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Numerical Simulations on Cracking Behavior of Rock-Like Specimens with Single Flaws under Conditions of Uniaxial and Biaxial Compressions Downloaded from ascelibrary.org by University of Nottingham on 10/06/19. Copyright ASCE. For personal use only; all rights reserved.
Cheng Zhao 1; Jialun Niu, Ph.D. 2; Qingzhao Zhang 3; Songbo Yu 4; and Chihiro Morita 5
Abstract: The expanded distinct element method based on a strain strength criterion was applied herein with the aim of analyzing and comparing the cracking behaviors of rock-like specimens with single flaws under uniaxial and biaxial compressions. The cracking characteristics in uniaxial and biaxial compressions were compared and the effects of confining pressure on the tensile/shear behaviors were analyzed. The confining pressure obviously changed the boundary condition and the cracking behaviors in biaxial compression. The initiation stress, the peak strength, and the number of shear cracks in biaxial compression were obviously larger than in uniaxial compression. The confining pressure affected the initiation and propagation of the secondary cracks, the failure mechanisms, and the distribution of horizontal displacement fields in a large extent. The initiation and propagation of the tensile cracks were inhibited by the confining pressure. An increase in confining pressure had no obvious effects on the initiation time and the propagation rates of the shear cracks but mainly affected the numbers of shear cracks. DOI: 10.1061/(ASCE)MT.1943-5533.0002967. © 2019 American Society of Civil Engineers. Author keywords: Cracking behaviors; Strain strength criterion; Expanded distinct element method (EDEM); Single flaw; Confining pressure.
Introduction Thorough investigation of the damage and failure of rock mass plays an important role in predicting the unstable failure of jointed rock engineering. The mechanical behavior of rock mass is largely controlled by microflaws. The cracking behaviors of rock mass have been studied theoretically (e.g., Ewy and Cook 1990; Griffith 1921; Horii and NematNasser 1985; Liu et al. 2015; Sammis and Ashby 1986) and experimentally (e.g., Brace and Bombolakis 1983; Bobet and Einstein 1998; Petit and Barquins 1988; Wong and Chau 1998; Wong and Einstein 2009a, b; Yang et al. 2016; Zou and Wong 2014) on a number of brittle materials. The initiation, propagation, and coalescence of rock mass and rock-like materials have been studied thoroughly. However, the internal stress and strain states, as well as the quantitative features during the cracking process, are worth studying numerically. 1 Professor, Dept. of Geotechnical Engineering, Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji Univ., Shanghai 200092, China; College of Engineering, Tibet Univ., Lasa 850000, China. Email: [email protected] 2 Dept. of Geotechnical Engineering, Tongji Univ., Shanghai 200092, China. Email: [email protected] 3 Associate Professor, Dept. of Geotechnical Engineering, Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji Univ., Shanghai 200092, China (corresponding author). Email: [email protected] 4 Engineer, Dept. of Geotechnical Engineering, Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji Univ., Shanghai 200092, China. Email: [email protected] 5 Professor, Dept. of Civil and Environmental Engineering, Univ. of Miyazaki, Miyazaki 889-2192, Japan. Email: [email protected] -u.ac.jp Note. This manuscript was submitted on July 27, 2018; approved on June 14, 2019; published online on September 30, 2019. Discussion period open until February 29, 2020; separate discussions must be submitted for individual papers. This paper is part of the Journal of Materials in Civil Engineering, © ASCE, ISSN 0899-1561.
© ASCE
Several numerical methods have been proposed to investigate the damage and failure behaviors of brittle materials. The most common numerical methods adopted in rock mechanics include mainly the finite-element method (FEM) (Guven and Madenci 2011; Tang et al. 2001), boundary element method (BEM) (Chen et al. 1998), bonded-particle model (BPM) (Zhang and Wong 2012), numerical manifold method (NMM) (Wu and Wong 2013), displacement discontinuity method (DDM) (Gonçalves and Einstein 2013), and discrete element method (DEM) (Vesga et al. 2008; Jiang et al. 2009; Gui et al. 2017). The DEM was proposed by Cundall and Strack (1979), in which the finite displacements and rotations of discrete bodies are allowed, and new contacts during the deformation processes can be identified automatically. In the numerical simulations of rock mass and rock-like materials, simulated specimens are treated as an assemblage of independent rigid or deformable blocks, or particles with certain particle friction and bond strength to reproduce the strength of brittle material (Cundall and Hart 1993). Vesga et al. (2008) employed DEM to simulate the failure process of brittle clay specimens and revealed good agreement with experimental results. As a kind of DEM, the Universal Distinct Element Code (UDEC) has obvious advantages in handling the stress-strain behavior within each grain and capturing the interaction of adjacent irregular grains through predefined contact constitutive models (Gui et al. 2017). To solve the problem of the initiation and propagation trajectories of macroscopic cracks, Jiang et al. (2009) developed an expanded distinct element method (EDEM) based on UDEC, and this numerical method was applied to simulate crack initiation and propagation due to the shear and tension failures in matrix rock blocks. Owing to the complexity of fracture distributions at the macrostructural and microstructural levels, the failure modes of rock mass and rock-like materials are complex and difficult to quantify or predict. The cracking processes still need to be assessed more comprehensively. The accurate estimation of crack initiation and
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propagation, as well as the tensile/shear behaviors of rock mass and rock-like materials, remain a challenge. In this study, the strain strength criterion proposed by Zhao et al. (2017) was employed in UDEC to comprehensively study the failure modes and cracking behaviors of rock-like specimens with single flaws in compression tests. The single-flawed specimens with different inclination angles were loaded to fail under uniaxial compression and biaxial compression in five confining pressure conditions (1.5, 3, 4.5, 6, and 9 MPa). The cracking behaviors, such as initiation stress, peak strength, propagation trajectories, and distribution of displacement fields, were thoroughly analyzed. In addition, the effects of confining pressure on the tensile/shear behaviors were studied.
Numerical Method and Model The EDEM is adopted in this paper because of its advantages in accurately simulating the initiation and propagation path of macroscopic cracks. The EDEM is based on UDEC, which was implemented with a strain strength criterion that combines tension and shear behaviors of rock mass. The cracking processes of rock-like specimens with single flaws under uniaxial and biaxial compressions were simulated by EDEM. The strain strength criterion proposed by Zhao et al. (2017) was employed to comprehensively describe tensile and shear cracks. Based on such a criterion, Zhao et al. (2017, 2018a) simulated the cracking behavior of rock-like specimens containing a single flaw or flaw pairs. It was shown that the stress-strain curves and crack trajectories obtained from their numerical simulation based on this criterion were highly consistent with experimental results. Virtual cracks were set up in all potential cracking areas to determine the crack trajectories. During the entire cracking process, the tensile cracks were judged by principal strain and the shear cracks were judged by the Mohr– Coulomb criterion. The critical state can be expressed by the following equation: ε3 ¼ −εt τ ¼ c þ σ tan φ
ð1Þ
where ε3 = principal strain; εt = critical tensile strain; τ = shear strength; c = cohesion; σ = normal stress on shear plane; and φ = internal friction angle. The tensile cracks were assumed to initiate when the principal strain reached a critical value and expand in the direction of the maximum principal strain. The shear cracks were assumed to initiate if Mohr’s circle touched the failure envelope and expanded in the direction of the most dangerous stress state. At the limit equilibrium state for a linear elastic material, the critical states can be expressed as σ1 ¼ ðσ3 þ Eεt Þ=ν sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ sin φ 1 þ sin φ σ1 ¼ 2c þ σ 1 − sin φ 1 − sin φ 3
ð2Þ
where σ1 and σ3 = principal stresses, respectively; E = elastic modulus; and ν = Poisson’s ratio. For the determination of the tensile and shear cracks in numerical simulation, the main factors are defined as f t ¼ σ3 − vσ1 þ Eεt
ð3Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ sin φ 1 þ sin φ f s ¼ σ1 − σ − 2c 1 − sin φ 3 1 − sin φ © ASCE
ð4Þ
Fig. 1. Numerical model.
where ft = factor to determine a tensile crack; and f s = factor to determine a shear crack. The tensile crack initiates if f t > 0, and the shear crack initiates if f s > 0. To compare the present work with a series of experimental results of single-flawed specimens conducted by Zhao et al. (2017, 2018a) the geometries of specimen for numerical simulations were the same as that for experimental specimens. The dimensions of the used specimen were 100 mm high and 40 mm wide (Fig. 1). The length of the prefabricated flaw was 14 mm, and the inclination angles α were 15°, 30°, 45°, 60°, and 75°. Based on the experimental results, the crack trajectory and mechanical properties of the specimen at 45° was the most complicated. Thus, the present work mainly focused on the specimen with inclination angles of 45° and compared the simulations with the experimental results of other specimens. The specimens were divided into hexagonal blocks by virtual cracks, and each hexagonal block had a side length dimension of 1.5 mm. To obtain the stress and strain of each block during the entire cracking process, the hexagonal blocks were divided into triangular grids where the maximum length of the grid was less than 1 mm. The material parameters used in the numerical simulations are the same as in Zhao et al. (2018b) where the simulations were conducted according to the compression experiments of gypsum specimens containing prefabricated cracks. For the EDEM, there are virtual cracks, precracks, and tensile and shear cracks converted from the virtual cracks. To obtain an accurate result, the chosen parameters of the precracks and the real cracks should be close to the practical situation. Therefore, for virtual cracks, the unit blocks should be bonded to each other and cannot slip or open. As a result, the cohesive force and tensile strength of the virtual cracks can be sufficiently high. The specimens contain single flaws with inclination angles of 45° were loaded to fail under uniaxial and biaxial compressions. In the uniaxial compression test, the vertical displacement at the bottom of the specimen was fixed, and the top surface was under a displacement load. Fig. 2 shows the loading process in the biaxial compression test. To keep the confining pressure constant, a horizontal load was applied by stress loading at the beginning of the simulations. The vertical load was applied to the upper end of the specimen after the stress of the specimen was stabilized. The vertical displacement of the lower end was fixed and the confining
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Fig. 2. Loading process in biaxial compression: (a) one loading process; and (b) another loading process.
pressure kept constant during the vertical loading process. In this study, to simulate crack initiation and propagation more accurately and effectively, the vertical load was applied using the displacement methodology, and the loading rate was 0.001 m=s.
Simulation Results under Uniaxial and Biaxial Compression In this study, the cracking behaviors from initiation to coalescence of single-flawed rock-like specimens was numerically simulated. Different cracking behaviors under uniaxial and biaxial compression were compared, such as stress-strain curves, tensile and shear crack numbers, crack trajectories, and horizontal displacement fields. Compared with the uniaxial compression test, there were several differences in the biaxial simulation including the stress of crack initiation, peak strength, propagation trajectories, and the distribution of displacement fields.
Fig. 3. Stress-strain curve and number of tensile/shear cracks in uniaxial compression. © ASCE
Stress-Strain Curves and Crack Numbers The entire stress-strain curve and the number of tensile and shear cracks of single-flawed specimens in the uniaxial compression test are shown in Fig. 3. The tensile cracks initiated when the stress reached approximately 24.6 MPa at the end of the elasticity stage (Point 1). After the initiation of cracking, the number of tensile cracks increased gradually. When the internal stress accumulated to a certain point, the tensile cracks suddenly expanded rapidly, and the corresponding stress-strain curve entered the unstable stage (Point 2). The number of tensile cracks increased slowly after the stress-strain curve entered the stable stage (Point 3), but the number of shear cracks almost remained unchanged. The tensile and shear cracks expanded rapidly almost at the same time when the stress reached a peak strength of approximately 30.78 MPa (Point 4). It can be seen from the stress-strain curve that the uniaxial compression specimen had a ratio of initiation stress to peak strength of 79.9%. The number of tensile cracks during the entire cracking process was significantly more than the shear cracks. The entire stress-strain curve and the number of tensile and shear cracks of single-flawed specimens under a confining pressure of 3 MPa in biaxial compression tests are shown in Fig. 4. As in uniaxial compression, the tensile cracks initiated earlier than the shear cracks, and the specimen eventually failed under the combined effect of tension and shear. However, compared with uniaxial compression, the cracking process under biaxial compression had several obvious differences. The initiation stress increased to 35.36 MPa (Point 1), the peak strength increased to 53 MPa (Point 4), and the confining pressure largely improved the strength of single-flawed specimen. The ratio of initiation stress to peak strength was 66.7%, less than that under uniaxial compression. The number of shear cracks exceeded the tensile cracks between Points 2 and 3 and was significantly more than the tensile cracks at the end of the cracking process. The application of horizontal confining pressure changed the boundary condition and cracking mechanism. In terms of the boundary condition, the confining pressure restricts the lateral deformation of the specimen. As for the cracking mechanism, the initiation of wing cracks is mainly due to the stress concentration at the crack tips, which cause the principal strain to exceed the critical value. The direction of maximum principal strain is basically horizontal in
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Fig. 4. Stress-strain curve and number of tensile/shear cracks in biaxial compression.
uniaxial compression, while in biaxial compression, the horizontal pressure reduces the horizontal principal tensile strain so as to increase the initiation stress and the peak strength. Compared with uniaxial compression, the confining pressure in the biaxial compression test caused the specimen to exhibit ductile failure characteristics, that is, the failure occurs after a relatively long period of time after crack initiation. Crack Trajectories Fig. 5 shows the crack trajectories of single-flawed specimens with an inclination angle of 45° during uniaxial compression. The dark oblique lines at the center of the specimens represent flaws, the dotted lines in hexagonal blocks represent virtual cracks, the cracks directed by T are tensile cracks, the cracks directed by S are shear cracks. The wing cracks, which generated under the tensile effect, initiated from the flaw tips, then propagated along the direction of the maximum compressive stress, and finally penetrated the upper and lower ends of the specimen. The secondary cracks generated mainly under the shear effect initiated from the flaw tips if the wing cracks were close to the ends of the specimen, and the angle of propagation was about 60° from the horizontal. The specimen failed when tension and shear deformation were combined, and the tensile effect was dominant.
To study the influence of inclination angle, crack trajectories before the failure of specimens with different inclination angles in experimental and numerical study are shown in Fig. 6. If the crack inclination angle was less than 45° (α ¼ 15° and 30°), the failure of a specimen was dominated by wing cracks, while shear cracks were not obvious during the loading process. Although the shear cracks did not sufficiently propagate, their shear stress fields affected the trajectories of the wing cracks. As a result, the new cracks propagated along the direction of maximum principal stress. If the crack inclination angle was larger than 45° (α ¼ 60° and 75°), the cracks mostly generated under a shear effect, leading to failure. The propagation trajectories were alternately distributed by the mixed cracks during the loading process, and the macroscopic cracks showed a mixture of tensile and shear characteristics. Despite the interaction of tensile and shear stress, shear cracks played a significant role in the cracking process. It is worth noting that the crack trajectories in the specimen with an inclination angle of 75° were slightly different from the experimental results. Such a difference arose probably because this inclination angle of flaw was close to the shear failure angle of specimens without flaws. Further numerical simulation showed obvious characteristics of shear failure, and the crack extended at both ends of the specimen along the 65° inclination direction. The crack trajectories of single-flawed specimens with an inclination angle of 45° during biaxial compression with a confining pressure of 3 MPa are shown in Fig. 7. As with uniaxial compression, the wing cracks initiated from the flaw tips. Compared with uniaxial compression, the failure process exhibited significant shear characteristics. From the crack trajectories and the number of tensile/shear cracks (Fig. 4), it can be found that secondary cracks under shear effect appeared shortly after the propagation of wing cracks (Point 2). Then secondary cracks rapidly propagated along the flaw direction to the left and right ends of the specimen; in addition, some cracks generated near the bottom of the specimen under the combined effects of horizontal stress and fixed bottom displacement (Point 3). At this time, the wing cracks had not propagated to the vicinity of both ends of the specimen. During the stage from Point 2 to Point 3, the propagation rate of tensile cracks was obviously slower than that of shear cracks, and the stress stagnated and fluctuated. The shear cracks continued to propagate and spread to the angled region between the wing cracks and the secondary cracks. The specimen exhibited significant plastic damage characteristics because of the confining pressure. During the stage from Point 3 to Point 4, the propagation of wing cracks ended, and the number of shear cracks was more than the number of tensile cracks.
Fig. 5. Crack trajectories in numerical simulation of specimen under uniaxial compression. © ASCE
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Fig. 6. Crack trajectories of specimens with different inclination angles in (a, b, c, d, and e) experimental study; and (f, g, h, i, and j) numerical study: (a) α ¼ 15°; (b) α ¼ 30°; (c) α ¼ 45°; (d) α ¼ 60°; (e) α ¼ 75°; (f) α ¼ 15°; (g) α ¼ 30°; (h) α ¼ 45°; (i) α ¼ 60°; (j) α ¼ 75°.
Fig. 7. Crack trajectories in numerical simulation of specimen under biaxial compression.
When the specimen failed, the secondary cracks penetrated the left and right sides of the specimen. Therefore, the entire cracking process of the single-flawed specimen under biaxial compression can be summarized as the propagation of wing cracks, the propagation of both wing cracks and secondary cracks, and the propagation of secondary cracks. The collection of blocks set in the EDEM are used to simulate rock-like material, which can accurately reflect the initiation process of microcracks and the brittleness of the specimen, ensuring the effective simulation of shear cracks. The trajectories of wing cracks generated under the tensile effect were similar in both uniaxial and biaxial compression tests. The differences were mainly © ASCE
reflected in the initiation time, the trajectories, and the numbers of secondary cracks under uniaxial and biaxial compression. Horizontal Displacement Fields Fig. 8 shows the horizontal displacement fields in the numerical simulation of biaxial compression with a confining pressure of 3 MPa. Compared with the horizontal displacement fields in the initial stage simulated under uniaxial compression, in the initial stage of biaxial compression, owing to the effect of confining pressure, the horizontal displacement generated from both sides of the specimen and toward the inside of the specimen. With the increase
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Fig. 8. Evolution of horizontal displacement fields under biaxial compression.
of vertical load, the horizontal displacement was gradually dominated by the horizontal expansion effect from the sides of the wing cracks. If the wing cracks initiated and the secondary cracks had not yet initiated (Point 1), the distribution of the horizontal displacement field was similar to that of uniaxial compression. However, compared with uniaxial compression, shearing dislocations generated in the horizontal displacement field when the secondary cracks initiated (Point 2). Especially after a large number of shear cracks generated (Points 3 and 4), the displacement contours divided the specimen into several regions, and the specimen was subjected to more shear effect than uniaxial compression.
Cracking Behaviors under Different Confining Pressures The specimens under different confining pressures were loaded to failure in the biaxial test. The confining pressures were 1.5, 3, 4.5, 6, 7.5, and 9 MPa. For convenience of presentation, the specimen under a confining pressure of, for example, 1.5 MPa was abbreviated as “1.5 MPa specimen,” and this system was followed for the other confining pressures. The development curves of tensile cracks under different confining pressure conditions, which focused on the variable number of tensile cracks during the simulations, were compared.
Effects of Confining Pressure on Shear Properties Fig. 10 shows the development curves of shear cracks under five confining pressures. With the increase in confining pressure, the initiation points of shear cracks under different loading conditions are basically unchanged. Moreover, the propagation rates of shear cracks are nearly the same. These phenomena indicate that confining pressure has no obvious effect on the initiation and propagation of shear cracks. With reference to the total number of shear cracks and tensile cracks, it is found that the proportion of shear cracks in new cracks increases gradually with increasing confining pressure. When the confining pressure is small, the failure behavior of the specimen is dominated by both tensile cracks and shear cracks. When the confining pressure is large, the tensile properties are replaced by shear properties, and the propagation of shear cracks significantly leads to the fracture of the specimen. The failure behaviors of the specimen exhibit shear properties. The development curves of tensile and shear cracks also directly reflect the crack initiation sequence. Under a low confining pressure (1.5 and 3 MPa), tensile cracks initiate earlier than shear cracks. Under a medium confining pressure (4.5 and 6 MPa), tensile cracks and shear cracks initiate almost simultaneously. Under a
Effects of Confining Pressure on Tensile Properties As shown in Fig. 9, the initiation time of tensile cracks are delayed with the increase of confining pressure. Apart from the crack initiation time, the confining pressure also inhibits the numbers and propagation rates of tensile cracks. With increasing confining pressure, the number of tensile cracks, as well as the slopes of the tensile crack development curves, decreases gradually. Compared with the uniaxial compression test, the confining pressure plays a major role in inhibiting the initiation and propagation of tensile cracks in the single-flawed specimen, and the inhibition is enhanced as the confining pressure increases. The mechanisms of the inhibition in tensile cracks can be summarized as two aspects. On the one hand, the tensile cracks mainly generate to the sides of the specimen horizontally, while the horizontal deformation of the specimen is limited by the confining pressure. On the other hand, the initiation of the tensile cracks requires a certain amount of energy, while the concentration of horizontal tensile strain is reduced by the confining pressure. © ASCE
Fig. 9. Development curves of tensile cracks under five confining pressures.
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of tensile cracks decreased. Compared to tensile cracks, there were no obvious effects of different confining pressures on the initiation time and the propagation rates of shear cracks. The increase in confining pressure mainly affected the number of shear cracks.
Data Availability Statement
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All data, models, and code generated or used during the study appear in the submitted article.
Acknowledgments
Fig. 10. Development curves of shear cracks under five confining pressures.
high confining pressure (9 MPa), shear cracks initiate earlier than tensile cracks. These outcomes are consistent with the theoretical analysis and the observation of failure status.
Conclusions The objective of this study was to analyze and compare the cracking behaviors of rock-like specimens with single flaws under uniaxial and biaxial compression. Using the EDEM, cracking behaviors under uniaxial and biaxial compression were studied, such as initiation stress, peak strength, propagation trajectories, and distribution of displacement fields. In addition, biaxial compression tests under different confining pressures were conducted, and the effects of confining pressure on tensile/shear behaviors were analyzed. Based on the simulations of a single-flawed specimen with an inclination angle of 45°, the following observations and findings were obtained: 1. Tensile cracks initiated earlier than shear cracks, and the specimens failed under the combined effect of tension and shear in both the uniaxial and biaxial tests. Under biaxial compression, the confining pressure increased the initiation stress and peak strength of the single-flawed specimen. If the specimen failed, the number of shear cracks under biaxial compression greatly exceeded the number of tensile cracks, in contrast to uniaxial compression. The confining pressure obviously changed the boundary condition and the cracking characteristics under biaxial compression. 2. The initiation and propagation of secondary cracks and the distribution of horizontal displacement fields were affected by the confining pressure to a great extent. Under biaxial compression, secondary cracks generated earlier than under uniaxial compression. The failures of the uniaxial and biaxial specimens were mainly controlled by wing cracks and secondary cracks, respectively. The cracking process of a biaxial specimen can be summarized as the propagation of wing cracks, the propagation of both wing cracks and secondary cracks, and the propagation of secondary cracks. 3. As a result of the changes in the boundary condition and the cracking characteristics under biaxial compression, the confining pressure inhibited the initiation and propagation of tensile cracks. With increases in the confining pressure, the initiation times were delayed, and the numbers and propagation rates © ASCE
The authors would like to acknowledge the financial support of the National Key R and D Program of China (2017YFC0806000), the National Natural Science Foundation of China (41202193, 41572262, and 41502275), the Shanghai Rising-Star Program (17QC1400600), the Fundamental Research Funds for the Central Universities (0200219207), the Shanghai Municipal Science and Technology Major Project (2017SHZDZX02), and the Key Laboratory of Rock Mechanics and Geohazards of Zhejiang Province.
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Liu, R., Y. Jiang, B. Li, and X. Wang. 2015. “A fractal model for characterizing fluid flow in fractured rock masses based on randomly distributed rock fracture networks.” Comput. Geotech. 65 (Apr): 45–55. https://doi .org/10.1016/j.compgeo.2014.11.004. Petit, J., and M. Barquins. 1988. “Can natural faults propagate under mode II conditions?” Tectonics 7 (6): 1243–1256. https://doi.org/10.1029 /TC007i006p01243. Sammis, G., and M. F. Ashby. 1986. “The failure of brittle porous solids under compressive stress states.” Acta. Metall. 34 (3): 511–526. https:// doi.org/10.1016/0001-6160(86)90087-8. Tang, A., P. Lin, R. H. C. Wong, and K. T. Chau. 2001. “Analysis of crack coalescence in rock-like materials containing three flaws—Part II: Numerical approach.” Int. J. Rock Mech. Min. Sci. 38 (7): 925–939. https://doi.org/10.1016/S1365-1609(01)00065-X. Vesga, L. F., L. E. Vallejo, and S. Lobo-Guerrero. 2008. “DEM analysis of the crack propagation in brittle clays under uniaxial compression tests.” Int. J. Numer. Anal. Met. 32 (11): 1405–1415. https://doi.org/10.1002 /nag.665. Wong, L. N. Y., and H. H. Einstein. 2009a. “Crack coalescence in molded gypsum and Carrara marble: Part 1. Macroscopic observations and interpretation.” Rock Mech. Rock Eng. 42 (3): 475–511. https://doi .org/10.1007/s00603-008-0002-4. Wong, L. N. Y., and H. H. Einstein. 2009b. “Crack coalescence in molded gypsum and Carrara marble: Part 2. Microscopic observations and interpretation.” Rock Mech. Rock Eng. 42 (3): 513–545. https://doi.org/10 .1007/s00603-008-0003-3. Wong, R. H. C., and K. T. Chau. 1998. “Crack coalescence in a rock-like material containing two cracks.” Int. J. Rock Mech. Min. Sci. 35 (2): 147–164. https://doi.org/10.1016/S0148-9062(97)00303-3.
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Wu, Z. J., and L. N. Y. Wong. 2013. “Modeling cracking behavior of rock mass containing inclusions using the enriched numerical manifold method.” Eng. Geol. 162 (Jul): 1–13. https://doi.org/10.1016/j.enggeo .2013.05.001. Yang, S. Q., W. L. Tian, Y. H. Huang, P. G. Ranjith, and Y. Ju. 2016. “An experimental and numerical study on cracking behavior of brittle sandstone containing two non-coplanar fissures under uniaxial compression.” Rock Mech. Rock Eng. 49 (4): 1497–1515. https://doi.org/10 .1007/s00603-015-0838-3. Zhang, X. P., and L. N. Y. Wong. 2012. “Cracking processes in rock-like material containing a single flaw under uniaxial compression: A numerical study based on parallel bonded-particle model approach.” Rock Mech. Rock Eng. 45 (5): 711–737. https://doi.org/10.1007/s00603 -011-0176-z. Zhao, C., C. C. Ma, C. F. Zhao, S. G. Du, and C. Bao. 2017. “Crack propagation simulation of rock-like specimen using strain criterion.” Eur. J. Environ. Civ. Eng. 22 (1): 228–245. https://doi.org/10.1080/19648189 .2017.1359677. Zhao, C., J. L. Niu, Q. Z. Zhang, C. F. Zhao, and Y. M. Zhou. 2018a. “Failure characteristics of rock-like materials with single flaws under uniaxial compression.” B. Eng. Geol. Environ. 78 (1): 593–603. https://doi.org/10.1007/s10064-018-1379-2. Zhao, C., Y. M. Zhou, C. F. Zhao, and C. Bao. 2018b. “Cracking processes and coalescence modes in rock-like specimens with two parallel preexisting cracks.” Rock Mech. Rock Eng. 51 (11): 3377–3393. https://doi .org/10.1007/s00603-018-1525-y. Zou, Z., and L. N. Y. Wong. 2014. “Experimental studies on cracking processes and failure in marble under dynamic loading.” Eng. Geol. 173 (May): 19–31. https://doi.org/10.1016/j.enggeo.2014.02.003.
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Cheng Zhao 1; Jialun Niu, Ph.D. 2; Qingzhao Zhang 3; Songbo Yu 4; and Chihiro Morita 5
Abstract: The expanded distinct element method based on a strain strength criterion was applied herein with the aim of analyzing and comparing the cracking behaviors of rock-like specimens with single flaws under uniaxial and biaxial compressions. The cracking characteristics in uniaxial and biaxial compressions were compared and the effects of confining pressure on the tensile/shear behaviors were analyzed. The confining pressure obviously changed the boundary condition and the cracking behaviors in biaxial compression. The initiation stress, the peak strength, and the number of shear cracks in biaxial compression were obviously larger than in uniaxial compression. The confining pressure affected the initiation and propagation of the secondary cracks, the failure mechanisms, and the distribution of horizontal displacement fields in a large extent. The initiation and propagation of the tensile cracks were inhibited by the confining pressure. An increase in confining pressure had no obvious effects on the initiation time and the propagation rates of the shear cracks but mainly affected the numbers of shear cracks. DOI: 10.1061/(ASCE)MT.1943-5533.0002967. © 2019 American Society of Civil Engineers. Author keywords: Cracking behaviors; Strain strength criterion; Expanded distinct element method (EDEM); Single flaw; Confining pressure.
Introduction Thorough investigation of the damage and failure of rock mass plays an important role in predicting the unstable failure of jointed rock engineering. The mechanical behavior of rock mass is largely controlled by microflaws. The cracking behaviors of rock mass have been studied theoretically (e.g., Ewy and Cook 1990; Griffith 1921; Horii and NematNasser 1985; Liu et al. 2015; Sammis and Ashby 1986) and experimentally (e.g., Brace and Bombolakis 1983; Bobet and Einstein 1998; Petit and Barquins 1988; Wong and Chau 1998; Wong and Einstein 2009a, b; Yang et al. 2016; Zou and Wong 2014) on a number of brittle materials. The initiation, propagation, and coalescence of rock mass and rock-like materials have been studied thoroughly. However, the internal stress and strain states, as well as the quantitative features during the cracking process, are worth studying numerically. 1 Professor, Dept. of Geotechnical Engineering, Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji Univ., Shanghai 200092, China; College of Engineering, Tibet Univ., Lasa 850000, China. Email: [email protected] 2 Dept. of Geotechnical Engineering, Tongji Univ., Shanghai 200092, China. Email: [email protected] 3 Associate Professor, Dept. of Geotechnical Engineering, Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji Univ., Shanghai 200092, China (corresponding author). Email: [email protected] 4 Engineer, Dept. of Geotechnical Engineering, Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji Univ., Shanghai 200092, China. Email: [email protected] 5 Professor, Dept. of Civil and Environmental Engineering, Univ. of Miyazaki, Miyazaki 889-2192, Japan. Email: [email protected] -u.ac.jp Note. This manuscript was submitted on July 27, 2018; approved on June 14, 2019; published online on September 30, 2019. Discussion period open until February 29, 2020; separate discussions must be submitted for individual papers. This paper is part of the Journal of Materials in Civil Engineering, © ASCE, ISSN 0899-1561.
© ASCE
Several numerical methods have been proposed to investigate the damage and failure behaviors of brittle materials. The most common numerical methods adopted in rock mechanics include mainly the finite-element method (FEM) (Guven and Madenci 2011; Tang et al. 2001), boundary element method (BEM) (Chen et al. 1998), bonded-particle model (BPM) (Zhang and Wong 2012), numerical manifold method (NMM) (Wu and Wong 2013), displacement discontinuity method (DDM) (Gonçalves and Einstein 2013), and discrete element method (DEM) (Vesga et al. 2008; Jiang et al. 2009; Gui et al. 2017). The DEM was proposed by Cundall and Strack (1979), in which the finite displacements and rotations of discrete bodies are allowed, and new contacts during the deformation processes can be identified automatically. In the numerical simulations of rock mass and rock-like materials, simulated specimens are treated as an assemblage of independent rigid or deformable blocks, or particles with certain particle friction and bond strength to reproduce the strength of brittle material (Cundall and Hart 1993). Vesga et al. (2008) employed DEM to simulate the failure process of brittle clay specimens and revealed good agreement with experimental results. As a kind of DEM, the Universal Distinct Element Code (UDEC) has obvious advantages in handling the stress-strain behavior within each grain and capturing the interaction of adjacent irregular grains through predefined contact constitutive models (Gui et al. 2017). To solve the problem of the initiation and propagation trajectories of macroscopic cracks, Jiang et al. (2009) developed an expanded distinct element method (EDEM) based on UDEC, and this numerical method was applied to simulate crack initiation and propagation due to the shear and tension failures in matrix rock blocks. Owing to the complexity of fracture distributions at the macrostructural and microstructural levels, the failure modes of rock mass and rock-like materials are complex and difficult to quantify or predict. The cracking processes still need to be assessed more comprehensively. The accurate estimation of crack initiation and
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propagation, as well as the tensile/shear behaviors of rock mass and rock-like materials, remain a challenge. In this study, the strain strength criterion proposed by Zhao et al. (2017) was employed in UDEC to comprehensively study the failure modes and cracking behaviors of rock-like specimens with single flaws in compression tests. The single-flawed specimens with different inclination angles were loaded to fail under uniaxial compression and biaxial compression in five confining pressure conditions (1.5, 3, 4.5, 6, and 9 MPa). The cracking behaviors, such as initiation stress, peak strength, propagation trajectories, and distribution of displacement fields, were thoroughly analyzed. In addition, the effects of confining pressure on the tensile/shear behaviors were studied.
Numerical Method and Model The EDEM is adopted in this paper because of its advantages in accurately simulating the initiation and propagation path of macroscopic cracks. The EDEM is based on UDEC, which was implemented with a strain strength criterion that combines tension and shear behaviors of rock mass. The cracking processes of rock-like specimens with single flaws under uniaxial and biaxial compressions were simulated by EDEM. The strain strength criterion proposed by Zhao et al. (2017) was employed to comprehensively describe tensile and shear cracks. Based on such a criterion, Zhao et al. (2017, 2018a) simulated the cracking behavior of rock-like specimens containing a single flaw or flaw pairs. It was shown that the stress-strain curves and crack trajectories obtained from their numerical simulation based on this criterion were highly consistent with experimental results. Virtual cracks were set up in all potential cracking areas to determine the crack trajectories. During the entire cracking process, the tensile cracks were judged by principal strain and the shear cracks were judged by the Mohr– Coulomb criterion. The critical state can be expressed by the following equation: ε3 ¼ −εt τ ¼ c þ σ tan φ
ð1Þ
where ε3 = principal strain; εt = critical tensile strain; τ = shear strength; c = cohesion; σ = normal stress on shear plane; and φ = internal friction angle. The tensile cracks were assumed to initiate when the principal strain reached a critical value and expand in the direction of the maximum principal strain. The shear cracks were assumed to initiate if Mohr’s circle touched the failure envelope and expanded in the direction of the most dangerous stress state. At the limit equilibrium state for a linear elastic material, the critical states can be expressed as σ1 ¼ ðσ3 þ Eεt Þ=ν sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ sin φ 1 þ sin φ σ1 ¼ 2c þ σ 1 − sin φ 1 − sin φ 3
ð2Þ
where σ1 and σ3 = principal stresses, respectively; E = elastic modulus; and ν = Poisson’s ratio. For the determination of the tensile and shear cracks in numerical simulation, the main factors are defined as f t ¼ σ3 − vσ1 þ Eεt
ð3Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ sin φ 1 þ sin φ f s ¼ σ1 − σ − 2c 1 − sin φ 3 1 − sin φ © ASCE
ð4Þ
Fig. 1. Numerical model.
where ft = factor to determine a tensile crack; and f s = factor to determine a shear crack. The tensile crack initiates if f t > 0, and the shear crack initiates if f s > 0. To compare the present work with a series of experimental results of single-flawed specimens conducted by Zhao et al. (2017, 2018a) the geometries of specimen for numerical simulations were the same as that for experimental specimens. The dimensions of the used specimen were 100 mm high and 40 mm wide (Fig. 1). The length of the prefabricated flaw was 14 mm, and the inclination angles α were 15°, 30°, 45°, 60°, and 75°. Based on the experimental results, the crack trajectory and mechanical properties of the specimen at 45° was the most complicated. Thus, the present work mainly focused on the specimen with inclination angles of 45° and compared the simulations with the experimental results of other specimens. The specimens were divided into hexagonal blocks by virtual cracks, and each hexagonal block had a side length dimension of 1.5 mm. To obtain the stress and strain of each block during the entire cracking process, the hexagonal blocks were divided into triangular grids where the maximum length of the grid was less than 1 mm. The material parameters used in the numerical simulations are the same as in Zhao et al. (2018b) where the simulations were conducted according to the compression experiments of gypsum specimens containing prefabricated cracks. For the EDEM, there are virtual cracks, precracks, and tensile and shear cracks converted from the virtual cracks. To obtain an accurate result, the chosen parameters of the precracks and the real cracks should be close to the practical situation. Therefore, for virtual cracks, the unit blocks should be bonded to each other and cannot slip or open. As a result, the cohesive force and tensile strength of the virtual cracks can be sufficiently high. The specimens contain single flaws with inclination angles of 45° were loaded to fail under uniaxial and biaxial compressions. In the uniaxial compression test, the vertical displacement at the bottom of the specimen was fixed, and the top surface was under a displacement load. Fig. 2 shows the loading process in the biaxial compression test. To keep the confining pressure constant, a horizontal load was applied by stress loading at the beginning of the simulations. The vertical load was applied to the upper end of the specimen after the stress of the specimen was stabilized. The vertical displacement of the lower end was fixed and the confining
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Fig. 2. Loading process in biaxial compression: (a) one loading process; and (b) another loading process.
pressure kept constant during the vertical loading process. In this study, to simulate crack initiation and propagation more accurately and effectively, the vertical load was applied using the displacement methodology, and the loading rate was 0.001 m=s.
Simulation Results under Uniaxial and Biaxial Compression In this study, the cracking behaviors from initiation to coalescence of single-flawed rock-like specimens was numerically simulated. Different cracking behaviors under uniaxial and biaxial compression were compared, such as stress-strain curves, tensile and shear crack numbers, crack trajectories, and horizontal displacement fields. Compared with the uniaxial compression test, there were several differences in the biaxial simulation including the stress of crack initiation, peak strength, propagation trajectories, and the distribution of displacement fields.
Fig. 3. Stress-strain curve and number of tensile/shear cracks in uniaxial compression. © ASCE
Stress-Strain Curves and Crack Numbers The entire stress-strain curve and the number of tensile and shear cracks of single-flawed specimens in the uniaxial compression test are shown in Fig. 3. The tensile cracks initiated when the stress reached approximately 24.6 MPa at the end of the elasticity stage (Point 1). After the initiation of cracking, the number of tensile cracks increased gradually. When the internal stress accumulated to a certain point, the tensile cracks suddenly expanded rapidly, and the corresponding stress-strain curve entered the unstable stage (Point 2). The number of tensile cracks increased slowly after the stress-strain curve entered the stable stage (Point 3), but the number of shear cracks almost remained unchanged. The tensile and shear cracks expanded rapidly almost at the same time when the stress reached a peak strength of approximately 30.78 MPa (Point 4). It can be seen from the stress-strain curve that the uniaxial compression specimen had a ratio of initiation stress to peak strength of 79.9%. The number of tensile cracks during the entire cracking process was significantly more than the shear cracks. The entire stress-strain curve and the number of tensile and shear cracks of single-flawed specimens under a confining pressure of 3 MPa in biaxial compression tests are shown in Fig. 4. As in uniaxial compression, the tensile cracks initiated earlier than the shear cracks, and the specimen eventually failed under the combined effect of tension and shear. However, compared with uniaxial compression, the cracking process under biaxial compression had several obvious differences. The initiation stress increased to 35.36 MPa (Point 1), the peak strength increased to 53 MPa (Point 4), and the confining pressure largely improved the strength of single-flawed specimen. The ratio of initiation stress to peak strength was 66.7%, less than that under uniaxial compression. The number of shear cracks exceeded the tensile cracks between Points 2 and 3 and was significantly more than the tensile cracks at the end of the cracking process. The application of horizontal confining pressure changed the boundary condition and cracking mechanism. In terms of the boundary condition, the confining pressure restricts the lateral deformation of the specimen. As for the cracking mechanism, the initiation of wing cracks is mainly due to the stress concentration at the crack tips, which cause the principal strain to exceed the critical value. The direction of maximum principal strain is basically horizontal in
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Fig. 4. Stress-strain curve and number of tensile/shear cracks in biaxial compression.
uniaxial compression, while in biaxial compression, the horizontal pressure reduces the horizontal principal tensile strain so as to increase the initiation stress and the peak strength. Compared with uniaxial compression, the confining pressure in the biaxial compression test caused the specimen to exhibit ductile failure characteristics, that is, the failure occurs after a relatively long period of time after crack initiation. Crack Trajectories Fig. 5 shows the crack trajectories of single-flawed specimens with an inclination angle of 45° during uniaxial compression. The dark oblique lines at the center of the specimens represent flaws, the dotted lines in hexagonal blocks represent virtual cracks, the cracks directed by T are tensile cracks, the cracks directed by S are shear cracks. The wing cracks, which generated under the tensile effect, initiated from the flaw tips, then propagated along the direction of the maximum compressive stress, and finally penetrated the upper and lower ends of the specimen. The secondary cracks generated mainly under the shear effect initiated from the flaw tips if the wing cracks were close to the ends of the specimen, and the angle of propagation was about 60° from the horizontal. The specimen failed when tension and shear deformation were combined, and the tensile effect was dominant.
To study the influence of inclination angle, crack trajectories before the failure of specimens with different inclination angles in experimental and numerical study are shown in Fig. 6. If the crack inclination angle was less than 45° (α ¼ 15° and 30°), the failure of a specimen was dominated by wing cracks, while shear cracks were not obvious during the loading process. Although the shear cracks did not sufficiently propagate, their shear stress fields affected the trajectories of the wing cracks. As a result, the new cracks propagated along the direction of maximum principal stress. If the crack inclination angle was larger than 45° (α ¼ 60° and 75°), the cracks mostly generated under a shear effect, leading to failure. The propagation trajectories were alternately distributed by the mixed cracks during the loading process, and the macroscopic cracks showed a mixture of tensile and shear characteristics. Despite the interaction of tensile and shear stress, shear cracks played a significant role in the cracking process. It is worth noting that the crack trajectories in the specimen with an inclination angle of 75° were slightly different from the experimental results. Such a difference arose probably because this inclination angle of flaw was close to the shear failure angle of specimens without flaws. Further numerical simulation showed obvious characteristics of shear failure, and the crack extended at both ends of the specimen along the 65° inclination direction. The crack trajectories of single-flawed specimens with an inclination angle of 45° during biaxial compression with a confining pressure of 3 MPa are shown in Fig. 7. As with uniaxial compression, the wing cracks initiated from the flaw tips. Compared with uniaxial compression, the failure process exhibited significant shear characteristics. From the crack trajectories and the number of tensile/shear cracks (Fig. 4), it can be found that secondary cracks under shear effect appeared shortly after the propagation of wing cracks (Point 2). Then secondary cracks rapidly propagated along the flaw direction to the left and right ends of the specimen; in addition, some cracks generated near the bottom of the specimen under the combined effects of horizontal stress and fixed bottom displacement (Point 3). At this time, the wing cracks had not propagated to the vicinity of both ends of the specimen. During the stage from Point 2 to Point 3, the propagation rate of tensile cracks was obviously slower than that of shear cracks, and the stress stagnated and fluctuated. The shear cracks continued to propagate and spread to the angled region between the wing cracks and the secondary cracks. The specimen exhibited significant plastic damage characteristics because of the confining pressure. During the stage from Point 3 to Point 4, the propagation of wing cracks ended, and the number of shear cracks was more than the number of tensile cracks.
Fig. 5. Crack trajectories in numerical simulation of specimen under uniaxial compression. © ASCE
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Fig. 6. Crack trajectories of specimens with different inclination angles in (a, b, c, d, and e) experimental study; and (f, g, h, i, and j) numerical study: (a) α ¼ 15°; (b) α ¼ 30°; (c) α ¼ 45°; (d) α ¼ 60°; (e) α ¼ 75°; (f) α ¼ 15°; (g) α ¼ 30°; (h) α ¼ 45°; (i) α ¼ 60°; (j) α ¼ 75°.
Fig. 7. Crack trajectories in numerical simulation of specimen under biaxial compression.
When the specimen failed, the secondary cracks penetrated the left and right sides of the specimen. Therefore, the entire cracking process of the single-flawed specimen under biaxial compression can be summarized as the propagation of wing cracks, the propagation of both wing cracks and secondary cracks, and the propagation of secondary cracks. The collection of blocks set in the EDEM are used to simulate rock-like material, which can accurately reflect the initiation process of microcracks and the brittleness of the specimen, ensuring the effective simulation of shear cracks. The trajectories of wing cracks generated under the tensile effect were similar in both uniaxial and biaxial compression tests. The differences were mainly © ASCE
reflected in the initiation time, the trajectories, and the numbers of secondary cracks under uniaxial and biaxial compression. Horizontal Displacement Fields Fig. 8 shows the horizontal displacement fields in the numerical simulation of biaxial compression with a confining pressure of 3 MPa. Compared with the horizontal displacement fields in the initial stage simulated under uniaxial compression, in the initial stage of biaxial compression, owing to the effect of confining pressure, the horizontal displacement generated from both sides of the specimen and toward the inside of the specimen. With the increase
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Fig. 8. Evolution of horizontal displacement fields under biaxial compression.
of vertical load, the horizontal displacement was gradually dominated by the horizontal expansion effect from the sides of the wing cracks. If the wing cracks initiated and the secondary cracks had not yet initiated (Point 1), the distribution of the horizontal displacement field was similar to that of uniaxial compression. However, compared with uniaxial compression, shearing dislocations generated in the horizontal displacement field when the secondary cracks initiated (Point 2). Especially after a large number of shear cracks generated (Points 3 and 4), the displacement contours divided the specimen into several regions, and the specimen was subjected to more shear effect than uniaxial compression.
Cracking Behaviors under Different Confining Pressures The specimens under different confining pressures were loaded to failure in the biaxial test. The confining pressures were 1.5, 3, 4.5, 6, 7.5, and 9 MPa. For convenience of presentation, the specimen under a confining pressure of, for example, 1.5 MPa was abbreviated as “1.5 MPa specimen,” and this system was followed for the other confining pressures. The development curves of tensile cracks under different confining pressure conditions, which focused on the variable number of tensile cracks during the simulations, were compared.
Effects of Confining Pressure on Shear Properties Fig. 10 shows the development curves of shear cracks under five confining pressures. With the increase in confining pressure, the initiation points of shear cracks under different loading conditions are basically unchanged. Moreover, the propagation rates of shear cracks are nearly the same. These phenomena indicate that confining pressure has no obvious effect on the initiation and propagation of shear cracks. With reference to the total number of shear cracks and tensile cracks, it is found that the proportion of shear cracks in new cracks increases gradually with increasing confining pressure. When the confining pressure is small, the failure behavior of the specimen is dominated by both tensile cracks and shear cracks. When the confining pressure is large, the tensile properties are replaced by shear properties, and the propagation of shear cracks significantly leads to the fracture of the specimen. The failure behaviors of the specimen exhibit shear properties. The development curves of tensile and shear cracks also directly reflect the crack initiation sequence. Under a low confining pressure (1.5 and 3 MPa), tensile cracks initiate earlier than shear cracks. Under a medium confining pressure (4.5 and 6 MPa), tensile cracks and shear cracks initiate almost simultaneously. Under a
Effects of Confining Pressure on Tensile Properties As shown in Fig. 9, the initiation time of tensile cracks are delayed with the increase of confining pressure. Apart from the crack initiation time, the confining pressure also inhibits the numbers and propagation rates of tensile cracks. With increasing confining pressure, the number of tensile cracks, as well as the slopes of the tensile crack development curves, decreases gradually. Compared with the uniaxial compression test, the confining pressure plays a major role in inhibiting the initiation and propagation of tensile cracks in the single-flawed specimen, and the inhibition is enhanced as the confining pressure increases. The mechanisms of the inhibition in tensile cracks can be summarized as two aspects. On the one hand, the tensile cracks mainly generate to the sides of the specimen horizontally, while the horizontal deformation of the specimen is limited by the confining pressure. On the other hand, the initiation of the tensile cracks requires a certain amount of energy, while the concentration of horizontal tensile strain is reduced by the confining pressure. © ASCE
Fig. 9. Development curves of tensile cracks under five confining pressures.
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of tensile cracks decreased. Compared to tensile cracks, there were no obvious effects of different confining pressures on the initiation time and the propagation rates of shear cracks. The increase in confining pressure mainly affected the number of shear cracks.
Data Availability Statement
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All data, models, and code generated or used during the study appear in the submitted article.
Acknowledgments
Fig. 10. Development curves of shear cracks under five confining pressures.
high confining pressure (9 MPa), shear cracks initiate earlier than tensile cracks. These outcomes are consistent with the theoretical analysis and the observation of failure status.
Conclusions The objective of this study was to analyze and compare the cracking behaviors of rock-like specimens with single flaws under uniaxial and biaxial compression. Using the EDEM, cracking behaviors under uniaxial and biaxial compression were studied, such as initiation stress, peak strength, propagation trajectories, and distribution of displacement fields. In addition, biaxial compression tests under different confining pressures were conducted, and the effects of confining pressure on tensile/shear behaviors were analyzed. Based on the simulations of a single-flawed specimen with an inclination angle of 45°, the following observations and findings were obtained: 1. Tensile cracks initiated earlier than shear cracks, and the specimens failed under the combined effect of tension and shear in both the uniaxial and biaxial tests. Under biaxial compression, the confining pressure increased the initiation stress and peak strength of the single-flawed specimen. If the specimen failed, the number of shear cracks under biaxial compression greatly exceeded the number of tensile cracks, in contrast to uniaxial compression. The confining pressure obviously changed the boundary condition and the cracking characteristics under biaxial compression. 2. The initiation and propagation of secondary cracks and the distribution of horizontal displacement fields were affected by the confining pressure to a great extent. Under biaxial compression, secondary cracks generated earlier than under uniaxial compression. The failures of the uniaxial and biaxial specimens were mainly controlled by wing cracks and secondary cracks, respectively. The cracking process of a biaxial specimen can be summarized as the propagation of wing cracks, the propagation of both wing cracks and secondary cracks, and the propagation of secondary cracks. 3. As a result of the changes in the boundary condition and the cracking characteristics under biaxial compression, the confining pressure inhibited the initiation and propagation of tensile cracks. With increases in the confining pressure, the initiation times were delayed, and the numbers and propagation rates © ASCE
The authors would like to acknowledge the financial support of the National Key R and D Program of China (2017YFC0806000), the National Natural Science Foundation of China (41202193, 41572262, and 41502275), the Shanghai Rising-Star Program (17QC1400600), the Fundamental Research Funds for the Central Universities (0200219207), the Shanghai Municipal Science and Technology Major Project (2017SHZDZX02), and the Key Laboratory of Rock Mechanics and Geohazards of Zhejiang Province.
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