4.1 Numerical simulation
In the PFC program, the particles are round, rigid bodies of particular masses. In the simulation process, the particles are not limited by deformation variables and can effectively reflect the mechanism, process, and results of material damage. Therefore, this method has been widely used by researchers to compensate for mechanical problems associated with discontinuous phenomena such as rock bursting.
In this study, PFC2D was used to conduct numerical simulation of uniaxial compression tests on rock samples with symmetrical and asymmetrical dentate discontinuities, and a square model with dimensions of 100 mm ×100 mm was established that included 6468 circular particles of various sizes. The PFC materials are composed of rigid balls or clusters accumulating at the contact point; the contact between particles adopts a parallel bonded model. Before the simulation, the mesoparameters of the complete rock mass test block were calibrated, as listed in Table 4.
Table 4
Partial parameters of PFC material
Minimum particle radius
(mm)

Maximum particle radius
(mm)

Density
(g m− 3)

Effective modulus
(GPa)

Stiffness ratio

Coefficient
of friction

1.0

1.66

1.96

1.4

1.0

0.5

The microscopic parameters of the modified particles were written into the 2D compression test model, and numerical simulation was then conducted through compression testing of the 2D parallel bonded model. The experimental and numerical results of the uniaxial compressive strength of the rock mass specimens with symmetrical dentate discontinuity with various β values when n = 1 and γ = 0° are shown in Fig. 11.
Figure 11 (a)–(d) show the results of the uniaxial compressive strength experiment and numerical simulation of the rock mass specimens with symmetrical dentate discontinuity when the β values were 30°, 45°, 60°, and 75°, respectively. When n and γ remained unchanged, the numerical strength of the rock mass specimens with different β values was about 80% of the experimental strength.
Figure 12 shows the experimental and numerical results of failure characteristics of the rock mass specimens with dentate discontinuity when n = 1, γ = 60°, and β = 75°. The results were essentially consistent in terms of crack initiation, crack propagation, and crack coalescence.
In summary, the test results of the uniaxial compressive strength of the rock mass specimen were consistent with the change trend of the numerical results, as were the failure characteristics. Therefore, numerical simulation is essential for uniaxial compression specimens with dentate discontinuity. According to the test grouping, β = 15° was added to the compression experimental grouping of the original regular dentate discontinuity as its numerical simulation grouping, and β1 = 15° and β1 = 75° were added to that of the original irregular dentate discontinuity as the numerical simulation grouping.
4.2 Influence of γ on the compressive strength of the specimen numerical simulation
Figure 13 shows the influence of various γ values on the numerical simulation results of the uniaxial compressive strength of the rock mass specimens with dentate discontinuity when n, β, and β1 remained unchanged.
Figure 13 (a)–(e) show the numerical results of the uniaxial compressive strength of the rock mass specimens with symmetrical dentate discontinuity; (f)–(i) show those for asymmetrical dentate discontinuity. When n, β, and β1 remained unchanged, the compressive strength of the irregular dentate discontinuity rock mass specimens was maximized at γ = 90°. The uniaxial compressive strength of the specimens assumed γ = 90° as the symmetry axis and increased (decreased) with an increase in γ from 0° to 90° (90° to 180°). These results are consistent with the experimental results.
4.3 Effect of β on the numerical simulation results
Figure 14 shows the influence of various β and β1 values on the numerical simulation results of the uniaxial compressive strength of the rock mass specimens with dentate discontinuity when n and γ remained unchanged.
Figure 14 (a)–(d) show the numerical results of the uniaxial compressive strength of the rock mass specimens with symmetrical dentate discontinuity; (e)–(h) show those for asymmetrical dentate discontinuity. When γ and n remained unchanged, the simulated strength of the rock mass specimens with symmetrical dentate discontinuity decreased gradually as β increases from 15° to 75°, which is consistent with the experimental results of those with symmetrical dentate discontinuity. When γ and n of rock mass with asymmetrical dentate discontinuity remained unchanged, the simulated strength at β1 = 15° was close to that at β1 = 75°, and the simulated strength at β1 = 30° was close to that at β1 = 60°. However, the simulated strength of the rock mass with asymmetrical dentate discontinuity showed no obvious change when β1 increased from 15° to 75°.
Figure 15 shows the influence of various β and β1 values on the numerical simulation results of the peak strain of the rock mass specimens with dentate discontinuity when n and γ remained unchanged.
Figure 15 (a)–(d) show the numerical results of the peak strain of the rock mass specimens with symmetrical dentate discontinuity; (e)–(h) show those for asymmetrical dentate discontinuity. When γ and n remained unchanged, the simulated peak strain of the rock mass specimens with dentate discontinuity decreased gradually as β increased from 15° to 75°, which is consistent with the experimental results of those with symmetrical dentate discontinuity. When γ and n of the rock mass with asymmetrical dentate discontinuity remained unchanged, the simulated peak strain at β1 = 15° was close to that at β1 = 75°, and the simulated peak strain at β1 = 30° was close to that at β1 = 60°. However, the simulated peak strain of the rock mass with asymmetrical dentate discontinuity showed no obvious change when β1 increased from 15° to 75°.
Figure 16 shows the influence of various β and β1 values on the numerical simulation results of the elastic modulus of rock mass specimens with dentate discontinuity when n and γ remained unchanged.
Figure 16 (a)–(d) show the numerical results of the elastic modulus of the rock mass specimens with symmetrical dentate discontinuity; (e)–(h) show those for asymmetrical dentate discontinuity. When γ and n remained unchanged, the simulated elastic modulus of the rock mass specimens with dentate discontinuity decreased gradually as β increases from 15° to 75°, which is consistent with the experimental results of rock mass specimens with symmetrical dentate discontinuity. When γ and n in the rock mass with asymmetrical dentate discontinuity remained unchanged, the simulated elastic modulus at β1 = 15° (30°) was close to that at β1 = 75° (60°). However, the simulated elastic modulus of rock mass with asymmetrical dentate discontinuity showed no obvious change when β1 increased from 15° to 75°.
4.4 Effect of n on the numerical simulation results
Figure 17 shows the influence of various n values on the numerical simulation results of the uniaxial compressive strength of the rock mass specimens with dentate discontinuity when γ, β, and β1 remained the same.
Figure 17 (a)–(e) show the numerical results of the uniaxial compressive strength of the rock mass specimens with symmetrical dentate discontinuity; (f)–(i) show those for asymmetrical dentate discontinuity. When γ, β, and β1 remained unchanged, the simulated strength of the rock mass specimens with dentate discontinuity decreased gradually as n increased from 1 to 4, which is consistent with the experimental results.
Figure 18 shows the influence of various n values on the numerical simulation results of the peak strain of the rock mass specimens with dentate discontinuity when γ, β, and β1 remained the same.
Figure 18 (a)–(e) show the numerical results of peak strain of the rock mass specimens with symmetrical dentate discontinuity; (f)–(i) show those for asymmetrical dentate discontinuity. When γ, β, and β1 remained unchanged, the simulated peak strain of the rock mass specimens with dentate discontinuity decreased gradually as n increased from 1 to 4, which is consistent with the experimental results.
Figure 19 shows the influence of various n values on the numerical simulation results of the elastic modulus of the rock mass specimens with dentate discontinuity when γ, β, and β1 remained the same.
Figure 19 (a)–(e) show the numerical results of the elastic modulus of the rock mass specimens with symmetrical dentate discontinuity; (f)–(i) show those for asymmetrical dentate discontinuity. When γ, β and β1 remained unchanged, the simulated elastic modulus of the rock mass specimens with dentate discontinuity decreased gradually as n increased from 1 to 4, which is consistent with the experimental results.
4.5 Comparative analysis of failure characteristics of test and numerical simulation
To obtain more complete uniaxial compression stress–strain curves, the maximum load was decreased 80% at the end of the loading process in this study compared with that used in the PFC numerical simulation; thus, the failure characteristics in the test were more obvious than those in the simulation. In both the experiments and the numerical simulation, almost all new cracks were initiated at the tip of the prefabricated crack, with a few initiated at the middle of or far from the prefabricated crack. The types of crack propagation include tensile, shear, and tension–shear composite cracking.
Figure 20 shows the influence of various γ values on the failure characteristics of the specimens in the experiments and numerical simulations when n = 2 and β = 75°.
Figure 20 (a)–(i) show failure characteristics of the experiment specimen, and (a1)–(i1) show those observed in numerical simulation. When γ was 0°, 30°, 150°, and 180°, most of the new cracks originated at the tip of the prefabricated crack and propagated at an acute angle from the direction of the hypotenuse of the prefabricated crack. When γ was 90°, most of the new cracks originated at the middle of the prefabricated crack and propagated at a right angle from the direction of the hypotenuse of the prefabricated crack. When γ was 45°, 60°, 120°, and 135°, most of the new cracks began at the tip of the prefabricated crack and propagated a right angle from the direction of the hypotenuse of the prefabricated crack. The failure of the rock mass specimens was mainly tensile crack type, and the coalescence direction of cracks eventually tended to follow the axial direction. When γ was 30°, 45°, 60°, 90°, 120°, 135°, and 150°, several new cracks were initiated at a distance from the prefabricated crack.
Figure 21 shows the influence of various β and β1 values on the failure characteristics of the specimens in the experiments and numerical simulations when n = 4 and γ = 180°.
Figure 21 (a)–(f) show the failure characteristics in the experiment specimens, and (a1)–(f1) show those of the numerical simulation. When β was 30°, 45°, and 60° and β1 was 30° and 60°, the new cracks initiated mainly at the tip of the prefabricated crack. The failure of rock mass specimens was dominated by tensile cracks; the new cracks were distributed evenly and symmetrically. A few shear cracks formed at the prefabricated crack tip owing to the stress concentration at occurring at both ends of the prefabricated crack. When β = 75°, the failure form of the specimens was relatively simple, with only a small number of cracks initiating at the tip of the prefabricated crack.
Figure 22 shows the influence of various n values on the failure characteristics of the specimens in the experiments and numerical simulations when β = 75° and γ = 60°.
Figure 22 (a)–(d) show the failure characteristics of the experiment specimens, and (a1)–(d1) show those of the numerical simulation. When n was 1, 2, and 3, the tensile cracks were initiate mainly at the tip of the prefabricated crack, and the shear cracks began at the edges of the specimens far from the prefabricated crack. The new cracks propagated at a right angle from direction of the hypotenuse of the prefabricated crack. When n = 4, the coalescence of the new cracks was not obvious.