Uniaxial Compression Experiments And Numerical Simulations of Arti ﬁ cial Rock Samples With Regular Dentate Discontinuity

To study the uniaxial compression performance of articial rock samples with symmetrical and asymmetrical regular dentate discontinuities, uniaxial compression tests and Particle Flow Code (PFC) numerical simulation are conducted on cement mortar specimens, and the combined effects of dip angle, undulation angle, and number of undulating structures of cracks on the compressive strength, peak strain, elastic modulus, and crack propagation in the specimens are studied. Among these parameters, undulating structure is dened as a single regular dentate structure in the dentate discontinuity; dip angle is the angle between the bottom line of the undulating shape and the horizontal line; undulating angle is the angle between the bottom line of the undulating structure and the left line of the undulating shape; and the number of undulating structures is the number of undulating structures in the dentate discontinuity. The experimental and numerical simulation results show that when the number of the undulating structures and undulating angles remain unchanged, the uniaxial compressive strength of the specimens peak at a dip angle of 90°. In addition, when the dip and undulating angles remain unchanged, the compressive strength, peak strain, and elastic modulus of the specimens decrease with an increase in the number of undulating structures. Moreover, when the number of undulating structures and the dip angle remain unchanged, the compressive strength, peak strain, and elastic modulus of the specimens decrease with an increase in the undulating angle. Further, almost all of the new cracks in the specimens initiate at the tip of the prefabricated cracks.


Introduction
The numerous intricate joints and cracks present in natural rock reduce the mechanical properties of the rock mass, which creates challenges in geotechnical engineering practices.
Scholars worldwide have conducted extensive research on crack initiation, propagation, and coalescence in fractured rock mass under compression. In particular, related studies based on indoor modeling began at the end of the last century. Brace et al. (1963) conducted compression experiments on a single prefabricated rock mass specimen containing inclined fractures. They found that an increase in loading under no or low con ning pressure caused micro-cracks to appear at the tip of the prefabricated fracture that gradually bent toward the loading direction along the prefabricated crack and nally approached the principal stress direction. Some scholars also veri ed the law of wing crack initiation and propagation through theory and experiments and proposed a wing crack model based on their results (Horii et al. 1985(Horii et al. , 1986Ashby et al. 1986). Wong et al. simulations based on the shear mechanical properties of rock masses containing dentate cracks. As early as 1966, Patton (1966) used plaster to simulate the climbing angle effect of dentate discontinuity. Kodikara and Johnston (1994) studied the shear mechanical properties of regular and irregular dentate discontinuities in rock-concrete joints under constant normal stress and constant normal stiffness and found that the brittle damage of regular jagged joints under normal stiffness is stronger than that of irregular jagged joints. Haber eld and Johnson (1994) detailed the failure mechanism of rock mass specimens with dentate discontinuity during the shearing process and found that joint roughness plays a major role in the shear behavior of rock joints. Seidel JP et al. (1995) conducted a direct shear test on a regular dentate discontinuity and subsequent energy analysis of the discontinuity and explained the relationship among the normal strain, tangential strain, and effective normal stress of the dentate discontinuity during the direct shear process. Yang et al. (2000) used a plaster dentate discontinuity model to conduct a shear test and analyzed the combined in uence of shear direction and dentate protrusion angle on the shear characteristics of the model. Homand et al. (2001) conducted a cyclic direct shear test on granite with prefabricated dentate discontinuity by analyzing the morphological evolution of the dentate discontinuity area of before and after shearing to quantitatively determine that the damaged area of the discontinuity increased with an increase in normal stress under low normal stress cyclic shear.
Park and Song (2009) used the three-dimensional PFC (PFC3D) program to study the shear mechanics characteristics of a rock mass discontinuity affected by the microscopic parameters of the medium including the particle size ratio, particle elastic modulus, and other features. Kwon et al. (2010) used plaster materials with rectangular discontinuity specimens of different undulation heights to conduct shear tests to obtain the relationship between the shear strength of the rough structure of the unit and the shear displacement. Bahaaddini et al. (2013Bahaaddini et al. ( , 2014 used PFC2D to study the shear behavior of rock with dentate discontinuity and analyzed the in uence of joint length on the peak shear behavior of the rock mass. The nal numerical simulation and physical experiment were in good agreement with consistent shear intensity.
Although the literature includes abundant compression tests and numerical simulation studies on crack propagation in fractured rock masses as well as direct shear tests and numerical simulation on rock masses with dentate discontinuity, few researchers have conducted uniaxial compression tests on rock masses with dentate discontinuity. In this study, indoor uniaxial compression tests and PFC numerical simulations are conducted on mortar specimens with symmetrical and asymmetrical regular dentate discontinuities. The in uence of the dip angle, undulation angle, number of undulating structures, and symmetry of the dentate discontinuity on the compressive strength, peak strain, elastic modulus, and crack propagation in the rock mass is evaluated.

Experimental Design
Whereas conventional test-based designs involve numerous specimens and require pre-cracking, cracks are easily developed and propagated in mortar. Both types of fabrication create specimens that are similar in brittleness, homogeneity, and isotropy; thus, the present study adopted cement mortar to create rock-like specimens.

Specimen preparation
This study conducted uniaxial compression testing on rock mass specimens containing symmetrical and asymmetrical dentate discontinuities. The specimens were composed of cement mortar with a water-tocement ratio of 0.65 containing 325-type cement and ISO standard sand with particle sizes of 0.5-1.0 mm. Many of the complete specimens were subjected to uniaxial compression testing, and the relevant mechanical properties of the material were obtained as listed in Table 1. The dimensions of the specimens used in this study were 100 mm × 100 mm × 100 mm. The mortar was poured into the mold, and after su cient vibration, the required plastic plates of various widths and 1 mm thickness were inserted into the bottom of the mold. Then, the plastic plates were removed before the mortar initial was set, and the specimens were cured at room temperature for 24 h before demolding. After the production was completed, the specimens were placed in water for 28 days to cure before being dried for testing. Figures 1 and 2 present schematic diagrams of uniaxial compression for the specimens with symmetrical and asymmetrical regular dentate discontinuities, respectively.
In the gures, n and γ are the number of the rock masses with symmetrical and asymmetrical dentate discontinuities and their dip angle, respectively, and β and β 1 are the undulation angle of the rock mass with symmetrical and asymmetrical regular dentate discontinuity, respectively. Among them, the two sides of each undulating structure in the rock mass with symmetrical regular dentate discontinuity were equal, and the vertex angle of the undulating structure with an asymmetrical regular dentate discontinuity was a right angle. The length of the bottom edge of each undulating structure in the rock mass with both regular dentate discontinuities was 20 mm. Figures 1 shows the case in which γ and β are unchanged, and Fig. 2 shows that in which γ and β 1 are unchanged; n assumes 1, 2, 3, and 4 in both cases.
This test was conducted to determine the in uence of the combined changes in the dip angle, undulation angle, and number of undulations on the compressive strength, peak strain, elastic modulus, and crack propagation in the specimens.

Test grouping
In this test, according to the combined changes of n, β, and γ and those of n, β 1 and γ, the compression tests of the specimens with symmetrical and asymmetrical regular dentate discontinuities were grouped, with 144 and 108 cases in total, as shown in Tables 2 and 3, respectively. When β was 45°, the shape of prefabricated fractures in the symmetrical dentate discontinuity rock mass was exactly the same as that in the asymmetrical dentate discontinuity rock mass when β 1 is 45°.

Experiment procedure
This experiment was conducted using a CSS-44300 microcomputer-controlled electronic universal testing machine. During the experiment, the computer screen directly displayed the curves and the measured values of the test force, deformation, and displacement. The maximum axial pressure was 300 kN. The loading method of this test adopted displacement loading at a rate of 1 mm/min. For each case in the test group, three identical specimens were created; therefore, 648 specimens were fabricated in total.
Test Results Figure 3 shows the uniaxial compression stress-strain curves of the rock mass specimens with dentate discontinuity with various γ values under each selected β and β 1 when n = 1.

Analysis of stress-strain curves
As shown in the gure, the compression failure processes of the rock masses with symmetrical and asymmetrical dentate discontinuities include stages of slowly developing compaction, stably developing linear elasticity, peak, and rapid stress decrease following the peak. When γ and n of the rock mass with symmetrical regular dentate discontinuity remained unchanged and β increased from 30° to 75°, the peak stress for β = 30° was higher than that for β = 75°. However, when γ and n of that with asymmetrical regular dentate discontinuity remained unchanged and β 1 increased from 30° to 60°, the peak stress for β 1 = 30° was close to that for β 1 = 60°.
3.2 In uence of γ on the compressive strength of the specimens Figure 4 shows the in uence of various γ values on the uniaxial compressive strength of the rock mass specimens with dentate discontinuity when n, β, and β 1 remained the same. The compressive strength assumed the peak point of the stress-strain curves. When n, β, and β 1 of the specimens remained unchanged, the compressive strength of the rock mass reached the maximum, at γ = 90°. The uniaxial compressive strength of the rock mass with dentate discontinuity increased (decreased) with γ = 0°-90° (γ = 90°-180°). When γ increased from 0° to 180°, the peak stress showed a symmetrical phenomenon, with γ = 90° as the axis of symmetry.
3.3 In uence of β on the compressive strength, peak strain, and elastic modulus of the specimens Figure 5 shows the in uence of various β values on the uniaxial compressive strength of the rock mass specimens with dentate discontinuity when n and γ remained unchanged.
As shown in the gure, when γ and n of the specimens remained unchanged, the uniaxial compressive strength of the rock mass specimens with dentate discontinuity decreased gradually as β increased from 30° to 75°. Figure 6 shows the in uence of various β values under each selected γ and n on the peak strain of the rock mass specimens with dentate discontinuity. The peak strain assumed the strain corresponding to the peak stress in the stress-strain curves of uniaxial compression.
The gure shows that when γ and n of the specimens remained unchanged, the peak strain of the rock mass specimens with dentate discontinuity decreased gradually as β increased from 30° to 75°. Figure 7 shows the in uence of various β values under each selected γ and n on the elastic modulus of the rock mass specimens with dentate discontinuity. The elastic modulus is the ratio of stress to strain occurring at 50% of the peak stress of the specimens.
The gure shows that when γ and n of the specimens remained unchanged, the elastic modulus of the rock mass specimens with dentate discontinuity decreased gradually as β increased from 30° to 75°.
3.4 In uence of n on the compressive strength, peak strain, and elastic modulus of the specimens Figure 8 shows the in uence of n on the uniaxial compressive strength of the rock mass specimens with dentate discontinuity when γ, β, and β 1 remained the same. show the uniaxial compressive strength test results of the specimens with symmetrical dentate discontinuity; (e) and (f) show those for asymmetrical dentate discontinuity. The uniaxial compressive strength of the specimens with dentate discontinuity gradually decreased with an increase in n from 1 to 4 when γ, β, and β 1 of the specimens remained unchanged. Figure 9 shows the in uence of n on the peak strain of the rock mass specimens with dentate discontinuity when γ, β, and β 1 remained unchanged. Figure 9 (a)-(d) show the peak strain test results of the specimens with symmetrical dentate discontinuity; (e) and (f) show those for asymmetrical dentate discontinuity. The peak strain of the specimens with dentate discontinuity gradually decreased with an increase n from 1 to 4 when γ, β, and β 1 of the specimens remained unchanged. Figure 10 shows the in uence of n on the elastic modulus of the rock mass specimens with dentate discontinuity when γ, β, and β 1 remained unchanged. Figure 10 (a)-(d) show the elastic modulus test results of the specimens with symmetrical dentate discontinuity; (e) and (f) show those for asymmetrical dentate discontinuity. The elastic modulus of the specimens with dentate discontinuity gradually decreased with an increase of n from 1 to 4 when γ, β, and β 1 remained unchanged.

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 re ect 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 meso-parameters of the complete rock mass test block were calibrated, as listed in Table 4. The microscopic parameters of the modi ed 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 β 1 = 15° and β 1 = 75° were added to that of the original irregular dentate discontinuity as the numerical simulation grouping.

4.2
In uence of γ on the compressive strength of the specimen numerical simulation Figure 13 shows the in uence 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 β 1 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 β 1 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. Figure 14 shows the in uence of various β and β 1 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 β 1 = 15° was close to that at β 1 = 75°, and the simulated strength at β 1 = 30° was close to that at β 1 = 60°.

Effect of β on the numerical simulation results
However, the simulated strength of the rock mass with asymmetrical dentate discontinuity showed no obvious change when β 1 increased from 15° to 75°. Figure 15 shows the in uence of various β and β 1 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 β 1 = 15° was close to that at β 1 = 75°, and the simulated peak strain at β 1 = 30° was close to that at β 1 = 60°. However, the simulated peak strain of the rock mass with asymmetrical dentate discontinuity showed no obvious change when β 1 increased from 15° to 75°. Figure 16 shows the in uence of various β and β 1 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 β 1 = 15° (30°) was close to that at β 1 = 75° (60°). However, the simulated elastic modulus of rock mass with asymmetrical dentate discontinuity showed no obvious change when β 1 increased from 15° to 75°. Figure 17 shows the in uence 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 β 1 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 β 1 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 in uence 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 β 1 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 β 1 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 in uence 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 β 1 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 β 1 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.

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.      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.

Conclusion
(1) When the number of the undulating structures and the undulating angles remain unchanged, the uniaxial compressive strength of the rock mass specimens with symmetrical and asymmetrical regular dentate discontinuities peaks at a dip angle of 90°. As the dip angle of the rock mass specimens increases from 0° to 180°, their uniaxial compressive strength rst increases and then decreases, and the uniaxial compressive strength is distributed along an axis of symmetry at 90°.
(2) When the dip angle and undulating angle remain unchanged, the compressive strength, peak strain, and elastic modulus of the rock mass specimens with symmetrical and asymmetrical dentate discontinuities all decrease with an increase in the number of undulating structures.
(3) When the number and dip angle of undulating structures are known, the compressive strength, peak stress, and elastic modulus of the rock mass specimens containing symmetrical dentate discontinuity decrease with an increase in the uctuation angle. The compressive strength of the rock mass with specimens with asymmetrical dentate discontinuity depends more on the smaller angle of 90° for β 1 and β 1 than that with symmetrical dentate discontinuity with the same undulating angle.
(4) Almost all of the new cracks in the rock mass specimens with symmetrical and asymmetrical regular dentate discontinuities are initiated at the tip of the prefabricated crack, with a few beginning at the middle of or far from the prefabricated crack. The failure characteristics of the rock mass are associated mainly with tensile cracks accompanied by a few shear cracks.
(5) When the undulation angle and the number of undulating structures remain unchanged and the dip angle is 45°-135°, the crack in the rock mass with symmetrical dentate discontinuity propagates rst at a right angle from the direction of the hypotenuse of the preformed crack and nally propagates axially.
When the dip angle and the number of undulating structures remains unchanged and the crack propagation of the rock mass is rich when the undulating angle is less than or equal to 60°, the undulation angle and dip angle remain unchanged, and the number of undulating structures is less than four, the crack propagation is abundant, and shear cracks are initiated at the edge of the specimens far from the prefabricated crack. Declarations 6. Acknowledgments