Experimental and Numerical Study onMechanical Properties and Deformation Behavior of Beishan Granite considering Heterogeneity

State Key Laboratory of Mining Disaster Prevention and Control Cofounded By Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, Shandong, China College of Energy and Mining Engineering, Shandong University of Science and Technology, Qingdao 266590, Shandong, China Jinan Engineering Polytechnic, Jinan 250200, Shandong, China China Construction Eighth Engineering Division Corp. Ltd. Construction Branch, Qingdao 266590, Shandong, China Design and Research Institute of Tianyuan Construction Group Co. Ltd., Linyi 276000, Shandong, China


Introduction
High-level radioactive waste (HLW) is one of the most harmful pollutants in the world. Currently, the most feasible way to isolate the HLW from the biosphere is to seal them in a geological disposal repository underground for 500-1000 m. Granite, characterized by low permeability and high strength, is a potential host rock for HLW. It is reported that Beishan granite is chosen as the host rock of a HLW repository in China [1][2][3]. erefore, understanding the mechanical response and deformation behavior of Beishan granite is of significant importance for the safety and longterm stability of a repository.
As a natural material, rocks contain microcracks. It has been widely recognized that the growth of microcracks associated with crack propagation and coalescence can ultimately cause crystalline rock failure [4][5][6][7][8][9][10][11][12][13][14][15][16]. Wawersik and Brace conducted compression experiments on Westerly granite, indicating that at low confining pressure faulting in granite occurs only in the post-failure region beyond the peak of the stress-strain curves [5]. Hadley measured crack lengths and widths in virgin and pre-stressed Westerly granite by using the scanning electron microscope. It is found that the average aspect ratio of stress-induced cracks increases with stress [7]. Oda et al. studied the evolution of stress-induced cracks and failure mechanism in terms of the invariants of crack tensor. It demonstrates that the density of the stress-induced cracks increases linearly with increasing inelastic volumetric strain [11]. However, the crack propagation process is difficult to be captured under experimental tests.
From another view, the variation of rock volume is closely related to the propagation of microcracks. As a result, rocks will show the most direct performance of compaction or expansion [17][18][19][20]. Brace et al. measured volume change of granite during deformation in triaxial compression. It reveals that open fractures propagate parallel with the orientation of maximum compression [17]. Besides, the dilatancy of specimens is directly proportional to fracturing [18] and occurs within about 10 percent of the fracture stress difference [19]. In addition, the failure mode of crystalline rocks is highly related to cracking, resulting in brittle failure or ductile failure [21][22][23]. erefore, a comprehensive study on the crack propagation process under a compression load is needed as a basis for understanding the mechanism of the granite deterioration process. But the laboratory experiments can only reflect the strain change of rock as a whole, and it is difficult to study the strain inside the rock.
In order to solve the problems mentioned above, numerical simulation is extensively used to study the deformation behavior and fracturing process during loading. Besides, the heterogeneity of rock can be studied by using numerical simulation. Tang et al. conducted a numerical analysis to evaluate the effect of heterogeneity on the fracture processes and strength characterization of brittle materials [24][25][26][27][28][29]. It is shown that heterogeneity plays an important role in determining the mechanical behavior on both microscopic and macroscopic scale. Li and Li studied the influence of heterogeneity on mechanical and acoustic emission characteristics of rock specimen [30]. e results demonstrate that as the homogeneity increases, the peak strength and brittleness of rocks increase. Liu et al. investigated the fracture process of heterogeneous rock under various test techniques [31]. It indicates that rock heterogeneity has an important influence on the crack initiation location and the crack propagation path.
us, the heterogeneity is considered to study the crack propagation of Beishan granite through numerical simulation. Furthermore, the strain inside the rock can be investigated. erefore, in this study, experimental tests are conducted to study the basic properties of Beishan granite. Parameters used in the numerical simulation are determined. In addition, the Weibull distribution is introduced to characterize the heterogeneity of rock in numerical simulation. Afterwards, a heterogeneous model of Beishan granite is built to study the deformation behavior and fracturing evolution during the complete failure process.

Preparation of Rock Specimens.
e granite cores were taken from the Beishan area, Gansu Province, China. And it is mainly composed of approximately 33.7% ± 5.8% plagioclase, 30.7% ± 6.1% K-feldspar, 28.6% ± 8.0% quartz, and 7.1% ± 4.8% biotite [32]. All the specimens were prepared based on ISRM recommendations [33]. Five specimens were prepared for the direct tensile test, three specimens were prepared for the uniaxial compression test, and five specimens were prepared for the conventional triaxial compression (CTC) test. Figure 1, a MTS815 Flex Test GT rock mechanics machine is used to conduct all the tests. e MTS815 experimental machine can provide a maximum load of 4,700 kN and supply a maximum confining pressure up to 140 MPa. A pair of extensometers were used to measure axial and circumferential displacement during loading.

Testing Procedure.
For the direct tensile test, the two ends of the specimen are bonded with the instrument with strong glue. During the loading process, the axial displacement was increased with a constant displacement rate of 0.005 mm/min until the failure of the specimen.
For the uniaxial compression test, the axial stress was increased with a constant loading rate of 30 kN/min until the failure of the specimen.
For CTC tests, the confining pressures were set to 5, 10, 15, 20, and 30 MPa, respectively. First, to fix the position of the specimen, a vertical load of about 2 kN was applied.
en, a constant loading rate of 0.05 MPa/s was applied to reach the designated confining pressure. Afterwards, the axial stress was increased with a constant loading rate of 30 kN/min. Lateral deformation control was used when the axial stress approached the peak strength. Figure 2, the trends of stress-strain curves under tension are basically the same, concaving upward. It indicates that the change rate of stress increases gradually under tension. When the tensile peak strength is reached, the tensile stress decreases sharply and the specimen is destroyed suddenly. e strength results are listed in Table 1, and the average tensile strength is about 8.66 MPa. Figure 3, during the initial loading the axial stress increases slowly because of the closure of microcracks [34]. Afterwards, a linear relationship between axial stress and strain was observed during the following loading. When the peak stress is reached, macrocracks are formed along the axis direction, showing splitting failure. e obtained mechanical parameters are listed in Table 2. Figure 4 shows the stress-strain curves under CTC tests and also displays the failure patterns of Beishan granite. ere is no doubt that the peak strength increases linearly with increasing confining pressure. e volumetric strain shows that the dilatancy threshold is about 81% of the peak strength. It is worth noting that the specimen is still in a state of compaction after dilatancy threshold, compared with the initial specimen volume. e strength results and calculated mechanical parameters of Beishan granite are listed in Table 3.

Introduction of Heterogeneity
As an inhomogeneous material, rock is composed of different mineral particles. Besides, micropores and cracks are distributed randomly inside rocks. ese individual components behave in different mechanical responses, leading to different deformation behaviors. erefore, the stress and strain states can be dominated by the spatial distribution of these individual components during loading [35]. To solve this problem, the Weibull distribution has been introduced to characterize the rock heterogeneity in numerical simulation works by many researchers [36].
e Weibull distribution is a continuous probability distribution. e general expression of Weibull probability density function can be written as with x random variable, m shape parameter, and λ scale parameter. e mean and variance of a Weibull random variable can be expressed as where Γ is the Gamma function. e relationship between the scaled mathematical expectation E (x)/λ, dispersion D (x)/λ 2 , and shape parameter m is shown in Figure 5. As can be seen from Figure 5(a), with increasing shape parameter m (m ≥ 1), the scaled mathematical expectation approaches 1, which means the scale parameter λ is approximately the mean value of the Weibull distribution. When m goes towards infinity, the variance goes towards zero. erefore, the shape parameter m is also  Advances in Civil Engineering defined as the homogeneity index and the scale parameter λ is defined as the mean value of the physical parameter [37].
In the numerical simulation, mechanical parameters, such as the elastic modulus E, Poisson's ratio etc., obey the Weibull distribution with scale parameter λ and shape parameter m. Based on the probability distribution 0 ≤ F (x) ≤ 1, the Weibull distribution parameters can be generated from random numbers in [0, 1]: e inverse function is Based on equation (4), parameter x is calculated.

Model Description and Setup.
A cylindrical model (see Figure 6), characterized by heterogeneous, is built in FLAC 3D to study the stress-strain state and fracture development inside the specimen during the complete failure process. e strain-softening model is chosen to simulate the mechanical response. e constant compressive displacement rate is set to 5e-8 m/step, which will be applied on end faces of the model in the following simulation. Based on the tests in Section 3 and reference [14], the input parameters are determined, as shown in Table 4. e validation of this numerical method has been proven by Tan [37].

Determination of Homogeneity Index m of Beishan
Granite. First, the homogeneity index m of Beishan granite needs to be determined. erefore, based on equation (4), models with different homogeneity index m are established in FLAC 3D . For instance, the cohesion distribution with different homogeneity index m is displayed in Figure 7. Afterwards, these models are used to simulate the CTC test under confining pressure of 20 MPa. e cohesion distribution after failure is shown in Figure 8, and 9 shows the simulated stress-strain curves. As demonstrated in Figure 8, the model has more degraded elements after failure when the homogeneity index is relatively low. In return, the model has fewer degraded elements when the homogeneity index is relatively high. Besides, the destroyed elements are more easily concentrate locally with a higher homogeneity index. Compared with the experimental results, the homogeneity m � 15 is selected to conduct the CTC tests simulation. Figure 10 shows the obtained stressstrain curves, and the comparison between simulated strength and experimental strength is shown in Table 5. e      Advances in Civil Engineering results indicate that the simulation for the CTC test is reliable. Taking confining pressure of 20 MPa as an instance, the stress-strain states and fracture development observed at different points (shown in Figure 11) are displayed in Figures 12-15. Some conclusions can be drawn based on the simulation results. During the pre-peak period from point a to c, the model is undamaged. e volumetric strain value is negative and decreases all the time, which indicates that the element is in the state of compression. e shear strain has been increasing, but its value is relatively small. e stress distribution in the model is uniform, and the minimum principal stress increases gradually. e displacement develops basically along the axial direction, and there is no significant deviation. erefore, during the pre-peak period, rock mainly shows elastic deformation.

Failure Process.
At the peak stress (point d), near the model boundary a few elements begin to degrade. e volumetric strain of the elements near the model boundary starts to increase. e shear strain, stress distribution, and displacement also show slight changes at the same location, indicating the beginning of macrocrack generation.
In the stress drop stage from point e to h, more and more elements are destroyed. e volumetric strain of most elements changes from negative to positive and increases rapidly, forming serval bands, which develop from the model boundary to the inside. In the meantime, shear strain bands develop rapidly at the same area. At point h, the shear band penetrating the whole model is formed. e tensile region appears in the model and is distributed in blocks along the shear bands. e displacement vector develops along the shear bands and outward at the boundary of the model.
In the residual stage from point h to k, there are not new macrocracks. However, the volumetric strain shows sustained growth, so does shear strain. e formed strain bands                 is means that the model has been destroyed. Figures 16 and 17 show the distribution of volumetric strain and shear strain in the model after failure under different confining pressures. It can be seen from Figures 16 and 17 that the elements in the fracture zone have large volumetric strain and shear strain. And with increasing confining pressure, the maximum volumetric strain remains relatively stable. However, the maximum shear strain increases with confining pressure. In addition, as the confining pressure increases, the shear failure surface becomes more regular. It can be concluded that the larger the confining pressure, the clearer the destroy angle. e deformation ability of the element, located in the destroy angle area, is strengthened. As a result, with the increase of confining pressure, the failure mode transforms from brittle splitting failure to ductile shear failure, which is consistent with the experimental result.

Conclusion
In this research, the strength and deformation processes of Beishan granite are obtained by experimental tests. e stress-strain state during the complete failure process is analyzed by numerical simulation. Based on the results, the following conclusions can be drawn. e tensile strength and uniaxial compressive strength of Beishan granite are 8.66 and 162.9 MPa, respectively. Under triaxial compression, the dilatancy of Beishan granite appears when the stress reaches about 81% of the peak stress. e heterogeneity of rock can be well introduced by Weibull distribution. With the increase of homogenization degree in numerical simulation, the degraded elements are more easily to concentrate locally. e splitting failure dominates the destroy mode when the confining pressure is relatively low. With increasing confining pressure, more and more degraded elements are concentrated in the shear band, which develops from the surface to the interior of the sample during loading. Hence, the granite shows ductile mechanical response characteristics when the confining pressure is relatively high.
For the safety of HLW repository, more tests are needed for further investigating mechanical properties and fracturing characteristics of Beishan granite subjected to different conditions.

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

Conflicts of Interest
e authors declare that they have no competing financial interests or personal relationships that could have influenced the work reported in this paper.