1 Introduction

Joints, fissures, and weak surfaces are common in natural rocks. When a load is applied, new cracks develop near the tips of existing joints and may propagate or coalesce with other cracks. The propagation of new cracks and the coalescence of fractures lead to a degradation in the mechanical properties of the rock. In some cases, this will result in rock mass failure, a topic that has received much attention in rock mechanics and rock engineering.

Previous studies (Vallejo 1988; Vallejo et al. 2013; Dyskin et al. 2003; Hoek and Bieniawski 1965; Li et al. 2005; Park and Bobet 2009; Cao et al. 2015; Yang et al. 2013; Sagong and Bobet 2002; Tang et al. 2001; Wong 2008; Wong and Chau 1998 and so on) of crack coalescence between fractures utilized laboratory tests because of the difficulties of in situ tests. With recent rapid developments in computer science, many numerical methods have been suggested to simulate crack initiation and coalescence; most of the simulation results show good agreement with experimental results. For specimens that contain a single fracture, two types of cracks can develop from the tips of the existing fracture: wing cracks and secondary cracks. A wing crack is a tensile crack that originates from the tip of the fissure and propagates at an angle toward the direction of maximum compression, whereas a secondary crack is a shear crack (Shen 1995; Sahouryeh et al. 2002; Dyskin et al. 2003; Li et al. 2005; Wong and Einstein 2006; Park and Bobet 2010; Yang et al. 2012; Cao et al. 2015).

For a specimen containing two or three fractures, previous studies focused on crack coalescence between parallel fractures (Vallejo 1987; Bobet and Einstein 1998b; Li et al. 2005; Wong and Li 2013; Park and Bobet 2009; Sagong and Bobet 2002; Tang et al. 2001; Vasarhelyi and Bobet 2000; Wong 2008; Wong and Chau 1998; Wong et al. 2001; Yang 2011; Zhang and Wong 2013a, b; Zhou and Yang 2007; Zhou et al. 2014, 2015; Vallejo 1989). Significant advances were achieved in understanding the coalescence pattern between parallel fractures. There are mainly three types of failure modes for these specimens: tensile failure, shear failure, and mixed failure.

In addition to studies on parallel fractures, the mechanical properties and penetration mode of un-parallel fractures in a specimen are also well discussed in the literature. Combining numerical simulation using the two-dimensional particle flow code (PFC2D) and un-parallel fissures experiments using PMMA, Hwangdeung granite, and Diastone, Lee and Jeon (2011) investigated the penetration mode between two un-parallel fissures. Based on parallel bonded-particle models, Zhang et al. (2015a, b) studied crack coalescence between two non-parallel flaws and observed five types of linkage between two flaws: tensile crack linkage, tensile crack linkage with shear coalescence at the tip, shear crack linkage, mixed linkage, and indirect crack linkage. Using photographic monitoring and acoustic emissions monitoring techniques, Yang et al. (2013) investigated the relationship between the real-time crack coalescence process and axial stress–time behavior for red sandstone containing two un-parallel fissures.

While previous work has promoted the understanding of crack propagation, coalescence, and failure modes of brittle materials with fissures, the strength and failure patterns of ubiquitous-joint rock specimens have not been studied comprehensively. Compared with specimens containing one, two, or three fissures, the mechanical behaviors of those with multiple flaws or joints are expected to be more complicated. In practical engineering, the rock mass usually contains multiple joints. The number of joints will increase in the case of ubiquitous-joint rock mass (Fig. 1). Both the inclination and intersection angles between two set of joints affect the mechanical properties and failure mode of rock mass. In this paper, we combine experiments using rock-like material and a discrete element numerical method (PFC2D) to investigate the mechanical behavior and failure mode of ubiquitous-joint rock-like materials.

Fig. 1
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Idealized schematic showing the ubiquitous-joint rock mass

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2 Experimental Studies

2.1 Specimen Preparation

As mentioned above, laboratory tests have been widely used as an effective research method for crack coalescence between fractures in brittle materials. So far, many types of materials have been employed for such tests, including glass (Hoek and Bieniawski 1965), Columbia Resin 39 (Bombolakis 1968), and molded gypsum (Park 2001; Shen 1995; Bobet and Einstein 1998a, b; Reyes and Einstein 1991; Sagong and Bobet 2002; Wong and Einstein 2009a, b).

Because the main framework of the rock consists of mineral grains and silicon or argillaceous material between the grains, friction will increase in the failure plane when the specimens fail. This friction will then affect the strength of the specimens. Therefore, we selected cement mortar as the rock-like material for our experiment. The grains in the specimen are sand, which provides the frictional behavior of the modeling material. The adhesive material between the grains is cement.

The rock-like specimens are made of white cement, water, and sand. The volume proportions V water:V white cement:V silica sand in the specimens are 1:1:2. The dimensions (height × width × thickness) of each specimen are 200 × 150 × 30 mm. The existing fissures were created by inserting mica sheets (0.6 mm thick, 30 mm long) into the fresh cement mortar paste at the desired location of the fissures; the sheets were not removed before the modeling material set. The specimens were removed from the mold and soaked in water for 3 days. They were placed inside a standard curing box (with the temperature controlled at 20 ± 2 °C and humidity controlled at 80 %) for 25 days before being subjected to mechanical testing. The average values of the unit weight (γ m), Young’s modulus (E m), uniaxial compressive strength (UCS), and Poisson’s ratio (v) of the rock material are γ m = 2.159 g/cm3, E m = 3.242 GPa, UCS = 8.01 MPa, and v = 0.2371.

The joint geometry is defined by two geometrical parameters as shown in Fig. 2: joint-1 with an inclination angle α (Fig. 2a) and intersection angle γ (β − α) (Fig. 2c). The ubiquitous-joint rock-like specimens are created with various combinations of joint-1 and joint-2 angles. The experiments were performed for specimens with five different α values of joint-1: 0°, 30°, 45°, 60°, and 75°, while joint-2 revolved around the z-axis counterclockwise in increments of 15° until γ reached 75°.

Fig. 2
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Schematics of joint geometry configurations in the blocks. α: joint-1 orientation with respect to horizontal. β: joint-1 orientation with respect to horizontal γ: β − α. a Joint-1, b joint-2, and c jointed model

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Table 1 shows the fissure geometry information for all the specimens in this study. Each specimen was assigned an ID number using the notation S-a-b, where S stands for the sample, ‘a’ represents the angle of joint-1 in degrees, and b is the intersection angle γ in degrees. Figure 3 shows the joint geometry configurations generated for α = 30° (S-30-15, S-30-30, S-30-45, S-30-60, S-30-75).

Table 1 Sample numbers and fissure geometrical parameter values used for all specimens
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Fig. 3
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Ubiquitous-joint specimens with α = 30. a Intact model, b S-30-15, c S-30-30, d S-30-45, e S-30-60, and f S-30-75

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2.2 Equipment

Figure 4 shows the compression test setup. The specimen was placed between the two loading platforms in the servo control uniaxial loading instrument, and the loading was controlled by the loading control system (DCS-200). The loading rate was 0.3 mm/min. The top and bottom boundaries are fixed in the horizontal direction. To reduce the end effect, the contact area between the specimen and the loading platform was coated with butter. The specimens were loaded under compression until failure, and the failure process was recorded by a video camera.

Fig. 4
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View of the specimens ready to be tested and the layout of the loading system

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2.3 Failure Model of a Multi-fissure Specimen

Previous experimental and numerical results showed that when a load is applied to specimens with a single fissure, three types of cracks develop from the pre-existing fissure (Fig. 5): wing cracks, quasi-coplanar secondary cracks, and oblique secondary cracks (Shen 1995; Sahouryeh et al. 2002; Li et al. 2005; Wong and Einstein 2006; Park and Bobet 2010; Yang et al. 2012; Cao et al. 2015).

Fig. 5
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Crack types observed in pre-flawed specimens under compression (Bobet 2000)

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For ubiquitous-joint specimens, the mechanical behavior will be more complicated. When compression is applied on the existing flaws, tensile or shear cracks will develop from the tips of the existing fractures. As loading continues, these cracks will propagate and join with other cracks, penetrating through the rock. Thus, the pre-existing fissures will join with neighboring cracks, resulting in various types of failure patterns. Based on experimental observations, the failure patterns of ubiquitous-joint rock-like specimens can be generally classified into four categories: (1) stepped path failure; (2) planar failure; (3) shear-I failure; and (4) shear-II failure.

2.3.1 Stepped Path Failure

Specimen S-0-30 (Fig. 6) exhibits a typical stepped path failure pattern. In this type of failure mode, several ‘stepped failure planes’ occur in the specimen in various locations. The coalescence mode between the pre-existing flaws on the stepped failure planes is mainly tensile cracks. Under compression, tensile cracks develop from the pre-existing flaws and propagate along the direction of axial stress. With further loading, the tensile cracks join with other cracks, forming the observed coalescence.

Fig. 6
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Crack coalescence in stepped path failure

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In the failure pattern shown in Fig. 6, most of the cracks are approximately parallel to the loading direction. Notably, other cracks such as shear or mixed cracks appear besides the tensile cracks in the stepped path failure specimens.

2.3.2 Planar Failure

Specimen S-60-75 (Fig. 7) is a typical example of a planar failure pattern. Several coalescence modes appear between the pre-existing flaws (including tensile, shear, and mixed cracks). Most of the shear cracks between the pre-existing fissures are quasi-coplanar secondary shear cracks, which are similar to the category 4 coalescence discussed by Wong and Einstein (2009a).

Fig. 7
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Crack coalescence in planar failure

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In addition to shear cracks, some parallel and un-parallel flaws also connect with each other through tensile cracks. Under loading, tensile and shear cracks develop from the pre-existing flaws. As the loading continues, new cracks propagate and join with others and penetrate through the specimen. At the residual stage, the specimen appears fragmented—the pre-existing flaws and new cracks have split the specimen into many small blocks. Clearly, there are no far-field cracks seen in the specimen.

2.3.3 Shear Failure

The shear failure mode can be divided into two types based on the failure characteristics of the shear failure pattern: shear-I failure pattern and shear-II failure pattern.

2.3.3.1 Type I Shear Failure

This pattern has either a single shear failure plane or a set of parallel shear failure planes. Specimen S-60-30 (Fig. 8) is a typical example of this failure pattern. The coalescence mode between the pre-existing flaws on the main failure plane comprises quasi-coplanar secondary shear cracks that correspond to category 4 coalescence according to Wong and Einstein (2009a). Under loading, shear cracks develop from the fissure tips; with further loading, the shear cracks propagate, joining with other shear cracks and penetrating through the specimen.

Fig. 8
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Crack coalescence in shear-I failure

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During failure, the specimen is split into two or three blocks by the parallel shear failure plane. The shear failure plane consists of quasi-coplanar secondary shear cracks and pre-existing flaws. Indentations and friction debris can be clearly seen on the surface of the failure plane. After the shear failure plane forms in the specimen, with further loading, tensile cracks continue to propagate until the specimen reaches the residual stage. As shown in Fig. 8, there are also some obvious tensile cracks in this pattern.

2.3.3.2 Type II Shear Failure

Specimen S-60-60 (Fig. 9) is a typical example of the shear-II failure pattern. Compared with the parallel shear failure plane in the shear-I failure pattern, in this pattern, the specimen exhibits a set of cross-shear failure surfaces. The coalescence modes between the pre-existing flaws on the cross-shear failure plane are shear cracks.

Fig. 9
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Crack coalescence in shear-II failure

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Under loading, shear cracks develop from the fissure tips. With further loading, new cracks propagate and join with neighboring cracks or pre-existing flaws to form a cross-shear failure surface. In Fig. 9, although some of the blocks fall outside the specimen, the cross-shear failure plane in the experimental result is very clear. As in the shear-I failure pattern, indentations and friction debris are clearly visible on the shear failure plane.

In this paper, the rock-like specimens are made of white cement, water, and sand. And the existing fissures were created by inserting mica sheets (0.6 mm thick, 30 mm long) into the fresh cement mortar paste at the desired location of the fissures; the sheets were not removed before the modeling material set. Based on the experimental results, the cohesion (C j ) and friction angle (ϕ j ) between the cemented material and the mica sheets are 18.2 kPa and 11°, respectively. The C j and ϕ j values in our study are very close to those in Zhang et al. (2015a, b). Clearly, C j and ϕ j affect the initiation and coalescence stress. However, according to Park and Bobet (2009), ‘the cracking pattern and the cracking processes that occur in specimens with closed flaws are the same as with open flaws, and so it is expected that the fracture mechanisms and principles that apply to both types of flaws are the same.’ Moreover, based on XFEM, Xie et al. (2016) found that the initiation location and angle of the cracks are not influenced by friction. Because the friction angle between the cement and the mica sheet in this study is much smaller than that of the cement–cement face, therefore, we can assume that the friction angle between the cement and mica sheet has no considerable effect on the direction of propagation of the wing cracks or secondary cracks.

Table 2 summarizes the failure patterns of rock-like specimens containing multiple flaws with different α and γ values approximately at the residual strength. Specimens S-0-15, S-0-30, and S-30-15 underwent stepped path failure, with the failure of the specimen caused mainly by the propagation of tensile cracks. Specimens S-45-15, S-60-15, S-30-60, and S-30-75 show failure mode patterns between stepped path failure and shear-I failure; in these specimens, the pre-existing fissures are connected through tensile or coplanar shear cracks, resulting in overall failure. The planar failure pattern appears mainly in the specimens with γ = 75°, for example, specimens S-45-75, S-60-75, and S-75-75.

Table 2 Failure patterns of ubiquitous-joint rock-like specimens
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Apart from the specimens mentioned above, all the other specimens exhibit shear failure patterns. The shear failure pattern is also the most common pattern in the ubiquitous-joint specimens. Specimens S-45-60, S-60-60, S-75-30, and S-75-45 are typical examples of the shear-II failure pattern.

2.4 Peak Strength of the Rock-like Specimens Under Uniaxial Compressive Loads

Figure 10 shows the peak strength (UCSJ) of the multi-fissure specimens obtained from the ubiquitous-joint rock-like specimen compression tests versus α. For the specimens with γ = 15° and 30° (Fig. 10a), UCSJ varies considerably as α increases from 0° to 75°. For both γ values, UCSJ decreases as α increases from 0° to 45°, and increases for α > 45°.

Fig. 10
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Peak strengths of ubiquitous-joint rock-like specimens with different inclination angles (α)

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The UCSJ values for the specimens with γ = 45° and 60° (Fig. 10b) show similar tendencies when α rises from 0° to 75°, increasing as α increases from 0° to 30°, and decreasing for α > 30°. When α = 30°, the specimens (S-30-45, S-30-60) reach their peak UCSJ value. However, for the specimen with γ = 75°, the UCSJ peaks at α = 0° and then decreases until α = 60° and increases for 60° ≤ α ≤ 75°.

3 Numerical Modeling and Simulated Results

3.1 PFC2D, Model Generation, and Calibration of Micro-mechanical Parameters

Itasca’s upgraded version of PFC2D (based on the principle of the discrete element method) can be used to study the propagation of cracks in brittle material such as rocks, rock-like material (Lee and Jeon 2011; Yang et al. 2014; Bahaaddini et al. 2013; Wong and Zhang 2014; Fan et al. 2015 and so on), and clays (Vesga et al. 2008). In PFC2D, material is modeled as collections of particles (rigid circular disks) (Itasca Consulting Group 2002). Each particle is in contact with the neighboring particles via the contact-bond or parallel-bond model. In the parallel-bond model, the bond is depicted as a rectangle of cement-like material that can transmit both force and moment between particles (Potyondy 2007). In this model, bond breakage causes an immediate decrease in macro-stiffness because the stiffness is caused by both the contact stiffness and bond stiffness (Potyondy 2007; Cho et al. 2007). Hence, the parallel-bond model is viewed as a realistic method for modeling brittle materials and is often used to simulate crack initiation and propagation in rocks or rock-like materials (Cho et al. 2007; Lee and Jeon 2011; Yang et al. 2014; Wong and Zhang 2014; Manouchehrian and Marji 2012; Manouchehrian et al. 2014 and so on). In this study, we also use PFC2D (parallel-bond model) to carry out the numerical simulation for the ubiquitous-joint rock-like specimens under uniaxial compression.

It is very difficult to determine the micro-parameters of a material (such as the particle contact modulus, parallel-bond modulus, particle normal/shear stiffness, and the parallel-bond normal/shear strengths) based on macro-parameters such as Young’s modulus, UCS, and Poisson’s ratio. To better understand the effect of the micro-parameters on the PFC2D numerical results, intensive research on the relationship between the microscopic parameters and macroscopic properties of the specimens/material was carried out (Huang 1999; Potyondy and Cundall 2004; Yang et al. 2006; Cho et al. 2007; Koyama and Jing 2007; Yoon 2007). The ratio of particle normal to shear stiffness (k n/k s) has a considerable effect on the Young’s modulus: as k n/k s increases, the Young’s modulus decreases (Yang et al. 2006). For the UCS, it increases with the normal/shear bond strength and friction coefficient, while it exhibited small fluctuations and had no clear trend of increasing or decreasing when k n/k s and E c (contact modulus) increased (Yang et al. 2006). The Poisson’s ratio is affected mainly by k n/k s; it increases with higher k n/k s (Yang et al. 2006; Yoon 2007). In this paper, the micro-parameters were determined through the trial-and-error approach, which is an accepted method (Lee and Jeon 2011; Yang et al. 2014; Ghazvinian et al. 2012; Zhang and Wong 2013a; Zhang et al. 2015a, b; Manouchehrian et al. 2014; Bahaaddini et al. 2013; Fan et al. 2015; Cao et al. 2016a, b). Although this method is not considered state of the art, it can nonetheless lead to successful results. After calibration, the micro-parameters used in this study reflect the macro-mechanical behavior of the rock-like material. Table 3 lists the micro-parameters determined in this study by a series of ‘trial-and-error’ processes (the calibration is performed only under consideration of compressive stress states); a comparison between the experimental and numerical results is provided in Table 4. Table 4 indicates that the simulated UCS and Young’s modulus of the intact rock-like specimens are similar or equal to those obtained experimentally.

Table 3 Microscopic parameters for rock mass
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Table 4 Comparison between experimental and numerical results for intact material macro-mechanical parameters
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Previous studies (Huang 1999; Potyondy and Cundall 2004; Yang et al. 2006; Cho et al. 2007; Koyama and Jing 2007; Yoon 2007; Zhang and Wong 2013a) have considered particle size and size distribution. For example, Yang et al. (2006) systematically studied the influence of the model size (L/d) and the size of the microscopic parameters on the macroscopic properties, such as Young’s modulus, Poisson’s ratio, and peak strength (UCS). They pointed out that the Young’s modulus showed an increase trend with rising L/d until it reached 32.5, after which the Young’s modulus remained almost constant. However, the Poisson’s ratio decreased with increasing L/d and become stably when L/d was larger than 32.5. The UCS was almost constant when L/d increased from 5 to 62.5. Additionally, the d max/d min ratios (2, 2.4, 4.2, and 6.2) were also considered and the results indicated that the particle microstructure has no considerable effect on the Young’s modulus, Poisson’s ratio, or UCS. Based on a fixed dmax/dmin ratio of 1.66, Potyondy and Cundall (2004) investigated the influence of particle and model size on the simulation results of Lac du Bonnet granite using PFC2D. The results indicated that the UCS and Young’s modulus exhibited slight fluctuations and showed no clear trend of increasing or decreasing when the L/d ratio changed from 11 to 88, while the Poisson’s ratio remained about the same. On the whole, the previous studies show that the particle microstructure configuration has only a small influence on the macroscopic deformability and strength behavior in the simulations. Further, these studies covered a wide range of L/d values, from 5 to 200, and the results indicate that the macroscopic properties exhibit some fluctuation at low L/d values and stabilize once L/d is larger than 50. However, in this study, the mean radius was set as 0.25 mm, yielding a L/d rate of 600, which is sufficiently high to achieve a stable PFC simulation.

Figure 11 shows the numerical intact specimen generated by PFC2D. The scale of the numerical specimen is equal to that of the specimens in the experiment: 150 × 200 mm (width × height). The gray circles are the particles, and the fine green and pink lines represent the contacts and parallel bonds, respectively.

Fig. 11
figure 11

Numerical intact specimen generated in PFC2D in this research

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Usually, the micro-mechanical parameter values assigned for the particles that represent joints are smaller than those for the particles that represent intact material. In the numerical model, the joints are generated through changing the micro-mechanical parameter values of the particles on the areas of the joints. The joint-related particles are shown in white in Fig. 12.

Fig. 12
figure 12

Numerical specimens containing multiple joints where α is the angle of joint-1 (α = 30°), β is the angle of joint-2, γ is angle of fissure 1 and fissure 2 (γ = β − α, γ = 15°, 30°, 45°, 60°, 75°)

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As mentioned above, we created the joints in the rock-like specimens by inserting mica sheets during specimen production; the specimens were then removed from the mold and soaked in water for 3 days. During the curing period, the humidity was controlled at 80 %. Hence, the strength of the mica sheets is similar to that of paper. Moreover, because the mica sheet is composed of multilayer paper-like material, after absorbing water, it separates easily during the testing. Therefore, for the micro-mechanical parameter values of the joints (Table 5), the parallel-bond normal and shear strengths were set as 0, and the other micro-mechanical parameters were also far below those of the intact material particles.

Table 5 Microscopic parameters for joints
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3.2 Cracking Characteristics in Rock-like Specimens Containing Multiple Joints

As mentioned above, under uniaxial loading, there are four types of failure patterns in the ubiquitous-joint rock-like specimens. The numerical results show similar failure characteristic to the failure patterns of the experiments (Figs. 6, 7, 8, 9). The stress–strain curve and tensile and shear bond failures were calculated for the stepped path failure (Fig. 13a), planar failure (Fig. 13b), shear-I failure (Fig. 13c), and shear-II failure pattern (Fig. 13d). The numerical result (black line) and experimental result (maroon line) are shown along with the tensile and shear bond failure curves obtained from the numerical simulation with incremental axial strain. Note that ‘Crack num’ stands for the total number of bond failures, ‘Tensile crack’ stands for the number of normal bond failures and ‘Shear crack’ stands for the number of shear bond failures. The crack propagation and coalescence process of the ubiquitous-joint rock-like specimens simulated by PFC2D under uniaxial compression are shown in Figs. 14, 15, 16, and 17. The failure process corresponds in sequence to points a–e on the stress–strain curve obtained from the numerical simulation. Careful examination of all the failure patterns shows that the stress–strain curve can be divided into three stages: stage I, microscopic cracks initiation stage (from 0 to point a); stage II, stable cracks growth stage (from point a to point c); and stage III, macroscopic fracturing stage (from point c to point e).

Fig. 13
figure 13

Stress–strain curve and tensile and shear bond failures for a stepped path failure, b planar failure, c shear-I failure, and d shear-II failure patterns

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Fig. 14
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Failure process at different axial strain levels for a typical stepped path failure (S-0-30) (black lines and red lines indicate normal and shear bond failures, respectively). a Crack initiation, b crack development (84.3 % peak), c peak strength, d crack coalescence (30.76 % post-peak), and e residual strength (7.7 % post-peak) (color figure online)

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Fig. 15
figure 15

Failure process at different axial strain levels for a typical planar failure (S-60-75) (black lines and red lines indicate normal and shear bond failures, respectively). a Crack initiation, b crack development (93.17 % peak), c peak strength, d crack coalescence (55.7 % post-peak), and e residual strength (14.18 % post-peak)

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Fig. 16
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Failure process at different axial strain levels for a typical shear-I failure (S-60-30) (black lines and red lines indicate normal and shear bond failures, respectively). a Crack initiation, b crack development (82.15 % peak), c peak strength, d crack coalescence (53.3 % post-peak), and e residual strength (11.76 % post-peak)

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Fig. 17
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Failure process at different axial strain levels for a typical shear-II failure (S-60-60) (black lines and red lines indicate normal and shear bond failures, respectively). a Crack initiation, b crack development (75.36 % peak), c peak strength, d crack coalescence (57.1 % post-peak), and e residual strength (7.4 % post-peak) (color figure online)

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3.2.1 Stepped Path Failure

In stage 1 (Fig. 13a), the stress–strain curve shows nonlinearity as the axial strain increases. After reaching the crack initiation point (point a), micro-cracks develop from the pre-existing flaws (Fig. 14a), and no far-field cracks appear in the specimen.

After crack initiation (point a), the specimen enters the second stage, the ‘stable cracks growth’ stage. In stage II, only small fluctuations appear in the stress–strain curve because the micro-cracks increase stably during this stage, and some micro-cracks link to form macro-cracks. At point b, there are obvious wing cracks in the specimen (Fig. 14b); at the same time the axial stress reaches 84.3 % of the UCSJ.

After point c (Fig. 14c), the stress–strain curve drops to a low stress level, which indicates that the specimen enters the macroscopic fracturing stage. From point c to e, the bond breakages undergo accelerating growth. In Fig. 14d, the specimen undergoes macroscopic failure and there are several stepped failure planes. On the main failure plane, the pre-existing flaws link with the others through tensile cracks. During stage III, from point d to e, even though the stepped failure plane appears in the specimen, with further loading, the tensile cracks continue to propagate until reaching the residual strength.

3.2.2 Planar Failure

The planar failure pattern appears mainly in the specimens with γ = 75°. The stress–strain curve and tensile/shear bond failures seen in specimen S-60-75 are typical examples of the planar failure pattern (Fig. 13b). For both specimens S-0-30 and S-60-75 (Fig. 13a, b), the numerically simulated stress–strain curve is in good agreement with the experimental curve except at the initial stage of loading.

In the numerically simulated stress–strain curve (S-60-75), there is some fluctuation between the point of origin and point c. The first micro-crack appears when the axial strain reaches 0.18 % (point a in Fig. 13b). As shown in Fig. 15a, the micro-cracks originate from the pre-existing fissure tips. With continuous loading, at point b the micro-cracks cluster and form recognizable macro-cracks in the fissure tips, but the specimen still maintains integrity and stability as shown in Fig. 15b. At the peak stress (point c), the macro-cracks extend further (Fig. 15c), and some pre-existing flaws link with neighboring flaws via tensile or shear cracks.

In the macroscopic fracturing stage (from point c to e), for specimen S-60-75 the increase in micro-cracks is more obvious than in the microscopic cracks initiation stage (from 0 to pint a) and in stage II (the stable cracks growth stage, from point a to c). More macro-cracks can be observed as the axial strain increases. During this stage, the micro-cracks coalesced (Fig. 15d), leading to a continuous degradation in strength (from point c to point d). After point d, the micro-cracks remain almost stable, and finally, the axial stress (with minor fluctuations) reaches its residual strength (Fig. 15e).

3.2.3 Shear-I

In the shear-I failure pattern, a single or a set of parallel shear failure planes appear in the specimen. Figure 13c shows the stress–strain curves of the numerical result (black line), experimental result (maroon line), and the tensile and shear bond failure curves for specimen S-60-30. As with specimens S-60-75 and S-0-30, although there is some difference between the numerical and experimental curve, the two curves show similar tendencies.

Figure 16 shows the crack propagation and coalescence process of specimen S-60-30 as a typical example of the shear-I failure pattern. As with the stepped path and planar failure patterns, in stage I of Fig. 14c, the stress–strain curve shows nonlinearity as the axial strain increases. After the axial strain rises to 0.165 %, the specimen reaches the crack initiation point (point a) and micro-cracks develop from the pre-existing flaws as shown in Fig. 16a.

In the stable cracks growth stage (from point a to point c), the axial stress at point b is equal to 82.15 % of the UCSJ and wing cracks start to appear in the specimen as shown in Fig. 16b. With continuous loading, the micro-cracks increase stably, but the specimen is still able to maintain its mechanical integrity and stability. When the axial stress reaches the peak strength, crack coalescence occurs in the specimen as shown in Fig. 16c. Finally, the specimen enters the last stage and the micro-cracks grow rapidly. In this stage, some macro-failure appears in specimen S-60-30.

At point d, many pre-existing flaws coalesce as the micro-cracks combine (Fig. 16d); according to previous studies (Wong and Einstein 2009a; Sagong and Bobet 2002; Cao et al. 2016b), most of the coalescence modes between pre-existing flaws are of the quasi-coplanar secondary crack type. With further loading, the axial stress decreases with the growth of axial deformation until reaching the residual strength (Fig. 16e). Specimen S-60-30 (Fig. 16e) was split into three blocks by the parallel shear failure plane. Generally, the simulated stress–strain curve (Fig. 13c) is similar to the experimental stress–strain curve apart from some fluctuations.

3.2.4 Shear-II

As mentioned above, in the shear-II failure pattern, the specimen contains a set of cross-shear failure surfaces. Specimen S-60-60 is a typical example of this pattern. Figure 13d depicts the stress–strain curves of the numerical result (black line), experimental result (maroon line), and the tensile and shear bond failure curves for specimen S-60-60. Figure 17 shows the crack propagation and coalescence process of specimen S-60-60, derived from PFC2D. As in the case of specimen S-60-30, the numerically simulated stress–strain curve (black line) can be divided into three stages and it agrees well with the experimental curve (maroon line), including the post-peak region.

In stage I, the axial stress grows as the axial strain increases. When the specimen is loaded to point a (the axial strain equals 0.15 %), micro-cracks develop from the flaws as shown in Fig. 17a. As in the stepped path failure, planar failure, and shear-I failure patterns, at the point a, only some micro-cracks and no far-field cracks appear in specimen S-60-60. At point b, the axial stress reaches 75.36 % of the peak strength, and the wing cracks and other tensile cracks are clearly visible.

Generally, during stages I and II, the simulated stress–strain curve (Fig. 13d) is similar to the experimental stress–strain curve (maroon line) apart from some fluctuations. After the peak stress (point c), the axial stress drops dramatically and the specimen enters the macroscopic fracturing stage (from point c to point e). During stage III, the numerical curve looks very similar to the experimental stress–strain curve.

At point d, the pre-existing flaws are linked through a combination of coplanar shear cracks or mixed cracks, and many rock bridges are cut by macro-cracks; the cross-shear failure plane is obvious at point d (Fig. 17d). After point d, the axial stress continues to decrease until reaching the residual strength (Fig. 17e).

Notably, as shown in Fig. 13, there is axial stress at zero axial strain. This phenomenon can be explained as follows: In PFC2D, the rock-like material is modeled as collections of particles (rigid circular disks). In order to make each particle contact with their neighboring particles, we have to input a locked-in stress during the process of model generation, and in this research, the locked-in stress value is 0.1 MPa. Therefore, in the numerical curves (Fig. 13), there is axial stress around 0.1 MPa at zero axial strain.

In addition, the rock specimen at the pre-peak stage in numerical simulations (Fig. 13c, d) seems to have a softer response than that in experiments, which is caused by the following reasons. The stress–strain curves of the specimens exhibit significant nonlinear characteristics, rendering it difficult to determine the Young’s modulus of the rock-like specimen. So the Young’s modulus was taken as the secant modulus in this research. During the calibration process, the numerical property was also determined in this way. At the same time, two other reasons may also contribute to this phenomenon: (1) the imbalance in the curing will have an effect on the mechanical properties of specimens, and (2) the PFC2D analysis is in two dimensions, while the characteristics measured in the rock-like specimens are three-dimensional.

3.3 Comparison Between the Experimental and Numerical Results

Figure 18 shows the peak failure stresses in the ubiquitous-joint rock-like specimens derived from the numerical simulations and laboratory experiments. Figure 18a–e shows that the numerical results have a similar trend to the experimental results. In the numerical simulation, for intersection angles of 15° and 30°, the peak strength of the specimens decreases, while α increases from 0° to 45° but increases when α widens from 45° to 75°. This trend matches the observation in the experiments.

Fig. 18
figure 18

Effect of joint-1 inclination angle α and intersection angle γ on UCSJ

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For the specimens with γ = 45° and 60° and 0° ≤ α ≤ 60°, the numerical result shows similar trends with increasing α. The specimens (S-30-45, S-30-60) obtain the highest UCSJ value when α = 30°. However, for γ = 75° the UCSJ is highest when α = 0° and decreases until α = 45°. For 45° ≤ α ≤ 75°, the simulated UCSJ value rises with increasing α. The experimental and numerical results show similar UCSJ trends for the specimens with γ = 45°, 60°, and 75°.

As mentioned above, the peak failure stresses of the numerical and experimental results show a similar trend, but also some differences. This may be due to imbalances in the curing and the heterogeneity of the specimens.

Figures 19, 20, 21, and 22 show the four typical failure modes obtained by PFC2D and the corresponding failure modes obtained for the same specimens in the rock-like material experiments. The numerical results compare quite well with the experimental results. For the stepped path failure pattern (S-0-15 and S-0-30) (Fig. 19), the overall failure of the specimen is caused by the propagation of tensile cracks. Notably, micro-cracks that appear in various locations in the numerical specimen cannot be observed by the naked eye in the experiment.

Fig. 19
figure 19

Stepped path failure mode comparisons between experimental and numerical results: a S-0-15; b S-0-30

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Fig. 20
figure 20

Planar failure mode comparisons between experimental and numerical results: a S-60-75; b S-75-75

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Fig. 21
figure 21

Shearing-I failure mode comparisons between experimental and numerical results: a S-60-30; b S-75-15

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Fig. 22
figure 22

Shearing-II failure mode comparisons between experimental and numerical results: a S-75-45; b S-60-60

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Figure 20 shows a comparison of the numerical and experimental results for the planar failure pattern specimen. In both the numerical and experimental results, most of the pre-existing flaws in the specimens are connected to neighboring flaws via tensile cracks or shear cracks. Thus, the specimen is cut into many blocks by pre-existing flaws and new cracks; this is the main characteristic of the planar failure pattern.

Comparisons between the numerical and experimental results of typical shear-I and shear-II failure patterns are shown in Figs. 21 and 22, respectively. In Fig. 21, for specimen S-60-30, the shear band is near the diagonal line; two similar shear bands can be observed in specimen S-75-15. The numerical results of the shear-I failure pattern compare quite well to the experimental results. In Fig. 22, although some of the blocks fell outside the specimen, the ‘cross-shear failure plane’ in the experimental result is very clear and agrees well with the numerical specimens.

4 Conclusions

Combined with the laboratory tests of rock-like material and numerical simulation (PFC2D), the peak strength, crack propagation, coalescence, and failure patterns of ubiquitous-joint specimens under uniaxial loading were investigated in this study. From the simulated and experimental results, the following conclusions can be drawn:

  1. 1.

    Under compression, tensile cracks or shear cracks will propagate and link with other cracks to form penetration. As loading continues, the pre-existing flaws link with neighboring cracks to form various types of failure patterns. The failure patterns of ubiquitous-joint rock-like specimens can be generally classified into four categories: (1) stepped path failure, (2) planar failure, (3) shear-I failure, and (4) shear-II failure.

  2. 2.

    The failure patterns of rock-like specimens are affected by the joint-1 inclination angle (α) and intersection angle (γ). Specimens S-0-15, S-0-30, and S-30-15 exhibited a stepped path failure pattern; the failure of these specimens was caused mainly by the propagation of tensile cracks. The planar failure pattern appears mainly in the specimens with γ = 75° such as specimens S-45-75, S-60-75, and S-75-75. The shear failure pattern is the most common failure pattern in the ubiquitous-joint specimens. Specimens S-45-60, S-60-60, S-75-30, and S-75-45 are typical examples of the shear-II failure pattern.

  3. 3.

    The joint-1 inclination angle α has a very high influence on UCSJ, and the strength of the ubiquitous-joint rock-like specimens showed different tendencies when α varied from 0° to 75°. The UCSJ values of specimens with γ = 15° and 30° show similar trends when α changes from 0° to 75°: decreasing as α increases from 0° to 45° and increasing as α increases from 45° to 75°. The UCSJ values for specimens with γ = 45° and 60° show similar trends when α changes from 0° to 75°. In contrast, for specimens with γ = 45° or 60°, the UCSJ increases as α increases from 0° to 30° and decreases when α rises from 30° to 75°. For specimens with γ = 75°, the highest UCSJ value occurs when α = 0°, while when α increases from 60° to 75°, the UCSJ value increases.