Rock masses with open-closed cross-flaws are common in nature. The open-closed cross-flaws usually control the strength of rock masses. However, studies about the influence of open-closed cross-flaws on cracking behaviors and mechanical properties of rock masses are rare. In this study, rock-like samples containing open-closed cross-flaws with different geometries were fabricated to be conducted uniaxial compressive tests. The cracks observed in the tests were classified to nine types, and two new crack modes were described and identified. Two failure modes, failure caused by tensile cracks and failure caused by the combined effect of tensile and shear cracks, were observed. The failure caused by the penetration of the rock bridge is not observed, indicating that the rock bridge is not a potential penetration path for the samples with cross-flaws. Experimental results show that, when α=0°, the peak stress decreases from 32.2 MPa to 17.0 MPa as β increases from 0° to 90°. When α=90°, the peak stress increases from 22.5 MPa to 40.0 MPa as β increases from 0° to 90°. The inclination angle of the open flaw has an obvious effect on the strength of samples. When the inclination angle of the open flaw is 0°, the peak stress is the lowest (17.0 MPa). When the inclination angle of the open flaw is 90°, the peak stress is the largest (40.0 MPa). The samples with an open flaw of large inclination angle tend to have great compressive strength. For samples with open-closed cross-flaws, the open flaw has a greater influence on the strength and failure mode than the closed flaw.

The rock mass is a multifractured body composed of rock blocks and a large number of crisscross structural planes in nature. Flaws in the rock mass are the key factor affecting the stability of rock mass. Natural jointed rock masses are liable to generate new cracks under secondary conditions such as engineering construction, artificial excavation, and blasting unloading, which can be simplified to jointed rock masses with different flaw properties and cross-distribution. The secondary open flaws interact with the primarily closed flaws, affecting the physical and mechanical properties of the jointed rock (Figure 1).

A large number of experiments have been conducted in precracked specimens containing single open flaw or parallel open flaws [18]. Uniaxial compressive tests were conducted on rock masses with overlapped open fissures [9]. Crack initiation, propagation, and coalescence were systematically investigated. Besides, uniaxial compression tests were carried out on rock-like specimens with various arrangements of two parallel open flaws [10]. Considering the flaw geometry, Lee and Jeon [11] investigated the crack coalescence of specimens with a combination of a horizontal open flaw and an inclined open flaw underneath. Fan et al. [12] studied cracking behaviors and deformation of rock samples with a single nonpenetrating flaw under uniaxial compression. A small number of scholars have carried out compression tests on rock specimens with closed flaws. Uniaxial compression tests were conducted on prismatic gypsum specimens with three preexisting closed flaws [13]. The result showed that the geometry and layout of preexisting flaws have influence on fracturing processes. Many attempts using numerical tests have been made to investigate the crack mechanism of fractured rock mass [1418]. Zhang and Wong [19] investigated the different effects of the flaw inclination angle, the ligament length, and the bridging angle on the coalescence patterns, coalescence stresses, and the peak strength of specimens containing two parallel open flaws. Liu et al. [20] used PFC to characterize the fracturing behavior of shale containing multiple cemented veins and bedding planes through numerical semicircular bend (SCB) tests. The typical mechanical interaction modes between the multiple preexisting flaws were studied from a microcosmic viewpoint by numerical simulation. Through numerical tests, some researchers claimed that the rock bridge is a potential penetration path for the samples with multifractures [11, 21, 22].

At present, there are some studies on the influence of open cross-flaws on the cracking behaviors and mechanical properties of rock mass. The strength and crack coalescence of rock-like material specimens with two open cross-flaws were discussed [23]. The crack behavior of PMMA specimens containing two 3D cross-embedded flaws was researched experimentally [24]. Crack propagation evolution of samples with different dip angles of cross open flaws was analyzed by numerical simulation [25]. In nature, open-closed cross-flaws in rock mass are very common, and they are considered to have an impact on the strength and failure pattern of rock mass. However, there are few studies on the rock mass with open-closed cross-flaws. It is necessary to conduct systematic research on jointed rock mass with open-closed cross-flaws with different geometries.

This study was aimed at studying the influence of the geometry of open-closed cross-flaws on crack behavior and mechanical property of samples by laboratory tests. Limestone-like samples containing open-closed cross-flaws with different geometries were fabricated. Mechanical properties and crack propagation mechanism of the samples containing open-closed cross-flaws under uniaxial compression were obtained. The effects of the inclination angle of the open flaw and the angle between open flaw and closed flaw on the mechanical properties and cracking behaviors of limestone-like specimens containing open-closed cross-flaws were analyzed.

2.1. Modelling Material and Its Mechanical Properties

Repeated tests are required to ensure the reliability of experimental results. To ensure that the samples are exactly the same, modelling materials are commonly selected to make rock-like samples [26, 27]. In this study, uniaxial compression tests were conducted on standard cuboids of dimensions 50mm×50mm×100mm containing two flaws. The flaws are crossed, and one is open and the other is closed. The rock-like material used in this study was the same modelling material selected by Le et al. [28], which is a mixture of standard cement, fine sand (the particle size is between 0.075 and 0.25 mm), and water, with a mass ratio of 1 : 2 : 0.4. Le et al. [28] conducted uniaxial compression tests, direct shear tests, and Brazilian tests to obtain the mechanical properties of the modelling material and found that the mechanical properties of modelling material are similar to those of limestone. The uniaxial compressive strength, elastic modulus, and Poisson’s ratio of rock-like materials are 39.2 MPa, 21.2 GPa, and 0.18, respectively. Therefore, samples made by this modelling material can be treated as limestone-like samples.

2.2. Specimen Preparation

The mixture made of standard cement, fine sand, and water with a mass ratio of 1 : 2 : 0.4 were poured into a mold box of dimensions 50mm×50mm×100mm. In previous studies, open flaws were often prepared by inserting steel inserts directly into the mixture [2931]. The solidification and expansion of the material can promote the slow closure of the thin crack; therefore, pulling out the thin metal sheet in advance can form the closed crack. Inserting a steel sheet to make flaws is simple to operate and can prepare multiflaw specimens in different combinations by flexibly adjusting the parameters such as the length, thickness, width, and pull-out time of the metal insert. Therefore, in this study, inserting a steel sheet into the material was used to prepare the cross-flaws. In this test, α and β are changed to produce different geometries of preset flaws (Figure 2). α represents the angle between the closed flaw and maximum principal stress, which is set to three values: 0°, 45°, and 90°. β represents the angle between open flaw and closed flaw, which is set to four values: 30°, 45°, 60°, and 90°. The length of the closed flaw and open flaw is 20 mm, and the thickness of the open flaw is 1 mm. Due to the complexity of the geometry of cross-flaws, the direct insertion of the steel sheet may lead to the deviation of the position of the preset flaws. Therefore, the process of inserting a steel sheet is improved to make sure the shape of the preexisting flaws can meet the test requirements. The improved method is described as follows: first is using a 3D printer to print a plastic cover. Cross-flaws are reserved on the cover. Figure 3(a) is a 3D printed plastic cover plate with cross-flaws, and Figure 3(b) is a schematic diagram of the assembled cover plate and steel mold. Second is pouring the mixture into the steel mold. Third is inserting a steel sheet with a thickness of 0.5 mm, a length of 20 cm, and a width of 20 mm as well as two steel sheets with a thickness of 1 mm, a length of 20 cm, and a width of 10 mm into the designated position (Figure 3(c)). After 2 hours, the steel sheet with a thickness of 0.5 mm was pulled out to form a closed fissure (Figure 3(d)). After 12 hours, the steel sheet with a thickness of 1 mm was pulled out, and the cover was removed (Figure 3(e)). Finally, the steel mold was disassembled and the specimens were removed, and the specimens were placed in water for 28 days for curing.

2.3. Uniaxial Compression Test on Samples

The testing system is shown in Figure 4. The microcomputer-controlled rock uniaxial test instrument was used to conduct the uniaxial compression test, the loading speed was 0.2 kN/s, and the stress change in real time was recorded by the pressure sensing device. The digital dial indicator recorded the vertical displacement during the test. A high-speed camera (60 frames/s) records the process of crack initiation, development, and coalescence during the test.

3.1. Deformation Behavior

The typical axial stress-displacement curves of samples under uniaxial compressive tests are shown in Figure 5. The loading process can be divided into four stages according to the crack initiation propagation and the deformation characteristics of samples.

Stage I: at this stage, the microcracks existing in the rock-like specimens close. This phase starts from the loading point to point A. As the stress increases, the microcrack closes, the axial displacement increases substantially, and the displacement and stress curves are not linearly related. No new cracks were observed in the specimens

Stage II: at this stage, cracks were observed at the tip of the preset flaws. This phase starts from point A to point B. After the microcracks existing in the rock-like specimens close, the compressive stress is transmitted to the cross-flaws, and the force on the flaws tip is continuously accumulated, which eventually leads to the generation of cracks. Different from the compression curve of the complete specimen, the stress-strain curve at this stage is not linear, presumably affected by the prefabricated flaws

Stage III: stage III is the crack propagation stage. This stage goes from point B to point C. With the continuous increase of pressure, new cracks continue to propagate toward the loading direction. In the prophase of this stage, the stress is almost linearly related to displacement. Approaching the peak stress point, the slope of the stress-displacement curve decreases significantly. It indicates that the deformation of the sample is about to reach its limit when it is close to the peak point, and the stress is no longer greatly increased as in stages I and II

Stage IV: postpeak stage starts from point C to point D. The cracks propagated to the edge of the specimen, resulting in complete failure of the specimen. At this stage, the stress decreases substantially as the displacement increases

3.2. Peak Strength and Crack Initiation Stress

Figure 6 shows the variation of the peak strength and crack initiation stress of the specimen with the angle between open flaw and closed flaw β. According to Figure 6, when α=0°, the peak stress decreases from 32.2 MPa to 17.0 MPa as β increases from 0° to 90°. When α=90°, the peak stress increases from 22.5 MPa to 40.0 MPa as β increases from 0° to 90°. Figure 6 also shows that, when α=0°, the crack initiation stress drops from 13.0 MPa to 8.1 MPa as β increases from 0° to 90°. The crack initiation stress raises from 10.6 MPa to 20.2 MPa as β increases from 0° to 90° in the condition that α=90°. When α=0° and β=90°, the inclination angle of the open flaw is 0°, and the peak stress and the crack initiation stress are the lowest. When α=90° and β=0°, the inclination angle of the open flaw is 90°, and the peak stress and crack initiation stress are the largest. The samples with an open flaw of a larger inclination angle usually have greater compressive strength. It means that for the samples with cross-flaws, the inclination angle of the open flaw has an obvious effect on the strength of samples.

When α=0°, the final failure of the specimen is caused by the propagation and coalescence of the cracks initiated from the tips of the open flaw. This shows that the leading role in the failure of the specimen is the open flaw. When the inclination angle of the open flaw is 0° (β=90°), the compressive stress on the open flaw surface is the largest, the opening degree of the preexisting open flaw reduces, and large lateral deformation of the specimens occurs, resulting in the occurrence of tensile cracks at the tip of the open flaw. The specimen is damaged by tensile force and the strength is relatively low. With the decrease of β, the compressive stress at the tip of the open flaws decreases and the shear stress increases, the lateral deformation of the specimen decreases, the specimens are damaged by tensile and shear cracks, and the strength of the specimens increases. Type I fracture toughness is required to produce tensile crack, and type II fracture toughness is required to produce shear crack. Type II fracture toughness of rock is greater than type I fracture toughness. Therefore, the specimen damaged by tensile cracks usually has a relatively low peak strength, and the specimen damaged by tensile cracks and shear cracks usually has a relatively high peak strength.

When α=90°, the closed flaws are perpendicular to the direction of the maximum principal stress, and the closed flaws mainly bear compressive stress. Because the flaws are closed, it will not produce large vertical deformation and large lateral expansion. The strength of the specimen is mainly affected by the open flaw. With the increase of β, the compressive stress on the open flaw surface decreases. When β=30°, 45°, and 60°, the opening degree of flaw decreases during the loading process, leading to different degrees of lateral deformation. When β=90°, the opening degree of the open flaw is almost unchanged, the lateral deformation is also the smallest, and the strength of the specimens is the largest.

3.3. Displacement at Peak Stress and Displacement at Crack Initiation Stress

The variation of the displacement at peak stress and displacement at crack initiation stress of the specimens is shown in Figure 7. When α=0° and β=90°, the displacement at peak stress and displacement at crack initiation stress are the lowest. When α=90° and β=0°, the displacement at peak stress and displacement at crack initiation stress are the largest. When α=0°, the displacement at peak stress decreases from 0.83 mm to 0.40 mm as β increases from 0° to 90°. When α=90°, the peak stress increases from 0.63 mm to 0.96 mm as β increases from 0° to 90°. Figure 7 shows that, when α=0°, the displacement at crack initiation stress drops from 0.34 mm to 0.2 mm as β increases from 0° to 90°. The displacement at crack initiation stress raises from 0.29 mm to 0.51 mm as β increases from 0° to 90° in the condition that α=90°. The displacement at peak stress of the specimens is much larger than the displacement at crack initiation stress, indicating that the specimens experience a large axial deformation from crack initiation to failure. Comparing Figures 6 and 7, it can be seen that the specimen with greater strength has a greater displacement at peak stress, indicating the specimen with greater strength tends to have a greater ability to withstand deformation.

Figure 8 shows crack propagation behaviors of samples captured by a high-speed camera. Based on the location and mode of crack initiation in Figure 8, the cracks were classified to nine types, as shown in Figure 9. Seven types (mode 1, mode 2, mode 3, mode 4, mode 6, mode 7, and mode 8) have been found in the previous studies, in which a detailed description of these types is available [32], so there are not repeated interpretations in this study. Mode 5 and mode 9 are two types of cracks that have not been found and identified in previous studies. It is noted that mode 5 crack is a tensile one that does not start from the area near the preexisting flaw but far away and then extends to the area near the tip of the preexisting flaw. Mode 9 crack describes a shear crack that does not initiate from the tip of preexisting flaw but away from the tip, making it different from the shear crack of mode 8. As the occurrence order of cracks could not be quickly captured in Figure 8, schematic sketches of crack propagation behaviors for samples are drawn in Figure 10, and these symbols in Figure 10 indicate the type and occurrence order of crack. For example, the crack expressed by T4-3 is a tensile crack of mode 4 and the third one to occur.

It can be seen from Figures 8 and 10 that both tensile stress and shear stress are applied to the surface of the open flaw when α=0° and β=30°, 45°, and 60°. As the fracture toughness of type I is less than that of type II, tensile cracks are the first to develop from the tip of the preexisting flaw. With the increase of shear stress, shear cracks subsequently arise, and the specimen ends up in failure with the combined effect of tensile and shear cracks. When α=0° and β=90°, there is mainly compressive stress subjected to the open flaw and tensile stress to the closed flaw during the early loading, so the closed one is the first to develop a tensile crack at the tip. However, as the normal stress increases, the opening degree of the open flaw becomes less and less, resulting in an increase in the lateral deformation of the specimen, which leads to the initiation of a tensile crack at the tip of the open flaw. The development and propagation of tensile cracks eventually make the failure of the specimen. As shown in Figure 8, there is no significant decrease in the opening degree of the open flaw during loading when α=0° and β=30° and 45°; however, a clear reduction in the opening degree of the open flaw was found when α=0° and β=60° and 90°.

Figures 8 and 10 show that, when α=45° and β=30°, 45°, and 60°, the inclination angle of the open flaws is relatively small (15°, 0°, and 15°, respectively). During the early process of loading, the open flaw is predominantly subjected to pressure and the closed flaw is predominantly subjected to tensile stress, which contributes to tensile cracks initiated from the tip of the closed flaw first. There are only tensile cracks, the inclination angle of both the open and closed flaws are 45°, and tensile cracks first occur at the tip of the open flaw. It is indicated that the stress concentration at the tip of the open flaw is greater than that of the closed one, and the stress intensity factor at the tip of the open flaw is first to reach fracture toughness of type I. When α=45° and β=30°, 45°, and 60°, there is an obvious decrease in the opening degree of the open flaw during compression.

When α=90° and β=30°, 45°, 60°, and 90°, the closed flaw is perpendicular to the direction of the maximum principal stress, so it is mainly subjected to compressive stress and no tensile crack is found at the tip. But the open flaw is subjected to both tensile and shear stress, and tensile cracks initiated from the tip earlier than shear cracks because the fracture toughness of type I is less than that of type II. When α=90° and β=30°, 45°, and 60°, it is clear that the opening degree of the open flaw decreases with the action of compression.

Table 1 summarizes the types of cracks observed in all the specimens. Table 1 shows that most of the cracks that develop first are tensile cracks of mode 1 or mode 2 and secondly far-field tensile cracks of mode 4 or mode 5. As the axial stress increases, shear cracks of mode 8 or mode 9 begin to initiate from the tip of the preexisting flaw in some specimens. It can be seen in Table 1 that no shear cracks are found in any of the specimens with α=90° and no antiwing cracks of mode 3 are observed in any of the specimens with α=30°. The failure pattern of the specimens can be divided into two categories according to Figure 8 and Table 1: failure caused by tensile cracks and failure caused by the combined effect of tensile and shear cracks. Only tensile cracks but no shear cracks initiate from the tips of closed flaws, but both tensile cracks and shear cracks are observed at the tips of open flaws, which indicates that the open flaw plays a decisive role in the failure pattern of the specimen.

In this study, no rock bridge was penetrated in all specimens, and the cracks generated at the tip of the open and closed flaw did not overlap. This is different from the crack propagation pattern obtained by previous studies [20, 33, 34]. The previous studies show that, for rock-like samples with two nonintersecting flaws under uniaxial compressive stress, rock bridges are potential through-paths. However, in this study, the rock bridge with cross-flaws is not a potential through-path, which means that if the cross-flaws are encountered in the engineering project, it is not necessary to pay special attention to the failure caused by the damage of the rock bridge with cross-flaws.

In this study, it can be seen that many samples (sample 0_45, sample 45_45, and sample 90_60) have parallel tensile cracks with a very short extension. These tensile cracks are generated when the samples reach the peak strength. The arrangement of these tensile cracks is very similar to the echelon tension joints in nature (see Figure 11). It can be inferred that the causes of these tensile cracks are similar to those of the echelon tension joints, which are all tension joints formed under shearing action.

For the samples with cross-flaws, when α=90°, the variation law of the strength with the inclination angle of the open flaw is consistent with the variation law of the strength of the specimen containing single open flaw with the inclination angle of the single flaw. The description of the variation law of the strength of the specimen containing single open flaw with the inclination angle of the single flaw can be found in Jin et al. [35]. It shows that for the cross-flaw, the effect of the closed flaw on the variation of the strength with the inclination angle of the open flaw is very small. Moreover, for samples with cross-flaws, only tensile cracks occurred at the tip of the closed flaw, and no shear cracks appeared. Both tensile and shear cracks were observed at the tip of the open flaw. It shows that, for the samples with cross-flaws, the open flaw has a greater influence on the strength and failure mode than the closed flaw.

In this research, rock-like samples with cross-flaws of different geometries were fabricated. The angle between closed flaw and maximum principal stress and the angle between open flaw and closed flaw were set to different values. Uniaxial compressive tests were carried out on these samples. The influence of geometries of cross-flaws on the cracking behaviors and mechanical properties of the samples were studied. The following conclusions can be drawn:

  • (i)

    The geometry of cross-flaws affects the strength of the sample. The samples with an open flaw of a larger inclination angle usually have greater compressive strength

  • (ii)

    The cracks observed in the samples are described and classified, and nine modes of cracks are obtained, of which mode 5 and mode 9 cracks have not been fully reported in previous studies

  • (iii)

    Two failure types are summarized: failure caused by tensile cracks and failure caused by the combined action of tensile crack and shear crack. The failure of the sample caused by the penetration of the rock bridge is not observed, indicating that the rock bridge is not a potential penetration path for the samples with cross-flaws

  • (iv)

    For rock samples with cross-flaws, the influence of the open flaw on the strength and failure mode of samples is much greater than that of the closed flaw, and the open flaw plays a leading role

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

National Natural Science Foundation of China (Nos. 42007256 and 41672258) and the China Postdoctoral Science Foundation (No. 2021M690865) are greatly appreciated for funding this project.

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