Brazilian Disc Test Research Articles

Dynamic Tensile Failure Characteristics and Energy Dissipation of Red Sandstone under Dry–Wet Cycles

Studying the dynamic properties of rocks in complex environments is of great significance to the sustainable development of deep-sea metal mineral resource extraction. To investigate the influence of dry–wet cycles on the dynamic tensile properties and energy dissipation of red sandstone, a series of dynamic Brazilian disc tests was carried out through the split Hopkinson pressure bar (SHPB) apparatus. The dynamic tensile behaviors and energy dissipation distribution of the red sandstone specimens after different dry–wet cycles (0, 10, 20, 30 and 40 cycles) were analyzed in this study. The degree of dynamic tensile fragmentation and energy dissipation of red sandstone is significantly affected by the loading rate. Specifically, when the number of dry–wet cycles remains constant, an increase in loading rate results in a significant reduction in the average fragment size, while the energy consumption density exhibits an approximately linear increase. At a fixed loading rate, the energy consumption density decreases approximately linearly with the increase in dry–wet cycles, and the higher the loading rate, the more sensitive the energy consumption density is to the dry–wet cycle. Under a fixed number of dry–wet cycles, the dynamic tensile strength has an exponential relation with the increase in energy consumption density.

Effect of a singular planar heterogeneity on tensile failure

Many rocks contain planar heterogeneities, in the form of open fractures, veins and/or stylolites, but scarce data exist on how strength and fracture pattern formation is affected by the presence of a singular planar heterogeneity in an otherwise uniform matrix. The mechanics of stylolite-bearing and/or fractured limestone is of interest to several engineering applications, from quarries to subsurface gas or geothermal reservoirs. We have performed Brazilian Disc tests on pre-fractured Indiana limestone samples and Treuchtlinger Marmor discs which contain cohesive stylolites, investigating Brazilian test Strength and the resulting fracture pattern. All experiments were filmed, and where possible analyzed with particle image velocimetry. When viewed in 2D, the planar discontinuity was set at different rotation angles compared to the principal loading direction, where perpendicular to the loading direction is defined as 0⁰. The results show that all samples are weaker than their intact counterparts. For the pre-fractured Indiana limestone, there is 10–75% angle-dependent weakening. However, in the samples with a stylolite, strength is weakened by 35–75%, independent of direction. Several new cracks appeared when fracturing a stylolite-sample, where the orientation is heavily influenced by the stylolite orientation. The fracture pattern and associated stress drops are more complex for high angles. In these samples always more than one fracture formed, whereas in pre-fractured samples usually only one new fracture formed. This suggests a potential for more permeability increase when hydrofracturing a stylolite-rich interval. Comparison with Finite Element Models indicates that this difference in fracture pattern is caused by the strength contrast between the anastomosing stylolite zone and the matrix material, leading to stress concentrations effects. This causes (micro-) fracture nucleation to occur locally, promotes fracture coalescence and fracture growth at lower overall sample-load conditions compared to intact samples.

Effect of Bi-modulus characteristic on the Brazilian disc test: Insights from numerical study

The materials with different elastic behavior under tension and compression are called bi-modulus materials, and this type of materials widely exists in nature, such as rock, concrete, and cohesive soil. However, due to the lack of a constitutive model that is suitable for bi-modulus materials, the bi-modulus property of materials was often ignored in previous studies. In this study, the uncoupled strain-driven constitutive model for bi-modulus materials proposed by Latorre and Montáns1 is firstly embedded into the finite element software FssiCAS. Then the distribution characteristics of stress, strain, and displacement in Brazilian disc are systematically compared under the two conditions, i.e., considering the bi-modulus property and ignoring them. Besides, the constitutive model is used to verify the reliability of the measurement methods proposed by Ye et al.2,3 for the tensile elastic modulus. The results show that the bi-modulus property of materials has a significant effect on the magnitude, and the distribution of stress, strain, and displacement in Brazilian disc. The measurement methods proposed by Ye et al.2,3 own acceptable reliability only under certain conditions.

A 2D adaptive finite-discrete element method for simulating fracture and fragmentation in geomaterials

This study proposes a new two-dimensional finite-discrete element method (FDEM) using dynamically inserted cohesive elements. Unlike the conventional FDEM, which sets up cohesive elements between adjacent solid elements to simulate fracture and fragmentation in the geomaterials, the new FDEM adaptively inserts the cohesive elements when the stress exceeds a critical value. The new method is consistent with the conventional FDEM except that it uses a post-peak constitutive model for cohesive elements and a new nodal updating scheme. The accuracy and efficiency of the new method are verified through a continuum mechanics example with an analytical solution. Examples of Brazilian disc and triaxial compression tests are used to verify the performance of this new method in simulating fracturing in geomaterials. Finally, dynamic loading on the Brazilian disc by a split Hopkinson pressure bar (SHPB) system is simulated, confirming that the new method effectively simulates dynamic fracturing problems. The study shows that computational results of the conventional FDEM highly depend on the penalty parameters of the cohesive elements. The adaptive FDEM overcomes these limitations with improved accuracy and efficiency by using adaptively inserted cohesive elements.

A SVM-based method for identifying fracture modes of rock using WVD spectrogram features of AE signals

In order to achieve the highly efficient and accurate identification of fracture modes including tension or shear fractures during rock failure, an intelligent identification method based on Wigner-Ville distribution (WVD) spectrogram features of acoustic emission (AE) signals was proposed. This method was mainly constructed by the following steps: Firstly, AE hits corre-sponding to tension and shear fractures were obtained through conducting the Brazilian disc test (tension fracture) and direct shear test (shear fracture) of limestone. Secondly, the WVD spectro-grams of these tensile-type and shear-type AE hits were respectively extracted and then trans-formed into the image features of relatively low-dimension as the sample set based on the gray-level cooccurrence matrix (GLCM) and histogram of oriented gradient (HOG). Finally, on the basis of the processed and classified sample set of the WVD spectrogram features, an identifica-tion model of rock fracture modes was established by a support vector machine (SVM) learning algorithm. To verify this method, the fracture modes of limestone subjected to biaxial compres-sion were identified by the method. The results showed that the method not only can greatly re-veal the fracture modes change from tension-dominated to shear-dominated fractures, but also has advantages over the RA-AF value method, such as applicability, accuracy and practicality.

Mixed mode I/II fracture behavior of CSTBD sandstone specimen under different loading angles

Cracks are often exposed to mixed mode I/II loading because of the random direction of crack relative to the external load, causing rock failure in rock engineering. However, many mixed mode I/II fracture properties of rock with different loading angles (β) are still poorly understood. To investigate the influence of β on mixed mode I/II fracture behavior, cracked straight through Brazilian disc tests were performed on sandstone under various β at the loading rate of 0.2 kN/s. The results show that the effect of β on the peak load and crack propagation velocity is slight. The mode I stress intensity factor (SIF) decreases from positive to negative with increasing β, while mode II SIF first increases and then decreases as β exceeds approximately 30°. The mixed mode I/II fracture toughness increases linearly with increasing β. When β exceeds 60°, the crack initiation location shifts from the center of the semi-circular notch tip to the surface of preset crack. For non-tip cracking, the crack propagates toward the loading point in a direction approximately perpendicular to the preset crack. The threshold β for it decreases with increasing internal friction coefficient and relative crack length. Moreover, the mixed-mode fracture surface becomes smoother as the mode I component contribution drops. There are remarkable differences in the prediction of fracture initiation angles and SIFs of various rock types applying the generalized maximum tangential stress criterion. The findings of this study could help understand the mixed mode I/II fracture observed in rock engineering.

Dynamic tensile mechanical properties of water-saturated and frozen sandstone after freeze-thaw fatigue damage treatment

To study the physical properties and dynamic mechanical response characteristics of rock materials in seasonally frozen regions, density, P-wave velocity, nuclear magnetic resonance, and dynamics Brazilian disc tests were carried out on sandstones treated with different numbers of freeze-thaw cycles. The experimental results show that the water-saturated density and P-wave velocity of sandstone decrease and the Nuclear magnetic resonance porosity increases as the number of freeze-thaw cycles increases. Under a certain loading rate, the dynamic tensile strength of water-saturated sandstone gradually decreases and that of frozen sandstone gradually increases as the number of freeze-thaw cycles increases; at the same time, the dynamic tensile strength of both water-saturated and frozen sandstone increases as the loading rate increases under the same number of freeze-thaw cycles. The reason for the variability of dynamic tensile strength between water-saturated and frozen sandstones with the increase in the number of freeze-thaw cycles can be attributed to the change in porosity. On this basis, the concept of relative dynamic tensile strength of sandstones was introduced and regression analysis was carried out with the increase ratio of porosity, and it was found that the correlation between the relative dynamic tensile strength and the increase ratio of porosity was strong. The relative dynamic tensile strength of water-saturated sandstone decreases exponentially with the increase ratio of porosity; the relative dynamic tensile strength of frozen sandstone increases exponentially with the increase ratio of porosity.

Experimental and numerical study on the mechanical behaviours of two rocks with circular openings

Abstract The presence of discontinuities in rocks can significantly compromise the performance of the rocks in terms of strength and stiffness, thereby threatening the reliability and safety of engineering structures built in or on the rocks, for example, tunnels, and slopes. In this paper, a comprehensive set of experimental tests, including Brazilian disc tests and uniaxial compression tests, were performed for two distinct rocks including granite and sandstone with seven patterns of opening holes, in addition to intact rock specimens. To investigate the micro-cracks initiation and propagation, a hybrid continuum-discrete element method incorporating a cohesive fracture model was adopted to simulate the laboratory testing. The stress–strain relations, micro-cracks initiation and propagation, Young’s modulus, compressive strength, tensile strength and micro-cracks rate and orientation were discussed. The results showed that the presence of the holes could significantly influence the tensile strength and compressive strength. However, its influence on the Young’s modulus was found to be relatively insignificant.

Research on Pure Modes I and II and Mixed-Mode (I/II) Fracture Toughness of Geopolymer Fiber-Reinforced Concrete

The unsustainable use of resources and the rising demand for traditional concrete have disrupted ecological equilibrium, necessitating the adoption of a more appropriate and long-lasting alternative. One such substitute for cement in concrete production is geopolymer concrete, although this material is prone to cracking and fracturing due to its low tensile strength, leading to costly repairs or even structural collapse. Fiber-reinforced concrete has recently been widely adopted as a construction material to counteract these issues. This research examined the crack proliferation and fracture toughness of geopolymer concrete comprising different fibers using a cracked Brazilian disc. Four different fibers were used, such as polypropylene and steel fiber (short and long), at a dosage of 1.5% by volume. Fracture toughness was computed for various modes (I, II, and I/II) of fractures, and crack propagation from cracked specimens was studied. A different angle of inclination (0, 15, 28, 83, 60, 75, and 90 degrees) was used to conduct the Brazilian disc test on the specimens with respect to the preexisting crack direction. The findings indicate that the increasing loading angle increased the load-carrying capacity. The fracture toughness values of specimens under all three modes ranged from 0.26 to 1.75 MPa.m1/2. Additionally, long polypropylene and steel fibers exhibit higher fracture toughness than short fibers.

Experimental Investigation on Cracking Characteristics of Dry and Saturated Shales in Nitrogen Fracturing after Liquid Nitrogen (LN2) Injection

Cryogenic LN2 fracturing is one of the environmentally friendly waterless fracturing technologies that promote the fracture complexity of shale gas reservoir. The water-ice phase transition under freezing condition causes frost heave in saturated shale. The effect of moisture in shale should be taken into account during cryogenic damage process. Therefore, the differences of cracking characteristics between dry and saturated shales were studied in this paper. A laboratory triaxial and high temperature fracturing system was developed for nitrogen fracturing dry and saturated shale after LN2 injection. The influence of moisture on breakdown pressure was studied under different confining pressures (3 MPa, 6 MPa, 9 MPa, and 12 MPa) and bedding directions (parallel bedding and vertical bedding). The experimental results demonstrated that when the confining pressure increased from 3 MPa to 12 MPa, the breakdown pressure of dry parallel bedding after LN2 preconditioning decreased 7.12 MPa, 6.06 MPa, 4.58 MPa, and 3.11 MPa, respectively. Therefore, LN2 preconditioning could damage shale resulting in a lower breakdown pressure, but the effect of cryogenic damage decreased with the confining pressure increasing. The moisture in shale had little impact on nitrogen fracturing without LN2 injection because the breakdown pressure difference between dry and saturated shales was small. However, the breakdown pressure of saturated shale after LN2 preconditioning was always lower than that of dry shale. The breakdown pressure of saturated parallel bedding shale after LN2 injection decreased 8.62 MPa, 7.67 MPa, 6.08 MPa, and 4.63 MPa, respectively, with the confining pressure increasing from 3 MPa to 12 MPa. The breakdown pressure difference between dry and saturated shales was impacted by the migration of unfrozen water and frost heave. In addition, the extent of cryogenic damage varied substantially between different bedding directions. When the confining pressure was 3 MPa, the breakdown pressure of saturated parallel bedding shale reduced by 69.18% after LN2 preconditioning, but that of saturated vertical bedding shale only decreased by 22.49%. The tensile strength of shale had a greater influence on the breakdown pressure. According to the Brazilian disc test results, the tensile strength of matrix was much higher than that of bedding planes. As a result, it is useful to wet the shale in order to reduce the breakdown pressure. The fracturing direction of horizontal drilling should be consistent with the bedding direction for better cryogenic fracturing effect.

Structural properties of volcanic precursors-based geopolymers before and after natural weathering

The structural properties of geopolymers based on locally available volcanic feedstocks were here investigated in the optics of the development of sustainable non-structural building materials. Geopolymer binders and mortars based on two volcanic pyroclastic residues from Mt. Etna (ejected ash and a paleosoil, named “ghiara”) were studied to assess the effects of natural weathering.The strength and durability of the resultant products were examined before and after six months of outdoor exposure by comparing ultrasound pulse velocity (UPV), specific weight, Brazilian Disk test (BD) and Digital Image Correlation (DIC) together with Dynamic Vapour Sorption (DVS) results. DVS data were also compared to those of pure salts (e.g. sulphates and carbonates) commonly occurring in geopolymers as efflorescences.The results obtained on unexposed and exposed samples and the relationship between the moisture transportation, efflorescence development, and structure degradation have shown a better response of the volcanic ash-based geopolymers to weathering in the hot summer Mediterranean climate zone than ghiara-based products. The latter are more affected by exposure conditions as revealed by the decrease of UPV, the influence of efflorescences on moisture adsorption-desorption curves and the consequent worsening of mechanical performances.The correlation between UPV and DVS applied for the first time to the study of volcanic precursors-based geopolymers exposed to natural weathering has proven to be an effective and useful tool for assessing the durability of materials.

A new analytical method for impact splitting strain rate in Brazilian disc test based on SHPB

A new analytical method for impact splitting strain rate in Brazilian disc test based on SHPB

Model-free damage prediction of brittle materials based on particle swarm optimization coupled with a probabilistic fission method

This study aims to develop a model-free damage prediction algorithm to predict directly local mesoscopic elasticity modulus and damage distribution of brittle materials based on only the monitored global structural response such as force–displacement curve, which can overcome complex theoretical modeling material behavior. The algorithm is established based on the modification of particle swarm optimization algorithm by establishing a probabilistic fission method, which can consider the physical mechanism that damage is correlatedpositivelywiththe stress level in the optimization process. A representative numerical example of a Brazilian disk test is implemented to verify the algorithm. The damage prediction result is compared with the result obtained from the traditional particle swarm optimization algorithm and the corresponding experimental result. As can be seen from the comparison results, the ability and effectiveness of the algorithm are supported. Furthermore, the effect of different divisions of representativevolumeelementsin the algorithm is analyzed to support the conclusion that fine division of representativevolumeelementscan obtain refined prediction results, but it can significantly increase the prediction error and computation time. The recommended size of representativevolumeelementis about 2 mm for the numerical case in this study.

Experimental and numerical analysis of dynamic splitting mechanical properties and crack propagation for frozen sandstone

To reveal the dynamic tensile properties and crack growth law of frozen sandstone under impact loading, dynamic splitting tests (−30–20 °C) with different strain rates were carried out using a split Hopkinson pressure bar (SHPB), and the crack growth process of the specimens was captured using a high-speed camera. The study focused on the analysis of dynamic tensile mechanical properties, energy dissipation, crack growth, and failure modes of frozen sandstone. Numerical simulation analyses were also performed using LS-DYNA. The results indicated that: (1) The dynamic tensile strength has an obvious strain rate enhancement effect, and the strain rate sensitivity coefficient increased with a decrease in temperature. The Logistic function can characterize the relationship between the dynamic strain rate enhancement factor and the strain rate of frozen sandstone. (2) In the dynamic Brazilian disk (BD) test, the energy absorption rate showed a transition between the strain rates of 40 s−1 and 50 s−1, and there were two main failure modes of the specimen. The Y-direction secondary crack in mode II was caused by tensile stress in the Z-direction. (3) In the dynamic BD test, the central main crack (CMC) did not occur at the central position first, and its Z-direction propagation rate was linear with the strain rate. In the dynamic flattened Brazilian disk (FBD) test, the condition of CMC initiation was that the platform loading angle (2α) was greater than 10°. The dynamic tensile strength increased with an increase in 2α, and the correction coefficient was introduced in the calculation. This study can serve as a theoretical reference and basis for the study of the dynamic tensile properties of rock-like materials.

The actual boundary conditions in the Flattened Brazilian Disc test

In the present study an attempt is made to assess a recently presented analytical approach to the problem of the contact stresses developed along the Flattened Brazilian Disc (FBD) – loading platen interface, during the implementation of the homonymous test. In this direction, a finite element model was constructed, resembling both the FBD and the platen, using the commercially available software ABAQUS. As it is seen, the predictions of the numerical analysis for the stresses induced at the interface and inside FBD are in a satisfactory agreement with the analytical ones, justifying the proposed analytic approach for the FBD configuration. In this context, it is concluded that FBD does significantly reduce the stress severity at contact region, however, the end points of the flat edges of the FBD now become points of increased stress magnitudes so that care should be taken during the preparation of the specimens regarding the cutting process at these points.

Effect of layer orientation on behaviour of 3D-printed rock specimens in indirect tensile testing

The mineralogical composition, texture and fracture of natural rocks can be complicated due to the inherent heterogeneity of geomechanical properties, making it challenging to understand their behaviour. Due to the intrinsic layer by layer manufacturing process, 3D printing technology enables to create rock-like materials and specimens with known and controllable inhomogeneity. In this study, disc specimens of a rock-like material are fabricated using 3D printing of cement mortar for testing to understand fabric orientation effect on fracture behaviour better. For the first time, a novel technique, named AUSBIT (Adelaide University Snap-back Indirect Tensile test), for Brazilian disc tests is applied for obtaining both strength and snap-back post-peak behaviour of the 3D printed rock-like materials. Adopting Digital Image Correlation (DIC) technique, in conjunction with the use of AUSBIT, allows obtaining full-field strains and their evolutions, showing strong effects of fabric orientation on strain distributions and both pre-and post-peak responses.

Quasi-Static and Dynamic Tensile Behavior of Water-Bearing Sandstone Subjected to Microwave Irradiation

Microwave irradiation on rocks before excavation is an effective method to reduce equipment wear and energy consumption during mechanical cutting. Rock mass excavation is usually carried out in a water-rich environment and exposed to dynamic loads, thus understanding the coupled effects of water content and loading rate on the mechanical behavior of rocks under microwave radiation is essential. In this study, sandstone samples with five levels of water content (from oven-dried to water-saturated) were exposed to microwave irradiation at a power of 700 W for 10 min. Brazilian disc tests were conducted on sandstone samples after microwave radiation under both quasi-static and dynamic loading conditions. Test results revealed that, with the increase of the initial water content, the microwave heating capacity of the rock is significantly improved. The surface temperature of the saturated samples is approximately 1.38 times higher than that of the dry ones. Moreover, weight, P-wave velocity, quasi-static and dynamic tensile strength of the rock decrease, while porosity and damage factor exhibit a similar growth law. Before microwave irradiation, the average value of the P-wave velocity and the quasi-static tensile strength of sandstone were about 2521.3 m·s−1 and 4.65 MPa. However, after microwave treatment, when the initial water content was 2%, 3%, 4% and 5.4%, the P-wave velocity decreased by 6.1%, 9.8%, 16.4% and 30.2%, while that quasi-static tensile strength reduced by 9.2%, 16.7%, 30.6% and 48.9%, respectively. For water-saturated samples under microwave irradiation, the porosity increases from 13.02% to 18.12% (showing an increase of 39.2%), and the damage value rises to 0.51. In addition, the dynamic tensile strength shows a significant loading rate dependence, and as the initial water content increases, also the dynamic increase factor (DIF) increases. At a given loading rate, the energy dissipation decreases with the increase of the initial water content, which indicates that the presence of water cause more significant damage to the rock when subjected to microwave radiation. Scanning electron microscopy (SEM) results indicate that the internal damage of the rock after microwave radiation is dominated by intergranular cracks, and crack density increases with increasing initial water content of the samples. The underlying damage mechanisms of microwave radiation on water-bearing sandstone were interpreted with the theory of pore water pressure and structural thermal stresses.

Fracture Process and Failure Mode of Brazilian Discs with Cracks of Different Angles: A Numerical Study

In order to determine the effect of internal cracks on the tensile failure of materials, a hybrid finite–discrete element method was used to analyze the Brazilian disc test with cracks of different angles. When the pre-crack angle is between 0° and 60°, the wing crack is initiated from the pre-crack end. When the pre-crack is 90°, the crack initiated from the pre-crack center. When the pre-crack angle is between 0° and 60°, the maximum principal stress and plastic strain are concentrated at the pre-crack end. When the pre-crack angle is 90°, the maximum principal stress and plastic strain are concentrated in the pre-crack center. As the crack angle increased from 0° to 90°, the failure mode of Brazilian discs with cracks transits from splitting into two parts to splitting into four parts. The influence of crack length is further studied. When the crack length is less than 5 mm, the crack angle has little influence on the disc failure mode; Brazilian discs with cracks of different angles undergoes splitting failure along the loading axis. When the crack length is larger than 5 mm, the crack angle has a great effect on the disc failure mode.

Experimental investigation on strength-weakening effect and fracture behavior of red sandstone under coupled static and dynamic cyclic loadings

Underground rock masses are often exposed to dynamic disturbances due to seismic waves and blasting excavation disturbances, resulting in significant changes in their physical and mechanical properties. To reveal the strength-weakening effect of rock under the coupled stress, the Brazilian disc tests of red sandstone were carried out by coupling pre-static and dynamic disturbing. The impacts of preloading, loading wave amplitude, dynamic disturbance frequency, and cycle number were studied on the weakening of Brazilian disc splitting strength. Furthermore, the fracture behaviors were characterized and discussed from the fracture morphology and fractal dimension. The results show that the loss ratios are positive under dynamic disturbance loading, indicating that the Brazilian splitting strength is weakened. For the same load wave amplitude, the loss of strength and P-wave velocity under 70 % preloading is more than twice that under 55 % preloading. At high preloading, the maximum strength loss and wave velocity losses are 13.7 % and 19.2 %, respectively. Comparing the disturbance degree of the sample with the increase of frequency by 1 Hz, cycle number by one time, and load wave amplitude by 1 % PS, it is concluded that the decline of wave velocity for the rock sample with the increase by 1 % PS in load wave amplitude is the largest, followed by the frequency and cycle. Only micro-cracks appear on the fracture surface under pure static loading. However, when the dynamic load disturbance is applied, the cement wrapped with debris falls off, besides micro-cracks. And with the increase of amplitude, the area of bare debris also progressively intensifies. Meanwhile, the width and number of micro-cracks gradually increased, with the broadest micro-crack reaching 18.2 μm. Preloading, loading wave amplitude, loading frequency, and cycles are positively correlated with fractal dimension D. The higher the D value, the more broken the sample.

Failure strength and fracture characteristics of rock with discontinuity under indirect tension

Large-scale discontinuities can significantly affect the mechanical properties of rock masses. However, the tensile behavior of rock discontinuities is often less investigated. To study the statistical characteristics of failure strength and fracture characteristics of rock discontinuities, Brazilian disc tests were conducted on limestone specimens with a single natural discontinuity at different load-discontinuity angles ( β ). In this study, β = 0 ° and β = 90 ° correspond to the discontinuity parallel and perpendicular to loading direction, respectively. The results show that Brazilian failure strength (BFS) can reasonably represent the tensile strength of rock with discontinuities, by comparing the BFS and tensile stress in the disc center at peak force. The two-parameter Weibull distribution can capture the statistical BFS characteristics of rock discontinuities parallel to loading direction ( β = 0 ° ) and at different load-discontinuity angles ( β ≠ 0 ° ). All specimens with discontinuity at different load-discontinuity angles show more plastic deformational behaviour than intact rock specimen. With increasing β , the mean BFS of limestone with discontinuity increases before reaching a plateau at β = 45 ° . The single plane of weakness theory best explains the BFS of fractured limestone with β . Only a specific segment of pre-existing rock discontinuity could affect the fracture process. When β = 0 ° , interfacial cracks and alternative cracks formed. When β ≠ 0 ° , mixed failure mode with shear and tensile failure occurred, particularly when β = 30 ° and β = 60 ° . The findings can contribute to better understanding the failure and fracture characteristics of rock with discontinuities, particularly the interaction of pre-existing discontinuities with stress-induced fracturing.