Study on mechanical properties and fracture modes of sandstone with cracks under triaxial compression

This study examines the mechanical properties and fracture modes of sandstone with single and double cracks under varying confining pressures through triaxial compression tests. The stress–strain curves, mechanical parameters, and fracture modes of both intact and cracked sandstone are analyzed based on experimental results. The study finds that the presence of cracks has a significant impact on the mechanical properties of sandstone, with greater damage observed with an increase in the number of cracks. The peak strength and strain of a cracked rock sample increase with the increase in crack angle. The fracture mode of sandstone is influenced by both crack angle and confining pressure. When the crack angle is small, the fracture mode is a combination of shear and tensile failure, while a large crack angle results in shear failure. Moreover, as the confining pressure increases, the degree of damage to the sandstone also increases.


INTRODUCTION
Natural rock formations are abundant on the Earth's surface and possess exceptional physical and mechanical properties, making them a popular choice in rock mechanics engineering.However, the presence of cracks in rock formations can significantly affect their mechanical properties and fracture patterns.Therefore, studying cracks in rock formations is crucial for designing and constructing related engineering fields.][3][4][5] Rock masses are often subjected to triaxial loading conditions, making it important to study their mechanical properties and failure modes under such conditions.In order to investigate the effect of water content and freeze-thaw cycles on the mechanical properties of rock, triaxial compression tests were conducted on layered composite rock. 6,7The tests provided mechanical parameters, which were then used to analyze the deformation, strength, and failure characteristics of the rock. 8,9Triaxial compression tests have been used to study the mechanical properties of various types of intact rock,

(A) (B)
F I G U R E 1 Schematic diagram of single and double cracks specimens (Unit in mm).][12][13] Natural rock masses typically contain numerous joint cracks, which can significantly affect the mechanical properties of the rock.The extension of these cracks is related to the stress intensity factor (SIF) and can be studied through experimental, theoretical, and numerical methods, with analytical methods being widely used. 14,157][18][19] This study investigated the mechanical properties of rocks with varying crack lengths and dip angles through indoor uniaxial compression tests.1][22][23] Furthermore, numerical simulations were conducted to determine the mechanical parameters and failure mode of rock specimens with preexisting closed cracks of different sizes, contributing to the ongoing development of numerical experiments. 24,25umerous studies have been conducted on the failure process of intact rocks under uniaxial and triaxial compression, as well as the mechanical mechanism of fractured rocks under uniaxial compression.However, there is a relative dearth of research on the mechanical properties and failure modes of fractured sandstone under triaxial compression conditions, with limited investigation into the impact of the number of cracks.This study aims to explore the impact of crack number and angle on the mechanical properties and damage modes of sandstone through triaxial compression tests.Additionally, the study will investigate the mechanical properties of red sandstone under various confining pressures and summarize the corresponding change laws.The focus will be on the effect of confining pressures and fracture crack angle on the compressive strength of red sandstone.The study will analyze the damage modes under different crack angles by observing the crack propagation pattern of the specimens.The results of this study will provide valuable insights for relevant engineering fields.

SAMPLE PREPARATION AND PROCEDURE
This study utilized red sandstone samples from Yunnan Province, China.The samples were prepared into standard cylindrical specimens with a diameter of 50 mm and a height of 100 mm, following the International Society for Rock Mechanics (ISRM) test procedures.To investigate the impact of crack angles and numbers on the failure mode of the red sandstone, the specimens were further processed using a water cutting machine.Single and double cracks with a length of 15 mm, a width of 2 mm, and crack angles of 0 • , 30 • , 45 • , and 60 • were produced.The crack angle, denoted as α, is the angle between the crack and the horizontal direction.Figure 1 displays a schematic diagram of the single and double cracks specimens.Red sandstone with a uniform texture and no macroscopic cracks was selected as the bedrock and cored using a core drill.Once the sandstone had been cored, the cored sample was processed using a rock cutter to a standard cylindrical sample of 50 mm diameter and 100 mm height in accordance with the ISRM test protocol.The two end faces of the sample were leveled using a rock grinder and finally, using the center of the cylinder as a reference point, 15 mm long and 2 mm wide cracks were cut with a rock cutter at 0 • , 30 • , 45 • and 60 • angles to the horizontal.The triaxial compression test apparatus for intact, single cracked, and double cracked rock specimens is the ZTRE-210 micro-computer controlled rock triaxial test system, the test equipment and rock specimens are shown in Figure 2. The loading rate was set at 0.04 mm/min with a target deformation of 2 mm.The confining pressure was controlled at 10, 15, and 20 Mpa, respectively, with a loading rate of 300 N/s.This article presents triaxial compression tests on cracked rock with principal stress  1 in the axial direction of the specimen and principal stresses  2 and  3 in a plane perpendicular to the axis of the specimen. 2 and  3 are equal in magnitude and perpendicular in direction to each other.The specimens were subjected to a deformation controlled loading procedure.

MECHANICAL RESPONSE ANALYSIS
To accurately investigate the mechanical properties of sandstone and eliminate errors due to specimen differences, three randomly selected intact sandstone specimens underwent triaxial compression tests.Figure 3 displays the stress-strain curves of the three intact rock specimens at a confining pressure of 15 MPa.It is evident from the figure that the three curves overlap well and the mechanical parameters are closely related.The variability of the three values, defined as the ratio of the difference between the maximum and minimum values to the mean value, Stress-strain curves for intact rock samples was used to reflect the variability of the specimens.The peak axial stress values of rock samples I 1 , I 2 , and I 3 were 116.58, 114.63, and 117.55 MPa, respectively, with a variability of 2.51%.The peak axial strain values of the rock specimens were 3.21 × 10 −3 , 3.19 × 10 −3 , and 3.22 × 10 −3 , respectively, with a variability of 0.94%.This indicates that the rock specimens have good consistency and can be used for processing and testing fractured rock specimens.

Stress-strain curves
Figure 4 displays the stress-strain curves of rock specimens with intact, single crack, and double cracks under varying confining pressures during triaxial compression tests.The stress-strain curve of rock under triaxial compression consists of four stages: micro-crack closure stage, elastic stage, yielding stage, and post-peak stage.At a confining pressure of 20 MPa in Figure 4A shows that during the initial loading phase, micro-cracks within the specimen are closed, resulting in a nonlinear compression phase.As the load increases, the curve shows linear growth and enters the elastic stage.Once the external load exceeds the rock's elastic limit, the stress increases slowly with strain and enters the yield stage.After reaching the peak stress, the stress decreases rapidly to reach the residual strength.The stress-strain curves for intact and cracked rocks under triaxial compression are broadly similar.
The length of the compression stage decreases as the confining pressure increases, and when the pressure reaches 20 MPa, the compression stage becomes almost invisible.This is because higher confining pressure causes micro-cracks within the rock to close completely before stress loading, resulting in a shorter compression stage.
During the elastic and yield stages of rock samples, the stress-strain curves of intact and fractured rocks exhibit different shapes due to the influence of cracks.The stress-strain curve of fractured rock samples is more complex and can be observed in Figure 4B,C.When the inclination angle of the crack is 0 • , the stress-strain curve of the double-cracked rock displays a bimodal phenomenon, where the stress drops significantly before increasing again to the stress peak.This bimodal phenomenon is also observed when the inclination angle of the crack is 30 • and the confining pressure is 10 and 15 MPa.The bimodal phenomenon is mainly observed in specimens with lower double crack strength, indicating that the probability of the bimodal phenomenon increases with decreasing strength.The preformed double cracks in rocks cause significant degradation.Additionally, communication and stress concentration at the tip of these cracks result in local transient damage, leading to a sudden release of energy and a sudden drop in stress. 26,27This process promotes the closure of rock cracks and the redistribution of stress, ultimately leading to an increase in stress as loading progresses.
During the post-peak stage, the stress-strain curves for both intact and fractured rock samples exhibit similar behavior.A stress drop is observed, and with increasing confining pressure, the magnitude of the stress drop decreases.This suggests that confining pressure can mitigate the effect of the stress drop.This is because higher confining pressure leads to a more uniform stress distribution within the fractured rock and smaller stress differences between cracks.This slows down the degree of stress concentration and results in a slower stress drop for the fractured rock.

Mechanical parameter characteristics
The relationship between fracture dip and compressive strength under triaxial compression is shown in Figure 5.
The figure demonstrates the notable influence of cracks on the peak strength of rocks, with fractured rock samples exhibiting significantly lower peak strength compared to intact rock samples.Moreover, under similar conditions, the peak strength of double-cracked rock samples is lower than that of single-cracked rock samples, indicating that preexisting cracks cause initial damage to rock samples, and the extent of damage increases with the number of cracks.Additionally, the peak strength of fractured rock samples increases as the inclination angle of the crack increases, suggesting a close relationship between the strength of fractured rock samples and the inclination angle of the crack, given the same confining pressure.
The relationship between the triaxial compressive strength of intact and fractured rock samples and the confining pressure is shown in Figure 6.In Figure 6A, the strength-confined pressure curves of intact and single crack rock samples are presented.The graph indicates that the strength of the rock samples increases with an increase in the confining pressure, given that the inclination angle of the crack remains the same.Figure 6B shows the strength-confined pressure curves of intact and double crack rock samples, and the trend of strength variation is consistent with that of single-cracked rock samples.The data is subjected to linear regression to express the relationship between strength and confining pressure.
where σ 1 is the compressive strength, σ 3 is the confining pressure, R 2 is the linear correlation coefficient, and α is the crack angle.Equation ( 1) represents intact rock samples with a single crack, while Equation ( 2) represents rock samples with double cracks.The linear correlation coefficients R 2 for both equations are close to 1, indicating a strong linear correlation between the triaxial compressive strength of intact and fractured rock samples and the confining pressure.The slope of the fit equation indicates that intact rock samples are more sensitive to confining pressure than fractured rock samples, and this sensitivity decreases as the fracture dip angle increases.
The Mohr-Coulomb theory is a popular method in geotechnical engineering for determining the internal friction angle and cohesion force of rock samples.This theory utilizes a strength-fitting relationship to obtain these values for both intact and fractured rock samples, as illustrated in Figure 7.
The figure demonstrates that intact rock samples have higher internal friction angle and cohesion force compared to fractured rock samples.Additionally, the cohesion force increases while the internal friction angle decreases with an increase in the fracture inclination angle.The relationship between peak axial strain and confining pressure for both intact and fractured rock samples is presented in Figure 8.The figure illustrates that as the confining pressure increases, the peak strain of both intact and fractured rock samples also increases.The linear fit analysis reveals that the slope of intact rock samples is notably higher than that of fractured rock samples, indicating that intact rock samples are more sensitive to confining pressure in terms of peak strain.

FRACTURE MODE AND ANALYSIS
Under triaxial compression, jointed rock samples will experience crack propagation as the load increases.The crack propagates from the tip due to stress concentration.This article classifies the crack propagation at the tip of prefabricated cracks into three types: wing crack, coplanar secondary cracks, and oblique secondary cracks, as shown in Figure 9. Wing crack is a tensile crack at a certain angle to the crack, coplanar secondary cracks are shear cracks roughly coplanar with the crack, and oblique secondary cracks are shear cracks at a certain angle to the crack and in the opposite direction to the wing crack. 28n Figure 10, the failure modes of 0 • fractured rock samples under varying confining pressures are depicted.The failure mode of single crack rock samples is observed propagate from the crack tip to the periphery along the crack tip.The wing tensile crack and the inclined shear crack develop and propagate to varying extents under different confining pressures, with their development becoming more pronounced as the confining pressure increases.At 20 MPa confining pressure, the shear crack and the tensile crack penetrate along the extension direction, resulting in a shear-tensile composite failure mode of the rock sample.This study observed the failure mode of double-cracked rock samples.It was found that due to stress concentration at the crack tip, the crack tip penetrates vertically, resulting in a crack propagation shape similar to that of single crack rock samples.However, the degree of damage and crack development in double-cracked rock samples is more significant under the same confining pressure.The failure mode of the rock sample is also a shear-tensile composite failure.In Figure 11, the failure modes of 30 • fractured rock samples are presented at varying confining pressures.Upon observing the failure mode of single crack rock samples, it is evident that the wing crack, coplanar secondary cracks, and oblique secondary cracks all propagate outward from the crack tip.As the confining pressure increases, the propagation becomes more pronounced.At a confining pressure of 20 MPa, both the coplanar secondary cracks and the oblique secondary cracks propagate to the edge of the rock sample, resulting in a shear-tensile composite failure mode.This study observed the failure mode of double cracks rock samples and found that stress concentration at the crack tip caused the two cracks to penetrate along the diagonal direction.The upper wing crack gradually changed to a shear crack under 10 MPa confining pressure, while coplanar secondary cracks and secondary oblique cracks developed.As the confining pressure increased to 15 and 20 MPa, the wing crack became almost absent and coplanar and oblique secondary cracks extended to the edge of the rock sample.The failure mode changed from a shear-tensile composite failure to a shear failure with increasing confining pressure.
Figures 12 and 13 depict the failure modes of fractured rock samples with angles of 45 • and 60 • , respectively, under varying confining pressures.The figures reveal that the failure modes of these samples are largely similar.In the case of single crack rock samples, the main failure mode is shear failure due to the development of coplanar and oblique secondary cracks.As the confining pressure increases, crack propagation becomes more extensive and reaches the edge of the rock sample at 20 MPa.In double cracked rock samples, the crack penetrates diagonally due to stress concentration at the crack tip.Coplanar and oblique secondary cracks develop more extensively than in single crack rock samples.As the confining pressure increases, the rock sample failure becomes more pronounced with a shear failure mode.
The triaxial compression failure mode of fractured rock samples is affected by the inclination angle of the fracture and the confining pressure.An increase in the inclination angle leads to a change in the failure mode from a shear-tension composite failure to a shear failure.Additionally, double crack rock samples exhibit more efficient crack propagation F I G U R E 13 Failure modes of the 60 • fractured rock samples compared to single crack rock samples for the same confining pressure and angle of inclination.The failure of the rock sample becomes more significant with an increase in the confining pressure, as crack propagation becomes more efficient.

CONCLUSION
The stress-strain curves of intact and cracked rock samples are similar in the micro-fracture closure phase, elastic phase, yield phase, and post-peak phase.However, in the post-peak stage, there is a stress drop that decreases in magnitude with increasing confining pressure.This suggests that confining pressure reduces stress concentration, which in turn slows down the stress drop.
Fractures have a notable impact on the peak strength and strain of rocks.Fractured rock samples exhibit significantly lower peak strength and strain compared to intact rock samples, with double cracks leading to even lower strength and strain than single cracks.This suggests that the presence of cracks causes initial damage to rock samples, with increasing cracks resulting in greater damage.The inclination angle of fractures is closely linked to the peak strength and strain of rock samples.Fractured rock samples experience an increase in peak strength and strain as the fracture inclination angle increases, at the same confining pressure.
The mechanical properties of rock samples are significantly affected by the confining pressure.As the confining pressure increases, the strength of the rock sample increases for the same fracture inclination angle.The sensitivity to confining pressure is highest in intact rock samples, and decreases as the fracture inclination angle increases.Double cracks rock samples are more sensitive to confining pressure for the same fracture inclination angle.In general, the effect of confining pressure on the mechanical properties of rock is substantial.
The failure mode of fractured rock samples under triaxial compression is influenced by both the inclination angle of the fracture and the confining pressure.As the inclination angle of the fracture increases, the failure mode transitions from shear-tensile composite failure to shear failure.Additionally, increasing the confining pressure results in more preexisting fractures and a greater propagation of fractures.

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Test equipment and rock specimens.(A) Test equipment and (B) rock specimens

F I G U R E 4 F 6
Stress-strain curves for rock samples under triaxial compression.(A) Intact rock specimens, (B) 0 • crack angle rock specimens, (C) 30 • crack angle rock specimens, (D) 45 • crack angle rock specimens, and (E) 60 • crack angle rock specimens Compressive strength versus confining pressure.(A) Intact and single crack rock samples and (B) intact and double cracks rock samples

F I G U R E 7
Internal friction angle and cohesion force of rock samples F I G U R E 8 Peak axial strain versus confining pressure

F I G U R E 9
Crack propagation mode F I G U R E 10 Failure modes of the 0 • fractured rock samples F I G U R E 11 Failure modes of the 30 • fractured rock samples F I G U R E 12 Failure modes of the 45 • fractured rock samples