Two-dimensional Particle Flow Code Research Articles

Critical Failure Characteristics of a Straight-Walled Arched Tunnel Constructed in Sandstone under Biaxial Loading

To characterize the failure of rock mass surrounding underground tunnels, biaxial compression tests were conducted on a real sandstone model with a straight-walled arched hole. The acoustic emission (AE) system and digital image correlation (DIC) optical inspection equipment were used to investigate the crack evolution process and failure precursors of the tunnel. A two-dimensional particle flow code (PFC2D) was used to conduct numerical simulations on the sample, so as to investigate the mesoscopic failure mechanism of rock mass. The results show that the failure of the single tunnel constructed in sandstone occurs mainly in the walls on both sides (between the spandrels and arch feet), showing slabbing failure characteristics and a certain abruptness. The crack initiation in sandstone in early stage is not obvious, and the crack propagation in rock mass is rapid when acoustic emissions are enhanced. The small increments in the AE count and amplitude and the continuous reduction in the b-value can be used as precursors for the failure of rock mass. When the height–span ratio is 0.8 and 1.0, the stress distribution around the chamber is more uniform, and when the height–span ratio is greater than 1.0, the stress is mainly concentrated in the vault and arch bottom. In the PFC simulations, tensile fractures firstly initiate in the middle of walls and at the arch feet, arcuate fracture concentration zones are then formed, in which shear fractures appear and a few particles spall from the surfaces. When approaching the ultimate bearing capacity, rock masses on both sides of the tunnel are fractured over large areas, and the slender coalesced fractured zone develops to the deep part of rock mass, causing failure of the sample.

Mechanical properties and failure mechanism of rock-like specimens with arc-shaped flaws under freeze–thaw cycles

To investigate the failure characteristics of jointed rock masses in cold regions after freeze–thaw (F-T) cycles, uniaxial compressive tests are conducted on rock-like specimens (a type of cement mortar) with prefabricated arc-shape flaws subjected to 0, 25, 50 and 100F-T cycles to determine their mechanical properties. Test results reveal that F-T cycles greatly affect the uniaxial compressive strengths (UCS) of the specimens, and the relationship between the peak strength and the number of freeze–thaw cycles is exponentially fitted. Notably, the initial 50 cycles have a more pronounced effect on UCS reduction compared to the subsequent 50 cycles, indicating that early F-T cycles are more detrimental to UCS. To further investigate the failure mechanism of the specimens, two-dimensional particle flow code (PFC2D) is utilized to simulate the F-T cycle process of the specimens. The mechanical properties and failure mechanism of specimens can be obtained after F-T cycles under uniaxial compression through numerical simulations. The findings indicate that F-T cycles can obviously deteriorate the ductility of the specimens, and to some extent, they also affect crack propagation during specimen failures. Through crack classification statistics and stress contour analysis of the simulated specimens, it reveals that different prefabricated flaw morphologies lead to distinct stress distributions, subsequently affecting crack propagation and specimen failure. It is also worth mentioning that the large curvature and inclination angle of the flaws may generate additional tensile and shear stress zones at the top of the flaw vault, which in turn may contribute to the development of crack propagation. The relevant experimental and numerical results are useful for investigating mechanical properties and failure mechanisms of rock masses in cold regions subjected to F-T cycles.

Investigation on failure behavior and hydraulic fracturing mechanism of Longmaxi shale with different bedding properties

Research on the failure behavior and hydraulic fracturing mechanism of Longmaxi shale is vital for deep shale gas exploitation. The anisotropic properties of shale, especially bedding properties, significantly affect its cracking behavior. In this study, by embedding the smooth joint model (SJM) into the parallel bond model (PBM) in two-dimensional particle flow code (PFC2D), uniaxial compression and fluid–solid coupling hydraulic fracturing models with different bedding properties were quantitatively established, and the effects of bedding properties on the failure behavior and hydraulic fracturing mechanism of Longmaxi shale were investigated. The results demonstrated that bedding properties significantly affect the failure pattern, strength response and hydraulic fracture propagation of shale. Shale failure is controlled by bedding inclination, bedding strength, and maximum principal stress, which mainly depends on their competition mechanism. In addition, it is found that the secondary induced fractures associated with primary through-bedding fractures tend to have larger fracture networks compared with those associated with primary along-bedding fractures.

Numerical Analysis of Interbedded Anti-Dip Rock Slopes Based on Discrete Element Modeling: A Case Study

Varying geological conditions and different rock types lead to complex failure modes and instability of interbedded anti-dip rock slopes. To study the characteristics of failure evolution of interbedded anti-dip slopes, a two-dimensional particle flow code (PFC2D) based on the discrete element method (DEM) was utilized to establish an interbedded anti-dip rock slope numerical model for the Fushun West Open-pit Mine based on the true geological conditions and field investigations. The slope model with an irregular surface consists of interbedded mudstone and brown shale as two different rock layers, and a number of small-scale rock joints are randomly distributed in the rock layers. The influence of different inclination angles (20° and 70°) of the rock layer and slope angles (60° and 80°) on the stability of interbedded anti-dip rock slopes was considered. The evolution of the failure progress was monitored by the displacement field and force field. The simulation results showed that the rock joints in the rock stratum promoted crack initiation and increased the crack density but did not change its shear-slip failure mode. A large inclination angle of the rock layers and slope angle can lead to topping slip failure along the slip zone. However, shear-slip instability generally occurs in interbedded anti-dip rock slopes with small inclination angles of the rock layer and small slope angles. These results can contribute to a better understanding of the failure mechanism of interbedded anti-dip rock slopes under different geological conditions and provide a reference for disaster prevention.

Experimental and numerical investigation on meso-fracture behavior of Beishan granite subjected to grain size heterogeneity

Research on crack evolution mechanism and failure pattern of Beishan granite is vital for evaluation of the engineering stability of disposal repository. The mechanical properties, especially the strong heterogeneity, significantly affect the cracking behavior of granite. To investigate the fracture behavior of heterogeneous granite, mesoscopic three-point bending tests were conducted on Beishan granite using an in situ scanning electron microscope (SEM) with a loading device, and the entire cracking process was observed in real time. Additionally, the effect of grain size heterogeneity on strength and deformation behavior of granite was quantitatively investigated by introducing a statistically defined heterogeneity index into the grain-based model in two-dimensional Particle Flow Code (PFC-GBM), and the crack development was characterized through fractal theory. The results demonstrated that fracture mechanism of granite is mainly tensile and shear failure, and the fracture surface of tensile failure is rough and irregular, which is mainly caused by the tearing between various mineral crystals, whereas the fracture surface of shear failure is relatively smooth, which is mainly caused by the dislocation and slip between minerals. The crack edge can be regarded as the damage zone, and the development of the main crack is promoted by the continuous generation and combination of micro-damage at the crack tip. The crack propagation path from a mesoscale perspective is a process of continuous crack orientation, which depends on the competition mechanism between mineral properties and maximum tensile stress. Generally, the granite strength tends to increase with the increase of heterogeneity.

Mechanical Properties of Composite Silty Soil Modified with Cement and Zirconia-Based Nanopowder.

This study assessed the modification effects of zirconia-based nanopowder and cement contents and curing age on the mechanical properties of silty soil. The orthogonal test design was applied to derive the best combination of each influencing factor using the lateral unconfined compressive test. Two-dimensional particle flow code (PFC2D) distinct-element modeling software was also used to fit and analyze the test curves, as well as simulate the triaxial test with the derived parameters. The test results reveal the optimal combination of 20% cement, 2% zirconia-based nanopowder, and 28 d curing age. The extreme difference table was used to plot the orthogonal trend diagram, and cement content was found to be the most significant factor controlling the silty soil strength. The maximum peak stress was 2196.33 kPa under the optimum combination of factors, which could be obtained through the index estimation, and these results were experimentally verified. According to the predicted strength envelope, the cohesive force of nanopowder-cement-modified silty soil in the optimal proportion was 717.11 kPa, and the internal friction angle was 21.05°. Nano zirconium dioxide will accelerate the hydration reaction of cement, the flocculent structure produced by the hydration of cement and soil particles connected to each other, play the role of filling and anchoring, and thus increase the strength of the nano-zirconium dioxide, and the optimal dosage of nano-zirconium dioxide is 2%.

Study on Muck Pile Shape of Open-Pit Bench Blasting Based on PFC

To study the influence of different resistance lines and rows delay intervals on the shape of the muck pile and the throwing distance of open-pit bench blasting based on the open-pit bench blasting test, two-dimensional particle flow code (PFC2D) is used for numerical simulation. The experimental and simulation data were compared and analyzed, and the error value was acceptable. Meanwhile, different resistance lines and row delay intervals are used to conduct numerical simulation analysis. The results show that the height change of the muck pile is directly proportional to the resistance line, and the rock throwing distance is inversely proportional to the resistance line change. The best resistance line is 1.75 m in the actual working condition, according to the effect of the muck pile. By fitting the relationship between the resistance line and rock throwing distance and blasting muck pile’s height, it is found that the relationship is nonlinear. By simulating different row delays, the optimal row delay time difference in the bench blasting model is determined to be 25 ms under similar working conditions. It provides a guiding significance for the blasting parameter design under similar working conditions.

Scale Effects on Shear Strength of Rough Rock Joints Caused by Normal Stress Conditions

Scale effects on the mechanical behavior of rock joints have been extensively studied in rocks and rock-like materials. However, limited attention has been paid to understanding scale effects on the shear strength of rock joints in relation to normal stress σn applied to rock samples under direct shear tests. In this research, a two-dimensional particle flow code (PFC2D) is adopted to build a synthetic sandstone rock model with a standard joint roughness coefficient (JRC) profile. The manufactured rock model, which is adjusted by the experiment data and tested by the empirical Barton’s shear strength criterion, is then used to research scale effects on the shear strength of rock joints caused by normal stresses. It is found that the failure type can be affected by JRC and σn. Therefore, a scale effect index (SEI) that is equal to JRC plus two times σn (MPa) is proposed to identify the types of shear failure. Overall, shearing off asperities is the main failure mechanism for rock samples with SEI > 14, which leads to negative scale effects. It is also found that the degree of scale effects on the shear strength of rock joints is more obvious at low normal stress conditions, where σn < 2 MPa.

Influence of arch shaped notch angle, length and opening on the failure mechanism of rock like material and acoustic emission properties: Experimental test and numerical simulation

The uniaxial compressive failure mechanism of rock like specimens with arched shaped notches was studied experimentally and numerically considering the effects of cracks’ properties on the mechanical behavior of the rocks. The rock like material specimens were provided in laboratory in form of cubes with dimensions of 150 mm × 150 mm × 50 mm containing one arch shaped notch at the center of the specimen where, the notch diameter was changed as 20 mm, 40 mm, and 60 mm. the notch apperture was also varied as 2 mm. 4 mm and 6 mm. The axisymetric axis of the arched notch made angles changed from 0° to 90° by increasing the 15°. In this research 63 different models of the compression tests were made by the two-dimensional particle flow code (PFC2D). The experimental tests and their numerical simulations performed simultaneously to compare them for the arch shaped notch tests under compression. The results of these analyses revealed that the geometric characteristics of the arch shape notch such as the notch diameter, angle and its opening (aperture) governe the failure process of the models and tests under compressive loading conditions. It is also shown that the failure mechanisms of the arch shaped notches and their fracture modes are associated with the compressive strengths of the testing samples. The changes in notch diameter, notch angle and notch opening may make some significant changes in AE hits during the failure process of the specimens. The AE hits grow rapidly after the loading process and before the peak load. Any stress drop may be detected by changes in AE hits during the test. It is concluded that the fracture pattern and failure of the experimental tests and their corresponding numerical simulations are similar to one another which verify the validity and accuracy of the proposed approaches in this research.

Tensile behavior of Brazilian disks containing non-persistent joint sets subjected to diametral loading: experimental and numerical investigations

Structural defects are part of the inherent characteristics of rock masses. They can be found in fishers, joints, and beddings and can be divided into persistent or non-persistent ones. The coalescence of non-persistent cracks may lead to the formation of persistent joints under the tensile stress field, leading to rock mass instability. The mechanical behavior of non-persistent jointed disks under tensile stress has important implications for rock structures. In this paper, experimental and Discrete Element Method (DEM) numerical simulation approaches were adopted to investigate the effect of joint continuity factor (the relationship between joint length and rock bridge length), bridge angle, joint spacing, and loading direction in relation to joint angle on the tensile strength and stiffness as well as failure pattern. Cement-mortar-based Brazilian disks containing open non-persistent joints were constructed and subjected to diametral loading and simulated using a two-dimensional Particle Flow Code (PFC2D). The investigations revealed that the tensile strength, stiffness, and failure pattern of Brazilian disks are highly affected by non-persistent pre-existing cracks. The increase of joint continuity factor and loading direction leads to the reduction of both tensile strength and stiffness. However, when spacing increases, tensile strength and stiffens have a minimum at the average spacing, but it was concluded that bridge angle does not affect both parameters. Finally, the results showed that the experimental and numerical approaches are in good agreement, and almost all the parameters affect the failure pattern, and some failure patterns such as step-path failure, splitting, or sliding may occur as a function of non-persistent joint sets.

Numerical study on the rock breaking mechanism of high-pressure water jet-assisted TBM digging technique based on 2D-DEM modelling

In order to improve the digging efficiency of tunnel boring machine (TBM) in high-strength and highly abrasive formations, high-pressure water jet-assisted tunnel boring machine rock breaking technology has been developed and applied in steps. In this study, rock breaking mechanism by the new technology is investigated based on two-dimensional particle flow code (PFC2D) modelling. The force chain field distribution law and crack extension evolution characteristics of three typical rock breaking models are studied, and the influence of precutting slits parameters on force chain field distribution, rock sample rupture pattern, and peak load are further analysed. The results show that: 1) The rock breaking processes of the three typical modelling types (i.e., complete cutting model, same trajectory cutting model, and different trajectory cutting model) are different. Among them, the different trajectory cutting model is more likely to produce tensile failure and effectively reduce the penetration depth required for rock breaking. 2) The percentages of tension cracks to the total cracks in the three typical modellings are 90.8%, 93.9%, and 89.8%, respectively, indicating that the above three models are dominated by tension damage in the mesoscopic view. 3) With the increase of the depth of the precutting slit, the depth of the stress concentration zone beneath the disc cutter increases, inducing the increase of the angle between the edge of the stress concentration zone and the upper surface of the rock sample. Meanwhile, the peak load decreases, hence the difficulty of the tunnel boring machine disc cutter penetration is gradually reduced.

Discrete Element Method Simulation of Borehole Breakout Based on the Strain Energy Concept

This study focused on studying the V-shaped rock failure around borehole walls and reproducing borehole breakout using numerical simulations. Discrete element method (DEM) is adopted, and a two-dimensional particle flow code (PFC2D) model is built based on Tenino sandstone. The simulations are conducted by applying different maximum horizontal stress values under the same minimum horizontal stress condition, and the results are compared with the true-triaxial breakout experiments on the same rock properties and loading conditions. Due to the absence of vertical stress in 2D simulations, the obtained angular spans are generally larger than the experimental values. On the other hand, the breakout depths are found to be significantly smaller than the experimental results since the failed particles within the breakout zone remained to withstand internal loads and constrain the damage zone. In order to better simulate the rock detachment process during breakout formation, a particle removal algorithm using the strain energy release concept is developed and embedded into the model. With the implementation of the algorithm, the breakout propagation process in the vicinity of the borehole shows a good agreement with the experimental observations, suggesting that the effective removal of failed particles is essential for accurately simulating borehole breakout. The results from this study can provide new insights into the V-shaped breakout formation mechanism and strain energy changes during borehole breakout initiation and propagation.

Effect of geogrid reinforcement on tensile failure of high-strength self-compacted concrete

In this study, the tensile strength, failure mechanism and ductile behaviour of geogrid-reinforced high-strength self-compacting concrete discs subjected to both the Brazilian tensile strength test and a biaxial compressive test are studied. To determine the combined effects of geogrid layer numbers and inclination angle on the ultimate tensile strength of concrete samples, 21 experiments were conducted with up to three layers of geogrids inclined at angles of 0° to 90°, at increments of 15°. In addition, discrete-element numerical simulations were conducted using two-dimensional particle flow code to examine the failure behaviour of geogrid-reinforced high-strength self-compacting concrete discs. The numerical models were first calibrated by the experimental results and then the failure behaviour of models containing geogrids was investigated. Both experimental and numerical results demonstrate that augmenting the concrete discs with geogrids increases the ductility of specimens, especially after failure. As the number of geogrid layers increased, the tensile strength of specimens also increased, whereas the tensile strength and absorbed energy were the same for specimens with different numbers of geogrid layers and inclination angles of 75° and 90°. The specimen with three horizontal geogrid layers had the highest tensile strength, biaxial compression strength and ductility of all specimens tested.

Prediction of the transitional normal stress of rock joints under shear

The transitional normal stress of a rock joint subject to shear refers to the critical normal stress under which the normal dilation is completely suppressed. The transitional normal stress is involved in many shear strength/constitutive models of joints as a key material constant; however, this parameter is poorly constrained and its determination is mostly empirical. Here we propose a simple formulation to predict the transitional normal stress based on the systematic, well-calibrated PFC2D (two-dimensional particle flow code) simulation of the shear characteristics of both sawtooth and JRC -profiled rock joints. In the PFC2D modelling, rock joints are confined by low to high normal stresses approximating the uniaxial compressive strength (joint wall compressive strength of fresh, dry and closely matched joints) of the simulated rock. The formulation can satisfactorily quantify the transitional normal stress of regular and irregular joint surfaces as a function of the rock strength and the joint surface roughness , as validated by the laboratory data. Therefore, it can be readily introduced to the shear strength criteria and constitutive models of rock joints, which could significantly promote the accurate quantification of rock joint shear behaviour .

Numerical simulation research on impact mechanical properties of frozen soil based on discrete element method

Many engineering activities have been conducted on permafrost. Frozen soil is subjected to impact loading, particularly during blasting and excavation projects. Thus, it is essential to study the impact mechanical properties of frozen soil. A split Hopkinson pressure bar (SHPB) experiment was conducted to investigate the mechanical responses of frozen soil specimens subjected to impact loading under different strain rates at different temperatures. Evident strain rate and temperature effects were also observed. As crack development in the specimens could not be observed experimentally, a two-dimensional particle flow code was utilized to numerically simulate SHPB impact experiments on frozen soil. A contact bonding model was used for the simulation. We assumed that only temperature could change the particle parameters. The parameters and establishment of the geometric model in the simulation were calibrated and validated by comparing them with the experimentally obtained impact stress–strain curves and wave signals. More influences of strain rate and temperature on crack development were presented intuitively in this study, in combination with the propagation of stress waves. The results of the numerical simulation demonstrated that when the frozen soil specimen was subjected to impact loading, shear failure was the primary failure in the specimen. For a given temperature, a lower strain rate decreased the number of cracks generated, increased the duration of crack generation, and delayed the formation of cracks. For a given strain rate, a lower temperature decreased the number of cracks generated and the duration of crack generation; however, the number of tensile cracks was negligibly affected by changes in temperature.

Experimental and Numerical Investigations of Dynamic Failure Mechanisms of Underground Roadway Induced by Incident Stress Wave

The mechanisms of dynamic disasters around underground roadways/tunnels were examined by adopting split Hopkinson pressure bar (SHPB) laboratory tests to reproduce the failure process of the surrounding rock subjected to incident stress waves. On the basis of ensuring the consistency of numerical simulations with the experimental results, the failure mechanisms of the surrounding rock and spatiotemporal evolution of the hoop stress around the hole were studied by using a two-dimensional particle flow code (PFC2D). The results of the numerical simulation indicate that tensile stress and compressive stress concentrate along the horizontal and vertical directions around the hole, respectively, owing to the instantaneous incidence of compressive stress waves. The failure modes of surrounding rocks are significantly different when the hole is subjected to various intensities of incident stress waves. In addition, the stability of the surrounding rock of the hole is greatly affected by the amplitude and wavelength of the incident wave and the elastic modulus of the surrounding rock.

Numerical Simulation of Failure Behavior of Brittle Heterogeneous Rock under Uniaxial Compression Test.

Rocks have formed heterogeneous characteristics after experiencing complex natural geological processes. Studying the heterogeneity of rocks is significant for rock mechanics. In this study, a linear parallel bond model with Weibull distribution in two-dimensional particle flow code (PFC2D) is adopted to study the mechanical characteristics and brittle failure mode of granite rock specimens with different heterogeneity. Firstly, we selected several combinations of key micro-parameters of the parallel bond model. Then, we subjected them to a Weibull distribution to satisfy heterogeneity, respectively. Finally, we chose one optimal combination plan after comparing the stress–strain curves of heterogeneous rock specimens. We analyzed the simulated results of heterogeneous rock specimens. The crack distribution of rock specimens under peak stress shows different characteristics: a diagonal shape in rock specimens with low heterogeneity indexes, or a rotated “y” shape in rock specimens with high heterogeneity indexes. As for failure mode, the numerical simulation results show high consistency with the laboratory experiment results. The rock specimen breaks down almost diagonally, and the whole specimen tends to form an x-shaped conjugate shear failure or the well-known “hour-glass” failure mode. With the increase of the homogeneity index of the rock specimen, the shear rupture angle becomes larger and larger. Generally, the crack number increases with time, and when the rock specimen reaches the peak failure point, the number of cracks increases sharply. The development of cracks in numerical rock specimens under compression test is a result of the coalescence of many microscopic cracks. Furthermore, tensile cracks formed initially, followed by shear behavior along the macroscopic crack plane. We also preliminarily study the mechanical characteristics of heterogeneous rock specimens with discontinuous structural planes. The discontinuous structural planes are simulated by the smooth-joint model. We can conclude that the discontinuous structural planes and the microscopic structural planes which contribute to the heterogeneity have a mutual influence on each other.

Numerical Investigation of Brittleness Effect on Strength and Microcracking Behavior of Crystalline Rock

Brittleness has a significant influence on rock failure under compression; however, the mechanism is rarely comprehensively discussed. This study numerically investigates the brittleness effect on microcracking behavior of crystalline rock using a grain-based model implemented into a two-dimensional particle flow code, with a focus on the discussion of how rock brittleness affects the failure mechanism. The simulated failure mode changes from tension to shear with decreasing rock brittleness, which is consistent with previous laboratory test results. As the brittleness gradually decreases in the model, the grain boundary (GB) tensile crack to shear crack ratio increases, and the corresponding fractures change from vertical or subvertical to an angle about 45° along the vertical direction. The propagation and coalescence of generated microcracks result in a transition of failure pattern from splitting to shear under uniaxial compression with a decreasing brittleness level in the rock. A transition from GB tensile crack to shear crack is also observed under direct tension when the brittleness index gradually decreases. The tension to shear transition mechanism is closely related to the relative strength of the mineral grain and mineral bonding. The relative strength of mineral and mineral bonding could be used as a parameter to characterize rock brittleness from a microscale viewpoint.

Modelling of dynamic tensile failure of inclusion-bearing rocks

Natural and synthetic inclusions may modify the tensile strength and the failure pattern of surrounding rocks. Understanding the failure mechanism could improve rock failure prediction and induced geohazard mitigation. Due to difficulties in controlling the geometrical and mechanical properties of embedded inclusions in the laboratory, we simulate the dynamic tensile failure of inclusion-bearing rock specimens using a two-dimensional particle flow code model. We investigate the effects of strength ratio, loading rate, and treatment temperature on the nominal tensile strength of inclusion-bearing specimens and discuss the brittleness index to interpret the strength evolution. We identify the dominant effect on the nominal tensile strength as strength ratio for a low tensile strength inclusion and loading rate and treatment temperature for a high tensile strength inclusion. The change in dominant effect is associated with induced cracks primarily formed in the rock/inclusion part with lower tensile strength and highlights key factors to control inclusion-induced rock failure. This study also reveals that low strength, large size inclusions promote a decrease in brittleness index and accelerate the degradation of inclusion-bearing rocks.

Numerical simulation of mesomechanical properties of limestone containing dissolved hole and persistent joint

In order to reveal the complex mechanical properties and mesoscopic failure behavior of limestone rock mass containing both dissolved hole defect and persistent joint, numerical models are developed using the two-dimensional particle flow code (PFC2D) to carry out the uniaxial compressive strength (UCS) tests. The stress–strain, contact force chain, vertical stress field, and crack evolution characteristics of limestone samples containing circular holes and different dip angles persistent joints were analyzed. The results show that the UCS and elastic modulus of the rock samples can be reduced by the circular hole and the persistent joint, and the persistent joints with steep dip angles have a more obvious control effect on the mechanical properties of the rock samples. With the increase in loading stress level, the contact force chain in the hole-joint combination model gradually deflects slightly along the joint dip angle from the left and right sides of the hole and gradually spreads outward. Under uniaxial compression, the initiation cracks in persistent joints and rock blocks are primarily tensile cracks. The initiation positions first appear in the persistent joint, and the initiation positions in the rock block change from the upper and lower vertices of the circular hole to the junction below the joint with the increase of the joint dip angle. The crack initiation stress in the persistent joint is smaller than that in the corresponding rock block, and the control effect of the joint dip angle on the crack initiation stress of the whole rock sample is reduced.