Fractal Characteristics of Coal and Sandstone Failure under Different Unloading Confining Pressure Tests

State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China College of Water Conservancy and Civil Engineering, Shandong Agricultural University, Tai’an 271018, China College of Mining and Safety Engineering, Shandong University of Science and Technology, Qingdao 266590, China Key Laboratory of Mining Disaster Prevention and Control, Qingdao 266590, China


Introduction
In the past decade, large-scale excavation of rock mass for engineering projects under high in situ stress conditions has presented rock mechanics problems for engineering geologists involved in infrastructure construction and mining in China. General engineering practice shows that large-scale underground excavation can lead to unloading of the rock mass. In particular, the mechanical behavior of the highly stressed rock under loading is significantly different from that under unloading conditions [1,2]. Mines in China are extracting coal from deeper mining depths, and coal mines enter the deep mining stage gradually. is results in a series of adverse conditions, including high rock pressure, large water inflow, and high ground temperature.
us, it is important to develop deep coal resources safely and efficiently [3][4][5][6]. In the coal mining process, confining pressure is unloaded from the rock mass, which often leads to coal mine disasters such as structural instability and rock burst [7,8]. erefore, there is an urgent need to investigate the mechanical behavior of coal and rock during the unloading process. To date, researchers have analyzed various factors affecting the unloading mechanical behaviors of coal and rock, including the unloading path, unloading rate [9,10], and initial unloading confining pressure [11,12]; many significant results have been obtained [13][14][15].
Fractal geometry is used to describe irregular and disorderly phenomena in nature. Since Mandelbrot firstly established fractal geometry in 1982, our understanding of natural phenomena has developed beyond conventional geometry such as lines (one-dimensional), surfaces (twodimensional), and bodies (three-dimensional). Moreover, the fractal geometry has been shown to be an efficient method for studying rock fragmentation [16]. Li et al. [17] discussed the relationship between the fractal characteristics of cracks in the rock and rock burst. ey suggested that low fractal characteristics could promote rock burst. Wang et al. [18] found that the fractal dimension of rock fragmentation reflects not only the fracture properties but also the stress state of rock and the loading and unloading paths. Many studies of the fractal characteristics of rock failure have been published (e.g., [19,20]). However, few experimental studies have been carried out on the fractal characteristics of rock failure under triaxial unloading confining pressure. For example, He et al. [21] simulated rock burst caused by excavation under high-stress conditions by unilaterally unloading granite in a 3D stress state. ey also analyzed the fractal laws of rock burst fragmentation in detail.
In this study, we analyzed surface cracking and fragmentation of coal and sandstone by adopting fractal theory. To illustrate the effects of the unloading rate and initial unloading confining pressure on the fractal dimension of surface cracking and fragmentation in coal and sandstone under unloading conditions, a series of triaxial unloading confining pressure tests with different unloading rates and initial unloading confining pressures were conducted using an MTS815.02 electrohydraulic servo rock mechanics testing system. We focused on addressing the fractal dimension of surface cracking and fragmentation in coal and sandstone under different conditions of unloading. Our investigation can provide guidelines for evaluating the stability of rock mass and predicting the failure properties of coal and sandstone during underground excavation under high in situ stress conditions. Our results also contribute to the theoretical understanding of the failure response and mechanism of coal and sandstone under high-stress conditions during loading and unloading.

Sample Selection.
e coal and rock samples used in the tests were taken from the No. 3 coal and its sandstone roof in Yangcun Coal Mine, Jining, Shandong, China ( Figure 1). e structure of No. 3 coal is relatively simple, with 1-2 layers of the intercalated rock comprising carbonaceous mudstone and pyrite nodules, with a thickness of 0.02-0.44 m. e direct roof of No. 3 coal is composed mainly of sandstone and sandy mudstone, overlain by mudstone, with a general thickness of 0.35-11.36 m.
According to engineering rock mass test standards, the extracted coal block was cut into 50 × 100 mm (diameter-× height) cylinders; three samples were prepared for each test run, with a total of 102 samples, including a number of spares. Before testing, the samples were firstly subjected to ultrasonic testing, and the specimens that showed high wave velocities were removed, leaving those specimens with similar velocities, as shown in Table 1.

Test Program.
e tests were carried out using an MTS815.02 electrohydraulic servo rock mechanics test system. e testing system can meet the test requirements under a variety of complex loading and unloading paths. e test plan in this study was the same as that of Yang et al. [5] (see Figure 2). And the testing apparatus and posttest samples are shown in Figure 3.
e results of the triaxial unloading confining pressure tests with different unloading rates and initial unloading confining pressures are shown in Tables 2 and 3, where ε 1 is the axial strain (MPa), ε 3 is the radial strain (MPa), σ 3 is the initial unloading confining pressure (MPa), v σ3 is the unloading rate (MPa·s − 1 ), σ 1 -σ 3 represents the principal stress difference when the sample is at peak stress (MPa), and σ 3 ʹ is the confining pressure when the sample fails (MPa).

Calculating the Fractal Dimension.
To date, considerable methods have been used for obtaining the fractal dimension such as the Hausdorff dimension, information dimension, and box dimension. e box dimension, which is adopted in this paper, can be expressed as follows: where D s is the fractal dimension of the surface cracks, r is the size of a box covering the fractal figures (m), and N(r) is the number of boxes. In practical applications, because r is finite, the scaling rule between N(r) and r needs to be determined. e box dimension method can be used in both simple and complex cases.

Obtaining the Surface Crack Image.
To obtain an image of the surface cracks in the rock samples after the triaxial unloading confining pressure tests, a transparent film paper was attached to the surface of the rock sample and the outline of the surface cracks was traced with an oil marker pen, as shown in Figure 4. en, the surface crack sketch was scanned and converted into a digital image ( Figure 5).

Fractal Characteristics of the Surface Cracks.
e D s curve for the tested sample calculated by the box dimension method is illustrated in Figure 6. e slope of the curve represents the fractal dimension. e correlation coefficient of the fitting curve lgN(r) − lg(1/r) is about 0.94, which indicates that the surface cracks of the failed sample have fractal characteristics. D s in the coal and sandstone samples under different unloading conditions is shown in Table 4.
From Figure 7, it can be seen that, at the same unloading rate, D s in the coal and sandstone samples decreased with 2 Advances in Materials Science and Engineering the initial unloading confining pressure (Figure 7(a)). is indicates that when the initial confining pressure is relatively low, the samples undergo splitting and shear failure; when the initial confining pressure is higher, the samples are mainly in shear failure. e shear failure of the samples is more obvious when the rock is under a higher confining pressure. e relationship between D s in the coal and sandstone samples and the initial unloading confining pressure can be expressed by two linear equations as follows: where D Coal.s represents D s of the coal samples and D Sandstone.s represents D s of the sandstone samples. In Figure 7(b), at the same initial unloading confining pressure, D s in the coal and sandstone increases with the unloading rate. is is because when the unloading rate is relatively high, the failure of the sample is accelerated, and the stored energy in the sample cannot be expended in time, which results in rapid energy release when the sample fails.

Advances in Materials Science and Engineering
In this respect, the failure form of the rock sample under high unloading rate is similar to that under uniaxial compression. e relationships between D s in the coal and sandstone samples and the unloading rate can also be expressed in the form of linear equations as Equations (2)-(5) suggest that the surface cracks of different lithologies show different fractal characteristics. Moreover, the fractal dimension of the surface cracks in coal is higher than that in sandstone. For example, when the initial unloading confining pressure is 7 MPa and the unloading rate is 0.05 MPa/s, the D s values in the coal and sandstone samples are 1.07 and 1.03, respectively. us, the lithology has a notable effect on the fractal characteristics of the surface cracks, and a rock sample with more internal fissures presents more surface cracks when the sample fails compared with a sample with less fissures under the same unloading conditions.

Calculating the Fractal Dimension.
e fractal dimension of the fragmentation can be calculated by the following method.
Assuming the mass of the rock fragments follows the Weibull distribution, we can use the following equation based on Weibull statistics: where M(r) is the mass of the fragments that are smaller than r (kg), M 0 is the total mass of all the rock fragments within the scale criterion (kg), ε is a parameter related to the average size of the rock fragments, and k is the slope of the fitted logarithmic curve. Based on equation (6), we obtain To describe D f , the fractal dimension of fragmentation in coal and sandstone, we derive Figure 3: Testing apparatus and rock samples after testing.
e values of M(r) and M 0 were obtained by sieving the fragments and calculating the value of k. In this study, we set a fragmentation threshold of 19 mm when calculating D f . us, only fragments smaller than 19 mm were used for the calculation, while the effect of the larger fragments on the fractal dimension was neglected. e sieved fragments of the coal and sandstone samples are shown in Figures 8 and 9, respectively.

Fractal Characteristics of Fragmentation.
e mass fractions of the fragments for the different fragment scales derived from the coal and sandstone samples under different unloading conditions are shown in Tables 5-7. e lgr − lg(M(r)/M 0 ) logarithmic curves under different unloading conditions are shown in Figures 10 and 11. e results of the fitting slopes of the logarithmic curves and D f are shown in Tables 8 and 9. Figure 12 shows the effects of the unloading condition on D f . Figure 12(a) shows that, at the same unloading rate, D f in the coal and sandstone samples decreases as the initial unloading confining pressure increases.
is can be explained as follows: when the initial confining pressure is relatively high, the samples are mainly in shear failure, which results in a higher number of larger fragments in the rock when the sample fails. In other words, the fragmentation of the samples is less homogeneous when the initial confining pressure is higher. Here also, the relationship between D f in the coal and sandstone samples and the initial unloading confining pressure can be expressed by two linear equations: where D Coal.f represents D f in coal and D Sandstone.f represents D f in sandstone. Under the same initial unloading confining pressure, D f in the coal and sandstone samples decreases with the increasing unloading rate (Figure 12(b)). us, the degree of fragmentation of the sample has a positive correlation with the unloading rate, and the inhomogeneity of the fragmentation in the coal and sandstone samples has the same trend. Moreover, the relationship between D f in the coal and sandstone samples and the unloading rate can be expressed by two additional linear equations, as follows:  Advances in Materials Science and Engineering Equations (9)- (12) indicate that the fragmentation of different lithologies shows different fractal characteristics. Moreover, the fractal dimension of the surface cracks in the coal samples is higher than that in the sandstone

Conclusions
A series of triaxial unloading confining pressure tests with different unloading rates and initial unloading confining pressures were conducted in this study. e following conclusion were drawn: (1) At the same unloading rate, the fractal dimension of the surface cracks in coal and sandstone decreased with higher initial unloading confining pressures. When the initial confining pressure was relatively low, the samples underwent splitting and shear failure; when the initial confining pressure was higher, the samples presented mainly shear failure.    (3) At the same unloading rate, the fractal dimension of fragmentation in the coal and sandstone samples decreased with rising initial unloading confining pressure. When the initial confining pressure was higher, the rock presented larger fragments and the fragmentation of the samples was less homogeneous when the sample failed. (4) Under the same initial unloading confining pressure, the fractal dimension of fragmentation in the coal and sandstone samples decreased with the increasing unloading rate. e degree of fragmentation of the samples and the inhomogeneity of fragmentation in the coal and sandstone samples had a positive correlation with the unloading rate. (5) e rock lithology had a notable effect on the fractal dimension of the surface cracks and the fragmentation in the coal and sandstone samples. Samples with more internal fissures had more surface cracks and less homogeneous fragments when the sample failed under the same unloading conditions.

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