Effect of Loading Conditions on the Shear Behaviors of Rock-like Materials Containing Circular Holes, with the CZM Method

Abstract

The mechanical behaviors of rock masses are significantly affected by the distribution and shape of the holes in it. In this research, the fracture mechanism and the shear properties of rock masses containing holes were investigated by the cohesive zone model (CZM) method. At first, a uniaxial compression laboratory test was carried out, and some mechanical parameters were obtained, and then the obtained mechanical parameters were used to build a mathematical model with zero-thickness elements. Subsequently, the numerical shear test was performed under a mixed-mode test, using the obtained mechanical parameters. Finally, the mechanical characteristics and crack behaviors were discussed separately. The results exhibit that the shear process in this research could be determined and identified as the elastic, strengthening, plastic, and residual stress stages, respectively. Note that the shear rate and normal stress of rock masses containing holes are significantly and positively correlated with their shear strength and dilatancy. In addition, the cracking behavior and mechanical properties of the specimens are closely related to the shear rate and normal stress. Overall, the results of this study have great significance in guiding future, in-depth research of rock masses containing holes in rock slopes.

1. Introduction

Many macro and micro-cracks, holes, and other defects are included in rock masses as a natural, heterogeneous, engineering geological material [1]. The rock mass defects that affect the mechanical properties of rock masses mainly include the shape, size, and geometric distribution. The crack initiation, propagation, and coalescence are greatly affected by defects in the rock masses. Therefore, deeply researching the effects of these defects on rock crack mechanisms and evolution laws has always been one of the focuses of rock mechanics.

In researching the mechanical properties of cracked rocks, people study the strength, deformation characteristics, and the cracking evolution law of rocks with defects, by laboratory tests. Fractured rock materials mainly contain the following two defects: cracks and holes [2]. Rock-like materials or real rocks, which include crack defects, have been extensively studied by many people [3,4,5,6,7,8,9,10,11], who have usually focused on the mechanical behavior and cracking mechanisms. In terms of the mechanical properties of rock masses with holes, the crack propagation mechanism of the block with prefabricated holes, under uniaxial compression was observed by Li et al. [12] and Zhu et al. [13]. They studied the mechanical properties of sandstone specimens with two holes, under uniaxial compression. Wong et al. [14] conducted a series of physical and numerical experiments on a heterogeneous solid, containing a single hole and proposed the crack coalescence mechanism of heterogeneous solids with a single hole, under uniaxial compression. Zhao et al. [15] improved the cracks of brittle rocks, containing pre-existing cylindrical cavities by a granite model and AE (acoustic emission) technology and showed that the effects of tensile stress and compressive stress on the fracture evolution around the cavity in brittle rocks are relatively significant. Yin et al. [16] investigated sandstone specimens’ mechanical properties, damage modes, and cracking characteristics with a single hole, under high-temperature conditions.

The crack propagation mechanism inside the rock masses plays a crucial role in the rock mass failure, so studying the change of the stress–strain field in the rock masses from the crack generation to the propagation of the rock mass is important. However, it cannot be performed clearly by experimental observation alone. Moreover, the influence of these mineral contents and the spatial distribution produced by the geophysical process in crust formation was inevitable during the experiment. Furthermore, any small change during the experiment, including the placement of the specimen, the contact condition between the sample and loading platens, and the application of the load, may lead to differences in the test results. In such circumstances, compared with the laboratory test, the numerical model can maintain the same loading conditions and has great advantages in eliminating the individual differences of specimens and human operation errors. Thus, numerical methods, including the discrete element method (DEM) [17,18], the boundary element method (BEM) [19,20], the extended finite element method (XFEM) [21,22], and the finite element method (FEM), have been widely used to replicate the rock-breaking process. It indicated that FEM is, perhaps, the most diffusely applied numerical method in conducted rock mechanics problems. Furthermore, the finite element method is more advantageous for heterogeneous materials and nonlinearity applications.

However, in the numerical simulation studies of rock masses with holes, not many studies have been conducted on the fracture mechanism of shearing. Most of the previous numerical studies focused on uniaxial compression, biaxial compression, or triaxial compression; however, high-rock-slope engineering disasters are often induced by shear failure (Figure 1). In addition, compared with the DEM, the FEM can obtain good approximate simulations for arbitrary complex structures by the subdivision element method, and the cohesion model is more advantageous in simulating multi-fracture development. In a study by Chang et al. [23], the complex crack behavior of layered discs with interfacial cracks was simulated by the cohesive zone model (CZM) method to establish a numerical model and to verify the correctness of the simulation results obtained through Brazilian experiments. To obtain the fast crack propagation characteristics by CZM, Valoroso et al. [24] modified the critical energy release rate to convert the effect of micro-cracks to the macroscopic level and to take into account the velocity sensitivity. The numerical results show that the study reduces the discrepancy between the experiment and the simulation and can better agree with the experimental data. In addition, the CZM method can clarify the continuity and discontinuity of rock, however, it has not been widely used in the study of intermittent joint behavior. It is necessary to explore and verify the scope of its application. Therefore, the FEM–CZM method that was developed in this study was proposed to reveal the mechanical and cracking mechanisms of rock masses with two holes. The results of this study have great significance in guiding further in-depth research of rock masses containing holes, in rock slopes.

2. Materials and Methods

2.1. Traction Separation Criterion

Generally, the cohesive element model follows the traction force separation criterion. This research developed the tangential and normal traction into a mixed-mode test to meet the model’s accuracy. The mixed-mode traction separation criterion of the cohesive element is plotted in Figure 2, which mainly includes the elastic and the damage evolution. An elastic constitutive matrix, applied in the elastic stage, can be utilized to describe the nominal stresses to the nominal strains through the interface. When the external force on the cohesive element does not reach the initial damage stress, the cohesive element is at the elastic stage, and the corresponding stress and separation vectors can be expressed as follows:

$$\mathrm{t}=\left\{\begin{array}{c}{\mathrm{t}}_{\mathrm{n}}\\ {\mathrm{t}}_{\mathrm{s}}\\ {\mathrm{t}}_{\mathrm{t}}\end{array}\right\}=\left[\begin{array}{ccc}{\mathrm{E}}_{\mathrm{nn}}& {\mathrm{E}}_{\mathrm{ns}}& {\mathrm{E}}_{\mathrm{nt}}\\ {\mathrm{E}}_{\mathrm{sn}}& {\mathrm{E}}_{\mathrm{ss}}& {\mathrm{E}}_{\mathrm{st}}\\ {\mathrm{E}}_{\mathrm{nt}}& {\mathrm{E}}_{\mathrm{st}}& {\mathrm{E}}_{\mathrm{tt}}\end{array}\right]\left\{\begin{array}{c}{\mathsf{\epsilon }}_{\mathrm{n}}\\ {\mathsf{\epsilon }}_{\mathrm{s}}\\ {\mathsf{\epsilon }}_{\mathrm{t}}\end{array}\right\}=\frac{1}{\mathrm{L}}\left[\begin{array}{ccc}{\mathrm{E}}_{\mathrm{nn}}& {\mathrm{E}}_{\mathrm{ns}}& {\mathrm{E}}_{\mathrm{nt}}\\ {\mathrm{E}}_{\mathrm{ns}}& {\mathrm{E}}_{\mathrm{ss}}& {\mathrm{E}}_{\mathrm{st}}\\ {\mathrm{E}}_{\mathrm{nt}}& {\mathrm{E}}_{\mathrm{st}}& {\mathrm{E}}_{\mathrm{tt}}\end{array}\right]\left\{\begin{array}{c}{\mathsf{\delta }}_{\mathrm{n}}\\ {\mathsf{\delta }}_{\mathrm{s}}\\ {\mathsf{\delta }}_{\mathrm{t}}\end{array}\right\}=\frac{1}{\mathrm{L}}\mathrm{K}\mathsf{\delta }$$

(1)

where t is the nominal traction stress vector, it has three components, including the normal traction (t_n) and two tangential tractions (t_s and t_t). ε_n, ε_s and ε_t are the three components of the nominal strain, and L is the cohesive element’s initial thickness. δ_n, δ_s and δ_t are the corresponding displacement components for the three components, and K is the stiffness matrix for the cohesive element.

Generally, the stiffness of the cohesive element will decrease linearly after reaching the damage evolution. Scalar D indicates the process of damage. The value of the damage in scalar D is increased from 0 to 1. The function of the effective displacement (δ_m) is introduced to describe the damage evolution of the cohesive element, as follows:

$${\mathsf{\delta }}_{\mathrm{m}}=\sqrt{\langle {\mathsf{\delta }}_{\mathrm{n}}\rangle {}^2+{\mathsf{\delta }}_{\mathrm{s}}^2+{\mathsf{\delta }}_{\mathrm{t}}^2}$$

(2)

When δ_n ≥ 0 indicates tension, <δ_n> is equal to δ_n. In addition, when δ_n ≤ 0 indicates compression, <δ_n> equals zero.

Furthermore, the damage variable (D) evolves according to the following:

$$\mathrm{D}=\frac{{\mathsf{\delta }}_{\mathrm{mf}}\left({\mathsf{\delta }}_{\mathrm{mm}}-{\mathsf{\delta }}_{\mathrm{mo}}\right)}{{\mathsf{\delta }}_{\mathrm{mm}}\left({\mathsf{\delta }}_{\mathrm{mf}}-{\mathsf{\delta }}_{\mathrm{mo}}\right)}$$

(3)

where δ_mf is the effective displacement when the tractions diminish, δ_mm is the maximum pure displacement during the loading process. δ_mo is the effective displacement when damage initiates. The normal stress components of the traction–separation model change as follows, due to the damage accumulation:

$${\mathrm{t}}_{\mathrm{n}}=\{\begin{array}{l}\left(1-\mathrm{D}\right){\mathrm{t}}_{\mathrm{n}0},{\mathrm{t}}_{\mathrm{n}0}\ge 0\\ {\mathrm{t}}_{\mathrm{n}0},{\mathrm{t}}_{\mathrm{n}0}<0\end{array}$$

(4)

$${\mathrm{t}}_{\mathrm{s}}=\left(1-\mathrm{D}\right){\mathrm{t}}_{\mathrm{s}0}$$

(5)

$${\mathrm{t}}_{\mathrm{t}}=\left(1-\mathrm{D}\right){\mathrm{t}}_{\mathrm{t}0}$$

(6)

Similarly, the stiffness of the cohesive element can be described as follows:

$${\mathrm{K}}_{\mathrm{n}}=\left(1-\mathrm{D}\right){\mathrm{K}}_{\mathrm{n}0}$$

(7)

$${\mathrm{K}}_{\mathrm{s}}=\left(1-\mathrm{D}\right){\mathrm{K}}_{\mathrm{s}0}$$

(8)

$${\mathrm{K}}_{\mathrm{t}}=\left(1-\mathrm{D}\right){\mathrm{K}}_{\mathrm{t}0}$$

(9)

2.2. The Inserting Process of Zero-Thickness Cohesive Elements

In this paper, the numerical method embedded the zero-thickness cohesive element on the boundary surface of each solid element and realized the process of rock fracture initiation, propagation, and coalescence through the destruction of the zero-thickness cohesive element. The initial finite element mesh needs to be processed to achieve this process. Figure 3 is a schematic diagram of the finite element mesh and the crack elements before and after the zero-thickness cohesive element was embedded. It should be noted that the thickness of the inserted zero-thickness cohesive element was zero, geometrically. To indicate the position of the cohesive element, an element with a certain thickness is shown in Figure 3.

The process of generating zero-thickness elements is to discretize the solid elements first, to read the total information of elements and nodes into the program (Figure 3a), and to make each element be composed of nodes that are not shared with others, by rearranging the element nodes. Next, the cohesive element nodes are sorted. Each element and its neighboring elements are searched for two overlapping element faces, through coordinate matching, and the nodes on the two faces are sorted by requirements of LS-DYNA, which are requirements for the arrangement order of nodes of the zero-thickness cohesive element. For the six-node zero-thickness cohesive element, the first three of the six nodes must belong to one surface, and the last three must belong to the other corresponding surface. The sequence of the first three nodes is consistent according to the right-hand spiral rule, as shown in Figure 3b. Finally, the six nodes on the coincident surface are output in the order of arrangement and a new element number and element type are assigned, thus completing the insertion of a zero-thickness cohesive element, as shown in Figure 3c.

This paper adopted the linear elasticity criterion in the solid element and the traction-separation criterion was adopted for the cohesive force element. In the model building, solid elements were connected in a series through cohesive elements. Moreover, under the action of the external load, solid elements and cohesive elements were used to bear the deformation in the elastic stage. The deformations in the plastic damage stage were determined by the cohesive element. The interaction between the two eventually achieved the effect of replacing the elastoplastic simulation. The types of cracks produced by the mesoscopic interaction between the elements in the shearing process were mainly divided into tensile cracks, compression shear cracks, and tensile shear cracks. The schematic diagram of crack evolution is shown in Figure 4.

3. Model Establishment

3.1. Determination of Parameters

First, cylindrical specimens of sandstone-like rocks were fabricated. The dimensions were set to a height of 100 mm and a diameter of 50 mm, according to the recommendations of the International Society of Rock Mechanics, as shown in Figure 5a. The compression rate was set to 0.01 mm/min. The experiment stopped when the specimen was obviously broken and the loading curve exceeded the peak value. Other variables were kept constant during the test, and no confining pressure was set.

Second, a three-dimensional uniaxial compression numerical model was developed with the same test dimensions. Under the same mesh size, compared with other shapes (wedge, hexahedron, etc.), the tetrahedral mesh could divide the model more finely, resulting in finer simulation results. In addition, since the simulated crack propagation and mode depended on the initial finite mesh, the tetrahedral mesh could have produced oblique failure, so the crack generation effect was better. Consequently, we referred to previous similar studies (Jiang et al. [25], Wu et al. [26], etc.) and finally selected the tetrahedral mesh. In the numerical simulation of the uniaxial compression of a cylinder, there were a total of 121,450 elements, including 76,230 cohesive elements (COH3D6) and 45,220 solid elements (C3D4). The use of the finite element plug-in to insert the cohesive element, and to set the loading plate and the stiffness of the loading plate was much greater than the stiffness of the specimen, to ensure that the load could be applied uniformly on the top of the specimen, as shown in Figure 5b. The lower loading plate was fixed, and the upper loading plate was moved downward with a speed of 0.1 mm/s, until the specimen was damaged. If the loading speed of the numerical simulation is the same as the test, it is often necessary to set a smaller time step to ensure the convergence of the calculation, and a longer calculation time is required. Therefore, without affecting the simulation effect, the loading speed of the simulation was set to be ten times higher than the loading speed of the test. There are two methods for determining the parameters of the cohesive element. The first method is error reduction: the elastic modulus, Poisson’s ratio, and other rock mass parameters are obtained through experiments. The creation of two uniaxial compression models that the cohesive elements have been inserted. Numerical simulations are carried out under the same boundary conditions. When the difference between the two stress fields tends to be stable, it is considered that the parameters are determined. The second method is the experimental comparison method: multiple sets of parameters are used for simulation, and the numerical simulation is compared with the experimental results. If the two are similar, the parameters used for the numerical simulation are considered feasible. In this study, we used the second method.

In order to obtain accurate fracture damage parameters in the shear numerical simulation of joint planes, we referred to the research of Jiang et al. [25], and the uniaxial compression test was simulated by the method of the global embedding of the cohesion element, which was used to calibrate the parameters of cohesive elements. The macroscopic physical and mechanical parameters, such as Young’s modulus and Poisson’s ratio, obtained in the uniaxial compression test, were brought into the numerical model as solid element parameters. The quadratic nominal stress criterion (QUADS) was used for the material behavior of the cohesive elements, to judge the onset of damage evolution, and the BK law was selected for damage propagation. In our study, the type I fracture energy, GfI, and the type II fracture energy, GfⅡ, were used to describe the fracture behavior of rock materials, which can be calculated from the traction-separation constitutive curve. The initial tensile stiffness and shear stiffness can be obtained by using the trial-and-error method, based on previous research experience, and by checking multiple sets of parameters. From Figure 6, it can be seen that the stress–strain curve of the uniaxial compression laboratory experiment was very similar to the stress–strain curve of the numerical simulation. The peak compressive stress of the numerical simulation and the experiment were both approximately 45 MPa; the strain at the peak compressive stress was also approximately 0.42%; and the decreasing trend after the peak compressive stress was also similar. The two stress–strain curve trends and the failure characteristics were highly consistent, as can be seen by comparing the numerical simulation results with the experimental results. Therefore, this paper’s numerical simulation mechanical parameters were reasonable, as shown in

Table 1. Numerical simulation material parameters.

Materials	Parameters	Value
Solid element	Density/kg·m−3	2500
	Young’s modulus/GPa	15
	Poisson’s ratio	0.3
Cohesive element	Initial tensile stiffness/GPa·m−1	15
	Initial shear stiffness/GPa·m−1	5.28
	Normal traction force/MPa	6
	Tangential traction force/MPa	22
	Model-I fracture energy/N·mm−1	0.06
	Model-II fracture energy/N·mm−1	0.165
Plate	Density/kg·m−3	7800
	Young’s modulus/GPa	210
	Poisson’s ratio	0.3

.

3.2. Establishment of the Direct Shear Test Model

The numerical model in this research is presented in Figure 7. In the previous studies, the effect perpendicular to the thickness direction of the loading surface was usually ignored, as such most of the previous studies were based on the 2D level. However, in this paper, the thickness perpendicular to the loading surface was not ignored, and a 3D model was established. The 3D model was 100 mm long, 1 mm thick, and 100 mm high. For the elements between the solid elements of the rock material, we defined the contact behavior by global contact in this study. This contact definition is only applicable to 3D surface contact. The length of the edge-notched crack and the hole radius was 10 mm and 8 mm, respectively. The lower and right boundaries were bounded and fixed boundary conditions, normal stress acted on the upper surface, and the shear direction was left to right. The main factors considered in this paper were the influence of the direct shear rate and the normal stress on the generation and development of cracks in porous rock materials. The simulation experiment, case A and case B, is set in

Table 2. Shear rate and normal stress of six cases in the numerical model.

Numerical Number	Shear Rate (mm/s)	Normal Stress (MPa)
AI	0.01	1
AII	0.02	1
AIII	0.04	1
BI	0.02	1
BII	0.02	2
BIII	0.02	4

.

4. Results of Numerical Simulation

4.1. Mechanical Properties

Figure 8 represents the shear stress–shear displacement curves of the specimens under different shear rates or different, normal stresses. As can be observed from Figure 8, the curves exhibit that the shear process could be determined and identified as the elastic, strengthening, plastic, and residual stress stages, respectively (I–IV). Figure 8b shows a comparison of the stress-displacement curves under three normal stresses, for the elastic stage (I) the change of normal stress has had a certain effect on the slope of the curve. Moreover, the cohesive elements unit did not reach the yield limit at the elastic stage, and no cracks occurred on the specimen. Under the shear load (Figure 8b), it gradually entered the strengthening stage (II). At this stage, the cohesive elements began to enter the linear damage stage, some of the cohesive elements reached the stress yield limit and disappeared immediately after a fracture. At this time, the initiation of cracks could be observed. It is worth noting that crack initiation led to stress change. Since the rock bridge was not coalesced by cracks at this time, the rock bridge was the main provider of the shear strength; therefore, the stress-displacement curve fluctuated at this stage, and the overall shear strength showed an upward trend. When the shear load was further increased, the specimen reached the peak shear strength, and it is noteworthy that the normal stress and shear rate were positively correlated with the peak shear stress. Immediately afterward, we entered into the plastic stage (III). At this stage, the crack expanded and coalescence cracks were generated, and the shear strength of the specimen decreased significantly. When the crack developed further, forming a continuous main crack, the curve entered the residual stress stage (IV). The residual shear strength was provided by the friction and mechanical occlusal forces of the upper and lower parts of the specimen at this stage. The larger the shearing rate was, the bigger the residual shear strength was.

4.2. Normal Displacement Characteristics

Figure 9 presents the normal displacement field diagram of group A, under the three conditions of crack initiation, crack propagation, and crack coalescence. The normal stress applied to AI, AII, and AIII was always in 1 MPa, and the shear rate was 0.01 mm/s, 0.02 mm/s, and 0.04 mm/s, respectively. In the crack initiation stage (Figure 9a(1)–a(3)), the crack initiation displacement was 0.04 mm, 0.05 mm, and 0.07 mm, in a(1), a(2), and a(3), respectively. By comparison, it was not difficult to find that the displacement of crack initiation increased with the development of the shear rate. In the crack propagation stage (Figure 9b(1)–b(3)), the displacement value on the specimen’s right side was negative, and the left side was positive. Moreover, the absolute displacement value was the largest at the edge-notched cracks on both sides. The reason for this phenomenon was that the shearing action produced a clockwise moment around the center of mass, marking M in Figure 9b(1), which caused opposite displacements on the left and right sides, with the largest displacement at the edge-notched cracks on both sides. In particular, the normal displacement of the right boundary was smaller than that of the left boundary, as seen in Figure 9b(1). This was because the sheer velocity loaded on the end of the left side of the rock bridge, between the holes and the fracture-hole, provided a certain degree of support to reduce the load transfer. Therefore, the right boundary, which was away from the loading end, had a smaller normal displacement than the left boundary. In the crack coalescence stage, the greater the shear rate was, the larger the displacement value of the upper half of the specimen was, reflecting that the specimen’s expansion was positively related to the shear rate.

Three groups of specimens with the same shear rate and normal stresses of 1 MPa, 2 MPa, and 4 MPa, respectively, and marked as case BI, BII, and BIII, were selected. The simulated experimental results of the normal displacement field, shown in Figure 10, were obtained. Three stages of crack initiation, propagation, and coalescence were selected for analysis. In the crack initiation stage (Figure 10a(1)–a(3)), the maximum displacement showed an increasing trend in the increase of normal stress, which was 0.08 mm, 0.10 mm, and 0.14 mm, respectively. It can also be seen that the upper part of the right edge-notched crack, at the stage where it was far away from the loading end of the shear rate (that is, on the right side), is the place with the largest displacement in a(1), a(2), and a(3). In the crack propagation stage (Figure 10b(1)–b(3)), under shear stress, an obvious clockwise moment around the center of the specimen was generated at BI (Figure 10b(1)). The action of the bending moment made the normal phase displacements at both ends of the specimen the opposite. During the crack coalescence stage, in the case of BI (Figure 10c(1)), the upper and lower parts were mainly subject to normal stress to produce normal displacement in the same direction.

4.3. Description of Cracking Behavior

4.3.1. Effect of Loading Rate

The effect of the shear rate on the specimen’s stress distribution and fracture process is mainly explored in this section. Moreover, the numerical model of the shearing rate in 0.01 mm/s, 0.02 mm/s, and 0.03 mm/s: cases AI, AII, and AIII (Figure 11a(1)–a(3)), were gathered and presented in Figure 11. It is noteworthy that the crack initiation started from the bottom-right tip of the left edge-notched crack during the crack initiation stage in Figure 11a(1),a(2), but the crack initiation angles for cases AI (Figure 11a(1)) and AII (Figure 11a(2)) were different. However, the crack initiation started from the top-left corner of the left circular holes for case AIII (Figure 11a(3)). Specifically, when the shearing rate was under 0.01 mm/s and 0.02 mm/s, the crack initiation angle was 45° (Figure 11a(1),a(2)), while for 0.03 mm/s, the crack initiation angle was 135° (Figure 11a(3)).

As summarized in Figure 11, from the observations recorded by the crack propagation (Figure 11b(1)–b(3)), the number of cracks and the path of the crack expansion pattern between edge-notched cracks and holes, or between two holes in the rock bridges were basically the same. However, the order of the cracks that appeared was obviously different. In case AII (Figure 11b(2)), crack 2 started from the left hole and propagated towards the left edge-notched crack. Furthermore, crack 2 appeared at the rock bridge in the middle of the two holes, initiated from the left hole and expanded to the right hole (Figure 11b(1),b(3)). Additionally, crack 3 started from the inner wall of the circular hole, which expanded to the end where the stress was applied (Figure 11b(1)).

In the crack coalescence stage (Figure 11c(1)–c(3)), the rock bridge between the holes and the edge-notch cracks on both sides were gradually penetrated by the cracks. It should be emphasized that there was no intersection between the generated cracks, and the cracks that penetrated the rock bridge from different paths. Additionally, the location of the spalling generated from the cracks’ coalescence was sensitive to the shearing rate. In the case of AI (Figure 11c(1)), three spalling areas could be clearly observed, located near the circular hole and right edge-notched crack. Specifically, in Figure 11c(2),c(3), five spallings appeared, respectively. However, their locations were different. For example, AII’s (Figure 11c(2)) spalling areas were mainly produced in the circular hole and the right edge-notched crack; however, in AIII (Figure 11c(3)), the circular hole and the left and right edge-notched crack, all had spalling areas.

4.3.2. Effect of Normal Stress

For this study, the group with the normal stress of 1 MPa, 2 MPa, and 4 MPa (named cases BI, BII, and BIII, respectively) were set up to verify the effect of normal stress during shear failure. The cracks started from the tip of the artificial crack on the left during the crack initiation stage for three cases, but the initiation angles differed. As can be seen from Figure 12a(1),a(2) cracked approximately along the 45° path, but a3 cracked along the 115° path.

In the crack propagation stage (Figure 12b(1)–b(3)), it should be noted that for the total number and location of cracks, six cracks were generated in all three cases, and their locations were approximately the same. Specifically, two cracks were produced at each of the three rock bridges, between the left edge-notched artificial crack and the hole. The artificial cracks and holes in the right edge-notch and the inside of the two holes had the same number of cracks, but the crack initiation sequence and propagation direction were different. For the inner rock bridge between the two holes, crack 3 gradually expanded to the right hole in cases BI (Figure 12b(1)) and BII (Figure 12b(2)), whereas in BIII, it gradually changed from being along the shear direction to being perpendicular to the shear direction (Figure 12b(3)). This transition was due to the tensile stress (Figure 12b(2)). A similar situation also occurred in crack 5, which is shown in Figure 12b(3). The crack did not directly form a connection with the right hole after initiation, but it continued to extend upwards, perpendicular to the shear direction until it reached the upper boundary of the specimen.

As the cracks continued to grow and intersected continuously, the crack coalescence stage of the specimen was started. The rock bridge between the two holes was passed through by the two non-intersecting cracks (Figure 12c(1)–c(3)). The spalling phenomenon was more obvious at the intersection of the crack and hole and at the edge of the artificial crack tip. For the edge-notched artificial cracks on the two sides, as the normal stress continued to increase, the cracks were increasingly dense at the tip and the spalling phenomenon was more obvious.

5. Conclusions

In this study, the FEM-CZM method that was developed was proposed to reveal the mechanical and cracking mechanism of rock masses with two holes, and the loading conditions (including the shear rate and normal stress) were mainly considered. The main conclusions are briefly described as follows:

(1)

The results exhibit that the shear process in this research could be determined and identified as the elastic, strengthening, plastic, and residual stress stages, respectively. The cracks’ initiation at the beginning of the crack strengthening stage gradually gathered and penetrated, mainly at the rock bridge.

(2)

It was observed that the crack initiation stress and the peak shear strength were significantly affected by the shearing rate and normal stress. The shear rate and normal stress were positively related to the peak shear strength. When the shear rate was 0.02 mm/s and the normal stress was 2 MPa, the crack initiation stress was the smallest, and it was easier than it was in the other conditions to bring about the crack initiation. This point should be considered in engineering practice.

(3)

The shear rate and normal stress significantly influenced rock masses’ cracking behavior and the mechanical properties of those with holes. Specifically, the maximum shear dilatancy was positively related to the shear rate, but negatively related to normal stress. The rock bridge played a reinforcing role in the load transfer process, as the vertical displacement near the load end was larger than that far from the load side. Hence, the displacement field of the specimen was non-centrosymmetric.

(4)

In this paper, a detailed analysis of the crack initiation, propagation, and coalescence behavior was carried out, and it was found that the shear rate and normal stress have significant effects on crack initiation, propagation, and coalescence. Under the load, the cohesive element at the end of the artificial crack in the edge-notched cracks were more likely to reach the damage evolution stage. The crack always started from the edge-notched, artificial crack tip, near the shear load end, and the stress concentration was more pronounced at the tip of the crack.