Effect of flaw inclination angle and crack arrest holes on mechanical behavior and failure mechanism of pre-cracked granite under uniaxial compression

PFC^2D can effectively simulate the interactions between material particles as a discrete element method, and its accuracy in simulating crack initiation and propagation in rocks and reflecting rock properties has been verified accordingly [31,32,33,34]. In PFC^2D, the rock model constructed through particle interaction and cementation binds adjacent particles together to form aggregates with elastic and fracture characteristics. Subsequently, through the movement of the wall, the particles move within the aggregate, and the bond between particles breaks, thus simulating the mechanical response of the rock. For the bond model between particles, the main types are the contact bond model and the parallel bond model, as shown in Figure 1. The parallel bond model can be viewed as a set of rectangular springs distributed around the contact points that transmit forces and moments. In contrast, the contact bond model is limited to transmitting forces on the contact surface. Yang et al. [35] simulated the progressive failure behavior of circular opening rock mass excavation by introducing a mechanical micro-parameter corrosion mechanism based on the parallel bond model. Ozturk and Altinpinar [36], Zhang et al. [37], and Huang et al. [38], respectively, used the parallel bond model to simulate uniaxial compression tests on volcanic sedimentary rocks, basalt, and sandstone. Related studies have shown that the parallel bond model is more suitable for simulating rock materials due to its ability to transmit forces and moments [39]. Therefore, the parallel bond model will be used for numerical simulation in this paper.

Figure 1

Schematic diagram of the cohesive model and its micromechanical behavior: (a) contact bond model and (b) parallel bond model.

In uniaxial compression simulation using PFC^2D, a closed-loop wall was established with dimensions consistent with the specimen, and random circular particles of various sizes were generated inside the wall. Adopting a parallel bonding model, the stiffness structure of the simulated specimen is formed through interparticle bonding. Subsequently, both the upper and lower walls move toward the specimen at the same speed to simulate the loading process. After reaching the set value, the loading ceases, and the uniaxial compression simulation ends. The rock material used in the test was G654 granite, sampled from Jinjiang, Fujian. Its density was approximately 2,800 kg/m³, and its length, height, and width were 80, 160, and 20 mm, respectively. Uniaxial compression tests were conducted using the MTS816 testing machine. Combined with the tests, a PFC^2D model with the same width, height and density was constructed, which consisted of 22,370 particles with different sizes and 58,400 contacts. The minimum particle radius was 0.3 mm, and the ratio of maximum to minimum radius was 1.66. Axial loading was controlled by displacement at a constant rate until the specimen failed.

Before simulation, determining the mesoscopic parameters of the parallel bond model is essential to ensure the accuracy of the simulation. Based on the study of Potyondy and Cundall [27], it is assumed that the contact bond modulus is equal to the parallel bond modulus, and so is their respective stiffness ratio. The parameter calibration process is as follows: First, it is necessary to fine-tune the contact bond modulus and parallel bond modulus in order to be consistent with the elastic modulus in the laboratory test. Second, the contact bond stiffness ratio and parallel bond stiffness ratio should be adjusted to correspond to Poisson’s ratio obtained in laboratory tests. Finally, the parallel bond tensile strength and parallel bond cohesion should be modified to simulate the peak strength obtained from laboratory tests, and their ratios should be adjusted to match the failure mode [40]. Through the correlation study of mesoscopic parameters in the parallel bond model [41,42,43] and similarity simulation test [44,45], a set of mesoscopic parameters reflecting granite characteristics can be obtained, as shown in Table 1.

Table 1

Mesoscopic parameters	Value
Density of ball, ρ/(kg m−3)	2,800
Porosity	0.05
Minimum particle radius, R min /mm {R}_{\min }\text{/mm}	0.3
Particle radius ratio, R rat {R}_{\text{rat}}	1.66
Particle friction coefficient, μ \mu	0.5
Contact bond modulus, E c /GPa {E}_{\text{c}}\text{/GPa}	18.0
Contact bond stiffness ratio, k n / k s {k}_{n}/{k}_{\text{s}}	1.8
Parallel bond modulus, E c ¯ /GPa \overline{{E}_{\text{c}}}\text{/GPa}	18.0
Parallel bond stiffness ratio, k n ¯ / k s ¯ \overline{{k}_{n}}/\overline{{k}_{\text{s}}}	1.8
Parallel bond tensile strength, σ n /MPa {\sigma }_{n}\text{/MPa}	48.5
Parallel bond cohesion, τ n /MPa {\tau }_{n}\text{/MPa}	97.0
Parallel bond radius multiplier, β ¯ \overline{\beta }	1.0
Parallel bond friction angle, φ / ( ° ) {\varphi }\text{/}(\text{°})	30

Figure 2 presents the comparison between simulation results and test results. The uniaxial compressive strength obtained by the test and simulation was 191.88 and 190.66 MPa, respectively, with an error of only 0.64%. In addition, the error between the elastic modulus was 0.48%. The error between the peak strain was −21.7%. This phenomenon is attributed to the existence of a compacting stage in laboratory tests during initial loading [46], while numerical simulations form interactive forces through preset particle cementation, where particles are already densely packed, entering the elastic stage directly after loading. During the simulation process, macrocracks were generated through the propagation of microcracks represented by fractures. These microcracks can be divided into two types: tensile cracks and shear cracks, represented by the purple and green line segments in Figure 2, respectively. The test results were in good agreement with the numerical results (Figure 2), indicating that the selected mesoscopic parameters are reasonable and effective.

Figure 2

Comparation between the numerical results and test results.

In order to investigate the effects of different FAs and crack arrest holes on the mechanical properties and crack propagation of pre-cracked granite under uniaxial compression, typical numerical models were designed accordingly. The model with FA α of 45° is shown in Figure 3, with a length of 80 mm, a height of 160 mm, and a diameter of 10 mm for the central circular hole. A straight crack is symmetrically distributed on both sides of the central circular hole, with a length of 10 mm and a width of 1.5 mm (Figure 3). The total length of the combined crack is 30 mm. A pair of crack arrest holes with a diameter of 2 mm are symmetrically arranged at a vertical distance of 5 mm from the tip of the straight crack. In order to distinguish the subsequent crack initiation angle (CA), the flaw inclination angle α is simplified as FA α, and the crack initiation angle is simplified as CA θ. This study was conducted to investigate the effects of different FA α (0°, 30°, 45°, 60°, 90°) and specific crack arrest holes on the mechanical behavior of pre-cracked specimens.

Figure 3

Typical numerical model and its dimensions.

Figure 4 shows the stress–strain curves of the pre-cracked specimen under uniaxial compression. The strain was determined by monitoring the displacements of the upper and lower walls, while the stress was calculated by measuring the contact forces between the walls and particles. Considering that the simulation using PFC^2D is difficult to reflect the initial compaction stage of the pre-cracked specimen, its stress–strain curve gradually transitions from the elastic deformation stage to the nonlinear stage and drops sharply after reaching the peak strength. As depicted in Figure 4, with the increase of FA α, the slope of the stress–strain curve increases, and the peak strength and apparent stiffness E of the specimen are both improved, but still lower than that of the intact specimen, indicating that the FA α has a significant impact on the compressive strength and apparent stiffness E. Compared with the specimen without crack arrest holes, the specimen with specific crack arrest holes exhibits higher peak strain, indicating that the crack arrest hole suppresses the crack propagation of the pre-cracked specimen.

Figure 4

Stress–strain curves of pre-cracked specimens: (a) no crack arrest hole and (b) set crack arrest hole.

In addition, some specimens with crack arrest holes exhibit a stress drop phenomenon on the stress–strain curve, characterized by a brief decrease in stress during loading, followed by a sustained increase to peak stress. This behavior is attributed to the interaction between the central circular hole, crack arrest holes, and straight crack, leading to the formation of stable new crack bands. This effect is particularly evident when crack arrest holes are set on specimens with FA α of 0° and 90°.

In order to further investigate the influence of crack arrest holes on the mechanical properties of pre-crack specimens, the changes in peak strength and apparent stiffness E of pre-crack specimens with crack arrest holes are analyzed, as shown in Figure 5. The results indicate that the presence of crack arrest holes has a minor effect on the apparent stiffness E of specimens, and its influence decreases with the increase of FA α. Specifically, a slight decrease in apparent stiffness E is observed when the FA α is less than or equal to 45°. The apparent stiffness E increases when the FA α is equal to 45°. When the FA α exceeds 45°, this effect becomes almost negligible. The comparison of peak strength reveals that the effect of crack arrest holes on the strength of the specimen varies with the change of FA α. Under the action of crack arrest holes, the strength of the specimen increases when the FA α increases from 0°, 30° to 45°, decreases when it increases from 60° to 75°, and remains almost unchanged when it reaches 90°. This indicates that, combined with the crack arrest mechanism, the reasonable arrangement of crack arrest holes can improve the strength of the specimen, but due to the crack structure of the specimen, the crack arrest holes may also weaken the strength. Therefore, it is of great significance to study the mechanism of crack arrest and design a suitable crack arrest scheme. Especially for underground rock engineering, the improvement of rock mass strength is essential to ensure its stability, and the reasonable arrangement of crack arrest holes is expected to become an effective method.

Figure 5

Strength and deformation of specimens: (a) peak stress and (b) apparent stiffness E.

The above analysis indicates that specimens with different FA α and crack arrest holes exhibit differences in mechanical behavior, including stress–strain curves, peak strength, and apparent stiffness E. These mechanical properties are closely related to the cracking and failure process of fractured rocks [47,48]. In this section, the initiation and failure processes of specimens under different characteristic stresses will be studied by combining PFC^2D numerical calculations. As shown in Figures 6 and 7, point a represents the crack initiation strength, which is defined as the strength corresponding to 1% of the number of cracks at the peak strength [27], point b represents the peak strength, and point c represents the stress point after point b when 70% of the peak strength is reached. The crack number was measured by the PFC fracture program.

Figure 6

The initiation and growth of cracks in specimens without crack arrest holes at different FA α under different characteristic stresses: (a) α = 0°, (b) α = 30°, (c) α = 45°, (d) α = 60°, and (e) α = 90°.

Figure 7

The initiation and growth of cracks in specimens with crack arrest holes at different flaw inclination FA α under different characteristic stresses: (a) α = 0°, (b) α = 30°, (c) α = 45°, (d) α = 60°, and (e) α = 90°.

The crack initiation process shown in Figures 6 and 7 indicates that the crack initiation modes can be divided into three types: (1) initiation at the upper and lower ends of the central circular hole, as seen in the specimen with α = 0°; (2) initiation at the straight crack tip, as depicted in the specimen with α = 30°, 45°, and 60°; and (3) initiation at multiple locations, including the straight crack tip and the surface of the central circular hole, as shown in the specimen with α = 90°. As the FA α increases, the crack initiation mode gradually transforms from tension on the circular hole to tension at the straight crack tip, and finally to compression on the combined crack surface. This change aligns with the findings of Liu et al. [49], thus verifying the reliability of the simulation results in this study. In addition, a comparison between specimens with and without crack arrest holes indicates that they did not significantly change the crack initiation mode in the specimens.

Figures 6 and 7 also illustrate that the failure mode of the specimen varies with the increase of FA α, showing a transition from tensile failure to tensile-shear failure and finally to shear failure. In detail, the specimen exhibited tensile failure when α = 0°, 30°, and 45°, tension-shear failure when α = 60°, and shear failure when α = 90°. Furthermore, statistical analysis of the crack numbers reveals that as the FA α increases, the growth rate of the crack number gradually decreases before the peak stress, but significantly accelerates after the peak stress. It is worth noting that the failure mode of the specimens with α = 30° and 90° changes when affected by the crack arrest hole, which is attributed to the suppression of crack growth and the reduction of stress concentration, promoting crack initiation and propagation on the side without the crack arrest hole. However, the effect of crack arrest holes on the failure modes of other types of specimens is minimal. Consequently, it can be seen that the crack arrest hole mainly affects the crack growth path by reducing the local stress concentration, and its reasonable arrangement can guide the changes in crack propagation and affect the failure mode of the specimen.

The above analysis shows that specimens with different FA α exhibit different failure modes. It is noteworthy that specimens with α = 30° and 90° are significantly affected when the crack arrest holes are arranged near the straight crack tip. The peak strength increases when the FA α = 0°, 30°, and 45°; the failure mode of the specimen does not change when α = 0° and 45°, but changes when α = 30°. The changes in macroscopic mechanical behavior of specimens inevitably conceal underlying mechanisms. Liu et al. [50] pointed out that the failure mode of rock materials is closely related to the internal stress field. Therefore, it is crucial to conduct an in-depth study on the distribution of stress fields in order to explore the failure mechanism of specimens with different FA α and the role of crack arrest holes.

In PFC^2D, local stress is evaluated by calculating the average stress within the measurement circle. This method is based on linking the contact force and parallel bond force applied to each particle within the measurement circle with the force and volume per unit length of the particle boundary in two planes, thereby constructing a stress field. The formula for calculating the average stress is as follows [27]: (1) σ ¯ i j = 1 − n ∑ N P V ( P ) ∑ N P ∑ N C ∣ x i ( C ) − x i ( P ) ∣ n i ( C,P ) F j ( C ) , {\bar{\sigma }}_{ij}=\left(\frac{1-n}{{\sum }_{{N}_{\text{P}}}{V}^{(\text{P})}}\right)\sum _{{N}_{\text{P}}}\sum _{{N}_{\text{C}}}| {x}_{i}^{(\text{C})}-{x}_{i}^{(\text{P})}| {n}_{i}^{(\text{C,P})}{F}_{j}^{(\text{C})}, where N P {N}_{\text{P}} represents particles containing the centroid inside the measuring circle; N C {N}_{\text{C}} represents the total contact between these particles; n n is the porosity within the measurement circle; V ( P ) {V}^{(\text{P})} is the area of the particles; x i ( C ) {x}_{i}^{(\text{C})} and x i ( P ) {x}_{i}^{(\text{P})} represent the positions of the particle’s centroid and contact, respectively; n i ( C,P ) {n}_{i}^{(\text{C,P})} is the unit normal vector from the particle’s centroid to the contact position; and F j ( C ) {F}_{j}^{(\text{C})} represents the force acting on the contact position.

800 measurement circles with a diameter of 4 mm are evenly distributed on the pre-cracked specimen with dimensions of 80 mm × 160 mm, as shown in Figure 8, which also illustrates the basic principle of using these measurement circles to calculate the average stress. Two force chains represent the distribution of contact forces within the measurement circles, with red representing compressive stress and blue representing tensile stress. According to equation (1), the average stress within each measurement circle can be calculated accordingly. The two curves in Figure 8 represent the average stress distribution of 20 measurement circles on line segment AB, which is vertically downward 2 mm from the center of the specimen, and 40 measurement circles on line segment CD, which is horizontally rightward 2 mm from the center of the specimen. Both curves exhibit a high degree of central axis symmetry, verifying the rationality of arranging the measurement circle.

Figure 8

Setting scheme and principle for measurement circles.

Figure 9 shows the stress field distribution of specimens with different FA α under crack initiation strength, which is obtained from the average stress data within the measurement circle and smoothed using contour lines. In the figure, positive values represent compressive stress, and negative values represent tensile stress. For specimens with FA α of 0°, 30°, 45°, and 60°, compressive stress concentration occurs at the straight crack tip, while tensile stress concentration occurs around the combined crack, and its distribution range gradually decreases with the increase of FA α. Due to a large number of subsequent occurrences, the tensile stress concentration zone is simplified to the TS-zone, and the compressive stress concentration area is simplified to the CS-zone. In particular, for the specimen with α = 90°, the CS-zone is mainly located at the left and right ends of the central circular hole, and the TS-zone becomes very small and can be ignored.

Figure 9

Effect of FA α and crack arrest hole on stress field distribution: (a) α = 0°, (b) α = 30°, (c) α = 45°, (d) α = 60°, and (e) α = 90°.

The variation in stress field distribution explains the differences in crack initiation modes between specimens with different FA α. The comparison between Figures 6 and 9 shows that the specimen with α = 0° starts to crack from the upper and lower ends of the central circular hole due to the concentration of tensile stress, which is attributed to the fact that the tensile strength of the rock is much lower than the compressive strength. As the FA α increases, the crack initiation position gradually shifts from the central circular hole to the straight crack tip. For the specimen with α = 90°, due to the relatively minimal impact of the flaw on rock strength, the initiation stress is relatively higher, and the crack initiation behavior occurs around the central circular hole and at the straight crack tip. In addition, the crack arrest hole has little effect on the distribution of stress contour, with no new stress concentration zones, indicating that the existence of crack arrest holes has no significant effect on stress distribution and crack initiation mode (Figure 9).

The tensile and compressive stress values on the surface of the central circular hole and the tip of the straight crack were compared to further analyze the subtle influence of crack arrest holes on stress concentration, as shown in Figure 10. This data was obtained based on the stress field distribution of the specimen by reading the maximum tensile and compressive stresses within the range of the central circular hole surface and straight crack tip. When α = 0°, the maximum tensile stress at the upper and lower ends of the central circular hole is 5.6 MPa. However, as the FA α increases to 30°, 45°, and 60°, the maximum tensile stress decreases to 1.03, 0.75, and 0.41 MPa, respectively. When FA α increases to 90°, the tensile stress at the upper and lower ends of the central circular hole disappears. The maximum tensile stress near the straight crack tip first increases and then decreases, with a maximum value of 8.08 MPa when α = 30°. Moreover, except for the specimen with α = 0°, the maximum tensile stress near the straight crack tip is greater than that at the upper and lower ends of the central circular hole. The maximum compressive stress near the straight crack tip also increases first and then decreases (Figure 10). When α = 90°, the CS-zone shifts from the straight crack tip to both sides of the central circular hole (Figure 9).

Figure 10

The variation law of tensile stress and compressive stress, as well as the influence of crack arrest holes: (a) maximum tensile stress at the central circular hole, (b) maximum compressive stress at the central circular hole, (c) maximum tensile stress near the straight crack tip, and (d) maximum compressive stress near the straight crack tip.

When the crack arrest holes are arranged as shown in Figure 3, except for the central circular hole area of the specimen with α = 90°, the crack arrest hole has a significant weakening effect on the tensile stress. Due to the fact that the crack arrest hole is closer to the TS-zone, the impact on compressive stress is relatively weak, resulting in a smaller weakening of the maximum compressive stress compared to tensile stress, but still beneficial for reducing stress concentration. For the specimen with α = 90°, crack initiation and propagation occur at multiple locations, and the weakening of stress concentration is not significant for the crack arrest holes arranged in this way.

Although crack arrest holes alleviate stress concentration, the magnitude relationship between tensile or compressive stresses of specimens with different FA α remains unchanged. In general, the influence of this type of crack arrest hole on the specimen gradually weakens as the FA α increases. For example, when FA α = 0°, the maximum tensile stress change rate is 14.1%, when FA α = 30°, 45°, and 60°, the maximum tensile stress change rates are 15.7, 12.9, and 7.2%, respectively. In summary, crack arrest holes can effectively alleviate stress concentration, but their impact on stress distribution is not significant.

In 1963, Erdogan and Sih proposed the maximum circumferential stress fracture criterion [51], which states that crack propagation is consistent with the direction of maximum circumferential stress. When the maximum circumferential stress reaches the critical value, the material fails and cracks begin to propagate here, as shown in the following equation: (2) ∂ σ θ θ ∂ θ = 0 ∂ 2 σ θ θ ∂ θ < 0 , \left\{\begin{array}{c}\frac{\partial {\sigma }_{\theta \theta }}{\partial \theta }=0\\ \frac{{\partial }^{2}{\sigma }_{\theta \theta }}{\partial \theta }\lt 0,\end{array}\right. where σ θ θ {\sigma }_{\theta \theta } is the circumferential stress at the crack tip in polar coordinate and θ is the polar angle in polar coordinate.

Under the combination of a central circular hole and straight crack, the determination of the stress intensity factor becomes the key to deriving the circumferential stress formula. However, the existing achievements mainly focus on pure type II closed cracks, and analytical and integral solutions for their stress intensity factors can be obtained [52,53,54]. In view of the non-closed crack in the numerical model, the pure type II formula is not applicable. However, the mechanism of crack initiation and failure can be viewed as the superposition of a central circular hole and a straight crack. For the problem of a central circular hole, it can be regarded as an infinite plate with a central circular hole and subjected to uniaxial compression load, as shown in Figure 11(a). The polar coordinate system can be established at the center of the circle, and the stress field is given by the following equation [55]: (3) σ r r = σ 2 1 − R 2 r 2 − 1 − 4 R 2 r 2 + 3 R 4 r 4 cos 2 θ σ θ θ = σ 2 1 + R 2 r 2 + 1 + 3 R 2 r 4 cos 2 θ σ r θ = σ 2 1 + 2 R 2 r 2 − 3 R 4 r 4 sin 2 θ , \left\{\begin{array}{c}{\sigma }_{rr}=\frac{\sigma }{2}\left[1-\frac{{R}^{2}}{{r}^{2}}-\left(1-\frac{4{R}^{2}}{{r}^{2}}+\frac{3{R}^{4}}{{r}^{4}}\right)\cos \hspace{.25em}2\theta \right]\\ {\sigma }_{\theta \theta }=\frac{\sigma }{2}\left[1+\frac{{R}^{2}}{{r}^{2}}+\left(1+\frac{3{R}^{2}}{{r}^{4}}\right)\cos \hspace{.25em}2\theta \right]\\ {\sigma }_{r\theta }=\frac{\sigma }{2}\left(1+\frac{2{R}^{2}}{{r}^{2}}-\frac{3{R}^{4}}{{r}^{4}}\right)\sin \hspace{.25em}2\theta ,\end{array}\right. where σ r r {\sigma }_{rr} represents the stress in the polar coordinate direction, σ θ θ {\sigma }_{\theta \theta } is the circumferential stress, σ r θ {\sigma }_{r\theta } is the tangential stress, R R is the radius of the circular hole, and r r θ represent the polar radius and polar angle in the polar coordinate system.

Figure 11

Principles of crack initiation and specimen failure: (a) schematic diagram of the specimen with a central circular hole, (b) crack initiation of the specimen with a central circular hole, (c) failure mode of the specimen with a central circular hole, (d) the crack initiation of the composite specimen with α = 0°, and (e) the crack initiation of the composite specimen with α = 90°.

Figure 11(b) shows that the central circular hole does not close under uniaxial compression, indicating that it is a non-closed crack, and the circumferential stress within its radius is zero. Therefore, the polar radius should be greater than the circular radius. Equation (3) shows that when the polar angle remains constant, the circumferential stress decreases with increasing polar radius. Additionally, equation (3) can be changed into (4) σ θ θ = σ [ 1 + 2 cos 2 θ ] . {\sigma }_{\theta \theta }=\sigma {[}1+2\hspace{.25em}\cos \hspace{.25em}2\theta ].

Obviously, σ θ θ {\sigma }_{\theta \theta } has a maximum value of 3 σ 3\sigma when θ = 0 ° \theta =0^\circ or 180 ° 180^\circ . It should also be noted that within the range of 60 ° < θ < 120 ° 60^\circ \lt \theta \lt 120^\circ , σ θ θ {\sigma }_{\theta \theta } is negative, indicating that the central circular hole is subjected to tensile stress and the tensile stress reaches its peak value of − σ -\sigma when θ = 90 ° \theta =90^\circ . Therefore, both compressive and tensile stresses should be considered comprehensively when analyzing the problem of the central circular hole. Figure 11(b) and (c) visually display the crack initiation form and failure mode of the central circular hole, which first cracks at the upper and lower ends of the hole under tensile stress. Subsequently, the crack transferred to the left and right ends of the central circular hole and extended toward the upper and lower ends of the specimen, ultimately leading to its failure. For the combination of the circular hole and the straight crack, according to the average stress distribution in Figure 9 and the crack initiation modes at FA α = 0° and 90° in Figure 11(d) and (e), it can be observed that when FA α = 0°, tensile stress concentration and crack initiation occur at the top and bottom of the circular hole. When FA α = 90°, the microcracks mainly initiates around the circular hole. The results confirmed the applicability of the crack initiation mechanism to specimens containing a central circular hole.

For a straight crack, the stress field at the crack tip in the polar coordinate system is as follows [56]: (5) σ r r = 1 2 2 π r K I ( 3 − cos θ ) cos θ 2 + K Π ( 3 cos θ − 1 ) sin θ 2 σ θ θ = 1 2 2 π r cos θ 2 [ K I ( 1 + cos θ ) − K Π sin θ ] σ r θ = 1 2 2 π r cos θ 2 [ K I sin θ + K Π ( 3 cos θ − 1 ) ] , \left\{\begin{array}{c}{\sigma }_{rr}=\frac{1}{2\sqrt{2\text{&#x03C0;}r}}\left[{K}_{{\rm I}}(3-\hspace{.25em}\cos \hspace{.25em}\theta )\cos \hspace{.25em}\frac{\theta }{2}+{K}_{\Pi }(3\hspace{.25em}\cos \hspace{.25em}\theta -1)\sin \hspace{.25em}\frac{\theta }{2}\right]\\ {\sigma }_{\theta \theta }=\frac{1}{2\sqrt{2\text{&#x03C0;}r}}\hspace{.25em}\cos \hspace{.25em}\frac{\theta }{2}{[}{K}_{{\rm I}}(1+\hspace{.25em}\cos \hspace{.25em}\theta )-{K}_{\Pi }\hspace{.25em}\sin \hspace{.25em}\theta ]\\ {\sigma }_{r\theta }=\frac{1}{2\sqrt{2\text{&#x03C0;}r}}\hspace{.25em}\cos \hspace{.25em}\frac{\theta }{2}{[}{K}_{{\rm I}}\hspace{.25em}\sin \hspace{.25em}\theta +{K}_{\Pi }(3\hspace{.25em}\cos \hspace{.25em}\theta -1)],\end{array}\right. where K _I is the mode I crack (opening mode) stress intensity factors and K _II is the mode II crack (sliding mode) stress intensity factors.

By substituting equation (5) into equation (2), the following can be obtained: (6) cos θ 2 [ K I sin θ + K Π ( 3 cos θ − 1 ) ] = 0 . \cos \hspace{.25em}\frac{\theta }{2}{[}{K}_{{\rm I}}\hspace{.25em}\sin \hspace{.25em}\theta +{K}_{\Pi }(3\hspace{.25em}\cos \hspace{.25em}\theta -1)]=0.

One solution of equation (6) is cos ( θ / 2 ) = 0 \cos (\theta /2)=0 , however, the minimum solution σ θ θ = 0 {\sigma }_{\theta \theta }=0 does not satisfy the condition of obtaining the maximum value in equation (2). Therefore, the crack initiation angle θ depends on (7) K I sin θ + K II ( 3 cos θ − 1 ) = 0 . {K}_{\text{I}}\hspace{.25em}\sin \hspace{.25em}\theta +{K}_{\text{II}}(3\hspace{.25em}\cos \hspace{.25em}\theta -1)=0.

When both K I {K}_{{\rm I}} and K Π {K}_{\Pi } are non-zero, the solution to equation (7) is as follows: (8) θ = 2 arctan 1 ± 1 + 8 ( K Π / K I ) 2 4 ( K Π / K I ) . \theta =2\arctan \left[\frac{1\pm \sqrt{1+8{({K}_{\Pi }/{K}_{{\rm I}})}^{2}}}{4({K}_{\Pi }/{K}_{{\rm I}})}\right].

The stress intensity factor for non-closed straight crack is as follows [56]: (9) K I = σ π a cos 2 α , {K}_{{\rm I}}=\sigma \sqrt{\text{&#x03C0;}a}{\cos }^{2}\alpha , (10) K Π = σ π a sin α cos α , {K}_{\Pi }=\sigma \sqrt{\text{&#x03C0;}a}\hspace{.25em}\sin \hspace{.25em}\alpha \hspace{.25em}\cos \hspace{.25em}\alpha , where σ \sigma is the uniaxial compressive strength, a a is the length of the straight crack, and α is the flaw inclination angle.

By substituting equations (9) and (10) into equation (8), the relationship between CA θ and FA α can be obtained: (11) θ = 2 arctan 1 ± 1 + 8 tan 2 α 4 tan α . \theta =2\arctan \frac{1\pm \sqrt{1+8{\tan }^{2}\alpha }}{4\hspace{.25em}\tan \hspace{.25em}\alpha }.

When FA α \alpha = 90°, it is a pure type II crack. CA θ can be obtained from equation (7), which is always 70.5° (12) θ = arccos 1 3 = 70.5 ° . \theta =\arccos \frac{1}{3}=70.5^\circ .

When FA α \alpha = 0°, it represents a pure type I crack, where CA θ = 180°, and the crack exhibits self-adaptive propagation.

From equation (11), it can be concluded that when FA α = 30°, 45°, and 60°, CA θ are 103° or −43°, 90° or −53°, and 81° or −60°, respectively. According to the conditions of equation (2), the maximum circumferential stress should be taken, and stresses with negative CA θ should be excluded. However, similar to the circumferential stress distribution around a circular hole, the positive CA θ of a straight crack corresponds to the maximum tensile stress in the tensile concentration zone, while the negative CA θ corresponds to the maximum compressive stress in the compressive concentration zone. Although the crack initiation is due to tensile stress, the final failure mode of the specimen is usually related to the CS-zone. Therefore, the arrangement of crack arrest holes should take into account both crack initiation and failure modes. In this study, the negative CA θ was retained, which corresponds to the CS-zone of the specimens with FA α = 30°, 45°, and 60° in Figure 9, while the positive CA θ corresponds to the tensile concentration area. Figure 12 illustrates a strong correlation between the crack initiation pattern of pure straight crack and the CA θ predicted by the maximum circumferential stress criterion, indicating that this criterion is applicable to the current analysis.

Figure 12

Influence of FA α on CA θ: (a) α = 30°, (b) α = 45°, and (c) α = 60°.

According to the analysis presented, for combined fractures, both the macroscopic crack initiation forms and failure modes can be reflected in the CS-zone and TS-zone. Therefore, the circumferential stress of combined fractures is closely related to the stress field distribution. Figure 13 illustrates the relationship between the stress concentration zones of specimens with different FA α and CA θ. It is evident from the figure that CA θ calculated based on the maximum circumferential stress criterion corresponds to the tensile and compressive stress concentration zones depicted in the stress cloud map. For specimens with FA α = 0°, cracks initiate from the upper and lower ends of the circular hole, and the crack develops at the tip of the straight crack. The TS-zone aligns with the crack initiation angle of the maximum circumferential tensile stress analyzed in the criterion, while the CS-zone corresponds to the CA θ when the straight crack is at 0°. For specimens with FA α = 0°, 45°, and 60°, the tensile and compressive stress concentration zones are located on both sides of the straight crack tip, demonstrating good consistency with CA θ derived from the maximum circumferential stress criterion. In specimens with FA α = 90°, there is almost no TS-zone, and failure occurs under compression at the left and right ends of the circular hole, corresponding to the CA θ of the maximum circumferential compressive stress of the circular hole.

Figure 13

Influence of stress field distribution on CA θ of specimens with different FA α: (a) α = 0°, (b) α = 30°, (c) α = 45°, (d) α = 60°, and (e) α = 90°.

Based on the theoretical analysis previously discussed in Section 4.1, the location of the concentrated zones of tensile and compressive stresses for the central circular hole and straight crack can be determined. Further determination of the effectiveness of crack arrest hole arrangement is required in two stress concentration zones or a single zone. In this study, taking the specimen with a central circular hole and the specimen with a straight crack (α = 45°) as examples, and considering that the distance between the center of the crack arrest hole and the straight crack is 5 mm, the position of the crack arrest hole is explored. The influence of three different arrangements of crack arrest holes on the peak strength of the specimen is depicted in Figure 14: (1) the crack arrest holes are arranged in the TS-zone (see specimen (1) in Figure 14(a) and (b)); (2) the crack arrest holes are arranged in the compressive stress concentrated zone (see specimen (2) in Figure 14(a) and (b)); and (3) the crack arrest holes are arranged in the concentrated zones of tensile and compressive stress (see specimen (3) in Figure 14(a) and (b)). The results showed that the crack arrest hole was arranged in the TS-zone of the specimen with a circular hole, and its peak strength was 177.54 MPa, significantly higher than the corresponding value of 146.33 MPa for the specimen without a crack arrest hole. For the specimen with a straight crack (FA α = 45°), if crack arrest holes are arranged in both the t CS-zone and TS-zone, the peak strength of the specimen can reach 157.74 MPa, significantly higher than the corresponding value of 142.16 MPa for the specimen without crack arrest holes. It can be seen that the reasonable arrangement of crack arrest holes has significantly improved the peak strength of the specimens.

Figure 14

The variation law of stress–strain curves under different arrangements of crack arrest holes: (a) specimen with a central circular hole and (b) specimen with a straight crack (α = 45°).

For the specimen with only a central circular hole, the crack stop hole is best arranged in the TS-zone e, while for the specimen with only a straight crack, it is more suitable to arrange crack arrest holes in the TS-zone and CS-zone. For the specimen with composite cracks, the circumferential stress between the circular hole and the straight crack affects each other. According to equations (3) and (5), for the circumferential stress around the central circular hole and the straight crack, the distance from the center of the circular hole to the straight crack tip is 15 mm. The influence of the polar radius on the circumferential stress of the central circular hole (approximately square and fourth power) is significant, while its effect on the straight crack is relatively insignificant (approximately first power). Therefore, the central circular hole is susceptible to the influence of the straight crack, leading to a higher likelihood of crack initiation, which is consistent with the crack initiation mode and stress field distribution in rock specimens. In addition, the peak strength increases when crack arrest holes are arranged in the TS-zone and CS-zone around the central circular hole. For composite cracks, it is most reasonable to arrange crack arrest holes in both TS-zone and CS-zone simultaneously.

Therefore, the optimization method for crack arrest hole arrangement has been proposed. First, according to the crack initiation mode of the specimen, whether the crack initiation is dominated by a circular hole or a straight crack. Then, the maximum circumferential stress criterion is used to analyze and calculate CA θ, and the crack arrest holes are set in the direction indicated by CA θ. Specifically, for the specimen of FA α = 0°, crack arrest holes are arranged in the direction of CA θ = 0° at the upper and lower ends of the circular hole and straight crack tip. For the specimen of FA α = 30°, crack arrest holes are arranged in the directions of CA θ = −43° and CA θ = 103° at the straight crack tip. For the specimen of FA α = 45°, crack arrest holes are arranged in the directions of CA θ = −53° and CA θ = 90° at the straight crack tip. For the specimen of FA α = 60°, crack arrest holes are arranged in the directions of CA θ = −60° and CA θ = 81° at the straight crack tip. For the specimen of FA α = 90°, crack arrest holes are arranged in the direction of CA θ = 0° at both ends of the circular hole.

Figure 15 shows the stress–strain curves of composite cracks subjected to three different crack arrest hole settings. From the figure, it can be seen that compared to a single straight crack or pure circular hole crack, the stress–strain curve of the combined crack with crack arrest hole exhibits a fluctuating trend compared to the strength of the original specimen. This fluctuation suggests that the straight crack and circular hole crack in the combined crack interact, resulting in a more complex stress distribution inside the specimen. Generally, with the exception of the specimen with FA α = 90°, which did not have a TS-zone, the best effect was achieved when two crack arrest holes were set in both TS-zone and CS-zone. This further substantiates the rationality and effectiveness of this crack arrest hole optimization method.

Figure 15

Stress–strain curves under three setting methods of crack arrest holes.

Specifically, when FA α = 0°, tensile stress in the specimen concentrates at the upper and lower ends of the circular hole, while compressive stress concentrates at the tips of the straight crack. After individual placement of crack arrest holes in these stress concentration zones, the peak strengths obtained were 113.45 and 107.72 MPa, respectively, both slightly lower than the peak strength of 114.04 MPa for the original specimen. However, when crack arrest holes were placed in both stress concentration zones simultaneously, the peak strength increased to 130.18 MPa, representing a 14.16% increase compared to the original.

For the specimen with FA α = 30°, tensile stress in the specimen concentrates at CA θ of 103° from the tip of the straight crack, while compressive stress concentrates at CA θ of −43°. Placing a crack arrest hole in the TS-zone, the peak strength obtained was 128.76 MPa, while in the CS-zone, it was 119.44 MPa. When crack arrest holes were placed in both zones, the peak strength was 129.98 MPa, showing limited improvement compared to the original specimen’s peak strength of 126.4 MPa. This is attributed to the TS-zone being in a transitional stage from concentrating around the circular hole to the tip of the straight crack, leading to a broad t CS-zone. Consequently, crack arrest holes did not effectively mitigate the tensile stress concentration, and the crack arrest effect is limited.

When FA α is 45°, tensile stress in the specimen concentrates at CA θ of 90° from the tip of the straight crack, while compressive stress concentrates at CA θ of −53°. By placement of a crack arrest hole in the TS-zone, the peak strength obtained was 126.04 MPa; in the CS-zone, it was 130.65 MPa. When crack arrest holes were placed in both zones, the peak strength increased to 146.30 MPa, representing a 14.12% increase compared to the original specimen’s peak strength of 128.19 MPa.

At FA α = 60°, tensile stress in the specimen concentrates at 81° from the tip of the straight crack, while compressive stress concentrates at −60°. By placing a crack arrest hole in the TS-zone, the peak strength obtained was 154.82 MPa; in the CS-zone, it was 157.09 MPa. When crack arrest holes were placed in both zones, the peak strength increased to 162.12 MPa, representing a 14.11% increase compared to the original specimen’s peak strength of 143.3 MPa.

When FA α = 90°, the TS-zone in the specimen virtually disappears, and compressive stress focuses on the left and right ends of the circular hole crack. By placing a crack arrest hole in the CS-zone, the peak strength obtained was 176.93 MPa, showing a modest increase compared to the original specimen’s peak strength of 172.01 MPa. This is because, at FA α = 90°, the crack initiation mode involves the formation of a crack band due to compression around the circular hole. The placement of crack arrest holes does not completely arrest the crack band, resulting in limited strength enhancement.

Furthermore, this study reveals that applying this method to set crack-arrest holes has the potential to become a rock mass reinforcement technique. Compared with traditional reinforcement methods, crack arrest holes demonstrate unique advantages in mechanical mechanisms, cost-effectiveness, and application scenarios. Through literature analysis [57,58,59], grouting reinforcement, a conventional method, shows lower strength enhancement rates when using gypsum fillings compared to crack arrest holes. Although cement and chemical resin materials provide higher strength improvement rates than crack arrest holes, they change the permeability of rock masses or cause chemical contamination, affecting intrinsic rock properties. Existing grouting studies predominantly focus on medium and low strength rock masses (30–57 MPa), whereas our simulations are based on granite with 190 MPa strength, indicating the method’s unique applicability potential in high-stress geological environments (deep mines, tunnels with strong tectonic stresses). Another traditional method, anchoring support, suffers from prestress attenuation under dynamic disturbances and construction limitations in jointed rock masses. In contrast, crack arrest holes mitigate stress concentration within rock masses and enhance specimen strength and anti-disturbance capacity, combining construction efficiency with economic benefits, making them suitable for rapid reinforcement demands in complex geological conditions.