Impact of Crack Inclination Angle on the Splitting Failure and Energy Analysis of Fine-Grained Sandstone

Abstract

To investigate the impact of crack inclination angle on the tensile strength and characteristics of splitting failure in rock, split tests were conducted on fine-grained sandstone with pre-existing cracks under different crack inclination angle conditions. Acoustic emission and digital image techniques were continuously monitored during the tests. The evolution of absorbed energy during the loading and failure processes was calculated and discussed, aiming to elucidate the interconnections among the maximum absorbed energy, the pre-existing crack inclination angle, the tensile strength, and the acoustic emission energy of the rock, which can provide a reference for the design and construction of tunnels or mines in rock formations with pre-existing cracks. The experimental findings indicate the following: (1) The tensile strength and failure displacement decrease first and then increase with the increase in the pre-existing crack inclination angle, demonstrating an approximate “V”-shaped alteration. (2) The failure modes of specimens with pre-existing cracks can be classified into three types: tensile failure along the center of the disk, tensile failure along the tip of the pre-existing crack, and tensile–shear composite failure along the tip of the pre-existing crack. (3) The crack inclination angle exerts a substantial influence on the evolution curve of energy absorption. The maximum energy absorption of the specimens first increases and then decreases with the increase in the crack inclination angle. Moreover, a corresponding nonlinear relationship is observed between the maximum energy absorption and the tensile strength, as well as the AE energy.

1. Introduction

As the world’s population and economy continue to expand, the demand for energy and raw materials is on the rise [1], leading to the further development of various mineral mining operations. Rock rupture is an inevitable problem during mineral mining, which poses a potential threat to construction safety. Therefore, understanding the propagation of cracks in rock masses is of great significance to ensure the stability of mining structures. There are many initial cracks of different scales inside the rock mass. These cracks not only alter the mechanical characteristics of the rock but also, under external forces, the propagation and connection of cracks lead to rock failure and ultimately result in engineering accidents [2,3,4]. The propagation and interconnection of cracks within the rock mass account for the majority of failure occurrences in rock and soil engineering, such as the landslide that occurred in Yanchihe Phosphorus Mine, Yichang City, Hubei Province, China, in 1980, and the water seepage accident during the construction of Foshan Subway Line 2 in 2018. In recent years, experts have conducted theoretical analysis, numerical simulation experiments, similar material, and mechanical tests on single-cracked and multi-cracked rock specimens, and have achieved rich research results. Wang et al. [5] carried out Brazilian splitting tests on shale disks incorporating pre-existing cracks that oriented at different angles. The findings revealed that the pre-existing cracks have a certain weakening influence on the crack resistance strength of the rock disks. Li et al. [6] performed uniaxial compression load tests on pre-cracked marble specimens. The results indicated that the cracks not only changed the mechanical properties of the rock but also the size of the rock bridge between the cracks, affecting the morphology of the fracture plane and the failure mode of the rock. Studies such as references [7,8,9] have investigated the influence of pre-existing cracks on the mechanical properties of rocks under compression conditions. Huang et al. [10] established a rock fracture stable propagation model based on crack strain, providing a comprehensive depiction of the evolution patterns of crack strain and wing crack length. Shi et al. [11] used acoustic emission (AE), digital image correlation (DIC), and scanning electron microscopy (SEM) techniques to explore the effects of creep loading on the fatigue behavior and acoustic emission characteristics of pre-cracked sandstone during fatigue loading. Due to the difficulty of conducting in situ loading tests on rock specimens containing cracks, similar material tests have become one of the main research methods for studying the mechanism of crack propagation and connection in rocks. Li et al. [12] used theoretical analysis and similar material methods to study and found that the connection between multiple cracks is an important cause of rock failure. Pu et al. [13] studied the influence of multiple cracks on the compressive strength of rock-like materials by using the method of embedding inserts in cement mortar specimens. Wang [14] used a certain proportion of cement mortar to make disc specimens containing pre-existing cracks for Brazilian splitting tests. By combining particle flow theory and numerical simulation, the study analyzed the influence of a solitary crack on the tensile properties and fracture patterns of the rock.

Given the intricate nature of crack propagation and interconnection in rocks, theoretical analysis alone faces significant challenges. Consequently, numerical simulation has become a commonly used research method among scholars [15,16]. Yuan et al. [17] conducted uniaxial compression tests on rock samples containing fissures using Particle Flow Code (PFC) 2D software, and systematically investigated the effects of different fissure lengths and inclinations on the failure modes and crack initiation and propagation of rock-like structures. Zhu et al. [18] and Wang et al. [19] studied the influence of pre-existing cracks at different angles on the propagation of cracks in rocks using the Realistic Failure Process Analysis (RFPA) system for rock fracture analysis. An et al. [20] proposed a hybrid finite–discrete element method to simulate rock fracture behavior under different loading rates and demonstrated the superiority of this approach. Jiang et al. [21] use the experimental analysis method and data collection method to study cloud computing-related technology and finite element method, rock tensile fracture, and numerical model. The researchers in [22,23,24] used the RFPA numerical analysis system to investigate the influence of intersecting cracks on the mechanical properties and deformation characteristics of rock under uniaxial compression and tension conditions.

In summary, although experts and scholars have conducted a considerable amount of experimental and numerical simulation research on the evolution of damage and failure modes in pre-cracked rock specimens, many of these experiments have employed concrete or similar rock-like materials instead of actual rocks. However, such materials possess notable disparities in physical and mechanical properties compared to real rocks. Consequently, using them to simulate the failure process of pre-cracked rocks may lead to inaccurate conclusions regarding the influence of pre-existing cracks on rock behavior. To address this limitation, in this study, we chose fine-grained sandstone as the experimental material and conducted splitting tests on sandstone specimens with pre-existing cracks oriented at different angles. The entire process was monitored in real time using acoustic emission, and the influence of pre-existing cracks on the strength and failure characteristics of the specimens was analyzed. The energy evolution patterns of specimens with pre-existing cracks under five distinct working conditions during the failure process were calculated and analyzed. Furthermore, the relationship between maximum energy absorption and other rock parameters was examined. Energy, as an important parameter for representing rock fracture characteristics, has been widely used in dynamic rock fracture testing research [25,26,27,28,29]. Therefore, in this study, energy theory was also employed to elucidate the failure mechanism of pre-cracked rocks and analyze the correlation between energy and the failure of such rocks. These findings offer reference points and theoretical guidance for the analysis of rock mass structures and their stability in mining and tunnel excavation projects, as well as for the monitoring and prevention of engineering disasters.

2. Materials and Methods

2.1. Sample Fabrication

To ensure the consistency of the test samples, the rock specimens used in this experiment were all selected from fine-grained sandstone in Jiaoxi Township, Hunan Province, China, and taken from the same rock block and drilled along the same direction. The rock samples are light gray in color, with relatively homogeneous particle size and texture, and good integrity. According to experimental measurements, the average density of the rock is 2.4 g/cm³, and the longitudinal wave velocity is on average 2333 m/s.

Firstly, a rock coring machine was used to drill a 100 mm diameter core from the same rock block, ensuring intact and non-fractured surfaces. Subsequently, the core was cut and polished using a cutting machine to make circular specimens with a diameter of 100 mm and a thickness of 50 mm. The parallelism error of the upper and lower surfaces was ∆x ≤ 0.2 mm, and the thickness error was ∆x ≤ 0.5 mm. Professional milling equipment was used to machine pre-existing cracks at the geometric center of the circular specimens, ensuring the machining precision of the pre-existing cracks and avoiding significant effects on the test results due to differences in crack size. The cracks in the specimens were all machined to be 20 mm long and 1 mm wide, and intact specimens were used as controls. The circular specimens, both intact and those with pre-existing crack, are shown in Figure 1.

2.2. Experimental Plan

The crack inclination angle refers to the angle between the crack plane and the horizontal plane. The crack inclination angle refers to the angle between the crack plane and the horizontal plane. Pre-existing cracks with inclination angles of 0°, 30°, 45°, 60°, and 90° were selected for the experiment. The specimen numbering format used is S-α, where S represents the fine-grained sandstone specimen, and α represents the inclination angle of the pre-existing crack. In addition, various parameters of each specimen were measured using devices such as vernier calipers and ultrasonic velocity meters, and the results are shown in

Table 1. Basic parameters of the rock specimens.

Specimen Number	Diameter/mm		Thickness/mm	Quality/g	Wave Velocity/(m·s−1)
	Upper Surface	Lower Surface
S-0	100.16	99.84	50.20	942.6	2336
S-30	100.10	99.80	50.30	943.1	2381
S-45	99.72	99.70	50.32	941.8	2296
S-60	99.82	99.82	49.82	941.5	2375
S-90	99.80	99.82	50.30	942.3	2278

. The schematic diagram and loading scheme of the splitting test are shown in Figure 2a,b, respectively. In Figure 2, 2a represents the length of the crack, α represents the angle between the crack and the horizontal direction, R represents the radius of the disc specimen, t represents the thickness of the disc specimen, and P represents the vertical line load.

The splitting test was conducted on a WDW-50 microcomputer-controlled universal testing machine in the Mining Disaster Monitoring Laboratory at the University of Science and Technology Beijing. The loading was displacement-controlled at a rate of 0.5 mm/min. To fix the circular specimen, a pre-load of 0.2 kN was applied to the specimen before loading in the vertical direction until the specimen failed. An acoustic emission detection system was used to monitor the entire process of rock instability and failure in real time during the entire test. The acoustic emission acquisition equipment used was the DS5 Full-Information Acoustic Emission Analysis System manufactured by Beijing Soft Island Company. To avoid noise interference, the monitoring threshold of the test was set to 40 dB, and the sampling frequency was 3 MHz. Figure 3 depicts the experimental system layout.

3. Results

3.1. Mechanical Properties

The tensile strength ratio refers to the ratio of the tensile strength of rock specimens with pre-existing cracks to that of intact specimens, while the failure displacement ratio refers to the ratio of the vertical displacement of rock specimens with pre-existing cracks to that of intact specimens when the peak load is reached. Figure 4 illustrates the variations in the tensile strength ratio and the failure displacement ratio as a function of the crack inclination angle for circular specimens with different inclination angles.

From Figure 4, it can be observed that the notable reductions in both the tensile strength and failure displacement of rock specimens with pre-existing cracks compared to intact specimens. The tensile strength of specimens with pre-existing cracks amounts to approximately 60% to 80% of the tensile strength of intact specimens, signifying significant strength degradation caused by the presence of pre-existing cracks. Furthermore, the tensile strength of specimens with pre-existing cracks exhibits considerable variability, with an approximate magnitude of variation of 18%. The tensile strength ratio of rock specimens with pre-existing cracks follows a “V” shape trend, initially decreasing and then increasing as the crack inclination angle increases. The minimum value is observed at a crack inclination angle of α = 45°, with a strength of only 60.3% of that of intact specimens, while the maximum value is obtained at a crack inclination angle of α = 90°, with a strength of 77.2% of that of intact specimens. Additionally, the difference in tensile strength is relatively minor within the crack inclination angle range of 30 to 60°.

Similarly, the failure displacement of specimens with pre-existing cracks also shows a decreasing-then-increasing trend with an increase in the crack inclination angle. However, in comparison to the tensile strength, the reduction in failure displacement is relatively small, and the variation range of the failure displacement ratio of specimens with pre-existing cracks is relatively small with an oscillation magnitude of approximately 12% as the crack inclination angle increases. The minimum value of the failure displacement ratio is observed at a crack inclination angle of α = 45°, corresponding to approximately 72.85% of that of intact circular specimens, indicating that the damage caused by the crack is the highest at this angle. Conversely, when the crack inclination angle is parallel to the loading direction (i.e., α = 90°), the damage caused by the crack is minimized, and the failure displacement is approximately 85% compared to intact circular specimens.

3.2. Failure Mode

Table 2. Failure mode and classification of failure patterns for specimens with pre-existing cracks.

Specimen Number	Failure Mode	Numerical Simulation Results	Failure Pattern
S-0			Tensile failure along the central axis of the circular specimen (Type I), and tensile–shear mixed failure along the tip of the pre-existing crack (Type III).
S-30			Tensile–shear mixed failure along the tip of the pre-existing crack (Type III).
S-45			Tensile–shear mixed failure along the tip of the pre-existing crack (Type III).
S-60			Tensile–shear mixed failure along the tip of the pre-existing crack (Type III).
S-90			Tensile failure along the central axis of the circular specimen (Type I), and tensile failure along the pre-existing crack tip (Type II).

displays the splitting failure modes observed in circular specimens with pre-existing cracks at different inclination angles. Under the splitting test conditions, the crack propagation and failure modes of specimens with pre-existing cracks are changed to a varying degree due to the change in the crack inclination angle.

As shown in Table 2, when the inclination angle α of the prefabricated crack is 0°, the loading direction is perpendicular to the crack surface, and the tensile failure of the sandstone specimen with through-going fissures is parallel to the loading direction. Furthermore, the tension cracks that occur during the failure pass through the center of the prefabricated crack, i.e., the center of the disk, resulting in a higher measured tensile strength. Secondary cracks extend from the tip of the prefabricated crack toward the edge of the disk, exhibiting symmetry about the height of the disk, indicating a higher degree of homogeneity within the sandstone.

When the inclination angle of the prefabricated crack is between 30 and 60°, at the macroscopic level, the failure mode of the sandstone specimens exhibits highly similar fracturing patterns. The main pattern is characterized by the formation of two through-going cracks with approximately the same orientation (primary) and secondary cracks. The cracks extend from the tip of the prefabricated crack toward the loading end and gradually become parallel to the loading direction.

When the inclination angle of the prefabricated crack is 90°, the loading direction is parallel to the crack surface, and the failure mode of the sandstone specimen is similar to that of an intact specimen. The through-going crack is parallel to the loading direction, and secondary cracks are also generated during the extension of the through-going crack in the upper part of the specimen. The upper part of the specimen is severely fragmented, and the vertical crack on the left side of the through-going crack may result from local tensile stress during the extension of the through-going crack at the center of the disk.

Based on the particle flow theory method, the PFC simulation was used to investigate the sample fracture process under different dip angles of joints. The simulation results exhibited good agreement with the experimental failure morphology, particularly in the through-crack region, further validating the experimental findings.

The cracks that appear during the rock sample’s failure mainly include three forms: tensile, shear, and combined cracks. Based on the morphology and formation mechanism of the specimen’s failure cracks, the failure modes can be classified into the following categories: tensile failure along the center of the disk (Type I), tensile failure along the tip of the prefabricated crack (Type II), and tensile–shear combined failure along the tip of the prefabricated crack (Type III), as shown in Table 2.

As shown in Table 2, due to the stress concentration caused by the prefabricated crack, the failure mode of the specimen is mainly in the form of a combination of cracks. Specifically, at a crack inclination angle of 0°, the failure mode predominantly comprises Type I and Type III, due to the presence of the horizontally oriented prefabricated crack, the specimen experiences a tensile–shear combined failure along the crack tip. In the range of crack inclination angles between 30 and 60°, a consistent Type III combined failure mode is observed, indicating intensified internal rock damage that leads to frictional sliding along particle boundaries during the fracturing process. Conversely, when the inclination angle of the prefabricated crack is 90°, the prefabricated crack has less influence on the failure mode, and the failure mode of the specimen is mainly Type I and Type II tensile failure.

Building upon the classification approach proposed in the study on the failure cracks of layered sandstone disks [30], the different inclination angle prefabricated crack disk splitting failure modes can also be categorized into three distinct crack forms, determined by the direction of crack propagation. This classification scheme is illustrated in Figure 5.

(1)

Central straight fissure: The fissure is located in the center of the disc, and its fracture surface is basically parallel to the vertical loading direction. The starting points of the upper and lower fissures are both near the center of the disc, and their distances to the center are much smaller than the diameter of the disc. The central straight fissure is less affected by prefabricated cracks, as shown in Figure 5a.

(2)

Crack tip wing fissure: The crack originates at the tip of a prefabricated crack, usually in a curved shape resembling that of a wing, and extends in the direction of loading while gradually becoming parallel to it, as shown in Figure 5b.

(3)

Crack tip edge straight fissure: The crack is located in the center of the disc, but it originates at the tip of a prefabricated crack rather than at the center of the disc, as shown in Figure 5c.

The presence of rock defects, such as cracks, exerts a notable influence on the propagation behavior of intersecting fissures. It leads to varying degrees of deviations from the loading baseline and forms curves or propagates through prefabricated cracks.

3.3. The Evolution Law of Displacement Field

The digital speckle correlation (DSC) method, also known as digital image correlation (DIC), is a solid material surface deformation measurement method based on digital image processing. It compares the digital images of the object material before and after deformation, and obtains the corresponding deformation information of the region of interest through correlation calculation. The underlying principle of DSC is to capture random speckle images of the object before and after deformation using a camera. The pre-deformation image is divided into a grid, and a specific subregion of interest is selected. Through a certain search method, a small image subregion corresponding to the sample subregion after deformation is found as the target subregion. The displacement and deformation information are contained in the position and shape differences between the sample subregion and the target subregion. In the calculation, the displacement and deformation of the image regions before and after deformation are characterized, and then the displacement and strain on the surface of the object are obtained through search algorithms such as the quasi-Newton iteration method. Calculating all of the divided image regions can obtain the deformation information of the entire field. Based on the load–displacement curves during the loading process of the rock specimens, the failure process of the specimens can be divided into four stages: compaction stage (displacement range: 0 to 0.40 times the total displacement), linear growth stage (displacement range: 0.41 to 0.80 times the total displacement), plastic deformation stage (displacement range: 0.81 to 0.93 times the total displacement), and failure stage (displacement range: 0.94 to 1.00 times the total displacement). At the critical points between the four stages, five node moments were selected as markers, which are distributed at the initial, compaction, stabilization, initiation of cracking, and failure moments during the entire loading process. The corresponding displacements for these markers are 0%, 40%, 80%, 93%, and 100% of the ultimate displacement, respectively. The digital image correlation method is utilized to compute displacement and strain fields of three typical crack patterns in specimens generated under splitting loading conditions, by analyzing photos captured by the DIC camera at corresponding identified points.

The digital speckle image acquisition system, as depicted in Figure 6, consists of several components. An LED light source is employed to provide illumination for the specimen during the experiment. To capture the planar displacement and strain variations of the specimen, a camera is utilized to acquire speckle images of the specimen’s surface. The acquired images are processed using digital image correlation (DIC) software for further analysis. The DIC software facilitates image acquisition and processing, allowing for the automatic triggering of image capture and configuration of parameters such as the camera’s frame rate and image resolution. In this experiment, the images can be automatically triggered for acquisition, and the camera’s image frame rate and other information can be set. The acquisition frame rate is set to five frames per second, and the image pixel size is set to 1080 × 1920. After the acquisition is completed, the software can be used to analyze the changes in the speckles and calculate the displacement and strain changes on the surface of the circular disk.

The surface displacement contour maps for the initiation and compact phases of different specimens exhibited similar features, and, hence, are omitted herein. The surface displacement contour maps of disk specimens that generate three typical crack patterns during stable, initiation, and failure stages are presented in Figure 7, Figure 8 and Figure 9. Figure 7 depicts the final failure mode characterized by a central straight fissure, taking a prefabricated crack angle of 0° as an illustrative example, the joint plane angle is perpendicular to the loading direction, and the displacement variation in the joint end region is minimal, exhibiting a vertically distributed state of displacement change. When it is loaded to the compaction moment, there is a small displacement change in the end region of the circular disk, and there is no obvious deformation feature on the surface of the circular disk. As the loading progresses to the stabilization moment, a significant increase in displacement is observed at the end of the circular disk surface, accompanied by small deformations appearing on both sides of the joint. The deformation increases from the center of the circular disk to the loading end. Upon reaching the cracking moment, the deformation area at the end of the circular disk surface continues to expand, extending to both sides of the joint, and the deformation on both sides of the joint notably intensifies. At the failure moment, the deformation area on both sides of the joint connects with the end region of the circular disk, and the overall displacement of the surface of the circular disk at the loading axis is the largest. Furthermore, deformation zones emerge at the joint ends, extending toward the edge of the circular disk. Nevertheless, these deformations remain relatively small in magnitude and lack macroscopic visibility.

Figure 8 exhibits the case where the final failure mode corresponds to a crack tip wing fissure, specifically considering a prefabricated crack angle of 30°. In this configuration, the displacement at the joint end region increases significantly compared to that of the 0° joint specimen. Upon reaching the compaction moment, the change is similar to the joint angle of 0°. At the stabilization moment, there are deformation zones at the joint end and loading end, with larger deformation at the loading end. As the loading progresses to the cracking moment, the deformation in both regions gradually increases and approaches each other, and the extension direction is close to the macroscopic crack. Finally, upon reaching the failure moment, the two deformation zones become fully connected, presenting a macroscopically visible wing-shaped crack.

Figure 9 depicts the scenario where the final failure mode corresponds to a crack tip edge straight fissure, specifically considering a prefabricated crack angle of 90°. In this case, the joint plane angle inclines toward the loading direction, and the displacement variation exhibits a vertically distributed state due to the presence of vertical joints. However, due to the presence of vertical joints, the displacement at the disk’s center point is relatively smaller. Upon reaching the compaction moment, a minor deformation zone emerges at the loading end of the circular disk, while no significant changes are observed in other areas. As the loading progresses to the stabilization moment, the deformation zone at the end of the circular disk extends toward the end of the vertical joint, with the deformation zone at the joint end exhibiting a smaller extent of deformation. Advancing to the cracking stage, the deformation zone at the joint end intensifies and gradually connects with the loading end. This progression continues until the failure moment when the two deformation zones expand and connect, and the overall displacement of the surface of the circular disk increases. Notably, the maximum displacement value is observed in the region between the loading end and the joint on the loading axis of the circular disk.

3.4. The Evolution Law of Strain Field

The presented figures (Figure 10, Figure 11 and Figure 12) exhibit the surface tensile strain contour maps of disk specimens that generate three typical crack patterns at different. Sub-figures a in Figure 10, Figure 11 and Figure 12 show the reference images collected during the initial compaction phase, where there is no significant change in the tensile strain of the circular disk. As the loading progresses and reaches the cracking stage, the pre-existing joint will undergo a certain degree of closure due to the effect of the axial load, However, the effect on the strain field distribution remains relatively minor. Sub-figures b and c in Figure 10, Figure 11 and Figure 12 correspond to the plastic failure stage of crack propagation. Due to the change in the dip angle of the joint, the area of maximum strain change shifts to a certain extent with the dip angle of the joint.

Figure 10 displays the strain distribution for the final failure mode characterized by a central straight fissure. A distinct strain band emerges along the loading axis, where the maximum change in tensile strain is observed. The macroscopic appearance is a standard tensile failure, and the strain field change is less affected by the joint. In Figure 11, when the final failure mode of the crack is a crack tip wing fissure, a clear strain band appears at the joint crack end, and it gradually extends toward the loading end with loading. The distribution of the tensile strain band from the joint tip clearly indicates the direction and propagation path of the crack, which eventually leads to the failure of the specimen. Figure 12 depicts the strain behavior for the final failure mode characterized by a crack tip edge straight fissure. Noticeably, the tensile strain in the joint region experiences a significant increase, accompanied by a clear strain band along the joint. As loading progresses, the strain band extends gradually from the joint end toward the loading end, resembling the strain band shape observed in the 0° rock sample. However, it is evident that the joint affects this strain band, radiating from the joint end.

The displacement contour map provides insights into the most pronounced changes in the displacement field during fracture, primarily observed at the loading end and the tip area of the pre-existing joint. Analysis of the displacement contour map reveals that the displacement change at the loading end increases first in the early loading stage. As loading progresses to the stable-cracking stage, a significant change in the relative displacement value occurs at the joint. For specimens with central straight fissures as the final failure mode, the displacement values at the joint tip do not change significantly and are mainly observed on both sides. In the case of specimens with crack tip edge straight fissures as the final failure mode, the displacement field changes significantly and is influenced by the joint angle. For specimens with crack tip edge straight fissures as the final failure mode, the displacement contour maps clearly show almost no displacement change on both sides of the joint. The displacement bands are concentrated between the joint tip and the loading end, reflecting a macroscopic vertical tensile failure mode. Notably, no significant secondary cracks are observed near the joint. Additionally, the strain contour map reveals the presence of substantial tensile strain near the tip of the pre-existing joint, aligning with the direction of macroscopic crack extension, mainly accompanied by the generation of secondary cracks and wing-shaped cracks.

3.5. The Evolution Law of Energy Absorption

In the process of rock failure under loading, there is absorption and a release of energy [31]. Studying the relevant energy characteristics and laws of rock fracture under loading can provide important information for a deeper understanding of rock mechanics [32]. By examining the impact of pre-existing cracks on the energy evolution law during the fracture of sandstone under loading, as well as the relationship between the dip angle of pre-existing cracks, the tensile strength of rock, the acoustic emission energy and the maximum energy absorption of rock, and the mechanical properties of sandstone can be further revealed.

The instability failure of the rock during loading is caused by the work performed by the testing machine on the rock, which is stored as energy inside the rock. The rock deforms and eventually fractures. Therefore, the energy absorbed by the rock during the entire process is equal to the work performed by the testing machine on the rock, which can be quantified by calculating the area under the load–displacement curve [33], as shown in Figure 13. This can be expressed using the following equation:

$$u={\int }_0^{l_i}{P}_{i}dl,$$

(1)

where u is the absorbed energy (J), P_i is the vertical load at a certain moment, and l_i is the vertical displacement increment at that moment.

Figure 14 presents the variation pattern of energy absorption in sandstones with different crack angles under various vertical displacement ratios, derived from the analysis of experimental data. It provides insights into the relationship between energy absorption and the extent of vertical displacement relative to failure displacement.

According to the observations from Figure 14, it is evident that the relationship between absorbed energy and vertical displacement ratio in all sandstone specimens exhibits nonlinear behavior during the loading and failure process. Initially, in the early loading stage, the absorbed energy increases slowly with the displacement growth rate, and the slope of the curve is small. As the vertical displacement ratio approaches approximately 40%, the slope of the curve increases rapidly, and the absorbed energy of the specimen increases with the displacement growth rate. This phenomenon can be attributed to the control displacement mode employed by the universal testing machine, ensuring uniform loading throughout the test. During the initial loading stage, the specimen absorbs less energy and the rate is slower, corresponding to the compaction stage of rock loading and failure initiation. Sandstone specimens are heterogeneous materials with pre-existing cracks, and most of the energy is dissipated due to the compaction of micro-cracks and pre-existing cracks inside the rock. In the later stage of loading, more energy is absorbed at a faster rate, indicating less energy dissipation, and the work performed by the testing machine is absorbed by the rock specimen after the pre-existing cracks and micro-cracks inside the rock are compacted. When the vertical displacement ratio reaches approximately 80%, the absorbed energy evolution curve approaches a linear relationship, and the growth rate stabilizes. This indicates that the absorbed energy and dissipated energy reach a relatively balanced state, and the rock will eventually become unstable and fail.

When the dip angle of the pre-existing crack α = 45°, the absorbed energy of the specimen increases at the slowest rate during the loading and failure process. On the other hand, the energy evolution curve of the specimens with the dip angles α = 30° and α = 60° exhibit similar trends. When the dip angle α = 90°, the absorbed energy of the specimen increases at the fastest rate, followed by the specimen with α = 0°. The rate of absorbed energy growth can reflect the severity of instability and failure during the rock loading process. Thus, when combined with the analysis of the failure morphology and mode of the specimens, it can be concluded that the severity of tensile sliding failure in sandstone decreases first and then increases with the increase in crack dip angle.

Table 3. Maximum absorbed energy and maximum absorbed energy ratio of the rock specimen.

Specimen	Maximum Absorbed Energy/J	Maximum Absorbed Energy Ratio/%
intact specimen	15.91	100
S-0	10.45	65.68
S-30	7.78	48.90
S-45	6.45	40.56
S-60	7.64	48.01
S-90	11.84	74.40

presents the maximum absorbed energy and the maximum absorbed energy ratio of specimens with different crack angles, as well as intact specimens. The data in the table reveal several key findings: the maximum absorbed energy of specimens with cracks is significantly lower than that of intact specimens, with the maximum absorbed energy of specimens with pre-existing cracks being about 40–75% of that of intact specimens. This indicates that pre-existing cracks induce substantial energy damage to the specimens. The range of maximum absorbed energy of specimens with cracks varies greatly, with a variation amplitude of about 34%, indicating that the crack angle has a significant effect on the energy required for rock failure. The specimen with the smallest maximum absorbed energy is the one with a crack angle of α = 45°, with a value of 6.45 J, which is only 40.56% of that of the intact specimen. The maximum value is obtained when the crack angle is α = 90°, with a value of 11.84 J, which is 74.4% of that of the intact specimen. The difference in maximum absorbed energy of rock specimens with crack angles of 0°, 45°, and 60° is relatively small, with an amplitude of variation of merely 9%.

Studying the relationship between the maximum absorbed energy (i.e., the total energy required for specimen failure) and other parameters of the rock can provide an important reference for certain rock engineering. Figure 15, Figure 16 and Figure 17 show the relationships between the maximum absorbed energy and the dip angle of pre-existing cracks, tensile strength of the rock, and acoustic emission energy, respectively.

From Figure 15, it can be seen that the dip angle of pre-existing cracks has a significant effect on the maximum absorbed energy of sandstone, with a large difference in absorbed energy among specimens with different crack angles. The maximum absorbed energy of the specimen decreases first and then increases with the increase in crack dip angle, and the variation law is consistent with the relationship between tensile strength and crack angle.

The occurrence of tunnel collapses and other instability phenomena in rock engineering can often be attributed to the initiation, propagation, and penetration of primary defects such as cracks and joints under the influence of loads, ultimately resulting in rock instability and failure. The entire failure process is closely related to the tensile strength and energy required for the failure of the rock. Figure 16 displays a scatter plot and fitted curve between the two, indicating that there is a nonlinear relationship between the tensile strength of the rock and the maximum absorbed energy. The plot demonstrates that as the maximum absorbed energy increases, the tensile strength of the rock also tends to rise.

In the field of rock engineering construction, microseismic monitoring is commonly employed to track the progression of rock mass cracks and assess the stability of rock formations in real time. In this study, indoor acoustic emission simulation tests were also conducted. The acoustic emission energy collected by the acoustic emission system during the test was statistically analyzed with the energy required for specimen failure. Figure 17 illustrates the findings, revealing that the cumulative acoustic emission energy also exhibits a nonlinear relationship with the maximum absorbed energy, mirroring the correlation observed between tensile strength and maximum absorbed energy. The cumulative acoustic emission energy also increases with the increase in the maximum absorbed energy. Moreover, the decreasing slope of the curve indicates that the energy storage loss inside the rock sample will increase with the increase in the maximum absorbed energy.

4. Conclusions

This paper conducted splitting tests on fine sandstone with pre-existing cracks of different dip angles, and investigated the effect of pre-existing cracks on the tensile strength and failure mode of fine sandstone. Based on the digital image correlation method, the displacement and strain evolution laws of different rock failure modes were analyzed. Additionally, the absorbed energy during the deformation and failure process was calculated and investigated. The relationship between maximum absorbed energy and crack dip angle, tensile strength, and acoustic emission energy was also studied, and the following conclusions were drawn. This provides a reference for the design and construction of tunnels or mines in rock formations with pre-existing cracks.

The tensile strength of sandstone specimens with pre-existing cracks is observed to be approximately 60% to 80% of that of intact specimens. The tensile strength first decreases and then increases with the increase in pre-existing crack dip angle, showing a “V” shape change. Specifically, at a crack dip angle of α = 45°, the tensile strength reaches its minimum, corresponding to approximately 60.3% of that of intact specimens. Conversely, at a crack dip angle of α = 90°, the strength is at its maximum, reaching approximately 77.2% of that of intact specimens. The difference in tensile strength is relatively minor when the crack dip angles are α = 30°, 45°, and 60°. Similarly, the failure displacement exhibits a similar pattern of initially decreasing and then increasing with the increase in crack dip angle. When the crack dip angle is α = 45°, the failure displacement is the minimum, approximately 72.85% of that of intact specimens, indicating the highest level of damage caused by the crack in the specimen. On the other hand, at a crack dip angle of α = 90°, the damage caused by the crack is minimal, and the failure displacement is approximately 85% of that of intact specimens.

The failure modes of sandstone specimens with pre-existing cracks can be categorized into three types based on the crack morphology and formation mechanism. These types are as follows: tensile failure along the central point of the disc (Type I), tensile failure along the tip of the pre-existing crack (Type II), and tensile–shear combined failure along the tip of the pre-existing crack (Type III). The failure crack morphology can be classified into three typical cracks: central crack, wing-shaped crack at the crack tip, and central crack at the crack tip. When the dip angle of the pre-existing crack is α = 0°, the failure mode of the fine sandstone specimen is tensile failure along the central point of the disc (Type I) and tensile failure along the tip of the pre-existing crack (Type II), and the tensile crack during failure passes through the center of the disc. When 30° ≤ α ≤ 60°, the failure crack extends along the tip of the pre-existing crack toward the loading end and expands in a wing shape; the failure mode is tensile–shear combined failure along the tip of the pre-existing crack (Type III). When the dip angle of the pre-existing crack is α = 90°, the failure crack of the specimen is parallel to the loading direction, passing through the crack tip; the failure mode is tensile failure parallel to the loading direction (Type I and Type II).

The absorbed energy and vertical displacement ratio of fine sandstone during loading deformation show a nonlinear relationship, and the dip angle of the pre-existing crack strongly affects the evolution curve of absorbed energy. Specifically, when the dip angle of the pre-existing crack is α = 45°, the growth rate of absorbed energy during the loading and failure process is the slowest; the energy evolution curve of specimens with dip angles α = 30° and α = 60° is almost the same; when the crack dip angle is α = 90°, the growth rate of absorbed energy is the fastest, followed by specimens with a dip angle of α = 0°. The presence of pre-existing cracks leads to considerable energy damage in the specimens. Notably, when the crack dip angle is α = 45°, the specimen experiences the smallest maximum absorbed energy of only 6.45 J, representing 40.56% of the intact specimen’s energy absorption. On the other hand, when the dip angle is α = 90°, the specimen exhibits the largest maximum absorbed energy of 11.84 J, which corresponds to 74.4% of the intact specimen’s energy absorption. The difference in maximum absorbed energy between specimens with crack dip angles of 0°, 45°, and 60° is relatively small, with a variation range of only 9%. The maximum absorbed energy of the specimen first decreases and then increases with the increase in crack dip angle, and the nonlinear correlation between the two is good. The maximum absorbed energy of the specimen also shows a corresponding nonlinear relationship with the tensile strength and acoustic emission energy. Moreover, the decrease in the slope of the curve indicates that as the maximum absorbed energy increases, the energy storage and consumption inside the rock sample will increase, and the corresponding increase in tensile strength will decrease.