1 Introduction

Different from the shallow underground engineering, the surrounding rock mass of the deep underground engineering can be divided as three zones: the elastic zone, plastic zone and broken zone [1]. In elastic zone, the rock mass is relatively intact, whose mechanical properties are close to those of the intact rock mass. In plastic zone, the plastic deformation occurs and the cracks (joints) are dispersed in the rock mass, whose mechanical properties are as similar as those of jointed rock mass. In broken zone, due to the propagation and coalescence of cracks (joints), the rock mass become a kind of heterogeneous and discontinuous material, which can be called the broken rock mass [2, 3]. For the broken rock mass, its structural integrity and bearing capacity are significantly reduced compared with those of intact rock mass and jointed rock mass. Moreover, the deformation characteristics, strength characteristics and failure modes of broken rock mass are very complex [4]. Nowadays, many researchers have studied the failure evolution mechanism of intact rock mass [5] and jointed rock mass [6] in the microscopic aspect by using experimental and numerical methods, but there are relatively few researches on the same aspect of broken rock mass. Compared with the intact rock mass and jointed rock mass, the broken rock mass contains a large number of irregular rock blocks and has more structural planes and fractures, which results in much poor mechanical properties [7]. Therefore, in order to understand the macroscopic mechanical properties of broken rock mass, it is necessary to study the micro behavior of broken rock mass deeply.

Nowadays, the commonly used research methods in rock mechanics include theoretical analysis, laboratory test and numerical simulation. Due to the complexity and discontinuity of the internal structure for broken rock mass, it is difficult to obtain results from theoretical research. Therefore, researchers mainly focus on the laboratory test and numerical simulation.

The broken rock mass contains many irregularly shaped blocks. It is difficult to prepare the standard specimen of broken rock mass for uniaxial compression test or triaxial compression test because of its loose structure. Therefore, the laboratory test of broken rock mass is mainly compaction test, which mainly studies the change of internal pores of the rock with the increase of load. In those studies, the researchers generally focus on the influence of particle strength, particle size, particle shape and other factors on compaction characteristics of broken rock mass [8,9,10]. In addition, the grouting effect of broken rock mass is also studied. Because the slurry can fill the cracks in broken rock mass and improve the integrity of broken rock mass [11], many engineering practices have proved that grouting reinforcement for broken rock mass is an effective technical method to maintain the engineering stability [12, 13].

In addition to studying the compaction effect and grouting effect of broken rock mass by laboratory tests, numerical simulation is another method to study the mechanical properties of broken rock mass. Numerical simulation can effectively overcome the difficulty of preparing real test specimens of broken rock mass and enable researchers to study the failure mechanism of broken rock mass in the microscopic aspect. However, due to the lack of real triaxial compression laboratory tests for broken rock mass and a hard work to conduct triaxial numerical test for the failure mechanism analysis of broken rock mass, there are no such studies at present. But researchers have done some similar researches. For example, Hosseininia and Mirghasemi [14] established a numerical model to simulate the breakage of two-dimensional polygon-shaped particles and study the influence of particle breakage on macro and micro mechanical parameters by using discrete element method (DEM). However, its research object is not a specific material, and its mesoscopic parameters are set beforehand rather than calibrated by tests. Moreover, Zhang et al. [15] carried out numerical compression tests of broken coal specimen based on Particle Flow Code software (PFC2D) which is also a DEM and studied the evolution characteristics of stress, strain and fracture during its compression. However, in that study, the mesoscopic parameters are calibrated based on the results of uniaxial compression test of complete coal. These research methods provide valuable references for the study of broken rock mass in this paper. However, the existing studies focus on the two-dimension model, in which the object is simplified to the plane strain or plane stress state, and those results are much different from the three-dimensional results [16]. The three-dimensional numerical model is closer to the real broken rock specimen than the two-dimensional numerical model, which is more convenient for researchers to analyze the failure mechanism of the broken rock specimens.

In this paper, the three-dimensional numerical models of broken rock mass are established by using PFC3D software. Based on the results of triaxial compression numerical test, the macro–micro mechanical properties of broken rock mass were studied. In order to further analyze the macro–micro failure characteristics of the broken rock mass, the microscopic failure characteristics of the broken rock mass are compared with those of the intact rock mass and jointed rock mass. Then, the results of acoustic emissions test of cemented waste rock backfill with poor cementing effect which is similar with broken rock mass are used to verify the numerical test. At last, the influence of rock block shape and size on the mechanical properties of broken rock mass is analyzed.

2 Numerical model establishment and parameter calibration

For the stone–soil mixture in which the gravel accounts for 30% to 70%, it is a kind of intermediate between soil and rock mass whose structure is as similar as that of the broken rock masses [17], because both of them are made up of blocks with different grades in discrete. The main difference of them is the fine-grained granules (particle size is less than 5 mm), which are rock debris for broken rock mass and clay soil for stone–soil mixtures. However, Zheng [17] pointed out that when the content of fine-grained soil (particle size is less than 5 mm) is less than 20%, the mechanical property of stone–soil mixtures is not affected by the fine-grained soil. Therefore, the laboratory test results of stone–soil mixtures with low content of fine-grained soil can be used to calibrate the triaxial compression numerical model of the broken rock mass. Zheng [17] conducted triaxial compression tests on stone–soil mixtures specimens of typical Chongqing highway roadbed to study the physical and mechanical properties of stone–soil mixtures. In this paper, based on the triaxial compression test results of the stone–soil specimen for which the content of fine-grained soil is less than 20% [17], the triaxial compression test of broken rock mass is simulated by using PFC3D. It must be noted that, in this model, the rigid clusters were used to figuratively represent the discrete rock blocks in the rock mass, which constitutes the broken rock mass model. The three-dimensional numerical model of broken rock mass is established as follows,

Firstly, according to the size, density and other parameters of the stone–soil specimen [17], a cylindrical rock mass numerical model was established, whose diameter, height and density are 300 mm, 600 mm and 2616 kg/m3, respectively. The cylindrical region of the numerical model was filled with particles. And then, the numerical model reaches the equilibrium by releasing the internal force.

Secondly, according to the particle size distribution of stone–soil mixtures (Fig. 1), the rigid clusters are generated in the rock specimen model to simulate the rock blocks, and the particles whose spatial position overlap with that of newly generated rigid clusters are identified and deleted. And then, the rigid clusters whose size is greater than 5 mm are generated according to the size distribution from 5 to 50 mm. The number of rigid clusters is 100, whose average level of particle overlap is 81.21%. Considering that the relative rotation may occur between the rock blocks or rock debris, the contact-bond model is used for the contact model of cluster-particle or cluster–cluster. Subsequently, the internal force caused by the generation of rigid clusters is released until the numerical model reaches the stress equilibrium state. The actual shape of small rock blocks contained in broken rock mass is irregular, which is one of the factors affecting its mechanical properties, and it will be discussed later in this paper. In order to facilitate the research, the real shaped rock blocks are simplified as spheres in the traditional DEM method [18], which is applied in this study. The numerical model of the broken rock mass is shown in Fig. 2. There are 1357 particles, 100 rigid clusters and 6063 contact bonds. The number of clusters with diameters of 5–15 mm, 15–25 mm, 25–35 mm and 35–50 mm is 44, 41, 13 and 2. It must be noted that, in Fig. 2, the orange spheres represent the rock blocks distributed in the specimen, and to obtain a more intuitive view of its internal structure, the particles of left part are hidden.

Fig. 1
figure 1

Particle size distribution curves of stone–soil mixtures [17]

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Fig. 2
figure 2

Numerical model of broken rock mass with spherical rock blocks

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Finally, to calibrate the model, the rigid servo method was used to apply compression and confining pressure on the numerical model of broken rock mass, which is realized by controlling the movement speed of upper and bottom walls, and at the same time adjusting the cylindrical wall to ensure the constant confining pressure. Here, the loading rate is 0.44 mm/min [17]. According to the stress–strain curves of stone–soil mixture specimens in different confining pressures [17], the micro-parameters of broken rock mass specimens are calibrated, as shown in Fig. 3. The mechanical parameters of the numerical test and real test are shown in Table 1.

Fig. 3
figure 3

Comparison of stress–strain curves of numerical study and real test

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Table 1 Comparison of mechanical parameters of numerical study and real test
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According to Fig. 3 and Table 1, there are little differences for the main macro parameters of the real and numerical models, and the numerical and experimental curves are agreement with each other too. Therefore, it can be concluded that the 3D numerical model of the broken rock mass here has similar macroscopic properties of the specimen in real test. Thus, the mesoscopic parameters used in the numerical model are suitable. The calibrated mesoscopic parameters of the numerical model are summarized in Table 2.

Table 2 Mesoscopic parameters of broken rock numerical model
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3 Study on mechanical properties of broken rock mass

3.1 Microscopic properties of rock failure

The triaxial compression test of the broken rock mass under 1.0 MPa confining pressure was taken as an example to analyze the failure mechanism of the broken rock mass. Because the development of microcrack number has a good agreement with the macro failure process for rock mass [19], in this study, the development of microcracks is analyzed (Fig. 4). Here, the microcracks of the broken rock mass refer to the bond breakage under loading.

Fig. 4
figure 4

Relationship between stress–strain curve and the number of microcracks

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From Fig. 4, according to the number and growth rate of microcracks, there are four characteristic stress points (i–iv) and by which the stress–strain curve can be divided into five stages (stage I ~ stage V). As shown in Fig. 4, there are few microcracks in the stage I. In the stage II, the number of microcracks begins to increase, but the increase rate is relatively slow, in which the strain increases from 0.014 to 0.026, and the number of microcracks increases from 30 to 170. As the strain continues to increase, the growth rate of microcracks increases suddenly and the stress tends to be stable. In the stage III, the growth rate of microcracks is nearly unchanged and the number of microcracks increases from 170 to 1200, where the strain increases from 0.026 to 0.040. In the stage IV, the growth rate of microcracks is decreased gradually and the stress decreases slowly. The total number of microcracks increases from 1200 to 4800, while the strain increases from 0.04 to 0.12. This phenomenon indicates that the structure of broken rock mass continues to be deteriorated in the post-peak stage. Finally, after the characteristic stress point iv, the stress–strain curve is into the residual stress stage (stage V) and the total number of microcracks increase slowly.

At the characteristic stress points of each stage, the vertical central axial planes of numerical model for the broken rock mass are shown in Fig. 5, which shows the microcracks distribution of broken rock mass. In Fig. 5, the orange part represents the microcracks caused by shear stress, while the green part represents the ones caused by tensile stress. As can be seen from Fig. 5, due to the existence of numerous randomly distributed small rock blocks in the broken rock mass, the initial microcracks are scattered in the broken rock mass without obvious regularity. This phenomenon is different from that of jointed rock mass, in which, the initial microcracks are concentrated at the joint, especially near the joint tip [20]. In the broken rock mass, there are many weak fracture surfaces by which the small rock blocks contact with each other. Under the loading, these weak fracture surfaces will be failure by tensile or shear stress, and produce numerous microcracks. At the point (iv) (Fig. 5d), in addition to the fracture of the weak structural plane, the rock block in the rock specimen is also destroyed. During the failure process, the number of microcracks in the broken rock mass increases with the increase of stress. However, these microcracks are randomly distributed and no penetrating main cracks are formed. While for the jointed rock mass, as the increase of stress, the microcracks will gradually expand and penetrate, and finally form the macroscopic main penetrating crack [20]. Moreover, it can be seen that there are more green parts than orange parts, especially in Fig. 5c, which indicates that there are more tensile cracks in the failure process of broken rock mass.

Fig. 5
figure 5

Microcrack distribution diagram at different characteristic points (Nt refers to the microcrack number of tensile crack, Ns refers to the microcrack number of shear crack)

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3.2 Analysis of microcracks number for different dip angles

In order to deeply study the micro failure mechanism of broken rock mass, by using numerical simulation method, the change of microcracks number for different the dip angles in rock failure has been researched here. In this study, a disk represents a microcrack, and the angle (0–90 degree) between the axial loading direction and the disk surface is defined as the dip angle of the microcrack according to the previous study [19]. The number statistical results of the microcracks for different dip angles under different stress states are shown in Table 3. Figure 6 shows the distribution of microcracks number for different dip angles under various stress states.

Table 3 Number statistical results of microcracks for different dip angles
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Fig. 6
figure 6

Distribution of microcracks number for different dip angles under different stress states (fractured rock mass)

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From Table 3 and Fig. 6, the following conclusions can be drawn. In the pre-peak stage, the broken rock mass is in the elastic–plastic state, and the microcracks gradually appear. At this stage, there are a few microcracks, and the dip angle of these microcracks is mostly small. There are only a few microcracks with large dip angle in pre-peak phase. With the stress increase, the number of microcracks with small dip angle increases slowly while the number of microcracks with large dip angle remains almost the same. When the stress in the pre-peak increases from the 0.75 times of peak stress to the peak stress, the total number of microcracks increases from 25 to 481. At the peak stress, the microcracks with different dip angle appear. At last, in the post-peak stage, the growth rate of microcracks with different dip angles increases obviously, and the microcracks number increases rapidly. Meanwhile, the stress decreases from the peak stress to 0.85 times of the peak stress, the total number of microcracks increases from 481 to 4425. And the microcracks with large dip angle are mainly generated in the post-peak stage.

In addition, it can be found from Table 3 that the dip angles for most of the microcracks are less than 40 degrees, and the number of those microcracks accounts for more than 70% of the total number of microcracks. However, as the stress increases, this ratio decreases gradually.

3.3 Comparison with other types of rock mass

For comparison the failure mechanism of three types of rock mass corresponding to those in the three zones of the surrounding rock for the deep underground engineering, the numerical models of intact rock mass and jointed rock mass are also established by PFC3D software. For the intact rock mass, the parallel-bond model is used for the contact model of particles. It must be noted that the “particles” in the numerical model represent the computing elements, not the mineral grain in the rock. For the jointed rock mass, the smooth-joint contact model is used to simulate the prefabricated joint of rock mass. The numerical models of intact rock mass and jointed rock mass are shown in Fig. 7. There are 6174 particles, 28,872 contacts and 28,530 parallel bonds in the intact rock mass model and 27,746 parallel bonds and 784 smooth-joint contacts in the jointed rock mass model. The particles in the numerical model of jointed rock mass are hidden to show the smooth-joint contact model. Figure 8 shows the microcrack evolution of the intact rock mass and jointed rock mass during the failure process. And the increase curve of microcrack number and the stress–strain curve for the intact rock mass and jointed rock mass are also shown in Fig. 8.

Fig. 7
figure 7

Numerical models of the intact rock mass and jointed rock mass

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Fig. 8
figure 8

Microcrack evolution of the intact rock mass and jointed rock mass

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By comparing Figs. 4 and 8, in the pre-peak stage, the numbers of microcracks for the three types of rock masses are all relatively small. When the stress reaches the peak point, the numbers of microcracks in the intact rock mass, jointed rock mass and broken rock mass are less than 200, 100 and 100, respectively. For the intact rock mass and jointed rock mass, the stress drop in the post-peak stage is obvious, and the number of microcracks in rock masses increases rapidly. Specifically, the number of microcracks of intact rock mass increases from 150 to 1600, and that of jointed rock mass increases from 80 to 650. For the intact rock mass and jointed rock mass, the increase rate of the microcracks number decreases and gradually stabilize in the residual strength stage. The number increase of microcracks for intact rock mass and jointed rock mass has an obvious decrease in the residual strength stage. However, for the broken rock mass, the stress drop is unobvious in the post-peak stage, and the increase rate of the microcrack number in the post-peak stage remains approximately unchanged. The number of microcracks of broken rock mass increases from 200 to 5000. At last, the numbers of microcracks of intact rock mass, jointed rock mass and broken rock mass are about 1700, 800 and 5000, respectively. Thus, the number of microcracks of the broken rock mass is much more than those of others. After the peak, the deviator stress of broken rock mass decreases less, the post-peak period lasts longer, and the microcracks develop continuously. On the contrary, the deviator stress of intact rock mass and jointed rock mass decreases rapidly in the post-peak stage. Microcracks of intact rock mass or jointed rock mass develop into macro-penetrating cracks, which leads to structural failure and causes the post-peak residual stage short. Thus, the total number of microcracks for intact rock mass and jointed rock mass is less.

In addition to the difference of the number of microcracks in each stage of the stress–strain curve, the spatial distribution of these microcracks also is different. By comparing Figs. 5 and 8, the following conclusions can be drawn.

Firstly, the initial microcracks of the three types of rock mass are all generated in the weak part of the rock specimen, but their spatial distribution is different. For broken rock mass, there are a lot of weak structural planes because it contains many small rock blocks. The generated initial microcracks are mostly distributed irregularly in these weak structural planes. For intact rock mass, there are relatively few weak structural planes, and most of them are grain defects of rock mass. These defects develop into microcracks, which are also randomly distributed in the whole intact rock mass. For jointed rock mass, the microcracks are always generated firstly at the joint tip of pre-exist joint. Secondly, with the gradual increase of stress, the microcracks of intact rock mass and jointed rock mass will gradually develop and merge to form macro-penetrating cracks. However, macro-penetrating cracks have never been formed in broken rock mass. With the increase of stress, the distribution of microcracks for intact rock mass will gradually concentrate in one area, while those of jointed rock mass are mainly distributed in the rock bridge between its prefabricated joints. Moreover, these microcracks whose distribution areas are relatively concentrated will develop to each other and combine to form larger cracks, which eventually form the macro-penetrating cracks and lead to the rock failure. In contrast, with the increase of stress, the number of microcracks in broken rock mass increases rapidly, but these microcracks are distributed in the whole region of the rock mass and no concentration phenomenon occur. The failure of broken rock mass is caused by too many microcracks rather than by macro-penetrating cracks.

For the number distribution of microcracks for different dip angles, the three types of rock masses are similar. The number statistical results of microcracks for different dip angles for intact rock specimens and jointed rock specimens are shown in Tables 4 and 5. Figure 9 shows the distribution of microcracks number for different dip angles under various stress states. As seen in Tables 3, 4 and 5, the number distribution of microcracks for different dip angles in three types of rock mass has the similar law that the number of microcrack decreases with the increase of dip angle within the range of 20–90 degrees. For the intact rock mass and jointed rock mass, the number of microcracks whose dip angles are in the range of 20–40 degrees are larger than those whose dip angles are other degrees in the pre-peak stage. For the intact rock mass, the angle between the macroscopic main through crack and the axial loading direction is about 35 degrees. Therefore, the microcracks in the range of 20–40 degrees develop firstly and eventually form the macroscopic main through crack. Moreover, for the three types of rock masses, the number of microcracks whose dip angle is less than 40 degrees account for more than 70% of the total number under different stress states, and this proportion decreases gradually as the stress increases. For the three types of rock masses, during the process of the generation and development of microcracks, microcracks with small dip angle have always been dominant, while only a few microcracks with large dip angle are generated. And the microcracks with large dip angle are mainly generated in the post-peak stage.

Table 4 Number statistical results of microcracks for different dip angles of intact rock mass
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Table 5 Number statistical results of microcracks for different dip angles of jointed rock mass
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Fig. 9
figure 9

Distribution of microcracks number for different dip angles under different stress states

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4 Verification by the experimental results

During the deformation and failure process of rock mass, the generated microcracks will result in acoustic emissions (AEs) [21]. Therefore, researchers usually use AE tests to monitor the failure process, failure mode, crack initiation and damage of rock-like or rock specimens [21, 22]. In AE tests, the failure process of the materials and microcrack distribution can be reflected by the hypocentre mapping of AE. The structure of broken rock mass is as similar as that of cemented backfill, because they all contain rock blocks and weak structural planes. Moreover, for the cemented waste rock backfill with poor cementing effect, its material strength is close to that of broken rock mass, and its structural composition is also very similar to that of broken rock mass. Because there are no suitable results of triaxial compression test or AE test of broken rock mass to verify the numerical tests, here, the results of AE test of cemented backfill are used to verify the numerical tests. Deng [22] studied the failure process of cemented rock backfill by AE test. The test results are shown in Fig. 10. The compressive strength of the cemented rock backfill is about 4.2 MPa, while the compressive strength of the cementing backfill with good cementing effect is about 14 MPa [23]. And the results of AE test show that the cementing effect of the cemented rock backfill is very poor [22]. Therefore, the results of this AE test can be used to verify the numerical test results in this paper. By comparison of Figs. 4 and 10, it can be found that the cumulative number curve of microcracks in Fig. 4 is highly as similar as the cumulative number curve of AE events in Fig. 10. And the increasing speed of the number of microcracks or the number of AE events is slow first, then increasing, and finally decelerating gradually. In the numerical tests, the increase rate of the microcracks number in the post-peak stage decreased more slowly compared with that of AE events. In the stage I, there are almost no AE events. The number of AE events increases slowly in the stage II. And the microcracks number in numerical model and the number of AE events were both small. In the stage III, AE events begin to increase rapidly. This is agreement with the phenomenon that the microcracks number increases rapidly in the post-peak stage of numerical test (Fig. 4). However, in the stages IV and V, the increase rate of AE events in the post-peak stage gradually decreases. In summary, the increase law of the number of microcracks in the numerical test is as similar as that of the AE events. It proves that it is feasible to use numerical simulation method to study the microscopic failure mechanism of broken rock mass, and the numerical results in this study are suitable enough.

Fig. 10
figure 10

AE parameters and stress for typical cemented rock backfill specimens [22]

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5 Discussion

5.1 Influence of rock block shape

The shape of rock block in real broken rock masses is irregular. In real engineering, the bearing capacity of broken rock mass is closely related to the interaction and friction of irregular blocks. Therefore, the influence of rock block shape on the mechanical properties of broken rock mass has been analyzed in this study. In the numerical model, the rock blocks are represented by the rigid clusters. Here, three types of numerical models containing different shapes of rigid clusters are established, which are spherical, approximate tetrahedral and approximate ellipsoidal (Fig. 11). The number of particles and contact bonds is 1389 and 6247 for model containing approximate tetrahedral rigid clusters, 1425 and 6283 for model containing approximate ellipsoidal rigid clusters.

Fig. 11
figure 11

Numerical models contained different shape rigid clusters

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To better analyze the influence of the three-dimensional shape of rock blocks, the shape characterization parameter “sphericity” [24] was used to quantify the shape, which is defined as,

$$ Q = {\raise0.7ex\hbox{$S$} \!\mathord{\left/ {\vphantom {S {S_{p} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${S_{p} }$}} $$
(1)

where Q is the sphericity of the rock block, S is the equivalent surface area of the sphere whose volume equals to that of a rock block, and Sp is the surface area of the rock block.

The volume and surface area of the sphere can be defined as,

$$ V_{p} = \frac{4}{3}\pi r^{3} $$
(2)
$$ S = 4\pi r^{2} $$
(3)

where \(V_{p} ,S_{s} ,r\) are, respectively, the volume, surface area and radius of the sphere.

Equations (2) and (3) are substituted into Eq. (1); there is,

$$ Q = \frac{4\pi }{{S_{p} }}\left( {\frac{{3V_{p} }}{4\pi }} \right)^{\frac{2}{3}} $$
(4)

The computed “sphericity” results for the three kinds of rock blocks are summarized in Table 6.

Table 6 Shape parameters of three types of rock blocks
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It can be seen from Table 6 that the sphericity “Q” of sphere, approximate tetrahedron and approximate ellipsoid shapes decreases successively. It indicates that the sharper the edges of the rocks are, the smaller of the sphericity. The smaller the sphericity is, the more angular and irregular of the shape.

By the same quantity and volumes of rigid clusters, three numerical models which contain three different shapes of rock blocks were established, respectively. The triaxial test curves of the three numerical models under the confining pressure of 1.0 MPa are shown in Fig. 12. The macro parameters of three numerical models are shown in Table 7.

Fig. 12
figure 12

Stress–strain curves for broken rock mass contained different shapes of blocks

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Table 7 Mechanical parameters of broken rock mass contained different shapes of blocks
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Here, the test results of broken rock masses which contain the same volume and different shapes of rock blocks were compared. The reason is that when the rigid clusters with the same volume and different shapes are randomly generated in the rock numerical models, the porosity of three rock numerical models is almost the same. Thus, the influence of porosity can be ignored.

From Table 7, for the broken rock masses which contain different shapes of rock blocks, the peak strength, elastic modulus and peak strain increase with the decrease of the rock blocks’ sphericity. The smaller the sphericity of the rock is or the more irregular the rock is, the stronger the interaction force between the rock blocks. And the sphericity of the rock blocks also affects the friction between the blocks, and there is the positive correlation relation between them. This is also found in the previous study [25]. In addition, the spherical rock blocks have the largest sphericity and they are more prone to relative slip and rotation; thus, the post-peak strength of the broken rock mass with spherical rock block is lowest.

The microcracks distribution of broken rock masses that contain different block shapes under different strains is shown in Fig. 13. It must be noted that, to see clearly, the images of central sections for the cylindrical numerical models are used here.

Fig. 13
figure 13

Microcrack distribution of different numerical model contained different shape of blocks under different strains (Nc refers to the microcrack number)

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From Fig. 13, when the strain is 0.02, no matter what the rock block shape is, the microcrack number of the three kinds of broken rock masses is almost the same. It is corresponding to the phenomenon that the stress–strain curves of the three broken rock masses are very close to each other in pre-peak stage. When the strain is 0.04 in the post-peak stage, the microcracks distributed in most space of the rock specimens. However, the broken rock mass which contains spherical block have the most microcracks, and its microcrack distribution area is larger. And the microcracks propagation and coalescence cause the generation of macrocracks in the rock specimen. With the increase of strain, the microcrack number in broken rock mass with the different rock block shape increases rapidly. Especially, the rock specimen is full of microcracks when the strain reaches 0.11. At this time, the microcrack distribution of three kinds of broken rock masses is basically the same.

5.2 Influence of rock block size

In real engineering, the sizes of rock blocks in broken rock mass are different. Therefore, the size of rock block is another factor affecting the mechanical properties of broken rock mass. In order to analyze the influence of rock block size, the volume of rock block in the numerical model is divided into three types, i.e., small volume (the volume of original spherical rigid clusters divided by two), medium volume (the volume of original spherical rigid clusters) and large volume (the volume of original spherical rigid clusters multiplied by two). Thus, the numerical models of broken rock masses which contain different volumes of spherical rigid clusters were established. The triaxial test results of the different numerical models under 1 MPa confining pressure are shown in Fig. 14. The macro parameters of the numerical model are shown in Table 8.

Fig. 14
figure 14

Stress–strain curves for broken rock mass contained different size of blocks

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Table 8 Mechanical parameters of broken rock mass contained different size of blocks
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From Table 8, for the broken rock masses which contain same shape of rock blocks, the peak strength and elastic modulus increase with the decrease of the block volume, and the peak strain decreases with the decrease of the block volume. The reason for this phenomenon is that the structural compactness of broken rock mass with small size of block is bigger and the strength of small rock blocks is greater than that of large rock blocks. Moreover, with the decrease of the block volume, the brittle failure of model becomes more obvious. In the post-peak stage, the stress of the broken rock masses which contain small volume of blocks decreases more quickly. As can be seen from Fig. 14, the residual strength of three types of broken rock masses is negatively correlated with the volume of the rock block. The larger the rock block is, the smaller the residual strength will be.

At last, the influence of block volume on the microcracks distribution is analyzed as shown in Fig. 15.

Fig. 15
figure 15

Microcrack distribution of the numerical model contained different size of blocks under different strains

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From Fig. 15a and b, when the strains are 0.02 and 0.04, the number of microcracks in the three numerical models decreases as the increase of rock block volume. This explains that for broken rock mass which contain small volume of blocks, the stress decreases more obviously in the post-peak stage. And when the strain is 0.11, as the decrease of the rock block volume, the number of microcracks in the broken rock mass increases. Moreover, the microcracks are randomly distributed in the broken rock masses which contain different volumes of rock block, and gradually accumulated with the increase of stress.

6 Conclusion

In this study, macro–micro mechanical behavior of broken rock mass was studied by the triaxial compression numerical test. The number and distribution of microcracks in broken rock mass were analyzed and compared with those of intact rock mass and jointed rock mass. Moreover, the results of numerical test were verified by the results of AE test. Finally, the influence of rock block shape and size on the mechanical properties of broken rock mass was analyzed. The main conclusions are as follows,

  1. (1)

    The microcrack evolution process of broken rock mass can be divided into five stages (stage I–stage V). In the stage I, there are few microcracks. From stage II to stage V, the increase rate of microcracks number increases first and then decreases. Moreover, the number of tensile cracks is obviously bigger than that of shear cracks.

  2. (2)

    The larger the microcrack dip angle is, the less the number of microcracks will be. And the number of microcracks whose dip angle is less than 40 degrees account for more than 70% of the total number of microcracks.

  3. (3)

    Comparing with the failure process of intact rock mass and jointed rock mass, for broken rock mass, the decrease of stress is unobvious after peak and no macro-penetrating cracks are generated, and the generated microcracks are mostly distributed irregularly in weak structural planes between the rock blocks.

  4. (4)

    The more irregular the shape of the rock block is, the smaller the sphericity is. The peak strength and elastic modulus increase with the decrease of the sphericity. When the strain is 0.04 in the post-peak stage, the microcracks in the broken rock mass with spherical block are most. For different rock block size, the peak strength and elastic modulus also increase with the decrease of the rock block size. And the microcracks in the broken rock mass with smallest rock block are most.

Although the influence of rock block shape and size is studied in this paper, in fact, the shape or size of rock blocks in real broken rock mass varies much with each other. Therefore, in future, the numerical model of broken rock mass whose rock blocks are randomly generated in different sizes and shapes will be established by the secondary development of PFC3D software. Moreover, due to lack of the experimental study on the failure of broken rock mass in the triaxial compression test, the verification of numerical results was based on the cemented waste rock backfill in the uniaxial compression test, which can only qualitatively verify the failure process of broken rock mass in the triaxial compression test. Therefore, to verify the numerical results comprehensively based on the real triaxial compression test of broken rock mass is also our next works.