Stressstrain characteristics of rock samples with different defect sizes. The stressstrain curves of each model under uniaxial compression are shown in Fig. 5(a), and the peak stress and elastic modulus changes are shown in Fig. 5(b). It can be obtained from Fig. 5 that the peak stress and elastic modulus of the intact rock sample’s model are the largest, at 125.05 MPa and 77.11 GPa, respectively. When the defect diameter is 2.5 mm, the effect on the stressstrain curve of the model is not obvious because the defect size is 1/20 of the bottom diameter of the rock sample. However, this is visible as the peak stress and elastic modulus are slightly reduced (by 1.81 MPa and 0.14 GPa, respectively). This indicates that even with relatively small defect sizes, there is still a degrading effect on the mechanical properties of the rock sample. With the increase in the circular defect’s diameter, the peak stress and elastic modulus of the rock samples decrease significantly. The peak stresses at the defect sizes of 1/10, 1/5, 3/10, 2/5, and 1/2 of the bottom diameter of the rock sample are 112.68 MPa, 100.14 MPa, 84.68 MPa, 68.01 MPa, and 52.67 MPa, respectively, and the elastic moduli are 76.04 GPa, 72.80 GPa, 66.68 GPa, 58.45 GPa, and 49.09 GPa, respectively. The UCS and elastic modulus show a gradual downward trend. Compared with the intact model, when the defect size reaches half of the rock sample’s bottom diameter, the UCS and elastic modulus decrease by 57.88% and 36.34%, respectively. It can also be seen in Fig. 5(a) that the peak strain of each model is gradually reduced as the defect size gradually increases. This indicates that the larger the defect size, the progressively lower the amount of deformation the rock samples can withstand, that is, the rock sample is damaged earlier.
The above analysis shows that circular hole defects have a direct effect on the mechanical properties of rock samples. If there are circular hole defects, regardless of the size of the defect, the mechanical properties of the rock sample will deteriorate. As the size of the circular hole increases, the UCS and elastic modulus gradually decrease, and the size of the defect is negatively correlated with the mechanical strength of the rock sample.
Contact force chain and stress field analysis. Based on the UCS of each model, the evolution characteristics of the contact force chain of each model were analyzed at the stress levels of 10%, 50%, and 100% of the peak stress and 60% of the peak stress in the postpeak stage, respectively. The corresponding vertical stress field distribution characteristics in the postpeak stage are also displayed correspondingly (rightmost pictures). The light cyan is the base color of the model area, the red force chains represent tensile stresses, and the black force chains represent compressive stresses. The more dense and darker the force chains in the parallel contact force distribution diagram, the greater the stress concentration. The evolution of the contact force chains and the distribution of the vertical stress field in the postpeak stage of each model are shown in Fig. 6.
In Fig. 6, the contact force chains of the complete model are relatively evenly distributed before the peak strength, with the tensile and compressive force chains interacting. When the peak stress is reached, the contact force chain appears to be in a relatively dispersed state. In the postpeak stage, the contact force chain is relatively loose on the whole, and the main concentrated area of the compressive force chain is formed near the failure position of the rock sample; also, a Vshaped distribution is formed roughly at the center of the bottom surface of the model. The distribution characteristics of the vertical stress field after the peak are consistent with the distribution of the force chain. In the model with a circular hole defect size of 2.5 mm in diameter, a slight concentration of compressive force chains can be seen on both sides of the circular hole before the peak stress, with a more even and dense distribution of tensile force chains in other areas of the model. In the postpeak stage, the concentrated area of the compressive force chain and the vertical stress is mainly in the vertical center of the model. In the defective rock mass models with hole diameters of 5 mm, 10 mm, 15 mm, 20 mm, and 25 mm, before the peak stress, the contact pressure chain concentration areas of each model are similar, being mainly concentrated near the circular hole defect. The compressive force chain is mainly concentrated on the left and right sides of the circular hole, and the tensile force chain is mainly concentrated near the upper and lower sides of the hole. When the peak stress is reached, there is still an obvious compressive force chain concentration phenomenon on the left and right sides of the circular hole in each model, and a compressive force chain distribution trend similar to an "X" shape is formed in the model with the circular hole as the center. The tensile force chain near the side disappears, almost forming a blank area of the force chain. In the postpeak stage, in each model, a diagonal compressive force chain concentration area mainly forms along the circular hole defect, and a sparse force chain area forms in the vertical middle area of the model, which also indicates the main failure mode of the model. The vertical stress field of each model with a circular hole defect diameter of 5 mm25 mm in the postpeak stage is mainly manifested in the form of a lowstress area or a low tensile stress area in the vertical middle of the model. The large stress and vertical stress concentration area are mainly on both sides of the model, and the range of the concentration areas is consistent with the range of the stress chain concentration area.
Comparing the evolution of the force chain of each model shows that the circular hole defect has a significant influence on the evolution of the force chain inside the rock sample. At the 10% peak stress level, with the increase in the defect size of the circular hole, the larger the defect size, the greater the concentration of the force chain near the circular hole, and the larger the range. For example, in the 5 mm diameter circular hole model, the concentration of force chains near the hole is only slightly increased, and the overall force chains in the model are more evenly distributed; however, in the 25 mm diameter circular hole model, the force chains are mainly concentrated near the hole, and the force chains in the other areas of the model are relatively sparse. This phenomenon, which affects the degree of concentration of the force chain in the vicinity of the circular hole, persists up to the peak stress. The degree of concentration and distribution of force chains near the round hole is positively correlated with the size of the circular hole defect. The larger the circular hole defect, the greater the relative stress concentration and the more likely the rock sample will break, which is consistent with the negative correlation between the compressive strength and the defect size in each model. Up to the postpeak stage, the larger the defect size in the middle of the model, the sparser the distribution of force chains above and below the circular hole, and the greater the extent of the lowstress region above and below the circular hole in the middle of the vertical stress field. In general, although the force chain distribution patterns are similar in the models with differentlysized circular hole defects, the defect size controls the concentration and distribution range of the force chain. The concentration and distribution range of the contact force chain is positively correlated with the size of the circular hole defect.
Crack initiation stresses and strains of each model. Tight rock samples will not generate new cracks until they are uniaxially loaded to a certain extent. At this time, the corresponding stress state is called the crack initiation stress, which is also an important characteristic of rock strength37–38. In the PFC2D code, the crack initiation stress can be determined by monitoring through the microcrack method39–40. Then, the influence of the size of the circular defect on the crack initiation stress of the rock sample is judged. The crack initiation stress and crack initiation strain in each model are shown in Fig. 7.
It can be obtained from Fig. 7 that the crack initiation stress and strain of the intact rock sample are about 48.16 MPa and 0.0625%, respectively, and the stress level at the crack initiation point is about 38.52% of the peak stress. The crack initiation stress and strain of the models with the 2.5 mm, 5 mm, and 10 mm diameter circular defects all decrease rapidly, and the decreasing trend is almost linear with the increase in the size of the circular hole defects. This indicates that the presence of the circular hole defects significantly reduces the crack initiation stress in the rock. In the models with the 15 mm and 20 mm diameter circular hole defects, the downward trend of the crack initiation stress and strain becomes slow, and in the model with the 25 mm diameter circular hole defect, there is an accelerated downward trend again. The crack initiation stress value of each model, the percentage of the UCS, and the relative decline rate are shown in Table 2. In general, with the increase in the size of the circular hole defect, the crack initiation stress of the model gradually decreases, and the rock sample is more prone to damage and deformation.
Table 2
Crack initiation stress value, percentage of the UCS, and the relative reduction rate
Circular defect models

Crack initiation stress
(MPa)

Percentage of UCS

Relative reduction rate

Intact model

48.16

38.52%

0%

2.5 mm diameter model

36.21

29.38%

24.81%

5 mm diameter model

25.14

22.31%

47.80%

10 mm diameter model

15.42

15.40%

67.99%

15 mm diameter model

13.05

15.41%

72.90%

20 mm diameter model

12.54

18.43%

73.97%

25 mm diameter model

6.91

13.12%

85.65%

Analysis of crack evolution and failure mode. Also based on UCS of each model, at the stress levels of 30%, 70%, 100% of the peak stress and 60% of the peak stress in postpeak stage, to analyze the crack evolution characteristics of each model and the corresponding postpeak stage displacement field distribution characteristics.The crack evolution and postpeak displacement fields within each model are shown in Fig. 8, where the blue cracks are shear cracks and the red cracks are tensile cracks.
In Fig. 8, at the 30% peak stress level, the intact model shows no crack generation. In the model with the 2.5 mm diameter circular hole, a crack has just been initiated, and the crack initiation position is near the circular hole. In each model containing the 5 mm25 mm diameter circular holes, cracks are generated at the upper and lower vertices of the circular holes, and as the size of the hole increases, the cracks expand outward along the upper and lower vertices of the hole, and the degree of crack penetration is greater. This shows that under a lower loading stress, the larger the defect size, the earlier the crack initiation in each model. At the 70% peak stress level, relatively scattered compression cracks are developed in the intact model. In each model with the circular hole defects, the crack at the upper and lower vertices of the circular hole continues to expand. Except for the model with the 2.5 mm diameter circular hole where new cracks are scattered within the model, there is only a slight concentration of cracks at the circular hole; all the other models with hole defects have cracks mainly concentrated near the hole, except for the vertical cracks developed at the top and bottom vertices of the circular hole which continue to expand by some amount; the main new cracks are distributed on the left and right sides of the circular hole and appear in a roughly Xshaped distribution with the circular hole at the center. At the peak stress, many scattered cracks are developed in the intact model and the model with the 2.5 mm diameter circular hole. In the other models with circular hole defects, the concentrated cracks with the original Xshaped distribution continue to expand and develop, and more shear cracks also appear at this time. In contrast, the tensile cracks at the top and bottom vertices of the circular hole develop more slowly, with only a small amount of extension. At post 60% of the peak stress, each model becomes damaged, the number of cracks in each model reaches the maximum, and the number of shear cracks develops rapidly in the postpeak stage.
The area of through cracks are the main area of model failure, which are consistent with the distribution characteristics of displacement field of each model. The intact model mainly undergoes conical splitting failure, while the model with the 2.5 mm diameter round hole mainly undergoes oblique splitting failure on the upper left and lower right side of the rock sample. In other circular hole diameter models, the area along the diagonal line of the model with the circular hole as the center is the main failure range. With the increase in the defect size of the circular hole, the degree of crack penetration at the diagonal area of the rock sample is higher, and the vertical cracks at the upper and lower vertices of the circular hole gradually penetrate the rock sample. To a certain extent, the number of cracks can represent the degree of fragmentation and the difficulty of failure of the rock sample. Table 3 shows the number of cracks in each model at post 60% of the peak stress. The crack counts show that the model with the 2.5 mm diameter round hole has more cracks than the intact model. However, all the other models have fewer cracks than the intact model, and the total number of cracks and the number of tensile and shear cracks decrease gradually as the size of the hole defect increases.
Table 3
Number of cracks developed in each model
Circular defect models

Total number of cracks

Tensile crack

Shear crack

Intact model (0 mm diameter)

14145

12325

1820

2.5 mm diameter model

16902

14789

2113

5 mm diameter model

12420

10950

1470

10 mm diameter model

10166

9163

1003

15 mm diameter model

9144

8270

874

20 mm diameter model

8507

7745

762

25 mm diameter model

6881

6306

575

The above analysis shows that the size of the circular hole defect influences the entire process of crack evolution in terms of crack initiation, extension, and penetration. This results in significant differences in the development, number, distribution range, and failure path of the cracks in each model. As the size of the circular hole defect increases, cracks at the top and bottom vertices of the circular hole start to develop earlier; in addition, they develop in greater numbers and have a higher degree of penetration after model failure. The total number of cracks developed and the number of tensile and shear cracks in each model in the postpeak stage tend to decrease gradually, indicating that the larger the size of the circular hole defect, the more likely the model is to fail.