Particle simulation of the failure process of brittle rock under triaxial compression

International Journal of Minerals, Metallurgy and MaterialsVolume 17, Number 5, October 2010, Page 507DOI: 10.1007/s12613-010-0350-4Particle simulation of the failure process of brittle rock undertriaxial compressionMing Xia.2) and Ke-ping Zhou)1) School of Resources and Safety Engineering, Central South University, Changsha 4 10083, China2) Ma'anshan Institute of Mining Research, Ma'anshan 243004, China(Received: 20 August 2009; revised: 17 September 2009; acepted: 3 October 2009)Abstract: In order to investigate the failure process of britle rock under triaxial compression through both experimental and numerical ap-proaches, the particle simulation method was used in numerical simulations and the simulated results were compared with those of the ex-periment. The numerical simulation results, such as fracture propagation, microcrack distribution, stress-strain response, and damage patterns,were discussed in dtail. The simulated results under various confining pressures (0-60 MPa) are in good agreement with the experimentalresults. The simulated results reveal that rock failure is caused by axial spitting under uniaxial compression. As the confining pressure in-creases, rock failure occurs in a few localized shear planes and the rock mechanical behavior is changed from brittle to ductile. Consequently,the peak failure strength, microcrack numbers, and the shear plane angle increase, but the ratio of tensile to shear microcracks decreases. Thedamage formation during the compression simulations indicates that the particle simulation method can produce similar behaviors as thoseobserved through laboratory compression tests.Keywords: rock mechanics; compression testing; failure; fracture modes; simulation; microcracks[This work was financially supported by the Graduate Degree Thesis Inmovation Foundation of Central South University (No. 2093st226).]1. Introductionused to simulate the crack initiation, propagation, and coa-lescence, associated with the rock failure. Recently, the par-Advanced techniques, such as scanning electron micros-ticle simulation method [13-17], in which the bonded parti-:opy (SEM), acoustic emission (AE), computerized tomo-cle model is commonly used [18], provides a new way tography (CT), and real-time laser holographic interferometry,reproduce many features related to rock behaviors, such asare applied to laboratory experiments for monitoring rockelasticity, fracturing, acoustic emission, material anisotropyfailure processes [1-4]. Although these studies enhance the due to damage accumulation, hysteresis, dilation, post-peakunderstanding of these processes under compression, the re-softening, and strength increase with confinement. For thislated failure mechanisms, including the microcrack initiation,reason, the particle simulation method has been used topropagation, coalescence, axial splitting, and shearing, arsolve a wide range of rock failure problems of sample-, en-not fully understood [5].gineering-, and geological-length scales [13-17].Many existing models, such as the lattice model [6] andIn this paper, the particle simulation method built in thethe rock failure process analysis (RFPA) model [7-8], havecommercial code (two-dimensional particle flow codesome considerable limitations for simulating rock failurePFC2D)) was adopted to simulate the failure process of brit-processes [9-10]. To overcome the limitations of the RFPAtle rock under the confining pressures of 0, 20, 40, and 60model, the local degradation approach [11] and theMPa, respectively. The comparison between experimentalelasto-plastic cellular automaton model (EPCA) [12] are and numerical results was carried out. In addition, the for-Corresponding author: Ming XiaE-mail: xiaming105@126.com◎University of Science and Technology Beiing and Springer-Verlag Berlin Heidelberg 2010中国煤化工包SpringerMYHCNM HG.508Int J. Miner. Metall. Mater, Vol.17, No.5, Oct 2010mation of microcracks, fracture patterns, and failure planeTable 1. Microscopic parameters of the PFCzD modelangles were carefully analyzed.Particle radius multiplier,..0Particle-particle contact modulus, E。/ GPa802. Experimental and numerical methodologyParticle siffness ratio, kn 1 ks.52.1. Test samples and experimental apparatusParllel-bond modulus, E。 /GPa30A total of 40 skarn samples were prepared to carry outParllel-bond siffnesss ratio, k。 1k。uniaxial and conventional triaxial compression tests (theParticle friction coefficient, μ).5confining pressure equaled to 0, 20, 40, and 60 MPa, re-Parllel-bond mean normal strength,元。/ MPa00spectively). The preparation and testing procedure of rockParallel-bond normal strength, standard deviation,25samples followed the standard testing methods by Interna-od /MPa .tional Society for Rock Mechanics (ISRM) [19].Parallel-bond mean shear strength,元。 / MPaThe triaxial compression experiments were carried out byParallel-bond shear strength, standard deviation,using an MTS815 full-digitally servo-controlled rock me-Taa /MPachanics testing machine at the Rock Mechanics Laboratory,Note: kn- -normal sifess; kg- shear sifness; kn - paral-Changsha Instute of Mining Research, China.lel-bond normal sifess, ks- parall-bond shear sifess.2.2. Brief introduction of PFC2DTable 2. Sample-level parameters of the skarn sampleThe two-dimensional particle flow code (PFC2) [20] isused to approximately simulate the physical tests. The codeParameterNunmerical Experimentalallows the user to explicitly model fracture damage directlyDimensions / (mmxmm)60*12060>120[18]. The microcrack initiation and propagation can be ex-Density 1 (kgm~3)3269pressed as a progressive breakage of contact bonds. TheElastic modulus / GPa117.2111.59 .crack pattermn is automatically determined without need forre-meshing and can be visualized directly during the simula-Poisson ratio0.177tion [18, 20]. As a result, the micromechanical damage canUniaxial compressive strength/ MPa 138.03139.8progressively evolve. Since typical PFC2D models requiresome micromechanical parameters, which cannot be meas-could be maintained. Fig. 1 shows the particle model of aured directly using existing laboratory tests, the trial and er-sample (about 60 mmx 120 mm in size) with 8884 particles,ror method is commonly used to determine these parametersthe minimum radius of the particle assembly is 0.35 mm,[13-17]. However, for the particle simulation ofand the particle size ratio is 1.66.two-dimensional problems, Zhao et al. [13-14, 17] presentedTop walladvanced theoretical methods recently to match parti-cle-level parameters and sample-level ones directly.Table 1 shows the microscopic parameters adopted in this. Particle assemblestudy to characterize the failure process of the sample sub-jected to a wide variety of confining pressures. These pa-rameters can lead to the sample-level parameters, as shownin Table 2.3. Numerical modelling of rock failure process8884 particles.using the particle simulation methodUniform size distributionParticle radi:60 mmR..-0.35 mm3.1. Numerical modelR=0.58 mmThe particle radii were chosen to have a uniform distribu-tion in the sample. For the biaxial test, the top and bottom↑↑↑↑Bottom wallwalls acted as loading platens. The velocities of two lateral中国煤化工walls were controlled so that a specified confining pressureFig.1. Pan4 particles.YHCNMHG.M. Xia et al, Particle simulation of the failure process of brittle rock under triaxial compression5093.2. Failure process of a specimen under various confin-out the specimen do not seem to be interacting. Most dam-ing pressuresage occurs at the post-peak stage (point c). Also, theFigs. 2-5 show the axial stress, total cracks, tensile cracks,stress-strain curve begins to descend. Finally, the failureand shear cracks plotted against axial strain and microcrackpattern of the specimen is axial splitting.distributions for the confining pressures of 0, 20, 40, and 60In Fig. 3, at the pre-peak stage, most of the microcracksMPa, respectively. In these figures, microcracks are de-are located through the specimen (point a). Then the micro-picted as bicolor lines in which black represents the ten-cracks can be found around the existing microcracks at bothsion-induced parallel-bond failure and red represents theends (point b, where the peak failure strength is 288.4 MPa).shear- induced parallel- bond failure.At the post-peak stage, microcracks tend to propagate andIn Fig.2, at the pre-peak stage (point a), the microcracksinteract (point c), and the macro fracture is formed by theare, in general, concentrated at the top of the model, and fewgrowth and coalescence of microcracks. The macro failuremicrocracks are observed. At the peak stage (point b, whereis the combination of axial splitting and local shearing.the uniaxial compression strength is 138.03 MPa), micro-In Fig. 4, at the pre-peak stage (point a), the microcrackscracks continue to increase, grow and coalesce at the localare random ditributed through the specimen. At the peakplace. The microcracks that form at the peak stage through-stage (point b, where the peak failure strength is 320.71407140030a 1750(a120Axial stress200号240Pa2%-1500t 12501001000Total cracks18(+ 100030 t+ 800差120一t 75050- 600Tensile cracksTensile cracks。0F0|t 400十50Shear cracks.oShear cracks25020.0 0.1 0.2 0.3 0.4 0.5 0.6 0.Axial strain/ 10-2(b,)(b2)↑(b)(b2)(b3)Fig. 2. Stress strain curves (a) and microcrack distributionsFig.3. Stress-strain curves (a) and microcrack distributionswithout confining pressure: (b) in the pre-peak stage at point aat a confining pressure of 20 MPa: (b,) in the pre-peak stage atin (a) (162 cracks, tensile/shear=94/68); (b2) in the peak stage atpoint a in (a) (105 cracks, tensile/shear=45/60); (b2) in the peakpoint b in (a) (529 cracks, tensile/shear- =352/177); (b3) in thestage at point b in (a) (828 cracks, tensile/shear- 434/394); (b3)post-peak stage at point c in (a) (1094 cracks, ten-in the post-peak. stape at noint C in. (a) (1664 cracks, ten-sile/shear=817/277).silelshear=1087/51中国煤化工MHCNMH G.510Int J. Miner. Metall. Mater, Vol.17, No.5, Oct 2010350, 2800(a50072003002400.400Axial stress1 64002000ε25t 5600Total cracks2001600. Axial stress4800e 200400050Tensile cracksF 1200罗100-3200Shear crackst 8002400o50 t.4000.1.12°0Axial strain/ 100.2 0.4 0.60.8 1.0 1.2(b)2)，(b)| 1b).2)《;’ (b3)Fig. 4. Stress-strain curves (a) and microcrack distributionsat a confining pressure of 40 MPa: (b) in the pre-peak stage atig. 5. Stress-strain curves (a) and microcrack distributionspoint a in (a) (115 cracks, tensilshear-33/82); (b2) in the peakat a confining pressure of 60 MPa: (b1) in the pre-peak stage atstage at point b in (a) (838 cracks, tensile/shear- -575/263); (bz)point a in (a) (765 cracks, tensile/shear 254/511); (b2) in thein the post-peak stage at point c in (a) (1675 cracks, ten-peak stage at point b in (a) (1523 cracks, tensile/shear- -572/951);sile/shear=1001/674).(b3) in the post-peak stage at point c in (a) (2992 cracks, ten-sile/shear=1537/1455).MPa), the microcracks develop along the diagonal line fromthe upper left cormner to the bottom right cormner. As the load-4. Discussioning increases, the stress changes a lttle (point c), so that the4.1. Microcrack formationformed macroscopic fracture plane propagates along the in-The failure patterns obtained from different confiningclined plane. The angle of the fracture plane is about 38*pressures have some common features. On the basis of thewith respect to the vertical axis.simulated microcrack distributions shown in Figs. 2-5, it canIn Fig. 5, at the pre-peak stage (point a), microcracksbe concluded that microcracks are primarily randomconcentrate and interact in the specimen. At the peak stagethroughout the specimen. As the loading increases, micro-(point b, where the peak failure strength is 432 MPa), thecracks concentrate and interact (shown as the peak stage).microcracks are still expanding around the previous crack-At the post-peak stage, microcracks result in the final col-ing. As the loading increases, the stress changes a lttlelapse of the specimen. From the cumulative crack plot in(point c), which is the same as what is observed in Fig.4.these figures, it is clear that most fracturing takes place inThe specimen has three inclined fracture planes, but thethe post-peak stage, so that the number of cracks increasesspecimen failure is caused by one major fracture. The anglesharply at the post peak stage.of the fracture plane is about 46° with respect to the verticalAlthough中国煤化工T tesile oraxis.shear-induced, t: more predomi-TYHCNMHG.M. Xia et al, Particle simulation of the failure process of brittle rock under triaxial compression511500nant than shear-induced cracking. The ratios of tensileExperimentalcracks to shear cracks in the post-peak stage for differentNumerical400confining pressures are 2.95, 1.88, 1.49, and 1.06, respec-tively. As the confining pressure increases, the ratio of ten-300sile microcracks to shear microcracks decreases. The con-fining pressure reduces the tensile forces that developed in a200direction perpendicular to the specimen axis and thereforecauses more shear microcracks to form [18].100 I4.2. Brittle to semi-brittle and ductile transitionsFig. 6 shows the experimental results of the sample undervarious confining pressures. These results indicate that theConfining pressure 1 MPaconfining pressureinfluencesthe non-linearityFig, 7. Variation of peak failure stress with confining pres-stress-strain curves. When the confining pressure is zero, at sure.the post-peak stage, the fracture process develops veryquickly, so the sample collapses over a very small straincompression test, the failure pattern of the specimen is axialrange. This indicates that the rock mechanical behavior issplitting. At the confining pressure of 20 MPa, the failurebrittle. When the confining pressure increases to 20 MPa,pattern of the specimen is manifested by a combination ofaxial splitting, and the inclined failure surface is observed.the stress -strain curve descends slowly, indicating that therock mechanical behavior transits to semibrittle. When theAt high confining pressures (40 and 60 MPa), the failureconfining pressure increases to 40 and 60 MPa, the rockpatterns of the specimen are mainly characterized by one ormore shear fracture planes. Compared with the laboratorymechanical behavior transits from semi-brittle to ductile.test, the numerical simulation in this paper indicates the es-sence of the features observed in the laboratory.5 60 MPa400 .Fig. 9 shows the variation of fracture plane angles withconfining pressures for both the experimental tests and nu-虽o=40 MPamerical results. In the uniaxial compression, axial spltting is。0:=-20 MPathe main fracture. The angle of the failure plane increases at20 MPa. When the confining pressure increases to 40 and 60MPa, the angle of the shear plane increases a lttle further.100Moreover, our numerical and experimental results are con-o,=0 MPasistent with those predicted by Liu et al. [8] and Fang et al.[11].0.00.20.40.60.81.01.21.4 1.6Axial strain /1025. ConclusionsFig. 6. Experimental results of the samples under variousconfining pressures (σ3 is the confining pressure).The particle simulation method built in the commercialcode (PFCD) was used to simulate the progressive failureFig. 7 shows the variation of the numerical and experi-process of brittle rock. In particular, microcrack distribu-mental peak stress with confining pressures. It shows that as .tions under various confining pressures were explored. Fromthe confining pressure increases, the peak strength increases.the related results, the following conclusions can be ob-The values of the peak failure strength between experimen-tained.tal and numerical results are very close, which indicates that(1) The particle model can be used to predict the failurethe simulated results can capture quantitative aspects of rockfailure in triaxial compression.process associated with development of tensile cracks andshear cracks. As the confining pressure increases, the peak4.3. Fracture patterns and failure plane anglesfailure strength and fracture plane angle increase. These re-Fig. 8 shows a comparison of the typical patterns of shearsults have a good agreement with laboratory test results. Asbands observed in laboratory tests and simulated by PFC'the confining p中国煤化工f tensile micro-corresponding to various confining pressures. In the uniaxialcracks to shearformation ofMYHCNMHG.512Int. J. Miner. MetalL. Mater, Vol.17, No.5, Oct 2010(a)0MPa;20 MPa40 MPa60 MPa(b) 0MPaFig. 8. Failure patterns of the samples under various confining pressures: (a) experimental results; (b) simulated results.5(crocracks.一Experimental一-1- NumericalAcknowledgementsC4The authors especially thank Prof. C.B. Zhao at Compu-3(tational Geosciences Research Centre and Dr. J. Yoon at Geo-g20ForschungsZentrum-Potsdam for their work on this paper.References101] X.Y. Wu, P. Baud, and TF. 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