the consumed elastic energy; $W_{te}$ is the total elastic energy	Gong et al., 2022 [54]

.

It can be seen from Table 2 that the dissipated energy, fracture energy and consumed elastic energy can reflect the brittleness characteristics of rocks. Combining different energies can reflect the ability of rock to resist inelastic deformation and crack propagation.

3. Verification and Optimization of Brittleness Index

It is found that the strength and plasticity of rock gradually increase with the increase of confining pressure, while the brittleness decreases accordingly [55,56,57]. With the increase of confining pressure, the rock has obvious brittle to ductile transformation characteristics [58]. In addition to a significant influence on reservoir strength and brittleness index, confining pressure also affects fracture propagation [59]. Therefore, it is of great importance to study the evaluation of rock brittleness under confining pressure.

In this paper, the variation law of various rock brittleness indexes with confining pressure is analyzed, so as to screen out the brittleness indexes with clear physical meaning, complete mathematical model and strong operability from the existing brittleness indexes, and provide reference for practical projects.

3.1. Basis for Rock Sample Selection

In excavation and crushing engineering, rock brittleness determines the drilling efficiency of the drilling rig during TBM tunneling [60,61]. Based on the Han to Wei River Diversion Project, Wang (2021) [62] considered the influence of surrounding rock and unfavorable geological conditions of caverns on TBM construction, in which the surrounding rock of the tunnel of the Han to Wei River Diversion Project is mainly granite. Tian (2020) [63] relied on the Yangtai Mountain Tunnel to study the tool wear and tunneling performance of TBM construction. According to the tunnel geological exploration data and the supplementary exploration core test report, the tunnel body of the Yangtai Mountain Tunnel is in the slightly weathered granite stratum. The surrounding rock of the tunnel body is hard granite with good integrity, undeveloped joints and high quartz content. Xu et al., (2014) [64] studied the influence of high temperature and confining pressure on the deformation and strength characteristics of granite and obtained the stress–strain curves of conventional triaxial compression under different confining pressures.

In hard rock tunneling and deep rock mass engineering, rock brittleness is an important internal factor of engineering disasters such as rock burst [65,66,67]. Several extremely strong rock bursts occurred in seven parallel tunnels in the Jinping-II hydropower station, and this consequently became a typical case in the analysis of related accidents [68]. The main rock types of Jinping mountain are hard and brittle marble and a small amount of slate and sandstone. Wang et al., (2019) [69] studied the effect of confining pressure on the mechanical properties of Jinping marble by triaxial compression test and proposed an elastoplastic brittle mechanical model. Zhang et al., (2010) [70] investigated the mechanical properties of deeply buried barite in the Jinping II hydropower station and carried out conventional triaxial compression tests under different confining pressures to obtain the corresponding stress–strain curves.

In the extraction engineering of unconventional oil and gas resources, rock brittleness, as a key indicator for reservoir evaluation, is closely related to the stability of the wall in vertical wells, the initiation and extension of hydraulic fractures and the effective communication of the final fracture network [71,72]. The Upper Ordovician Wufeng formation and the Lower Silurian Longmaxi formation in the Sichuan Basin and its surrounding areas are the only sets of shale gas strata that have been commercially developed in China [73]. He et al., (2017) [74] investigated the indoor macroscopic mechanical behavior of the Longmaxi formation outcrop shale in Sichuan and Chongqing and obtained multiple sets of stress–strain curves. Kivi et al., (2018) [51] collected several sets of shale samples from organic-rich shale formations in Iran, carried out conventional triaxial compression tests and obtained several sets of stress–strain curves.

In addition, red sandstone is a more widely distributed rock in East China, South China and Southwest China, and is a hard and brittle sedimentary rock. With the rapid development of regional engineering construction, there have been increasing problems with the foundations, road foundations, tunnels and slope works that consist of red sandstone, while red sandstone is often developed in large quantities as building stone, all of which indicate the need for in-depth study of the physical and mechanical properties of red sandstone. Su et al., (2014) [75] carried out triaxial compression tests on red sandstone and obtained stress–strain curves for different confining pressures.

Combining the above typical rock engineering fields, four common rock types, granite [64], marble [70], shale [51] and red sandstone [75], were selected for this paper, and their corresponding stress–strain curves are shown in Figure 3.

Based on their stress–strain curves, the mechanical parameters of various types of rocks under different confining pressures are calculated as shown in

Table 3. Mechanical parameters of various rocks under different confining pressures.

Rock Type	${\mathit{\sigma }}_{\mathbf{3}}$ /MPa	${\mathit{\epsilon }}_{\mathit{p}}$ /10−3	${\mathit{\sigma }}_{\mathit{p}}$ /MPa	${\mathit{\epsilon }}_{\mathit{r}}$ /10−3	${\mathit{\sigma }}_{\mathit{r}}$ /MPa	E /GPa	${\mathit{W}}_{\mathbf{pre}}$/ (kJ/m3)	${\mathit{W}}_{\mathit{total}}$/ (kJ/m3)
Granite [53]	0	4.229	120.251	4.934	12.025	37.622	236.683	256.840
	10	6.826	198.413	7.717	4.509	35.789	641.066	696.774
	20	7.531	284.843	8.125	98.455	41.867	1029.612	1161.984
	30	9.498	339.708	10.425	142.046	42.599	1637.621	1852.088
	40	11.464	326.931	13.319	172.860	35.371	1651.368	2069.166
Marble [59]	2	3.728	140.927	7.091	37.500	54.334	278.495	468.527
	10	6.056	164.516	14.740	78.024	40.135	629.765	1446.567
	15	6.256	179.032	14.077	97.379	40.421	685.228	1742.071
	30	8.480	217.742	19.185	172.984	55.656	1374.995	3515.755
	40	12.888	238.306	33.779	212.903	50.890	2649.103	7135.708
	50	15.571	258.871	32.914	232.863	50.131	2608.109	7517.521
Shale [40]	0	3.003	84.055	6.967	0.000	32.509	114.634	282.584
	10	5.526	183.827	8.889	90.888	37.816	544.030	966.653
	25	7.928	224.146	17.778	164.009	41.644	1188.820	3300.789
	40	9.249	260.364	19.940	191.344	42.818	1631.923	4284.461
Red sandstone [64]	0	6.067	67.683	6.290	16.463	17.143	191.788	207.505
	10	9.010	132.317	12.960	60.976	23.555	608.519	973.384
	20	11.999	173.780	14.404	130.488	22.818	1152.657	1373.155
	30	14.518	212.805	17.310	162.195	22.470	1617.703	2161.071
	45	14.518	264.024	18.673	200.610	22.967	2123.243	3028.882

.

3.2. Analysis of Brittleness Index Calculation Results

Table 1 and Table 2 summarize 20 commonly used brittleness indexes based on the stress–strain curve. Since the establishment of various brittleness indexes is based on different theoretical backgrounds, their expression forms are different, and the characteristic points of curves selected in application are also different. Li et al., (2022) [33] believed that there would be some human error in selecting characteristic parameters. For example, when the crack volume strain inflection point method is used to determine the crack initiation stress, the crack initiation stress will vary with the selected linear elastic stage. Therefore, in the brittleness index summarized in Section 2, six brittleness evaluation methods with clear mathematical model, simple calculation and strong operability are selected for calculation, and the corresponding calculation results are shown in Figure 4.

As can be seen from Figure 4, the brittleness indexes B₁ and B₅ basically reflect the trend of brittleness of various types of rocks with the confining pressure. However, except for granite, the brittleness curves of the remaining three types of rocks have a lot of crossover and overlap, and the degree of differentiation is not high. When B₉ is used to describe the brittleness of granite and red sandstone, they are basically coincident and cannot be effectively distinguished, and the brittleness characterization effect of marble and shale is also not well known. B₁₃ is not ideal for characterizing the brittleness of the other three types of rocks except for granite. In fact, the brittleness indices B₁ and B₅ are based on the post-peak softening model of the rock, B9 considers the pre-peak stress growth rate, and B₁₃ introduces the pre-peak energy ratio. However, only two state quantities, the peak point (or the onset of the yielding phase) and the residual point, are considered in the final expression, ignoring the specific curve form. Therefore, it is theoretically possible to obtain the same brittleness for different forms of the curve.

The evaluation results of B₂₀ on the brittleness of marble are relatively reasonable, but the calculation results of the other three types of rocks vary slightly. However, B₁₇ has a good representation effect on the brittleness trend of all kinds of rocks with confining pressure, and the brittleness curve has an obvious stratification phenomenon, indicating that the brittleness of different rock samples varies greatly. In conclusion, among the brittleness indexes based on the stress–strain curve, the brittleness index B₁₇, considering the whole process of energy evolution, has the best characterization effect and should be the preferred method for brittleness evaluation of conventional rock Class I curves.

4. Optimization and Supplement of Brittleness Index

4.1. Optimization of Brittleness Index with Confining Pressure

It was found that the stress–strain curves of certain rocks under high confining pressure conditions would show a significant yield plateau around the peak strength, exhibiting significant brittle-plastic transition characteristics, and eventually exhibiting strain softening with increasing deformation [40,69]. As shown in Figure 5, point A indicates the stress–strain curve just entering the yielding phase, point B indicates the end of the yield plateau and point C indicates the residual phase of plastic flow.

Zhang et al., (2017) [50] argued that for plastic rocks, as the fracture continues to expand, the block continues to rotate and collapse into smaller pieces, so that the resistance to shear gradually changes from rock cohesion to friction between the fracture surfaces. Due to the need to maintain the extension of the fracture in the material, the collapse process of the block requires the absorption of a large amount of energy, and this energy corresponds to the energy corresponding to the yield plateau before and after the peak in the stress–strain curve. Therefore, the brittleness calculation for such curves can be approximated by describing it with an elastic–plastic brittle model [69], considering the endpoint of the plastic yield plateau instead of the peak rock point as the starting point of the load carrying capacity fall when taking values.

Among the four types of rock samples selected in Section 3.1, marble and shale have significant brittle to plastic transition characteristics under high confining pressure, i.e., long peak horizontal sections. Therefore, based on these two groups of data, the calculation results of B₁₇ are compared with the results when replacing the peak point with the endpoint of the peak horizontal segment, as shown in Figure 6.

In the literature [70], it was pointed out that the deeply buried marble rocks of the Jinping Baishan formation showed obvious brittle characteristics at low envelope pressure; with the increase of the envelope pressure, the ductile characteristics were obviously enhanced. When the envelope pressure was further increased to 50 MPa, the post-yielding stage approached the mechanical response of an ideal plastic material, and the brittleness index tended to 0. Finally, the peak post-yielding characteristics were summarized by the brittle–ductile–plastic progressive transition law. In the literature [51], it was found that the strong brittle damage characteristics of shales develop at low envelope pressure states and the evolution to relatively ductile behavior develops at high envelope pressure states.

It can be seen from Figure 6 that the revised marble brittleness index decreases exponentially with the increase of confining pressure, and the curve fitting correlation coefficient is 0.993. The revised shale brittleness index decreases linearly with the increase of confining pressure before the confining pressure is 25 MPa, and then remains basically unchanged, with a curve fitting correlation coefficient of 0.975. The revised calculation result is consistent with the calculation result of B₁₇, which verifies the correctness of the revised model, though the fitting accuracy is higher.

4.2. Applicability Analysis of Brittleness Index B₁₇ to Rock Class II Curve

Class II stress–strain curve is the manifestation of the unstable failure of rock. After the external force exceeds the peak value, the testing machine does not need to do work. The accumulated elastic energy of the specimen itself makes the fracture continue to expand and leads to the failure of the whole rock sample. Chen et al., (2010) [76] believed that it is difficult to obtain a Class II curve for weak rock with low strength and elastic modulus regardless of the loading method used, indicating that the Class II curve itself indicates that the rock has high brittleness. Hou et al., (2015) [77] conducted uniaxial compression tests on Longmaxi formation shale in different bedding directions and obtained a Class II curve of unstable fracture type propagation, which also shows that rock with a Class II curve is relatively brittle.

The brittleness index B₁₇ is based on the Class I stress–strain curve of typical rocks. Whether the brittleness of Class II curve rocks can be reasonably characterized has not been reported and is analyzed theoretically first below. Assuming no energy dissipation before the peak, the brittleness index is equal to 1 when the rock exhibits ideal brittleness, i.e., when the post-peak stress falls vertically to 0. When the stress–strain curve is Class II, the mechanical energy input is no longer required after the peak and the rock sample undergoes self-sustaining damage. As shown in curves ABC, ABD and ABF in Figure 2, the excess energy released gradually decreases, the fracture energy fraction gradually increases but is always less than the elastic energy consumed, and the brittleness index decreases, in line with the objective rule that the brittleness index B17 is also valid for the brittleness characterization of Class II curves.

Ai et al., (2016) [38] obtained stress–strain curves for black shale at different confining pressures, as shown in Figure 7a. As can be seen from the graph, the stress–strain curve of the rock sample changes from Class I to Class II and then to Class I as the confining pressure increases, indicating that the degree of brittleness first increases and then decreases. The brittleness index B₁₇ was used to calculate and analyze it and the brittleness index variation curve was obtained as shown in Figure 7b; it can therefore be seen that the brittleness variation pattern is generally consistent with the findings of Ai et al., (2016) [49]. In addition, the calculated results are significantly greater for 60 MPa and 90 MPa confining pressure conditions, i.e., when the stress–strain relationship is a Class II curve, further demonstrating that the brittleness index B₁₇ can reasonably characterize the brittleness of the rock for both types of curve.

4.3. Supplement to Class II Curves with Incomplete Post-Peak Curves

It was found that the Type II curve is essentially obtained during the unloading of brittle rock specimens during compression damage. It comes with conditions: one is the loading method, in which a slower circumferential deformation is used as the loading control variable when the rock sample is damaged; the other is that the rock sample itself has brittle properties and is able to accumulate a certain amount of elastic energy before damage [76].

He et al., (1994) [78] derived the full stress–strain curves of Class II characteristic rocks based on a linear combination of force and displacement controlling rock tests; Ai et al., (2016) [52] obtained the full stress–strain curves of granite by axial-radial control. This shows that the displacement control system has significant advantages in obtaining the complete post-peak curve.

Nevertheless, in the uniaxial compression of Longmaxi Formation shale, Hou et al., (2015) [77] used a hydraulic servo rigid test material machine with high accuracy and stable performance, and adopted circumferential displacement control, and the stress–strain curve obtained was still severely distorted in the post-peak stage. He et al., (2017) [74] considered the effect of surrounding pressure and used a combination of axial and circumferential loading, but did not obtain a complete post-peak shale softening curve. According to the author’s analysis, the main reason for this is that the failure mode of shale includes sudden brittle failure, the rapid interaction between the equipment and the specimen in a very short period of time, the difficulty of rapid response of the equipment or the over-range of the displacement transducer, etc., all of which forced the experiment to stop before it could obtain the residual strength.

When characterizing the brittleness of such rocks, it can only be analyzed from the pre-peak stage. In combination with the research method of Kivi et al., (2018) [51], supplementing the brittleness index of such rocks from the perspective of energy evolution has been considered. Hucka and Das (1974) [17] believed that the proportion of pre-peak elastic properties to brittleness has a good characterization role, and proposed the classic brittleness index B₁₁ to describe the pre-peak brittleness characteristics. This model has a clear physical meaning and a concise mathematical model. However, the index is always less than 1, so this paper will supplement it in the numerical range.

5. Establishment of Joint Brittleness Index

Four types of stress–strain curve models summarized in Figure 1 are combined to establish the brittleness index of joint characterization from the perspective of energy evolution in this paper. The capacity distribution of each part of the rock during loading is shown in Figure 8.

From the above analysis, it can be seen that the brittleness index B₁₇ can have a better evaluation effect on the brittleness of type ② and type ③ curves.

The index considered the degree of self-sustained damage of the rock from the perspective of energy evolution on the one hand, and the proportion of consumed elastic energy in the fracture process on the other. It can be expressed as:

$$B_{17}=\frac{1}{2}\left(\frac{W_{\mathrm{ue}}}{W_{\mathrm{te}}+W_d}+\frac{W_{\mathrm{ue}}}{W_r}\right)$$

(1)

The corresponding strain energy can be expressed as:

$$W_{ue}=W_{te}-W_{re}=\frac{{\sigma }_p^2-{\sigma }_r^2}{2E}$$

(2)

$$W_{te}=\frac{{\sigma }_p^2}{2E}$$

(3)

$$W_d=W_{pre}-W_{te}={\displaystyle \underset{0}{\overset{{\epsilon }_p}{\int }}\sigma d\epsilon }-\frac{{\sigma }_p^2}{2E}$$

(4)

$$W_r=W_{te}+W_{post}-W_{re}+=\frac{{\sigma }_p^2-{\sigma }_r^2}{2E}+{\displaystyle \underset{{\epsilon }_p}{\overset{{\epsilon }_r}{\int }}\sigma d\epsilon }$$

(5)

The meanings of the relevant parameters are explained in Table 2.

This paper corrects the type ④ curve by adjusting the stress characteristic value. Therefore, it only needs to replace all ${\sigma }_p$ in Equation (2) to (5) with ${\sigma }_t$.

It is known from the analysis in Section 4.3 that the type ① curve needs to be supplemented on the basis of the brittleness index B11. The basic formula of brittleness index B11 is as follows.

$$B_{11}=\frac{W_{te}}{W_{te}+W_d}$$

(6)

The ideal brittle material does not have a plastic deformation stage before the peak, but the real rock dissipates part of its energy during deformation and has a yielding stage, resulting in B11 being necessarily less than 1. The brittleness index B₁₇ also takes values in the range of 0 to 1. Therefore, when describing the hard and brittle characteristics of type ① curves, there should be a more significant distinction in numerical results from the remaining types of curves.

It can be seen from Table 1 that many scholars have considered the influence of peak strain ${\epsilon }_p$ on brittleness [39,42,43,44,45]. According to the definition of brittleness, a small peak strain represents less axial deformation when the rock fails, which corresponds to a higher brittleness. Therefore, this paper supplements B₁₁ with the perspective of peak strain.

First, the peak strain is normalized, so that the data with a large range of changes can be assigned to the same level, which is expressed as follows:

$${\epsilon }_{p,nor}=\frac{{\epsilon }_p-{\epsilon }_{p,\min }}{{\epsilon }_{p,\max }-{\epsilon }_{p,\min }}$$

(7)

After normalization, the minimum peak strain drops to 0, which cannot effectively supplement the value range of B₁₁. A logistic function (or sigmoid function) with the maximum value of 2, which is monotonic between the independent variables of 0 and 1, is introduced and the interval of variation meets the requirements of this paper to complement the brittleness index. In addition, the peak strain is generally negatively correlated with the degree of brittleness, and the opposite of the normalized peak strain is used as the variable. Then, the added brittleness index is obtained, as follows:

$$B_{\mathrm{add}}=\frac{2}{1+e^{{\epsilon }_{p,nor}}}$$

(8)

The new brittleness index can be obtained by combining Equations (6)–(8):

$$B_{new}=\frac{W_{te}}{W_{te}+W_d}+\frac{2}{1+e^{{\epsilon }_{p,nor}}}$$

(9)

Figure 9a shows the test curve of Longmaxi formation shale. The curve at the post peak stage is severely incomplete and tortuous, so the residual strength cannot be obtained. Equation (9) can be verified based on this data.

Figure 9b shows the calculation results of the brittleness index B₁₁ and the revised brittleness index. It can be seen that the result of brittleness index B₁₁ is between 0.5 and 0.75, and the revised result is significantly higher than that of a Class I curve, which can better reflect the brittle characteristics of such rock samples. In addition, the brittleness results of shale show a linear decreasing trend with the increase of confining pressure, which is basically consistent with the research conclusions of He et al., (2014) [74] and also verifies the reliability of the conclusions.

In summary, the four types of stress–strain models proposed in this paper all have corresponding brittleness indices, which can be integrated to obtain a joint brittleness index:

$$\mathrm{BI}=\left\{\begin{array}{cc}\frac{W_{te}}{W_{te}+W_d}+\frac{2}{1+e^{{\epsilon }_{p,nor}}}=\frac{\frac{{\sigma }_p^2}{2E}}{{{\displaystyle \int }}_0^{{\epsilon }_p}{\sigma }_i\mathrm{d}{\epsilon }_i}+\frac{2}{1+\exp \left(\frac{{\epsilon }_p-{\epsilon }_{p,\min }}{{\epsilon }_{p,\max }-{\epsilon }_{p,\min }}\right)}\hfill & \hfill \mathrm{Type} ① \mathrm{curve}\\ \frac{1}{2}\left(\frac{W_{\mathrm{ue}}}{W_{\mathrm{r}}}+\frac{W_{\mathrm{ue}}}{W_{\mathrm{te}}+W_{\mathrm{d}}}\right)=\frac{1}{2}\left(\frac{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)}{2E}}{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)}{2E}+{{\displaystyle \int }}_{{\epsilon }_p}^{{\epsilon }_r}{\sigma }_i\mathrm{d}{\epsilon }_i}+\frac{\frac{\left({\sigma }_p^2-{\sigma }_r^2\right)}{2E}}{{{\displaystyle \int }}_0^{{\epsilon }_p}{\sigma }_i\mathrm{d}{\epsilon }_i}\right)\hfill & \hfill \mathrm{Type} ② \mathrm{and} ③ \mathrm{curve}\\ \frac{1}{2}\left(\frac{W_{\mathrm{ue}}}{W_{\mathrm{r}}}+\frac{W_{\mathrm{ue}}}{W_{\mathrm{te}}+W_{\mathrm{d}}}\right)=\frac{1}{2}\left(\frac{\frac{\left({\sigma }_t^2-{\sigma }_r^2\right)}{2E}}{\frac{\left({\sigma }_t^2-{\sigma }_r^2\right)}{2E}+{{\displaystyle \int }}_{{\epsilon }_t}^{{\epsilon }_r}{\sigma }_i\mathrm{d}{\epsilon }_i}+\frac{\frac{\left({\sigma }_t^2-{\sigma }_r^2\right)}{2E}}{{{\displaystyle \int }}_0^{{\epsilon }_t}{\sigma }_i\mathrm{d}{\epsilon }_i}\right)\hfill & \hfill \mathrm{Type} ④ \mathrm{curve} \end{array}\right.$$

(10)

6. Conclusions

In this paper, the existing commonly used evaluation indexes of rock brittleness are summarized from the perspective of the complete stress–strain curve, and the applicability of various brittleness indexes is verified and optimized by combining them with corresponding engineering cases. A new brittleness index was added for the incomplete post-peak curve, and a joint brittleness index which can characterize various curves was established. The results are as follows.

(1)

Among the brittleness indexes based on the stress–strain curve, the brittleness index B₁₇, which considers the whole process of energy evolution, has the best characterization effect on rock brittleness. It can not only reflect the variation law of brittleness of various rocks with confining pressure but also represent the brittleness differences between rocks of different lithologies. The evaluation results show good differentiation, and it is recommended as the preferred method for brittleness evaluation of conventional rock Class I curves.

(2)

Under high confining pressure, a significant yield plateau may appear in the stress–strain curve before and after the peak strength. Given this situation, the endpoint of the plastic yield platform is used to replace the original peak stress point as the starting point of the drop of the bearing capacity, and the optimized brittleness index is revised. The revised brittleness index is consistent with the original curve, which further verifies that the brittleness index B17 can also better represent the failure behavior of the rock Class II curve.

(3)

Considering that the residual strength of rock cannot be obtained in the short post-peak stage of some stress–strain curves, the existing brittleness index is modified and supplemented, and a joint brittleness evaluation index that can reflect the four types of curve rocks is established. The newly established brittleness index can better reflect the brittleness of various rocks, which is reasonable and referential and can provide a more general evaluation method for rock brittleness evaluation in engineering.