Assessing the Fracturing Process of Rocks Based on Burst–Brittleness Ratio (BBR) Governed by Point Load Testing

The determination of brittleness or so-called “Brittleness Index” is largely empirical, which measures the relative susceptibility of materials to deformation and fracture responses (Altindag 2003). In an early attempt to quantify the rock brittleness, the BI₁ was introduced as the ratio of UCS to tensile strength (TS) (Hucka and Das 1974). It is generally accepted that the difference between compressive and tensile strengths increases with an increase in brittleness (Hucka and Das 1974). The UCS is widely used for classification and design purposes in rock engineering practice. In general, unavoidable challenges associated with expensive testing equipment and recovery of core samples with the desired geometry from highly jointed brittle rocks are yet to be considered as major limitations in implementing different rocks testing techniques for brittleness assessment. Under point load testing (PLT), these challenges to high extent have been addressed in which the flexibility in sample size and shape, little preparation time, and ease of operation can be ensured and thus making it a popular strength indexing approach (Masoumi et al. 2012, 2018; Forbes et al. 2015). The index-to-strength conversion factor (k) that is relating UCS to the point load index (PLI) was proposed to estimate the UCS from the PLI indirectly (ISRM 1985). Numerous studies have been conducted on a wide range of rock types with different sizes, shapes, and point loading directions to establish reliable empirical conversion factors (ISRM 1985; Li and Wong 2013).

Broch and Franklin (1972) argued that the UCS is approximately 24 times (k) higher than the PLI of a standard core sample size (50 mm). A size correction chart was also developed for determining the cores’ UCS with various diameters (Broch and Franklin 1972). Bieniawski (1975) suggested a value of 23 as a universal conversion factor (k). Greminger (1982) and Forster (1983) showed that the conversion factor of 24 could not be adequately applied to anisotropic rocks. The International Society for Rock Mechanics (ISRM) stated that, on average, UCS is 20–25 times greater than PLI, although it has been highlighted that for different rock types, especially anisotropic rocks, k can range between 15 and 50 (ISRM 1985). Chau and Wong (1996) argued that k depends on the UCS/TS ratio, Poisson’s ratio, and length and diameter of rock samples. Hawkins (1998) indicated that the k varies from 7 to 68 for rocks originated from different lithologies. Kahraman (2001) found two separate trends for k values using the testing data obtained from 48 different rock types.

From various experimental and theoretical investigations, it has been noted that k is dependent on a range of physical and intrinsic rock properties, including origin and fabric (Fener et al. 2005; Kahraman et al. 2005; Kahraman and Gunaydin 2009), degree of anisotropy (Greminger 1982; Forster 1983), size and shape (Chau and Wong 1996; Masoumi et al. 2012, 2018), Poisson’s ratio (Chau and Wong 1996), water content (Broch and Franklin 1972; Hawkins 1998), porosity (Palchik and Hatzor 2004), and degree of weathering (Chau and Wong 1996). Although several forms of regression analysis have been used to correlate such parameters, the relationships between UCS and PLI were mainly established through linear regressions with a zero intercept (Broch and Franklin 1972; Bieniawski 1975; Hassani et al. 1980; ISRM 1985; Smith 1997; Sabatakakis et al. 2008; Singh et al. 2012; Li and Wong 2013).

In the same vein as that elaborated regarding the correlations between UCS and PLI, the so-called “Brittleness Index”, including BI₁ (UCS/TS), can be examined where the brittleness depends on a variety of rock physical and intrinsic characteristics, including mechanical properties (Hucka and Das 1974), size and geometry (Hajiabdolmajid and Kaiser 2003), water content (Kodama et al. 2013), temperature (Kodama et al. 2013), stress–strain responses (Tarasov and Potvin 2013; Kuang et al. 2021), energy transformation (Munoz et al. 2016b), mineral composition (Wang and Gale 2009), elastic parameters (Rickman et al. 2008), porosity (Heidari et al. 2014), and strain rate (Kodama et al. 2013). Thus, it can be hypothesized that in principle, k and BI₁ are dependent on the physical and intrinsic properties of rock materials given that both are functions of UCS and the dominant failure mode under point load and indirect tensile Brazilian tests is indeed tensile failure (Russell and Wood 2009). However, the indirect tensile Brazilian loading on strong rocks with potentially high brittleness has been proven to be unreliable due to the limitations associated with shearing at the contact platens and the disk sample that can lead to multiple cracking failure and overestimation of tensile strength (Erarslan et al. 2012; Masoumi et al. 2018; Khadivi Boroujeni et al. 2021; Khadivi et al. 2023). Such limitations can be avoided under point loading and, thus, making the PLI a suitable alternative to TS for burst–brittleness characterization of various rocks, particularly those with high brittleness.

The PLT involves compressing of a brittle sample between two conical steel platens (pointers) until failure to evaluate its point load index (PLI or \({I}_{\mathrm{s}}\)). In this study, for standard classification of such an index, the reference sample diameter of approximately 50 mm was used, and to obtain the optimum outcome, the rock samples were loaded under diametral and axial directions as suggested by ISRM (1985). The diametral \({I}_{\mathrm{s}}\) is defined according towhere P is the maximum applied load and D is the diameter of a sample which equals to the distance between the two pointers. Under axial point loading, the force is applied at the center of the end surfaces and the \({I}_{\mathrm{s}}\) is estimated using the following equation:where L is the length of a sample and LD represents the minimum cross-sectional area of a plane through the pointers. Equations (1) and (2) were developed by Franklin (1985) where the effect of contact surface area between the pointers and sample were not considered. Masoumi et al. (2016) argued that the radius of the conical platens under point loading is important for estimation of PLI inspiring by the work conducted by Russell and Wood (2009) who demonstrated that the contact area controls the stress intensity immediately below the contact points where the failure initiates. Thus, Masoumi et al. (2016) included the contact area based on Timoshenko and Goodier’s (1951) theory to introduce a more reliable estimation of point load index. The radius of contact area between two elastic spheres with different radii and mechanical properties can be calculated as follows:where R₁ and R₂ are radii of two spheres; and K₁ and K₂ are calculated by the following equations:where \({v}_{1}\) and \({v}_{2}\) represent Poisson’s ratios and E₁ and E₂ are Young’s moduli of two spheres. For simplicity, the surface roughness between the pointer and sample can be ignored as explained by Russell and Wood (2009). The pointers are usually made of tungsten carbide or hardened steel, with a smooth and spherically curved tip of 5 mm as suggested by ISRM (Franklin 1985). The elastic modulus and Poisson’s ratio of tungsten carbide are about 700 GPa and 0.25, respectively (Russell and Wood 2009). Under diametral point loading, the contact area is elliptical shape as a pointer with a spherical tip pushes on a cylindrical surface. For typical elastic properties, the major and minor diameters in the ellipse have a ratio of one; therefore, the contact area can be assumed as a circle for simplicity.

Under axial point loading, since the pointer pushes on a flat surface, the contact area is circular leading to R₂ \(\gg\) R₁ where Eq. (3) can be simplified as follows:

As a result, the new PLI was proposed by Masoumi et al. (2016) including the influence of contact area according towhere \({I}_{\mathrm{s}(\mathrm{n})}\) denotes the new PLI; A is the contact area that can be estimated based on the radii obtained from Eqs. (3) and (6), for point loadings under diametral and axial directions, respectively. As explained earlier, a conventional regression to correlate UCS with PLI iswhere k denotes index-to-strength conversion factor. The new correlations for relating UCS and PLI are derived through re-arranging Eq. (8) and incorporating Eqs. (1), (2) and (7) as follows:where \({I}_{\mathrm{s}(\mathrm{nd})}\) and \({I}_{\mathrm{s}(\mathrm{na})}\) represent new diametral and axial point load indices, respectively. As a result, the burst–brittleness ratio (BBR) is defined as follows;

The compression component is estimated through UCS, while the \({I}_{\mathrm{s}(\mathrm{na})}\) and \({I}_{\mathrm{s}(\mathrm{nd})}\) derived from Eqs. (9) and (10) are substituted to tensile component leading to BBR for two different point loading directions as follows:where BBR_d and BBR_a represent the Burst–Brittleness Ratio for diametral and axial loading conditions, respectively. In principle, the BBR has the same characteristics as BI₁, but its tensile component is replaced by the new PLI proposed by Masoumi et al. (2016). It is believed that incorporation of contact area in the proposed BBR can improve the accuracy of burst–brittleness assessment and can assist to make it more versatile for various rocks and different applications. Also, flexibility in performing point load testing is another major advantage of BBR over conventional BIs where PLI can be estimated very rapidly as opposed to other experiments, such as triaxial, direct/indirect tensile, and punch penetration tests. To assess the performance of the proposed BBR, three different rock types from various geological origins, including coal, granite, and sandstone, were selected for a comprehensive examination and their BBR, BI₁, and k are compared. Also, the failure processes of tested rocks under uniaxial compression and point loading are visualized through high-speed imaging technique to then analyze their level of violent failure with the estimated BBR and BI₁.

Gosford sandstone was sourced from Gosford quarry in Somersby, New South Wales (NSW) (Roshan et al. 2018), Coal blocks were obtained from Ulan seam in NSW and granite was sourced from Mount Martha Quarry, Victoria in Australia (Khadivi et al. 2022). All samples were cored at standard NX (54 mm) size followed by cutting into cylinders with three different slenderness (length-to-diameter) ratios of 0.5, 1, and 2 to be utilized for testing under uniaxial compressive, Brazilian and point loadings as shown in Fig. 2. The ends of the samples were cut parallel in accordance with the ASTM (2001) to reduce the end surface effects with an accuracy of ± 0.02 mm. A minimum of five repetitions were executed for each testing scenario at room temperature (21–25 °C) condition.

The uniaxial compressive tests were performed on the samples with a slenderness ratio of two at the displacement rate of 5 × 10^–3 mm/s to obtain the sample failure within the suggested period by ASTM (1992). A servo-controlled Instron 600DX testing frame with a maximum loading capacity of 60 tons was used to perform the experiments (see Fig. 3) and the results are presented in Table 1. To measure the Poisson’s ratios, necessary instrumentations including circumferential linear variable differential transducer (LVDT) were attached to the samples where the radial strains were computed according to Masoumi et al. (2015).

The stress–strain curves obtained from uniaxial compressive tests representing the strength and deformation characteristics of rocks in the pre- and post-failure stages are shown in Fig. 4.

The Brazilian tests were performed using an Instron 33R-4204 loading frame with a maximum loading capacity of 50 kN (see Fig. 5) on the cylindrical samples with a slenderness ratio of 0.5 according to the suggested method by the ASTM (2016). The same displacement rate as that used for uniaxial compressive tests was utilized here to obtain the suggested testing period as per the ASTM (2016). The maximum recorded load is used to determine the tensile strength according to (ISRM 1978; ASTM 2016)where P denotes the peak load at the failure, D is the disc’s diameter, and t represents the thickness of sample measured at the center. Table 2 lists the resulting data from the Brazilian tests. Also, the examples of load–displacement curves obtained from the Brazilian tests are shown in Fig. 6.

The NX size cylindrical samples with a slenderness ratio of one were tested according to the ISRM suggested method under axial and diametral loadings (ISRM 1985). The tests were conducted using a GCTS PLT-2W point loading apparatus (see Fig. 7) equipped with a high-precision load sensor with an accuracy of ± 0.05%. The PLT results are tabulated in Table 3 using Eqs. (1) and (2) to estimate PLI (I_S(50)) from diametral and axial loadings, respectively. Figure 8 demonstrates the examples of load–displacement curves obtained from point load tests on coal, granite, and sandstone samples under axial and diametral loading conditions.

Real-time study of the macro-fracturing process of brittle materials can help to identify and classify their failure mechanism. The high-speed imaging is an advanced laboratory technique that enables the real-time visualization of rock deformation and fragmentation at high resolution (Xing et al. 2017; Khadivi Boroujeni et al. 2021; Khadivi et al. 2023). In this study, a high-speed CMOS camera (Phantom V2511) was adopted with a resolution of 512 × 512 pixels at the frame rate of 200,000 fps with the lens of Nikon AF-S DX Nikkor 18–105 mm F/3.5–5.6 G ED to track the full fracturing behavior of tested rock types under uniaxial compressive and point loadings. The setup of high-speed imaging camera for uniaxial compressive testing is shown in Fig. 9.

Acoustic emission (AE) is a technique for monitoring of microcracks’ development in rock materials (Lockner et al. 1991; Chang and Lee 2004). In this study, such a technique was adopted to further investigate the brakeage process of samples under point loading.

The resulting high-speed imaging frames under axial and diametral point loadings are presented in Figs. 13, 14, 15, 16, 17 and 18 along with their cracking velocities written in the last frames. Such velocities were calculated by tracking the cracks’ development under pointers, from the initiation of the crack to complete propagation using high-speed frames. Single tensile failure on the loaded line is the dominant failure mechanism in almost all tested rock types. However, failure time, cracking velocity, and the subsequent failure intensity were different under each condition. For coal sample under axial loading, a tensile crack was instantly developed under loaded line and at the peak load (see Fig. 13) resulting in a tensile failure through splitting the sample into two halves. Such splitting was also observed under diametral loading on coal sample; however, the controlling failure mechanism was shearing or sliding along the pre-existing bedding plane, as shown in Fig. 14. Granite was considered isotropic and intact given that there were no observable defects in the tested samples. A similar failure mechanism as that reported for coal under axial loading is observable for granite under both axial and diametral loadings with relatively high cracking velocities (see Figs. 15, 16). Sandstone samples were also intact under both axial and diametral loadings which exhibited a progressive failure rather than an instant failure, as demonstrated in Figs. 17 and 18.

Under point loading, the load is applied through pointers on small local areas where a sample is failed in tension with less energy dissipation, whereas failure in compression due to a significant shearing mechanism can exhibit a high level of energy dissipation. Therefore, to further investigate such behavior, fracturing responses of all three rock types were tracked under uniaxial compression as one of the important experiments to assess the credibility of the proposed BBR. The chains of real-time fracturing processes under uniaxial compressive loading are shown in Figs. 19, 20 and 21.

Figure 19 demonstrates various stages of coal failure under uniaxial compression where a localized vertical crack initially formed within the sample (see Fig. 19b) and then propagated with an increase in the load. During a very short segment of time, intensive localized cracks were activated at various scales and orientations (see Fig. 19d, e). Such cracks were not necessarily correspondent to the initially formed cracks which could be due to the activation of the pre-existing fractures or so-called “cleat network”. Despite such an extensive crack network development, the sample was still under the cylindrical shape and took further load (see Fig. 19f). After just 2 ms, it failed in a strongly violent manner through a mixed mode of shear and tension with dominant tensile fractures. Fragmentation ranged from powder to two course conical-shaped fragments (see Fig. 19h, i). It is evident that the stable and unstable cracking occurred simultaneously in the coal sample where the complete failure process happened within 2 ms.

A similar failure process as that observed for coal sample was recorded for the granite sample (see Fig. 20) but with less severity. Cracking was started through a number of localized vertical tensile cracks which then propagated toward the end surfaces as load increased (see Fig. 20b, c). The multiple tensile cracks at different scales and orientations were generated within the sample and propagated (see Fig. 20d–f) leading to the strong violent failure (see Fig. 20g, h). In this case, unlike coal, the propagated cracks were mainly the continuation of the generated ones leading to a violent mixed mode of shear and tensile failure.

For the sandstone sample which was subjected to compressive loading (see Fig. 21), a vertical crack was initiated near the end surfaces of the sample (see Fig. 21b) and then propagated and coalesced (see Fig. 21c–e) to form a shear band (see Fig. 21f) as load increased. The sample failed through a single shear band into two approximately equal halves (see Fig. 21h) with a single tensile crack.

Coal demonstrated the highest level of cracking density during the failure process followed by granite and then sandstone, respectively. Stable and unstable cracking stages can be differentiated in sandstone during its progressive failure, while in coal and granite samples due to the sudden activation of intensive localized cracks, stable and unstable cracking happened at nearly the same time. The other major distinction between sandstone and the other two rock types is related to the failure mechanism where the former was primarily shearing and the latter was a mixed mode of shear and tension. The complete failure process of coal from the initial crack formation to complete failure happened at about 10 and 100 times faster than those in granite and sandstone, respectively, indicating high brittle failure in coal compared to granite and sandstone and endorsing the validity of the proposed BBR. The coal sample also exhibited the least duration of dynamic fracturing and the highest rate of strength loss, while the sandstone sample resulted in the greatest dynamic fracturing and lowest rate of strength loss. It is noteworthy that tested coal has the lowest compressive and tensile strengths among the examined rock types while possessing the highest relative burst-brittle level. Therefore, it is important to highlight that the burst–brittleness characteristic cannot be described by the UCS alone, and indeed, other strength factors like PLI with precise estimation are required.

The AE events obtained from point load tests were parametrized to characterize the microcracking process in the tested rocks. The cumulative scatter AF-RA data as well as the cumulative probability density maps for coal, granite, and sandstone samples are presented in Figs. 22 and 23. The data distribution changes from sparse to dense with the color changing from red to purple in the density maps. The associated scatter plots are provided next to each of the density maps showing the AE parameters’ distribution in blue dots. The AE parameters are split into two regions by the red benchmark line, differentiating tensile and shear microcracks with associated cracking percentages in pie charts. The AF ranged from 0 to 2000 kHz and RA varied between 0 and 20 ms/V. For consistency and making the results comparable, the area where the largest proportion of AE parameters concentrated (e.g., in the range of 0–250 kHz for AF and 0–1 ms/V for RA) was selected to be further investigated. The dense areas located above the diagonal benchmark line indicate that the tensile microcracks in all rock types are the dominant ones. Under both axial and diametral loadings, the AF values are mainly concentrated in the range of 0–100 kHz for the granite samples, which is larger than that for coal and sandstone samples with the interval of 0–50 kHz. These results confirm the failure of rocks in tension under point loading as highlighted by Russell and Wood (2009). The percentage of tensile microcracks can also be associated with the level of brittleness in the rocks where higher tensile cracks can lead to violent rock failure as has been indicated under compressive failure (Zhou et al. 2019; Zhang et al. 2021a; Peng et al. 2022).

Table 5 summarizes the distribution of tensile and shear microcracks obtained from this analysis along with the computed BBR and BI₁ for the tested samples. Coal samples revealed the maximum tensile microcracks’ portion among the tested rock types followed by granite and sandstone, respectively, endorsing the validity of the proposed BBR based on AE analysis. Furthermore, this effort aimed to unify the concepts of burst and brittleness by introducing a versatile ratio that can be utilized for the burst (coal and rock bursts) and brittleness assessments of a broad range of rocks in rock engineering projects. It is also suggested to investigate the applicability of BBR across a wide range of brittle rocks and rock-like martials in future studies.