fracture behavior of intact rock using acoustic...

Shuhong Wang,1 Runqiu Huang,2 Pengpeng Ni,3 Ranjith Pathegama Gamage,4 and Minsi Zhang5

Fracture Behavior of Intact Rock UsingAcoustic Emission: Experimental Observationand Realistic Modeling

REFERENCE: Wang, Shuhong, Huang, Runqiu, Ni, Pengpeng, Pathegama Gamage, Ranjith, and Zhang, Minsi, “Fracture Behavior of IntactRock Using Acoustic Emission: Experimental Observation and Realistic Modeling,” Geotechnical Testing Journal, Vol. 36, No. 6, 2013,pp. 903–914, doi:10.1520/GTJ20120086. ISSN 0149-6115.

ABSTRACT: It is well known that acoustic emission (AE) is a powerful nondestructive testing tool for examining the behavior of materialsdeforming under stress. One can use it to monitor fracture or damage in a rock mass by listening to AE events during failure under compressiveloads. In this paper, an experimental study on the AE source location in square cylinder granite specimens under uniaxial compression is reported.In order to determine the three-dimensional location of AE events, eight AE sensors were mounted on the specimen. The AE source location wasdetermined via the acquisition of eight channel AE sensors after filtering, processing, reporting, and visualizing seismic data. On the basis of thelaboratory experiment results, the granite sample was numerically simulated via the Burgers model using the discrete element program PFC2D (Par-ticle Flow Code in Two Dimensions) to further study the mechanism of fracture initiation and propagation in intact rock. In PFC2D, materials maybe modeled as either bonded (cemented) or unbonded (granular) assemblies of particles. It can describe nonlinear behavior and localization withaccuracy that cannot be matched by typical finite element programs. The consistency of stress-strain curves obtained with PFC2D and with the testresults shows that PFC2D is a practical tool for reproducing AE events in rock, and use of the Burgers model is feasible in the field of rock failureand provides an analysis of the microcracking activity inside the rock volume to predict rock fracture patterns under uniaxial loading conditions.

KEYWORDS: acoustic emission, microcracking, rock, numerical modeling, fracture process, initiation and propagation of fracture

Introduction

With the development of the national economy and defense con-struction, people gradually recognize that the uses of undergroundspace play an important role in the improvement of ground envi-ronments such as subways, tunnels, and underground caverns.The safety of excavation in underground projects attracts moreattention. The study of damage formation in jointed or bulk rockunder stress has been a subject of widespread interest, and theresults have led to a number of comprehensive texts (Eberhardt1998; Falls and Young 1998; Eberhardt et al. 1999; Chang andLee 2004; Golshani et al. 2007). The formation of the excavationdamaged/disturbed zone is affected by various factors, such as

stress redistribution and excavation methods (Wang et al. 2009).For many reasons, it is important to be able to predict the time,location, and intensity of potential rock fractures. Fracture devel-opment in stressed rock has been observed extensively in the labo-ratory via a number of methods. Nowadays, as an effectiveapproach, acoustic emission (AE) techniques are broadly appliedto rock in order to obtain information on crack initiation and prop-agation in rock engineering (Mlakar et al. 1993; Eberhardt et al.1998; Lavrov et al. 2002; Lei et al. 2004). The acquisition of cer-tain channel numbers of AE data can be analyzed to determine thethree-dimensional location of AE events.

In spite of the extensive work and the numerous successes inpredicting rock fracture behavior, our comprehension of the physi-cal processes that ultimately control fracture behavior is still weak(Landis 1999; Cai et al. 2004; Leguillon et al. 2007). Because ofthe heterogeneous inclusions (rigid or soft), the magnitude of localstress is significantly higher than the magnitude of applied stress,especially when the material is highly heterogeneous. As localstress and strength vary in a random fashion, the failure site in thematerial also varies randomly, and does not necessarily coincidewith the maximum stress location unless a strong interactive stagebetween microfractures is reached (Alkan et al. 2007). Diederichs(2003) has performed PFC analysis to achieve a good understand-ing of the effect of the heterogeneity of rock materials on the sub-sequent progressive fracture process. It logically follows that abetter conception of heterogeneity will place us in a better positionto formulate predictive models and predict rock fracture behavior.

Manuscript received May 23, 2012; accepted for publication July 25, 2013;published online September 12, 2013.

1Professor, Key Laboratory of Ministry of Education on Safe Mining ofDeep Metal Mines, College of Resource and Civil Engineering, NortheasternUniv., Shenyang, 110819, China, e-mail: [email protected]

2Professor, State Key Laboratory of Geohazard Prevention andGeoenvironment Protection, Chengdu Univ. of Technology, Chengdu,610059, China, e-mail: [email protected]

3Ph.D. Candidate, GeoEngineering Centre at Queen’s - RMC, Queen’sUniv., Kingston, ON, K7L 3N6 Canada, e-mail: [email protected]

4Associate Professor, Deep Earth Energy Lab, School of Civil Engineering,Monash Univ., Clayton, VIC3800, Australia, e-mail: [email protected]

5Ph.D. Candidate, College of Resource and Civil Engineering, NortheasternUniv., Shenyang, 110819, China, e-mail: [email protected]

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doi:10.1520/GTJ20120086

However, little attention has been paid to the detailed theoreticalinvestigation of the influence of rock heterogeneity on progressivefailure that leads to the collapse of a rock mass under uniaxialcompression. From this point of view, the numerical tool is agood means of gaining some insights into the problem of hetero-geneous and anisotropic rock fracture propagation. In fact, duringthe past few years, with the rapid development of computingpower, interactive computer graphics, and topological data struc-tures, a large number of numerical methods incorporating hetero-geneous material models have been developed for research on therock fracture process, such as in the studies of Blair and Cook(1998a), Tang and Kaiser (1998), Liu et al. (2002), Tang et al.(2007), Feng and Hudson (2010), Wang et al. (2011, 2013), andWang and Ni (2013). From the numerical methods and materialmodels introduced above, it seems that a numerical method devel-oped on the basis of a statistical material model is appropriate forresearch on the rock fracture problem.

In other papers (Cai et al. 2001; Fakhimi et al. 2002; Bertrandet al. 2005; An and Tannant 2007), a method to characterize therock mass damage based on microseismic monitoring is devel-oped and a damage-driven numerical model for rock mass behav-ior simulation is presented. Softening of the rock mass due tofracturing is considered in a constitutive model based on micro-mechanics. The damage-driven model takes the microseismicmonitoring information as inputs to determine the damage state(fracture density) of the rock mass and the resulting anisotropicsoftening. In this work, the acquisition of AE events and the AEsource location was carried out on granite specimens under uniax-ial compression. The numerical analysis was conducted using thenumerical tool PFC2D coupled with the Burgers model, which isused to characterize the viscoelastic-plastic contact characteristicsbetween different material units and fully considers normal andtangential direction contact (Bizjak and Zupanccic 2009). Thesimulation was compared with the laboratory test results to verifyits ability to model crack initiation and propagation and the pro-gressive fracture process of brittle rock.

Acoustic Emission Experiment

AE characteristics of rock undergoing deformation reflect all ofthe failure or damage behavior, which is very complicated for the-oretical and analytical explanation. In the laboratory experimentaltest, the complexity of AE events during a rock failure or damageprocess is due to many factors, such as the composition and self-structure of the rock material, the processing and precision of thespecimens, the loading control method, and test environments.This is the main reason for the study on an AE experiment thatcannot get the intrinsic properties.

Despite this, there are some significant characteristics derivedfrom laboratory testing under present conditions. Uniaxial com-pression testing is the favored means to measure the uniaxial com-pressive strength (UCS) and study its damage mode and state.Thus, a study on rock AE is done to observe the characteristics ofAE data under uniaxial compression. In the field of AE eventsunder uniaxial compression, extensive laboratory test research hasbeen conducted among domestic and foreign scholars (Weng

et al. 1992; Cox and Meredith 1993; Tam and Weng 1994; Eber-hardt et al. 1998; Watanabe et al. 2004; Nair 2006; Ishida et al.2010). In this study the AE experiments were conducted in theExperimental Centre of Rock Mechanics and Engineering atNortheastern University, China.

Sample Preparation

Granite samples from Quyang County in Hebei Province, China,were used; this granite is widely distributed in most of China’sprovinces. The granite specimens to be used in tests on physicalproperties were made into square prism samples. The samplepreparation and testing procedure were in accordance with thetesting standard of the International Society for Rock Mechanics.The determination of key experimental parameters such as Pois-son’s ratio followed previous research and the experiences of theauthors (Chang and Lee 2004; Wang et al. 2007). Theoretically,the UCS for granite material ranges from 120 to 250 MPa depend-ing on its constituent difference. For the specific selected granitesamples from the Quyang County region, the UCS strength is rec-ommended as 137 MPa based on the Chinese geotechnical testingcode. Other recommended physical and mechanical properties andcorresponding dimensions of tested rock samples are summarizedin Tables 1 and 2, respectively. However, the theoretical solutioncannot be exactly reproduced in laboratory experiments becauseof the existing imperfections in the samples (i.e., initial cracks).

The samples were prepared in the Experimental Centre ofRock Mechanics and Engineering at Northeastern University. In

TABLE 1—Properties of rock samples recommended by the Chinese geotech-nical testing code.

Property Units Granite

Specific gravity kg/m3 2670

Apparent porosity % 0.8

Uniaxial compressive strength MPa 137

Young’s modulus GPa 70

Poisson’s ratio 0.2

TABLE 2—Information on rock samples.

SpecimenNumber

Length,cm

Width,cm

Height,cm

P-wavevelocity, m/s Lithology

G1 10.1 10.2 20 3980 Granite

G2 10.2 10.1 20.1 3800 Granite

G3 10.1 10.2 20.1 4030 Granite

G4 9.9 10.2 20.2 4600 Granite

G5 10.1 10.1 20 4987 Granite

G6 7.1 7.1 15.1 4480 Granite

G7 7.1 7 15 4916 Granite

G8 7 6.9 15 4123 Granite

G9 7 7.1 15.1 4257 Granite

G10 7.1 7.2 15.1 4531 Granite

G11 10 10 15 4523 Granite

G12 10 10 15 4210 Granite

G13 10 10 15 3660 Granite

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total, 13 tests (in Table 2) have been classified into three groups.For each group, the replication of the experimental tests has beenchecked, and the representative test (i.e. G3, G6, or G13) falls inthe average AE rate range of the whole group. The presentedresults for G3, G6, and G13 are representative of all testresponses. Therefore, only the experimental results and corre-sponding numerical simulations of these three representativegroups (i.e. G3, G6, and G13) are presented in the remainder ofthis paper. The ratios of height to width or length of the specimenswere 2, 2.14, and 1.5, respectively (see Fig. 1). Both ends ofspecimens were ground parallel to within 0.02 mm, and specimenswere dried at 105�C before testing. Before the AE sensors wereattached, petroleum jelly was applied to the surface of the speci-mens to enhance the acoustic coupling between the sensor andsample so as to ensure high-quality AE data.

Testing Equipment

The loading was performed using a YAG-3000 MicrocomputerControlled Rock Stiffness Testing Machine, which is produced byHangzhou Popwil Instrument Co., Ltd. Its maximum loads for thevertical and transverse directions are 3000 kN and 400 kN, respec-tively, and the corresponding maximum compression testing dis-placements are 800 mm and 220 mm. The diameter of bothpressure plates is 300 mm with 1 % testing precision.

At the same time, an AE system (Hyperion Seismic SoftwareSuite, ESG Canada, Inc., Engineering Seismology Group) wasadopted to detect damage and fracture in rock samples. ESG’sHyperion Seismic Software package is a suite of programsdesigned for filtering, processing, reporting, and visualizing seis-mic data. The modular Windows-based software platform uses an

open database design that allows users to manage all aspects ofseismic data flow, from acquisition and processing to detailedwaveform analysis. Visualization and reporting are made easywith SeisVis, an interactive three-dimensional (3D) tool that out-lays the characteristics of detected seismic events (location, mag-nitude, source parameters) on a digital map of the specific mine,reservoir, or structure being monitored.

The data-acquisition system Ultrasonic Concrete Testing, pur-chased from Proceq UK Ltd., was used to record the load and dis-placement. Ultrasonic Concrete Testing is based on the pulsevelocity method and provides information on the uniformity ofconcrete, cavities, cracks, and defects. The pulse velocity in a ma-terial depends on its density and its elastic properties, which inturn are related to the quality and the compressive strength of thesample. It is therefore possible to obtain information about theproperties of components through sonic investigations and applyit to AE tests.

Figure 2 shows the AE experimental system of the rock uniax-ial compressive test.

The diameter and the thickness of the AE sensors (HagiSonic,Korea) were 3.6 and 2.4 mm, respectively. The main frequencyband of the AE sensors was between 100 kHz and 1 MHz. To eas-ily attach AE sensors to samples and obtain the same sensitivityfor each sensor, an electron wax was used. AE signals measuredin the sensors were amplified by 40 dB with pre-amplifiers (PACmodel 1220 A).

Figure 3 shows the sample before loading with attached sen-sors and strain gauges. The mechanical and AE measurementswere conducted in the conventional uniaxial compressive test con-figuration. Figure 1 shows the locations of eight sensors attachedto the specimen during the uniaxial compressive test to determinethe AE source location. Of these, four sensors (numbered 1, 2, 3,and 4) were attached near the bottom of the specimen, centeredrelative to each of the four sides, and the other four sensors (num-bered 5, 6, 7, and 8) were mounted on the upper part of the speci-men (Hou 2009).

Theoretically, one AE event location can be detected accu-rately by three sensors. In the current work, eight sensors wereemployed to guarantee that all the AE event locations could bedistinguished at a high confidence level (Wang et al. 2007).

Experimental Procedures

During the uniaxial compression test, the loading rate of displace-ment was controlled at 0.02 mm/min. In addition, the Ultrasonic

FIG. 1—Size of rock samples.

FIG. 2—AE experimental systems.

WANG ETAL. ON FRACTURE BEHAVIOR OF INTACT ROCK 905


Concrete Testing system manufactured was used for the AE meas-urements. Considering background noise, the AE trigger levelwas set to 40 dB. Time parameters for AE wave forms includedpeak definition time, hit definition time, and hit locking time.

Experimental Results

Rock Failure Pattern—Figure 4 displays the failure pat-tern for the three representative specimens tested. It can be clearlyobserved from the failure mode that despite the differences indimensions (samples ranged from 7 to 10 cm in width and 15 to20 cm in height), a similar damage pattern was observed involvinglongitudinal splitting parallel to the loading direction. No sizeeffect was detected with respect to the ultimate failure pattern forthe specimens tested.

Acoustic Emission Characteristics—The AE locationprocess for rock specimens of different sizes is shown in Figs. 5through 7.

The AE events first appeared in the smaller rock sample G6 (7by 7 by 15 cm3), and it is noteworthy that the AE events of G6 arein a dispersed state at the initial stage of acoustic emission (50 %of peak loading), rather than in the lower part of the sample as inthe larger rock specimens. A large amount of AE events occurred

in the middle of rock specimen G6, and with increasing loading,these became more dominant than those near the ends of the spec-imen. Eventually, the increasing loading led to the initiation andpropagation of microcracks. As a result of the large quantity ofmicrocracks, penetrating cracks appeared, which resulted in a dis-persed failure pattern.

AE events around the ends of samples were observed in earlystages in rock specimens with larger dimensions (G3 and G13).These AE events indicate the propagation of microcracks throughthe sample to the middle of them, which is most likely due to endeffects. Specimen end effects can dominate the AE location pat-tern significantly if stress concentrations develop. Such a phenom-enon is introduced because the sample preparation and grinding offlat parallel sample ends cannot be replicated perfectly in differentsamples, which is a limitation of rock testing. Even a slight differ-ence can result in significant variability in AE results.

As an interpretation of the variation of AE events with appliedloads in Figs. 5 through 7, one can infer that the AE characteristicsof the rock samples were similar for all sample sizes in terms ofthe following: (1) During the initial loading stage, there was littleAE activity. (2) As the load reached about 60 % of the failureload, the AE activity became more intense. (3) When the externalload approached the ultimate strength, the AE activity increasedrapidly.

FIG. 3—The testing apparatus and a rock sample.

FIG. 4—Failure modes of rock samples. FIG. 5—AE location process for G3.



To illustrate this observed phenomenon more clearly, the cor-relations between the occurrence of AE events and the appliedaxial stress of rock samples are shown in Fig. 8. The UCS of thetest specimens is expected to be lower than the value of 137 MParecommended in the Chinese geotechnical testing code, asexplained above. As shown in Fig. 8, around 70 to 110 MPa wasapplied in a sense that the failure mode of rock samples has beenobserved. The smaller testing UCS strength is partly due to theinitial microcracks and imperfections existing within the rocksample.

According to the relation between AE number and stress, nomatter what size the rock specimens are, at the initial stage ofloading there are barely any AE events. After the load reaches 50% of failure strength, AE events become obvious in the rock sam-ples. Sample G13 (10 by 10 by 15 cm3), with a height-to-width ra-tio of 3:2, entered a peak period of AE events when 60 % of thefailure strength was applied. At 70 % of stress, this trend transi-tioned to a period of slow increase, and another vertex occurred at80 % of stress. However, before the ultimate failure, the incrementof AE events for G13 was less than for the other two sizes of rocksamples. In contrast, with a height-to-width ratio of 2:1, the curvesof G3 and G6 (10 by 10 by 20 cm3 and 7 by 7 by 15 cm3, respec-tively) maintained stable growth after 50 % of peak stress. By thefinal stage of failure, AE events had increased significantly incomparison with the earlier stages.

The total AE numbers at the failure modes of different testingmodels are different. Thus the AE number for each specimen isnormalized against the total AE number so as to better comparethe increasing rate of AE events. Figure 9 reflects the relationsof such increasing rates (normalized AE numbers) and the

corresponding applied stress. G13 showed an obvious increase inAE rate when the loading increased from 60 % to 70 %, which isdifferent from the curves of G3 and G6. At the other stages, thethree kinds of rock specimens maintained consistency with nosignificant disparity.

According to Fig. 10, the loading time (loading phase) whenG6 and G13 underwent a significant increase in AE events wasearlier than that for G3. This can be interpreted as meaning thatthe slowly increasing stage of AE events is much shorter for thesmaller rock sample than for the relatively larger samples, whichmeans that the smaller sample enters into the stable increasingstage of AE events much earlier. However, all test samples exhib-ited consistency of loading time at 90 % of applied loads for therapid growth stage of AE events.

It can be observed from all of the comparisons among the threetypes of testing models that the differences in the AE curves forG3, G6, and G13 cannot be attributed to size effects. The AE datacan vary greatly from test to test because of end effects (as dis-cussed) as well as the presence of imperfections and planes ofweakness in the samples. The end effects could be influenced bythe slenderness ratio of the sample in such a way that the uniaxialstresses in the center of the sample would be altered. The free de-velopment of fractures in more slender samples is allowed whenconfining stresses are less. Then axial splitting of the rock wouldbe promoted, which leads to the brittle failure mechanism. In con-trast, larger developed confining stresses in the center of the sam-ple due to reduced slenderness would inhibit crack developmentand promote shearing and friction mobilization (i.e., plastic post-peak behavior). Table 2 shows that sample G13 had a significantly

FIG. 6—AE location process for G6.

FIG. 7—AE location process for G13.

FIG. 8—Relation between AE number and stress.

FIG. 9—Relation between normalized AE number and stress.



lower P-wave velocity than the rest of the samples tested. Thismay be an indication of a large, preexisting flaw/fracture in thesample, which might explain its different AE-versus-stressresponse. Even though small differences between testing modelsexist, the general behavior is consistent for all samples.

Numerical Modeling

Numerical Tool

PFC (Cundall and Strack 1999), from Itasca Consulting Group, isa powerful numerical tool for analyzing geotechnical problems.The geomaterials are reproduced by a random assembly of circularparticles (grains), and the bonding effect is modeled to present thecontact between particles. Different bonding strategies help engi-neers to solve various issues, from rapid flow to brittle fracture ofa stiff solid. The discrete element approach is employed in thePFC package, which enables one to locate microcracking withingeomaterials. Therefore, AE events and AE locations of heteroge-neous and anisotropic materials such as rock and concrete can beeasily distinguished by PFC (Hazzard et al. 2000; Simizu et al.2010).

Energy is stored in the element during the loading process andis released as AE through the onset of elemental failure. AE insimple terms is defined as a transient elastic wave generated as anoutcome of material deformation. This stress wave propagatesthrough solid due to the energy released during the deformationprocess. The amount of released acoustic energy depends primar-ily on the size and the speed of the local deformation process(Hazzard et al. 2000; Lai and Zhang 2008). During the whole fail-ure process in rock, the relationship connecting the rock strain, thenumber of AE events, and the degree of damage is as follows:e / N / D (Cundall and Strack 1999). Based on the AE eventnumbers, the constitutive equation of the degree of rock damagecan be expressed as

r ¼ E0 � e � exp � N

N0

� �m� �(1)

We define m as the homogeneity index of the rock (Huang1999), with a larger value of m implying a more homogeneousmaterial and smaller values suggesting less homogeneity.

In this numerical simulation, the model is constructed on thebasis of the heterogeneity of rock at a mesoscopic level by assum-ing that the material properties conform to Eq 1. The specimensundergo an external load that is applied slowly by the relativemotion of the upper and lower rigid plates at a constant rate of0.02 mm/min.

Brief Description of the Numerical Simulation

As the counterpart of the experiments on the rock failure process,numerical testing was conducted on three groups of specimens.The 3D numerical models should be utilized to reproduce 3D ex-perimental results. However, the computational efforts involved in3D full-scale simulations are expensive, such that simplificationwith two-dimensional (2D) models can be used to represent test-ing results to a relative credence extent with minimal costs. Basedon the fixed radii of the particles, the radius ratio, and the porosity,the PFC2D model is composed of particle elements under randomdistribution through four frictionless walls. At first, models werebuilt at the actual size: G3, 200 mm by 100 mm; G6, 150 mm by70 mm; and G13, 150 mm by 100 mm. Different ratios of particleswere adopted so as to simulate realistic heterogeneity and anisot-ropy, which conformed to a normal distribution from Rmin to Rmax

(Rmin¼ 1.75 mm and Rmax/Rmin¼ 1.66). Other basic parameters ofthe granite PFC2D model are shown in Table 3.

Rock material manifested typical viscoelastic-plastic character-istics. The normal Burgers model is used to characterize theviscoelastic-plastic contact characteristics between different mate-rial units and fully considers normal- and tangential-direction con-tact (Bizjak and Zupancic 2009). To simulate the mechanicalbehavior of a particular rock, Quyang granite (China), some nu-merical calibration tests were necessary. Calibration involvedfinding the micromechanical parameters of the assembly frommacroscopic properties such as Young’s modulus and the uniaxialcompressive strength. In order to perform this calibration, we esti-mated the micromechanical parameters using the work of Huang(1999). To model the contact interaction between particles, thebuilt-in bond model in Discrete Element software, PFC2D, wasselected to represent the bonding characteristics of the rock mate-rial. The physical meaning of the bond model is demonstrated bythe uncracked and cracked strength between elements. The differ-ent mechanical phenomena are designated as Living and Dying,where full and null strength are possessed by the bond model,respectively. The parameters in PFC2D are the normal and shearstiffnesses of the contact Kn and Ks, the radii of the particles, thenormal and shear strength of the contact bonds Tn and Ts, and thecoefficient of friction between particles l. The calibrated uncon-fined compressive strength, Young’s modulus, and Poisson’s ratioare listed in Table 4. Other microparameters of the damage modelare presented in Table 5.

FIG. 10—Relation between loading time and AE number.

TABLE 3—Basic parameters of granite PFC2D mode.

Density,kg/m3

Rmin,mm Rmax/Rmin

Particle ContactModulus, GPa

Granite 2670 1.75 1.66 70



The calibration of the parameters in Tables 4 and 5 can be seenin Fig. 11, where a comparison between the laboratory testingresults and the numerical simulation with respect to sample G6 ispresented. From this it can be observed that the damage model(Burgers model) produces a better fit of the data than the regularmodel. The small testing UCS strength, as a counterpart to the the-oretical UCS strength indicated in Table 1, is partly due to the ini-tial microcracks and imperfections existing within the rocksample.

In Fig. 12, the numerical model is presented. After the samplewas created, the lateral walls were adjusted to create a hydrostaticpressure equal to 7.5 MPa. Then the upper and lower walls weremoved slowly to simulate the biaxial test, while the movement oflateral boundaries was controlled to keep the lateral pressure con-stant and equal to 7.5 MPa. Then the load at a constant displace-ment increment of 0.02 mm/step was applied in the verticaldirection. The specimen was loaded up to failure, and the wholeforce-displacement curve was recorded.

Note that throughout this paper, following the conventions ofrock mechanics, compressive stresses and strains are taken as pos-itive, whereas tensile stresses and extensional strains are taken asnegative.

Numerical Results

The numerical simulation of the UCS test of rock samples is rela-tively simple and has been studied by many researchers (Blair andCook 1998b; Fakhimi et al. 2002; Backstrom et al. 2008). Thestress-strain curve of an idealized rock particle damage model canbe seen in Fig. 13. In general, the damage process can be dividedinto four phases, corresponding to phases OA, AB, BC, and CDin Fig. 13. The slope of the stress-strain curve for brittle failure(type 2) is nearly linear instead of nonlinear, which can beexplained by the initial particle distribution leading particles to bein contact with and bonded to each other. This is in contrast toplastic failure (type 1), in which a larger number of open flaws inthe rock are being simulated.

The four failure evolution phases are summarized as follows:(1) Phase OA: the slope of the stress-strain curve is graduallyincreasing, which can be seen in Fig. 13. In actual testing of rocksamples, inherent flaws within the rock (along grain boundaries,

through mineral grains, etc.) that are preferentially oriented rela-tive to the loading direction will close, resulting in increasing stiff-ness in the stress-strain curve. In the numerical model, unlesssome particles in the model are not bonded, representing an initialpresence of microcracks, this phase will not appear. In the currentstudy, the particles in the model were not in close contact witheach other; such microcrack closure can be modeled implicitly byshortening the distance between particles under external loads. (2)Phase AB: the rock remains in the elastic deformation path, andthe slope of the stress-strain curve is constant. The bond betweenparticles starts to weaken with increasing loading. All particlesfeature a load-dependent strength in terms of peak and residualstrength as defined in Tables 4 and 5. The residual strength will beactivated and employed after the bond has ruptured. For a detailedmodeling strategy, refer to Cundall and Strack (1999). The micro-crack initiation marks the end of phase AB. (3) Phase BC: rockstarts to deform in the plastic path, where the slope decreasesgradually. The localized cracks increase with the strain and lead tofailure of the sample. This phase can be recognized as the criticalfailure phase. The nonlinear deformation increases significantlyand more internal microcracks form, which result in macrocrackseventually. (4) Phase CD: the post-failure phase is characterizedby type 1 and type 2 post-failure mechanisms corresponding toplastic failure and brittle failure, respectively, according to Fig.13. For civil engineering projects, the plastic failure of material isfavorable because the post-peak strength can provide an additionalfactor of safety. In contrast, brittle failure occurs with no signs ofcollapse, which leads to a substantial loss of strength and cata-strophic consequences.

The numerical simulation of different failure evolution phasesby means of PFC2D is presented in Fig. 14 to show how the par-ticles are changing during each loading stage. The contact bondsafter the initial microcrack closure are shown in Fig. 14(a). Thereis not much more space for particles to move at this stage. Theblack lines define the positions of all particles at this stage. Withan increase in imposed loading along the axial direction of therock sample, particle bonds will start splitting apart as adjacentparticles are pressed between neighboring bonded particles. Suchstrength degradation is relevant to the particle movement. Negligi-ble cracks can be observed from particle arrangement inFig. 14(b), which corresponds to the initial linear loading stage.

TABLE 4—Numerical modeling parameters of uniaxial compression test.

Contact Modulus,GPa Kn/Ks

FrictionCoefficient l

Tn,MPa

Ts,MPa

G3 70 4 0.50 498 123

G6 78 4 0.50 640 168

G13 80 2 0.50 640 168

TABLE 5—Microparameters of damaged model.

c,MPa

/,deg

cres,MPa

/res,deg

ParallelBondingStrength,

GPa

ParallelBondingStiffness,1013 N/m

ParallelBonding

Radii,mm

0.56 30 0.29 26.0 2.70 3.0 0.20

FIG. 11—Uniaxial compression laboratory test results versus PFC2D results(G6 granite sample).



More localized cracks can be noticed from the particle distributionand the development of macrocracks during the initiation stage, asis evident in Fig. 14(c) in the form of more white voids within themodel. The failure mode is featured by gradual obvious macro-crack propagation in Fig. 14(d), in which the black and red linesrepresent the contact bonds and broken bonds, respectively. Forinterpretation of color in Fig. 14, the reader is referred to the webversion of this article. The difference between the contact bondsin the initial stage and the final stage can be noticed.

Figure 15 shows the simulated stress-strain curve for the cur-rent study and the associated cumulative AE events. The curvehas a substantial linear region (line before point A), a nonlinear(AB) region, and a clear post-peak region (BCD, strain softening).Only relatively larger energy losses can be monitored as AE sig-nals. During the initial deformation or linear-elastic phase (beforepoint A), little elastic energy is released, although some failureevents occur because of heterogeneity. The released energy issmall, which is not relevant. The circles in Fig. 15 represent largedistinguishable energy releases. An increasing rate of failureevents accompanies the inelastic phase (AB) and the failure stage(BCD). In comparison with the experimental data for the referencerock sample G6, the simulated uniaxial compressive strength andelastic modulus are relatively accurate (see Fig. 11).

With careful choice of intact rock and micromechanical param-eters, using the calibrated procedure, it has been shown that PFCcould be used in modeling the strength and fracture behavior ofintact rock.

Discussion

The dependence of a rock sample’s strength on its dimensions isgenerally attributed to the presence of fractures and cracks inspecimens. The exact nature of this dependence is poorly under-stood (Da Cunha 1990). The literature presents a number of con-tradictory opinions and observations on the complex problem ofscale effects in the determination of rock strength. Bernaix com-ments that not only the scatter in the testing results but also themean values are a function of the specimen sizes (Wang et al.2009). Li et al. (2011) found out that the failure mode of hardrock may be transformed from shear to slabbing when the height:-width ratio of the prism specimen is reduced. Other authorsreported that the specimen size has no influence (Wang et al.2009). The above-discussed laboratory testing results showed anegligible size effect. The mechanical difference in rock strengthand deformation characteristics for various sizes of samples isbelieved to be attributable to end effects. This research now is try-ing to discuss such controversies as whether it is size effects orend effects influencing the anisotropic deformation of heterogene-ous rock.

As an in-depth study of the phenomena, short parametric anal-yses were conducted using PFC2D based on the fact that the Bur-gers model and the available set of constitutive parameters arecapable of predicting the stress-strain behavior of rock samplesrelative to that in experimental tests. Different numerical modelswith various dimensions under plane strain conditions were

FIG. 12—The particle model (G6 granite sample).

FIG. 13—Schematic stress-strain curve: (a) plastic failure; (b) brittle failure.



investigated. A detailed discussion is presented in the remainderof this section.

The rock specimens are categorized into four groups (M1, M2,M3, and M4) here with dimensions of 100 mm by 50 mm,100 mm by 100 mm, 100 mm by 200 mm, and 100 mm by400 mm, respectively. The physical and mechanical parametersused here are presented in Table 6.

Figure 16 shows the stress-strain curves of four groups of rocksamples with different dimensions. It is apparent that the rockmodels present the brittle failure mechanism regardless of the as-pect ratio. A slight difference in secant stiffness until peak loadcan be detected. The model with a slenderness ratio of 1:1 has thehighest overall stiffness, followed by the models with aspect ratiosof 1:2, 1:0.5, and 1:4. There is no consistent pattern to indicate thecorrelation between secant stiffness and model dimension. Thepeak loads for the three models M1, M2, and M3 are very close,in the range of 26.5 to 30 MPa, wheras the M4 model has a muchsmaller peak strength of 20 MPa. The height of model M4 is morethan twice that of the other models, which means more particlesare assembled. The potential for failure would be increasing, asmore bonding elements are involved and it is easier for the cracksto develop along these bonding elements.

A detailed illustration of the M3 sample (slenderness ratio of1:2) is presented in Fig. 17. As the load increases, the slope of thestress-strain curve (modulus) decreases gradually. The interactionbetween particles becomes stronger when loads increase frompoints O to A. The stress-strain curve starts to present nonlinearitynear the peak applied load after point B. Such a nonlinear branchis very short for typical brittle material. Micro-level particle frac-ture starts to influence the macromechanical behavior of the sam-ple. With further interaction of particles, the moduli drop and thesoftening characteristics emerge. When the load increases beyondthe peak strength, the rock sample enters the post-failure phase.Macrocracks are concentrated, which leads to a loss of bearingcapacity in the rock model.

The role of end effects is identified here as a possible factorcausing a heterogeneous stress state. For more slender samples,the end effects do not influence the uniaxial stress state in the mid-dle of the sample, allowing axial cracks to develop freely. Whenthe sample width increases and the end effects interfere more withthe stresses that develop in the middle of the sample, a more con-fined stress state develops and the corresponding failure mode isaltered. Such alteration of failure modes can be noted in Fig. 16,where the stress-strain curves for all samples are following the

FIG. 14—The failure evolution phases of the PFC model.

FIG. 15—Stress-strain curves and associated AE events induced by the UCStest (G6 granite sample).

TABLE 6—Rock sample constitutive parameters for parametric analysis.

ModelAspectRatio

InitialVoid Ratio n0

Density,kg/m3

Rmin,mm

Rmax/Rmin

ContactModulus, GPa

M1 1:0.5 0.16 2630 0.25 1.66 78

M2 1:1 0.16 2630 0.25 1.66 78

M3 1:2 0.16 2630 0.25 1.66 78

M4 1:4 0.16 2630 0.25 1.66 78

Friction

Coefficient

Model

Kn/

Ks

Parallel Bonding

Strength, GPa Tn/Ts Particles Wall

Loading

Rate, mm/s

M1 4.0 49 1.0 0.5 0 0.02

M2 4.0 49 1.0 0.5 0 0.02

M3 4.0 49 1.0 0.5 0 0.02

M4 4.0 49 1.0 0.5 0 0.02



same trend but branch down to different post-failure paths. Thismeans that the size effects have no influence on the elastic defor-mation of rock samples. The M4 sample, with an aspect ratio of1:4, presents brittle failure when the stress reaches about 20 MPa.At the same time, the other samples start to enter the nonlinear de-formation range. The early failure of M4 is mainly due to the factthat no confining pressures are generated by the end effects to fur-ther support the stability of the rock sample. The stresses developfreely in the middle of the sample, which leads to lower stress rel-ative to the other samples. The stress hardening of three samples(M1, M2, and M3) could be directly related to the confining pres-sures provided by the end effects. With increasing slenderness,there would be reduced end effects and therefore more uniaxialstress in the center of the sample, allowing fractures to developmore easily and promoting axial splitting and a brittle post-peakresponse. With decreasing slenderness, the corresponding increasein confining stress in the center of the sample would inhibit crackdevelopment and promote shearing and plastic post-peak behav-ior. Intuitively, the M1 sample should present the highest confin-ing pressures due to the end effects and have the largest peak axialstress. However, the M2 sample, with an aspect ratio of 1:1, hasthe greatest peak stress in Fig. 16, which is assumed to be a limita-tion of the current numerical model. 2D modeling may neglectsome key characteristics of the model, and further improvementof the constitutive behavior in PDF2D should be carried out tofacilitate understanding of the phenomena.

Conclusion

Laboratory experiments and numerical simulations were per-formed to investigate the dependence of the fracture behavior ofintact rock on varying factors. Measurements of acoustic emission(AE) and the source location were conducted on granite speci-mens under uniaxial compression conditions in the laboratoryfirst. The Burgers model was then introduced to perform the nu-merical analysis using the numerical tool PFC2D to deal with rockcharacteristics of heterogeneity and anisotropy. Although the sim-ulations in this study were conducted in two dimensions, theresults of the crack initiation, propagation, and full progressivefracture process were quite comparable with experimentallyobserved results. The analysis of the constitutive stress-strain rela-tionship of rock samples verifies its effectiveness in predicting theexperimental results. However, this does not mean that such mod-eling is sufficiently realistic for all the problems it might beapplied to.

After the effectiveness of the numerical simulation had beenproven, further parametric studies were conducted to investigatethe influence of size effects on the stress-strain curves of rocksamples. It is concluded that the size effects do not exert signifi-cant influence on the mechanical characteristics of rock samples.The different failure evolutions for various models observed fromnumerical simulations are speculated to be due to end effectsrather than size effects. Various end dimensions of rock samplesmay lead to the development of different uniaxial stresses in thecenter of the sample. The altered fracture pattern along the samplewould be induced by confining stresses in such a way that crackswould be promoted or inhibited by these stresses. The limitationof the current analyses is associated with capturing the end effectsthrough clear evidence. Further improvements in extracting keyinformation to prove such a sophisticated end effect phenomenonin the PDF2D model should be made in order to facilitate ourunderstanding.

Acknowledgments

The research presented in this paper has been supported jointly bythe National Natural Science Foundation (NNSF) of China(Grant Nos. 51179031 and 51074042), Projects of InternationalCooperation and Exchanges NSFC (Grant No. 51250110531),Major State Basic Research Development Program (Grant No.2013CB227902), State Key Laboratory of Geohazard Preventionand Geoenvironment Protection (Grant No. SKLGP2012K009),Fundamental Research Funds for the Central Universities (GrantNos. N090401008 and N090101001), and the “985 project” ofNortheastern University, China. All of these are gratefullyacknowledged.

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