3.1.Numerical model set-up and its rupture-analysis method
A schematic of a triaxial test on the SRM specimen under a constant confining pressure is presented in Fig.8.The top and bottom walls act as loading plates, and the lateral walls are used to maintain a constant confining pressure using a servo-control mechanism [25]. All walls are 0.2 times larger than the SRM specimen. All sides of the SRM specimen are loaded first into the targeted confining pressure using a servo-control mechanism. Then a velocity of 0.1 m/s is applied to the loading walls to initiate the triaxial test, while the lateral confining pressures kept constant through servo-control walls during the test. Sensitivity studies have shown that this loading rate is sufficiently slow to ensure that the specimen remains in quasi-static equilibrium. The axial stress is obtained by dividing the average recorded reaction force on the loading walls by the cross-sectional area of the SRM specimen. The axial strain is calculated as the ratio of the sum of the displacement of both loading walls to the initial height of the SRM specimen. The lateral strain has a similar calculation method.
BPM damage is represented by shear and tension breakage of bonds that results in microcracks, which can be classified into shear and tension microcracks. The initiation, propagation, and coalescence of these microcracks form macro-matrix fractures, which are either of a tension or shear mechanism. A discrimination method for the macroscopic formation mechanism of matrix fractures was proposed, which differ from a comparison of velocity or displacement vectors of particles on the sides of a matrixfracturebyBewick et al.[32].If a matrix fracture is composed of only tension microcracks and opens with a large gap, the macro formation mechanism is considered as tension. If a matrix-fracture is composed of tension and shear microcracks and closes, the macro formation mechanism can be judged as shear, because shear always occurs between two conterminous objects accompanied by tension breakage of asperities[33]. ForSRM specimens, activated joints that result from the activation of pre-existing joints must be considered in rupture analysis. Fractures in the SRM specimen are always composed of activated joints and matrix fractures. With a large gap, the open fractures can be displayed easily and clearly by means of cut planes. However, cut planes do not display closed fractures, because the gap is very small. Microcracks can be used to depict the geometry of closed fractures, but they are inadequate when the fractures are complicated in spatial distribution. Particles on opposite sides of the closed fracture contact each other and form linear-contact models in the sliding process, thus the linear-contact models can be used to extract and display the closed shear fracture that results from triaxial compression on the jointed-coal SRM specimen.
All triaxial compression numerical tests were conducted by using code PFC3D5.0 on a 64-bit, two 2.30 GHz Intel(R) Xeon(R) processor computer with 64.0 GB RAM. Each case took 2–15 days to complete the calculations because of the large number of particles used (518,432 particles). The calculation time increases sharply with confining pressure because many microcracks grow and adjust their locations and orientations in the calculation under a high confining pressure.
3.2. Numerical triaxial compression tests for different loading orientations
3.2.1.Numerical triaxial compression test on intact coal with REV size
The triaxial compression numerical model for coal BPM with a REV size is shown in Fig.9A, with the microparameters listed in Table 2. After triaxial compression under a 1-MPa confining pressure, the specimen retains integrity, except for some microcrack bands along the surface (Fig. 9B). As illustrated in the cut planes (Fig. 9F, G, and H) at the Y = 0position, the microcrack bands form a cut-through fracture that is labeled by a red dotted line, which is closed and composed of tension and shear microcracks. According to the rupture-analysis method in Section 3.1, the fracture is a shear-rupture surface for intact coal that is subjected to triaxial compression. For a clear display, the shear-rupture surface that is denoted by linear-contact models is extracted and rotated twice (Fig. 9C, D, and E). The shear-rupture surface is of an approximate parallelogram with a missing corner. The total number of microcracks is 1,225,010, of which 1,072,842 are tensile microcracks and 152,168 are shear microcracks.
The curves of axial stress difference, confining pressure, and lateral strain against axial strain are plotted in Fig.10. The confining pressures that are imposed on the lateral sides fluctuate around 1 MPa, which indicates a good servo control on the lateral walls. The stress difference refers to the difference between the monitored axial stress and the targeted confining pressure. The pre-peak curve of stress difference remains relatively straight, and the post-peak curve drops rapidly to a steady condition, with an elastic modulus of 2.36 GPa, a triaxial compression strength (TCS) of 33.7 MPa, and residual strength of 10.3 MPa.The lateral strain increases sharply after the peak strength, stabilizes briefly at the residual strength stage, and shows a rapidly increasing trend, with a Poisson's ratio of 0.23.
3.2.2. Numerical test on jointed coal with loading direction perpendicular to bedding planes
TheSRM specimens of jointed coal to be loaded perpendicular to the bedding planes are presented in Fig.11A (joints geometry) and B (contact models). After the triaxial test, no fractures with a visible gap grow but an inclined band that consists of tension and shear microcracks cut through the specimen (Fig. 11E and F). A same inclined band of the linear-contact model that corresponds to the microcrack band also appears in the final contact models (Fig. 11C), which is shown in Fig. 11D.According to the rupture-analysis method, the inclined band is the shear fracture surface, which is a parallelogram.
The curves of microcrack number, axial stress difference, confining pressure, and lateral strain against axial strain are plotted in Fig.12. The confining pressures fluctuate around 1 MPa, which indicates a good servo control on lateral walls. In the pre-peak stage, the specimen transits gradually from elasticity to plasticity with the increase in axial stress, which produces the yield point. The sharp increase zone of the microcrack curves ranged from the yield point to the lowest point, where the coal matrix breaks heavily and results in a rapid growth of microcrack number. In the post-peak stage, the number of microcracks does not increase, but the lateral strain still increases, which indicates that the lateral deformation of the specimen originates from the increase in the gap of the existing fractures, rather than the formation of new fractures.The TCS, elastic modulus, Poisson's ratio,and residual strength are 26.6 MPa, 1.78GPa, 0.20,and4.0 MPa,respectively.
3.2.3. Numerical test on jointed coal with loading direction perpendicular to face cleats
The SRM specimen of jointed coal to be compressed triaxially perpendicular to the face cleats is presented in Fig. 13A. The corresponding contact model geometry is displayed in Fig. 13B. After the test, some linear-contact models were added to the specimen (Fig. 13C), which are extracted for a clear display in Fig. 13D. The distribution of microcracks agrees well with the linear-contact models (Fig. 13E and F). The combination bands of tension and shear microcracks represent closed fractures that result from triaxial compression. According to the rupture-analysis method for SRM specimens, bands of the linear-contact model are a shear-rupture surface. The geometry of shear-rupture surface differs from that under triaxial compression that is perpendicular to the bedding planes and is composed of the main fracture and a secondary fracture (Fig. 13D).
The curves of stress, strain, and microcrack number are plotted in Fig.14. The steady confining pressures also indicate a good servo control on the lateral walls. The TCS is 26.8 MPa at a confining pressure of 1 MPa, with an elastic modulus of 1.75GPa and a Poisson's ratio of 0.18. The sharp increase of microcrack number ranges from the yield point to the lowest point in the curve of the axial stress difference, where the coal matrix breaks heavily. In the residual strength stage, the axial stress is stable, but the lateral strain increases sharply. The microcrack number shows an increasing trend, which means that the lateral deformation results mainly from the increase in the gap of the existing fractures, and the formation of new fractures also provides some contribution. The residual strength is 7.9 MPa.
3.2.4. Numerical test on jointed coal with a loading direction perpendicular to butt cleats
The joint geometry in the SRM specimen to be triaxially compressed perpendicular to the butt cleats is shown in Fig. 15A and the corresponding contact model geometry is presented in Fig. 15B. After the test, some microcrack bands that consisted of tension and shear microcracks crossed through the specimen surface (Fig. 15E and F). This situation is the same as the two situations above, and many linear-contact models appear in the contact model geometry (Fig. 15C). These linear-contact models depict the geometry of the shear-rupture surface, which is extracted for clear display (Fig. 15D). The shear-rupture surface approximate diamond.
The numerical results are shown in Fig.16. Similar to the situation with triaxial compression perpendicular to the bedding planes and face cleats, the sharp increase in microcrack number ranges from the yield point to the lowest point in the curve of axial stress difference. The lateral deformation still increases significantly in the residual strength stage. The TCS, elastic modulus, Poisson's ratio, and residual strength are 26.1 MPa, 1.89 GPa,0.21 and 8.5 MPa, respectively.
3.3. Numerical triaxial compression tests under different confining pressures
A series of numerical triaxial compression tests on jointed coal under different confining pressures were performed to investigate the confining-pressure effect. Based on the jointed-coal SRM specimen with loading perpendicular to the butt cleats, four magnitudes of confining pressure: 0, 1, 5, and 10 MPa were involved.
The fracture geometries under different confining pressures are shown in Fig. 17. Without a confining pressure, the loading is a uniaxial compression test. After the uniaxial compression test, fractures initiated and propagated along the joints parallel to the loading direction (Fig. 17A). The fractures resulted from inactivation, and their formation was dominated by the joints. If confining pressure is imposed, the interaction of confining pressure and axial loading stress will result in shear stress in the specimen, which will lead to the formation of shear fracture. When the confining pressure increases to 1 MPa, the shear-rupture surface approximates a rhombus (Fig. 17B). Joints parallel to the loading direction can be activated partially and affect the local geometry of the shear-rupture surface under low confining pressure. When the confining pressure continues to increase to 5 MPa, joint activation becomes more difficult and the joints will have a weak effect on the shear fracture. The shear-rupture surface is of a regular rhombus shape under a middle confining pressure (Fig. 17C). Therefore, the formation of a shear-rupture surface is dominated by shear stress and joints for jointed coal at low and middle confining pressures, but the influence of the joints weakens with the confining pressure. Many microcracks exist over the specimen at a 10-MPa confining pressure, which indicates that the jointed-coal specimen ruptured by plastic flow (Fig. 17D). Fracture formation is dominated only by shear stress and is independent of the joints for jointed coal under high confining pressure.
The numerical results for jointed coal that is subjected to triaxial compression with different confining pressures are shown in Fig. 18. As illustrated in Fig.18A, the TCS is low and the residual strength is close to 0 without a confining pressure. When the confining pressure increases to 1 MPa, the TCS and residual strength increase significantly, but the axial stress drops sharply in the post-peak stage, which indicates a brittle behavior for jointed coal. When the confining pressure increases to 5 MPa, the TCS and residual strength also increase.The peak zone becomes smooth and wide, and the axial stress decreases gently, which shows a transition from brittleness to ductility for jointed coal at a 5-MPa confining pressure.When the confining pressure increases to 10 MPa, the TCS continues to increase but the peak zone disappears. After the TCS has been reached, the axial stress tends to be stable despite the axial strain increasing with loading. As shown in Fig. 17D, the specimen is in a plastic-flow state at a 10-MPa confining pressure.The estimated transiting pressure of brittleness to ductility for jointed coal in this study is 10 MPa. The elastic limit increases, but the elastic modulus remains stable with an increase in confining pressure, which means that the confining pressure has no effect on the elastic modulus, and the elastic modulus is an intrinsic property of jointed coal.As shown in Fig. 18B, the lateral deformation without confining pressure is severe and is far larger than that with confining pressure. The jointed-coal deformation is infinitely sensitive to a low confining pressure. The number of microcracks tends to be steady at the post-peak stage when the confining pressure is 0, 1, and 5 MPa, whereas the microcrack number keeps growing at a high rate during the entire stage under a 10-MPa confining pressure, which verifies that the specimen is in plastic-flow failure under a 10-MPa confining pressure. The number ratio of shear microcracks to tension microcracks increases with confining pressure, which shows that a transition from tension to shear for the macro rupture mechanism of the coal matrix occurs.