#### Abstract

Failure of rocks is commonly induced by compressive and shear coupling loading. Knowledge of the mechanism and process of deformation and failure of rocks under compressive shear loading condition is an important basis for the study of stability in rock engineering. Based on the nonlinear fractal theory, it is possible to examine the evolution rules of fractures in mudstone under compression shear load and the fractal characteristics of broken blocks using the shear compression test with variable angles of mudstone specimens in natural conditions. This research shows that the cohesion and friction angle parameters of rock samples are achieved by draw Mohr’s strength envelope according to the test date of variable-angle shear compression test. It also shows that the shape of load-displacement curves of rocks can be divided into four stages: compaction, elastic, plastic, and fracture, and the curve can accurately represent the transformation and breakage characteristics of rock during shear fracture. And the distribution of broken blocks shows a strong statistical resemblance to the fractal distribution, and the fractal dimension is able to reflect the distribution characteristics of broken blocks. With increasing the shear angle, the fractal dimension of broken blocks decreases in a logarithmic relationship.

#### 1. Introduction

Rock compression and shear fracture are the most common failure modes in nature and the subject of the rock mass-engineering field. Incidents include roadway and pillar cave-ins, landslides, and rock bursts, all of which can cause disasters and economic losses. The fracturing and crushing of the top coal caving of a thick coal seam should make full use of rock compression and shear fracture. The prediction, prevention, and use of rock compression and shear fracture have long been important issues in the engineering field. In an effort to reveal the inherent mechanisms of rock compression and shear fracture, many studies have analyzed these phenomena from different aspects: shear-failure features of rock material [1–5], evolution rules and mechanical mechanisms of microcracks and microdefects in rocks [6], local deformation and fracture characteristics [7], mathematical and mechanical approaches to clarify damage and failure mechanisms [8–10], determination of fracture criteria and deformation fracture mechanisms [11, 12], and numerical simulations of the rock fracture process [13].

The compressive strength of rock material is much larger than its tensile strength because of the compressive hardening, dilatancy, and isobaric yielding properties of rock material. Most rock fractures are shear failures because the principal stress is compressive in most circumstances in rock mass engineering. The shear strength of rock is one of the most important indicators of rock mechanical properties and is generally determined through specialized shearing tests. The shear strength of rock can be divided into several different kinds of strengths according to the different loading methods used in various shear tests [14].

The first kind is called lion-loaded shear strength. It equals the shear strength of rock when there is no normal loading on the shear plane. The adhesive force between rock particles on the shear plane is equal to the lion-loaded shear strength. The second kind is what is usually called shear strength, which is the maximum shear stress that the specimen can bear when normal loading is imposed to the shear plane. The shear strength of rock is a variable quantity related to normal stress upon the shear plane when the specimen is destroyed. The third kind is quality frictional strength. This is the maximum shearing stress that the specimen can bear under certain normal compressive stresses when there are already fractured faces in the specimen. It is often used to determine the shear strength when there are weak structural planes in the specimen.

Surrounding rock is always situated in a certain spatial state of stress fields. Although shear failure is the basic failure mode in rock engineering structures, such a failure is usually not caused by shear stress alone. It usually occurs when shear stress surpasses the certain limit equilibrium and the shear stress is related to normal stress. Therefore, it is necessary to describe the shear strength characteristics of rock under different stress states because shear strength alone is not enough to describe the stability of the rock.

The shear strength of rock is usually determined by drawing a Mohr strength envelope using the triaxial test [15]. However, this method is complex, and the equipment is expensive. Moreover, the calculated amount is numerically large, and the equipment is difficult to operate, especially given that the specimen must be formed into standard cylinders because the test is carried out in a bucket of confining pressure. These requirements are too difficult for rocks in coal measure strata because of the fluffy texture and fracture development of coal petrography. Bedding, joints, and cleats are well developed in coal petrography, and it is usually considered to be an orthotropic material. Such materials cause great difficulties in the gathering, processing, and testing of specimens. The cohesion and internal friction angle of such materials are important mechanical properties, which are required to have certain values in a large number of practical projects. In the variable-angle shear test, the indenter leads only to compressive stress and shear stress on the failure surface, thus avoiding the inadaptability of Mohr’s theory of failure in tensile crack areas. Moreover, Mohr’s failure theories consider the maximum and minimum principal stress irrespective of any intermediate principal stress, which leads to simplifying three-dimensional stress into plane stress. By this means, it is possible to obtain satisfactory results using the variable-angle shear die test instead of the triaxial test.

Under the guidance of the nonlinear fractional theory, and using the variable-angle shear compression test, this study will explore the fracture evolution laws of the mudstone compression shear-failure process and the fractal properties of rupture blocks.

#### 2. Variable-Angle Shear Compression Test

##### 2.1. Experimental Specimen

The specimen, which was taken from the roof of the 3410 tail entry of the mine at Gaoping (in the province of Shanxi, China), was a continental classic sedimentary rock from the lower Permian Shanxi formation. The specimen was sealed in the tail entry and processed into 15 cylindrical samples, each 68 mm in diameter and 70 mm in height, which were then sealed with wax in the laboratory. An example of the X-ray diffraction patterns generated by the samples is shown in Figure 1. The X-ray diffraction patterns could be used to qualitatively analyze the samples by comparing them with PDF cards of minerals using MDI Jade 6.0 software. Using these qualitatively analyzed results, the K content of all minerals was found in a standard PDF card, quartz was chosen as a reference, and the K content of other minerals was transformed. The mass fraction of the test mineral, , can be calculated with the following equation [16]:where *i* is the reference mineral, *j* is a test mineral, *N* is the total amount of all minerals, is the diffraction intensity of the strongest peak of the test mineral, and is the reference intensity of the test mineral.

According to the calculation, the contents of illite, kaolinite, quartz, and anorthite in the sample were 45%, 10%, 38%, and 7%, respectively. It is obvious that the main clay constituents are illite and kaolinite.

##### 2.2. Preparation of the Test Specimen

The key to the variable-angle shear compression test is to ensure that the rock specimen bears stress evenly and sustains damage along a predetermined shear plane. To this end, the machining accuracy must be strictly controlled, with adjacent surfaces perpendicular to each other at a deviation less than 0.25° and opposite sides parallel to each other at a deviation less than 0.05 mm. Too much deviation will cause uneven distribution of stress on the shear plane. Moreover, any stress concentration will lead to local failure and earlier failure of the entire specimen. To reduce the effect of individual data in the test, three different specimens should be tested at every angle, and every group of specimens should be tested at four or five different angles. The specimen is presented in a natural state.

##### 2.3. Experimental Apparatus

Variable-angle shear compression tests were performed on servo-controlled mechanical test equipment TYT-600 manufactured by Jinli Test Technology Co. Ltd. (Changchun, China). The maximum load capacity of the test equipment is 6.0 × 10^{4} kg, the minimum displacement rate is 1.0 × 10^{−3} mm/s, and the loading rates can be changed under the control of the servo-control system.

Using different shear dies with different angles, the angle between the specimen indenter and the plane can be changed in the variable-angle shear press test. If the angle is too large, a force-coupling effect will be generated, which can lead to tensile stresses. Moreover, under these circumstances, the die with the rock sample will turn over more easily. If the angle is too small, the fracture may not follow the required section as a result of too heavy pressure. The wedge shear test uses a shear die from four different angles (40°, 45°, 50°, and 55°) at a load-displacement speed of 2.0 × 10^{−3} mm/s.

##### 2.4. Experimental Data Processing

The normal stress and the shear stress on the fracture surface of sample are determined as follows:where is the failure load of the sample; is the area of the shear surface; is the angle of the shearing mold; is the friction coefficient of the roller, , where is the number of rollers; and is the diameter of the roller.

Obviously, the point with these two stress values as coordinates corresponds with a certain breaking point on the Mohr strength envelope. Therefore, it is possible to draw the Mohr strength envelope through the data from a group of variable-angle shear tests and thereby determine the cohesion and the internal friction angle of the rock material. Least-squares linear regression is normally used to determine the cohesion and the internal friction angle of the rock material:where is the number of samples and and represent the normal stress and shear stress of the sample, respectively, on the shear-failure surface.

##### 2.5. Experimental Results

The test results are listed in Table 1, and curves based on them are shown in Figure 2. Through the least-squares method, the linear regression equation was obtained, and the cohesion and the friction angle of the rock sample were determined to be 4.74 MPa and 21.9°, respectively.

#### 3. The Fracture Evolutionary Rules of Compressive Shear Damage

The load-displacement curve of rock reflects the failure characteristics of macroscopic deformation, which occurs in the process of pressure shear. Macroscopic deformation is related to rock type, initial damage status, stress states, and load path. In the interests of brevity, only specimen No. 2 is considered here, with an angle of 55°, as shown in Table 1. The fracture evolutionary rules of compressive shear damage in rocks will now be analyzed taking into account the load-displacement curve of compressive shear damage.

Figure 3 shows the load-displacement curve of the rock. Figure 4 shows the production and development process of fractures in the specimens, corresponding to the points marked in Figure 3. As apparent in Figure 3, the shape of the load-displacement curve of compressive shear damage can be divided into four stages: compaction (I), elastic (II), plastic (III), and fracture (IV), which is similar in shape to the complete stress-strain curve of rock under uniaxial compression.

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Fracture first appears in the plastic stage, accompanied by rock sound, when two cracking points appear at the contact point of the specimen and the die. As the load begins to accumulate, the fractures extend at a speed of 1.25 × 10^{3} *μ*m/s in the direction of 42° to the shear plane and a speed of 2.83 × 10^{3} *μ*m/s in the direction of 16° to the shear plane (these fractures can be seen in Figure 4(b)). Gradually, they extend to become a penetrating crack, which can be seen in Figure 4(c).

After this, the load decreased rapidly, and new cracks appeared along the direction of the diagonal line of the die (the new cracks can be seen in Figure 4(d)). Gradually, these cracks extended to become principal penetrating cracks. The load may increase slightly in this process because of frictional resistance along the failure surfaces. Meanwhile, large numbers of new cracks were produced, extended, and stretched in the specimen. With the addition of cavities caused by the slipping of fracture surfaces, the volume expanded to become much larger than before.

What needs to be explained is that the ultimate failure developed along the direction of principal stress (Figures 4(e) and 4(f)) rather than the expected longitudinal section (Figure 5). However, this phenomenon does not affect the reliability of the test. According to the definition of shear strength, the specimen can bear the maximum shear stress when loading is imposed normal to the shear plane. The maximum shear stress appears at the maximum load point C, and specimen failure develops along the longitudinal section (Figure 4(c)). Moreover, a fracture which is open along bedding plane orientation can be seen on the top corner of the sample (Figure 5(b)). This fracture development is caused by the broken block that rotates about the contact between dies and samples under compression shear load.

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In fact, the loading capacity of the rock consists of cohesion and internal friction, and friction is generated only when there is relative slippage. When stress reaches the limit of cohesion, yield slippage occurs and generates friction, and cohesion decreases with the increase of friction. This phenomenon usually occurs from the ends towards the inside. Stress will be released after the slippage, tension fracture will occur as a result of the instability of the shear fracture, and new cracks will appear at the same time. Intensive damage zones will be formed as a result of the interaction and combination of new cracks, which will lead to the phenomenon of strain localization and finally the occurrence of macroscopic fractures in the direction of the principle load stress.

#### 4. Fractal Characteristics of Compress Shearing Damaged Blocks

##### 4.1. Description of Fractals

Under compressive and shear stress, microcracks constantly appear, develop, extend, and connect. In fact, it is such a nonlinear kinetic process which finally leads to compression and shear damage in rocks. Both the distribution of cracks and the scale of the broken blocks show fractal characteristics and strongly resemble each other statistically [17, 18].

The fractal dimension is able to reflect the crack distribution and the scale of the broken-block distribution, which can be used as a standard to represent the fracture characteristics of rocks.

In fractal theory, the relationship between the length and the yardstick can be expressed as follows [19, 20]:where is a constant and *D* is the fractal dimension value of fractal curves, which is also a constant.

When the measured equations for fractal curves are extended to *n* dimensions, Equation (4) becomes

Equation (5) applies to measured fractal curves, fractal area, and fractal volume. *G* and refer to a line when , to an area when , and to a volume when .

Assuming that the density of the fracture block is constant, the weight measured by the sieving method can be used to study the distribution law. The relationship between the weight and the size of a fracture can be described as

Logarithm of both sides of Equation (6) can be obtained to givewhere is a constant; if the size-weight distribution of the fracture block is fractal, it should follow the distribution law in Equation (7) and should be expressed as a linear function in double logarithmic coordinates, with a gradient of .

In this theory, the fractal dimension of 0 is for a dot, i.e., rock mass is intact without crack, and not failure; fracture block is an ideal cubic particle when *D* = 3, both nothingness in reality. Therefore, the fractal dimension value of the size distribution of a fracture block ranges between 0 and 3. The weight of blocks per unit size is equal when ; the weight of larger blocks is relatively greater when , which will benefit stability control in geotechnical engineering; and the weight of smaller blocks is relatively greater when , which is harmful to stability control in geotechnical engineering.

##### 4.2. Distribution of Compression Shear-Damaged Blocks

Figure 6 shows the distribution curve of the fracture block. Table 2 gives the fractal dimension and the relationship of compression shear-damaged blocks in Table 1. From Table 2, it is apparent that the value of is between 0.9603 and 0.9963. This means that the value of the fractal dimension can represent the distribution characteristics of compression shear-failure blocks.

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Figure 7 shows the relationship between the value of the fractal dimension and the shear angle. From Figure 7, it is clear that with the increase of the shear angle, the fractal dimension of the fractured blocks decreases in a logarithmic relationship. In other words, fractured blocks tend to be relatively larger. The reason for such a result is that as the shear angle increases, the dominating reason for the fracture changes from tensile fracture to the instability of shear failure in the shear plane. Actually, the initial cracks formed in tensile fracture will not develop into large cracks. These cracks deform, deflect, and generate more small cracks, but finally some of the small cracks change into large cracks extending along the shear stress direction. Moreover, the level of granularity is relatively small. For example, a geologic shear fault zone is made up of a crush zone instead of a crack. This is because small cracks appear first, but as their numbers begin to increase, a large number of small cracks coalesce into one larger crack.

However, the initial cracks parallel to the shear plane change at a larger level of granularity. The specimen in the single shear test showing fracture along the shear plane could be a special case of small cracks extending into larger cracks in shear fracture.

#### 5. Conclusions

Based on analysis of the experimental results presented in this study and examination of related research reports, certain conclusions can be drawn as described below.(1)The indenter produces only compressive stress and shear stress on the failure surface in the variable-angle shear compression test, which makes it possible to avoid the inadaptability of Mohr’s failure theory when considering the tensile fracture zone. The coordinates of the points on the strength envelope of the rock represent the normal stress and the shear stress of the rock when it fractures. It is possible to determine the Mohr strength envelope using data from a series of variable-angle shear tests, after which the cohesion *c* and the internal friction angle can be determined.(2)The shape of the load-displacement curve of the rock can be divided into four stages: compaction (I), elastic (II), plastic (III), and fracture (IV), which makes this curve similar in shape to the complete stress-strain curve of rock under uniaxial compression. The curve can accurately represent the transformation and breakage characteristics of rock during shear fracture.(3)The shear failure of rocks is a nonlinear kinetic process. The distribution of broken blocks shows a strong statistical resemblance to the fractal distribution, which is in accordance with fractal rules. In addition, the fractal dimension is able to reflect the distribution characteristics of broken blocks. With increasing shear angle, the terminal fracture mode changes from tensile fracture to shear fracture and the fractal dimension of broken blocks decreases in a logarithmic relationship.

#### Data Availability

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

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant no. 51674173), by the Shanxi Province Science Foundation (grant no. 201601D102038), by the Research Project Supported by Shanxi Scholarship Council of China (grant no. 2016-040), and by the Shanxi Province Key Research and Development Program (international cooperation and exchange; grant no. 201803D421078).