Granular Matter (2015) 17:775–791DOI 10.1007/s10035-015-0594-9

ORIGINAL PAPER

A comprehensive study on the smooth joint model in DEMsimulation of jointed rock masses

Dan Huang1 · Jianfeng Wang2 · Su Liu2

Received: 22 January 2014 / Published online: 24 September 2015© Springer-Verlag Berlin Heidelberg 2015

Abstract This paper reports the results from a comprehen-sive numerical study on the effects of micro-parameters ofthe smooth joint model (SJM) on the macro-properties andthe associated failure modes of synthetic rock masses (SRM)under the uniaxial compression condition using the three-dimensional discrete element method. Important mechanicaland geometrical micro-parameters of SJM that show signif-icant effects on the macro-properties and the failure modesof SRM are identified. Strong coupling effects are found toexist between various important micro-parameters so thatthe eventual sample failure is a result of the complicatedinteraction among these micro-parameters. A limitation ofthe current stress-dilatancy relation accounting for the jointroughness effect is also identified. The numerical resultspresented in this paper are valuable for the evaluation ofthe current model capability in simulating and predictingthe shear failure behavior of rock masses, and the furtherimprovement of the model for its full application to the studyof the behavior of real rock masses at the laboratory or fieldscales.

Keywords Smooth joint model · Synthetic rock mass ·Discrete element method · Unconfined compression test ·Strain localization

B Jianfeng Wangjefwang@cityu.edu.hk

1 Department of Civil Engineering and Mechanics, HuazhongUniversity of Science and Technology, Wuhan, China

2 Department of Architecture and Civil Engineering, CityUniversity of Hong Kong, Kowloon, Hong Kong

1 Introduction

The field behavior of rock masses is governed by both thein-situ geological structure and stress conditions induced byengineering activities. From the engineering perspective, themajor features of the geological structure that are critical tothemechanical behavior of a rockmass include various typesof discontinuities existing at different length scales such asfaults, joints, bedding planes, foliations and fissures, etc.These discontinuities present weak structural planes whichmay yield, evolve and coalesce to lead to the failure of rockmasses under the influence of complex engineering-inducedstress path conditions. Obviously, evaluating and quantify-ing the effects of highly complex geometrical configurationsand spatial distributions of the discontinuity features on themechanical properties of rock masses remains a central issuein many practical rock engineering problems.

Traditional approaches for estimating the mechanicalproperties of a rockmass include empirical rock classificationmethods [1–6],mathematicalmodeling [7], large-scale labo-ratory testing and back-analysis methods [8,9]. Despite theirwide use in rock engineering industry, the rock classificationmethodswere largely derived frompractical observations andoften rely on individual property indexes to assess the rockquality and strength. Mathematical modeling usually repre-sents the rock mass as an idealized discrete system of intactmaterial and discontinuities and makes simplified assump-tions regarding the interactivemechanisms between differentdiscrete components. Although being the most straightfor-ward method for determining the rock mass strength, thelarge-scale laboratory testing ismuch limited by the high costand difficulties in sampling, manipulation and simulation ofin-situ stress path conditions. The back-analysis method isan interesting approach to determine the mechanical prop-erties of rock masses based on failure case histories but it

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requires a priori knowledge of the failure mechanism whichoften is not clearly understood. More importantly, all theabove conventional methods do not address explicitly therole of discontinuities in the failure event of rock massesand consequently encounter difficulties in handling criticalissues associated with the failure mechanism such as sam-ple size/scale effects, anisotropy behavior and progressivefailure, etc.

Over the last two decades, the discrete element method(DEM) has emerged as a powerful tool for exploring themechanical behavior of intact and jointed rock masses atboth laboratory- and field-scales. The major advantage ofDEM stems from its capability of capturing the discretenature of various constituents of rock masses, although theessential problem of how the actual discrete fracture network(DFN) should be represented in a tractable way to yield real-istic mechanical behavior sufficing the engineering purposesremains unresolved. Earlier works along this line attemptedtomodel a jointed rockmass as a 2Dor 3D systemofmultiplerigid blocks which are separated by persistent joint planes[10]. Frequently, many of these studies [11–13] employedthe discrete element stress analysis packages called UDEC(2D version) or 3DEC (3D version) developed by Itasca, Inc.[14,15], which can incorporate the information of DFN gen-erated from field-measured structural fracture sets. A fewback-analysis studies have shown that the fracture of intactrock bridges between non-persistent joints is mainly respon-sible for the failure of rock masses [16].

Recently, a new discrete element simulation method,called the synthetic rock mass (SRM) approach, has beendeveloped by Ivars [17] for the more realistic simulation ofjointed rock masses. The main idea of SRM is to treat theintact rock mass as a large assembly of cemented granu-lar particles in which both the particles and the cement candeform under external loading. The idea has been imple-mented via the development of bonded particlemodel (BPM)[18] created using PFC2D & 3D [19,20]. In BPM, paral-lel bonds are installed initially between all particle contactsto simulate the inter-particle cement and can break later toform new cracks during the loading process. Grain shapeeffects could be incorporated using the particle-clump tech-nique [21]. The inclusion of pre-existing joints into the BPMwas traditionally realized by debonding all the contacts alongthe joint plane and assigning low contact strength and stiff-ness values to the contacts [10,22] or by directly deleting theparticles in the vicinity around the joint plane [23]. However,this simple way of representing interfaces is problematic dueto the inherent roughness of the interface caused by local par-ticle contact geometry. To remove the artificial and unwantedeffects of local interface roughness, Cundall et al. [23] pro-posed the concept of smooth-joint contactmodel (SJM). SJMsimulates the behavior of a smooth interface created by ajoint plane, reflecting the predominant influence of joint ori-

entation and inclination on the joint and hence the wholerock mass behavior by neglecting the local particle contactgeometry surrounding the joint plane. This convenient tech-nique has recently proven to be effective in predicting, ina more realistic way, a range of important rock mass prop-erties including pre-peak modulus, damage threshold, peakstrength, post-peak dilation, fragmentation, brittleness andresidual strength, etc [24,25]. Moreover, it has also beendemonstrated to be capable of capturing the scale effects,anisotropy and progressive failure characteristics [26–28],and successfully used in the field-scale stability analysis ofvertical excavation in hard rock [29].

In light of the potential application of SRM to a morediversified areas of research including not only rock engi-neering but also fundamental material science research ofother natural or man-made polycrystalline materials, espe-cially the demonstrated capability of SJM to deal with theissues of microscale fracture initiation and propagation, acomprehensive investigation of the influence of microscopicmodel parameters of SJM on the model behavior of SRMis urgently demanded. The necessity of such an investiga-tion stems from the need of the further enhancement of SJM,based on a clear understanding of the capabilities and lim-itations of the current model, for modeling the fracturingbehavior of quasi-brittlematerials in amore physically soundand realistic manner. In this paper, we seek to fill this gap byconducting a detailed 3D parametric study of SJM, provid-ing insights into the relationship between the rock fracturemode, pattern of strain localization andmacroscopic strengthand stiffness behavior. To facilitate understanding the mech-anism of fracture initiation and propagation, we focus onthe simulations of SRM samples containing only one jointplane. Results from numerical unconfined compression (UC)tests are presented, accompanied with a comparison betweenmodel behavior of SRMs with and without (i.e., pure BPM)SJM. Due to the unavailability of the data of rock joint sets,model validation against published laboratory test data ofmarble [31] was only performed using BPM. The cali-brated micro-parameters of BPM were subsequently usedin all SRM simulations containing SJM. The novelty of thisstudy is delivered by its being perhaps the first of such a typein SJM-related studies and more importantly, a rich bodyof new insights into the fracture behavior of SRM from thesimulation results.

2 Establishment and validation of BPM

In the current study, the BPMwas constructed and calibratedagainst the laboratory test results ofmarble samples extractedfrom the site of Jinping II hydropower station located atYalong River of Sichuan, China. Involving the constructionof four 16km long headrace tunnels with themaximumdepth

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Fig. 1 a Bonded particle model; b middle vertical plane for straincalculation

Table 1 Tested values of BPM micro-parameters

Parameter Value

Dmin (mm) 1, 1.5, 2, 3

Ec (GPa) 30, 45, 60

ks/kn 0.25, 0.4, 1.0

pbn, pbs (MPa) 120 ± 24, 140 ± 0, 140 ± 14, 140 ± 28, 160 ± 32,

μ 0, 0.25, 0.5, 1.0

of 2525m, the Jinping II hydropower station is the largest ofits kind in the world.

The establishment of BPM was done by following thematerial-genesis procedure proposed by Potyondy and Cun-dall [18], which allowed the generation of a homogenoussample with a relatively uniform locked-in stress field. Themodel has a dimension of 50 × 50 × 100mm3, which isthe same as that of the real marble sample, as shown inFig. 1a. The calibration process involves the trial-and-erroradjustment of the followingmicro-parameters until themodelprediction matches the measured response of marble sam-ples: minimum particle diameter Dmin (with a fixed ratio ofDmax/Dmin = 1.66), particle Young’s modulus Ec, particlecontact stiffness ratio ks/kn , interparticle friction coefficientμ, parallel bond Young’s modulus Ec, parallel bond stiffnessratio ks/kn , and parallel bond strength pbn (normal compo-nent) and pbs (tangential component). All the tested values ofthe above micro-parameters are summarized in Table 1. Notethat in this study, Ec = Ec, ks/kn = ks/kn , pbn = pbs, andfor pbn and pbs, the first number is the mean value and thesecond one is the standard deviation. The loading rate usedin our simulations is 0.02 m/s, which has been chosen for thecomputational efficiency and found not to induce significantdynamic loading effects on the compression behavior of thesample (e.g., Young’s modulus and peak strength).

Table 2 Laboratory measurements [31] and model predictions ofmacro-properties of Jinping marble

Macro-properties Unconfined compression test

E (GPa) qu (MPa) εp (%)

Laboratory measurement 31.6 140 0.4

Model prediction 33.1 136.8 0.42

For damping dissipation, a local, non-viscous dampingmodel available in PFC3D (Itasca 2008) was used, with thelocal damping coefficient taking a value of 0.1 in this study.This value is much lower than the default value of 0.7 inPFC3D because some previous studies [32,48] have indicatedthat a high local damping coefficient may led to the over-damping, non-physical dissipation behavior. Hazzard et al.[32] showed that reducing the damping coefficient from0.7 to0.015 led to a reduction of about 15% in the unconfined com-pressive strength and a significant (about 45 times) increaseof the seismic quality factor of granite, with the failure beingmore severe and accompanied with much more rapid crackgrowth. This result suggests that a low damping coefficient ismore appropriate for a competent rock like marble in whichthe wave-attenuation is not very severe.

The macro-properties of Jinping marble used for modelcalibration include Young’s modulus E , unconfined com-pressive strength qu and the corresponding strain εp at thepeak strength from the unconfined compression test. The lab-oratory measurements of these macro-properties are listed inTable 2. These measurements were obtained from the uni-axial compression tests on standard rock samples extractedfrom Baishan stratum [31]. The samples were cylinders witha 50mm diameter and a 100mm height. They were testedat the MTS machine of Wuhan Institute of Rock and SoilMechanics with a typical loading rate of 0.2mm/min.

The major effects of various micro-parameters on differ-ent model-predicted macro-properties are shown in Fig. 2.Other results showing only weak or insignificant correla-tions are not presented. The following trends are evident inFig. 2: (1) qu increases with decreasing Dmin, and increasing(pbn)avg (or (pbs)avg) and increasingμ; (2) E increases withincreasing Ec (or Ec), increasing ks/kn (or ks/kn), increas-ing μ and decreasing Dmin, with the first correlation beinglinear and having the highest rate of change; (3) εp increaseswith decreasing Ec (or Ec), decreasing ks/kn (or ks/kn) andincreasing (pbn)avg (or (pbs)avg), with the first correlationhaving the highest rate of change. The above observationssuggest a generally good correspondence between themicro-parameters and their macro-counterparts.

However, the correlations are rarely in a simple linearform since the macro-properties are determined by the col-lective behavior of a large assembly of interactive bondedparticles in which the initial fabric (i.e., spatial distributions

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Fig. 2 Major effects of BPM micro-parameters on model-predicted macro-properties: a Dmin; b ks/kn ; c Ec; d μ; e (pbn)avg

of particle size and particle shape) plays an equally or evenmore important role than the individual micro-parameters.For BPMmaterials, however, the small-strain behavior (e.g.,Young’s modulus) is relatively simple since the behavior isfully controlled by the elastic deformation of the cement pro-vided that no bond breakage takes place. This is reflected in

the linear correlation between E and Ec. This scenario issimpler than uncemented granular materials for which evenvery small-strain deformation is not fully elastic due to inter-particle slips and rotations. Once the deformation of a BPMmaterial is large enough to cause some initial cracks (i.e.,bond breakages), the subsequent material response is pre-

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dominantly controlled by the propagation and coalescence ofthese initial cracks. Since then, the material enters a regimeof highly nonlinear behavior where the ultimate brittle fail-ure of the material is featured with the formation of a majorfracture plane or a local crushing zone containing a networkof multiple minor fracture planes, which is usually associ-ated with the instantaneous occurrence of a large number ofbroken bonds and release of a significant amount of strainenergy. The observed variations of qu and εp against differ-ent micro-parameters (Fig. 2) are the exact consequences ofthe above microscopic processes.

It is not intended in this paper to formulate a quantitativepredictive model of the macro-properties of BPM based ona comprehensive account of the relationship between micro-parameters and failure modes, which has been the subjectof a number of previous studies (e.g., [33–42]). The aboveparametric study of BPM was mainly conducted to ensurethat the concepts, principles and various issues of numer-ical implementation of this method are fully understood.This parametric study yielded a set of best-fit parametersas listed in Table 3, which resulted in the best predictions ofthe laboratory measured macro-properties of Jinping marblesamples from UC tests [31], as listed in Table 2. The cor-responding shear strain distribution within the sample afterits failure shown in Fig. 3. The shear strain distributions pre-sented throughout this paper were calculated on a 2D basis

Table 3 Best-fit parameters of BPM

Dmin (mm) Ec, Ec (GPa) ks/kn, ks/kn pbn, pbs (MPa) μ

2 35 0.4 124 ± 28; 124 ± 28 0.5

Fig. 3 Shear band in UC simulation using BPM

for the middle vertical plane of the sample (Fig. 1b) perpen-dicular to the X axis using the mesh-free strain calculationmethod previously developed by the second author [43–46].The effectiveness of this simplification has been proven inour previous 3D DEM simulation studies on granular soils[47–50], and shouldwork equallywell, given the geometricalsymmetry of the specimen and the joint plane, in represent-ing the strain localization occurring within the SRM in thecurrent study.

It is seen in Table 2 that the model predictions agreevery well with the laboratory measurements, with a model-predicted inclined shear band making an angle of about 50◦from the horizontal shown in Fig. 3. It should be mentionedthat the set of best-fit parameters shown in Table 3 is notthe unique one for producing the target material properties.As a matter of fact, to represent a real rock mass using asimplified material model like BPM, which disregards allmicrostructure features, it is extremely difficult, if not com-pletely impossible, to obtain the true set of micro-parametersfitting the various components of the rock mass. The limita-tion also comes from the imperfect model validation processwhich is unable to match the full range of the true materialbehavior.

In the next part of this paper, the BPM of Jinping marblewith the best-fit parameters established above will be usedfor a detailed study on the effects of pre-existing joints onthe rockmass behavior. This is achieved by incorporating theSJM into the above BPM.

3 Comprehensive investigation of SJM

In this section, the conceptual model of SJM is first reviewedand followed by a detailed parametric study on the micro-parameters of SJM and subsequently a comprehensive analy-sis of the effects of these micro-parameters on the behaviorof SRM under the uniaxial compression condition.

3.1 Brief review of SJM

The concept of SJM was proposed by Ivars et al. [23] tosimulate the behavior of a smooth interface created by a rockjoint in a BPM material. The joint geometry is idealized astwo parallel planes which initially coincide with each other,as schematically shown inFig. 4. In a discrete particle system,however, this continuous joint plane is virtually representedby a series of smooth-joint contacts prescribed at all pre-existing ball–ball contacts that are intersected by the jointplane. Specifically, a ball–ball contact is said to be intersectedby the joint planewhen the centers of the two balls are locatedat opposite sides of the joint plane (Fig. 4a). In 3DSJM, a jointplane is assumed to have a circular shape and its geometry isfully defined by three parameters, namely, dip angle θp, dip

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Fig. 4 a Schematic diagram of SJM (reproduced from ref. [15])

Fig. 5 a The definition of dip angle, dip direction; b the definition of dilatancy angle under shear test of joint

direction θd and joint radius rj. The definition of dip angleand dip direction are illustrated in Fig. 5a.

After any ordinary (bonded) ball–ball contact is changedinto a smooth-joint contact, the original contact orientationis replaced by the joint plane orientation, based on which therelevant contact force anddisplacement vectors are redefined.Accordingly, the constitutive behavior of the joint contact isgiven by:

Fn := Fn + kn A�Uen, (1)

Fs := Fs + ks A�Ues , (2)

where Fn and Fs are the contact normal force and shear forcevectors, respectively; kn and ks are the contact normal andshear stiffnesses, respectively; �Ue

n and �Ues are the elas-

tic components of the normal displacement increment vector�Un and shear displacement increment vector �Us, respec-tively; and A is the area of the smooth-joint contact given by

A = π R2, (3)

where smooth-joint radius R = λmin(R(1), R(2)

), with R(1)

and R(2) being the ball radii, and λ being the radius mul-tiplier. Similar to the conventional particle contact model,a Mohr–Coulomb type failure criterion applies to the shearbehavior of the smooth-joint contact expressed in Eq. (2).For an unbonded joint, it defines the following relationships:

⎧⎪⎨

⎪⎩

|Fs | =∣∣∣Fs

∣∣∣

(for

∣∣∣Fs

∣∣∣ ≤ F∗

s = μFn)

(4a)

|Fs | = F∗s

(for

∣∣∣Fs∣∣∣ > F∗

s = μFn)

; (4b)

where μ is the friction coefficient. It is assumed that slid-ing will occur when the condition in Eq. (4b) is satisfied.It should be noted that the friction coefficient of the jointcould be defined separately from the friction coefficient ofparticles. However, an essential difference of the constitutivebehavior of the smooth-joint contact from that of the conven-

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tional particle contact is the dilatancy law which involves thecoupling between the contact normal force and plastic sheardisplacement, expressed as follows:

Fn := Fn + kn A(�U∗

s tanψ), (5)

where ψ is the dilatancy angle and �U∗s is the plastic shear

displacement given by

�U∗s =

(|Fs | − F∗

s

)/ks A. (6)

The definition of the dilatancy angleψ is illustrated in Fig. 5b.The above dilatancy law reflects the influence of the jointroughness on the joint constitutive behavior through a sin-gle parameter ψ which compensates for the consequenceof neglecting the original local contact geometries of allsmooth-joint contacts before they are enforced. The exami-nation of the effects of ψ on the model behavior, therefore,is of particular interest and will be presented as a part of theSJM parametric study below.

The option of a bonded joint is also allowed in SJM andthe bond can exist in the normal or shear or both directions ofa smooth-joint contact. The force-displacement behavior ofa bonded joint is also described by Eqs. (1) and/or (2) beforethe bond normal strength σj and/or τj is reached. Once thebond breaks in any direction, the conventional mode of theunbonded smooth-joint contact is activated.

3.2 Simulation program of SJM

To achieve a comprehensive understanding of the perfor-mance characteristics of SJM, a detailed parametric studywas carried out on the effects of a series of micro-parametersof SJM including dip angle θp, dip direction θd, contact nor-mal stiffness kn and shear stiffness ks , friction coefficient μ,dilatancy angleψ, bond normal strength σj and shear strengthτj, and radius multiplier λ. To simplify our study, we focuson the simulations of SRM samples containing only one jointplane. The range of the tested values of the above SJMmicro-parameters is listed in Table 4. We are interested in how thevariations of these micro-parameters will affect E , qu and

εp, as well as the final failure patterns from the unconfinedcompression simulations. Note that the micro-parameters ofthe BPM take the values of the best-fit parameters shown inTable 3.

3.3 Effects of mechanical micro-parameters of SJM

Figure 6 shows the relationship between the mechanicalmicro-parameters of SJM and the macro-properties of SRMsamples fromUC simulations. Note that in these simulations,the joint plane has a dip angle of 30◦, making it deviate fromthe failure plane of the BPM sample observed in Fig. 3 sothat any effect of SJM mechanical micro-parameters can beclearly identified.

Simulation data in Fig. 6a–d indicate that ks/kn , μ andψ have varying levels of influence on the unconfined com-pression behavior of the SRM sample with an unbonded andnon-persistent joint. It is seen that an increase of ks/kn up to0.5 (with a fixed kn value of 1.3 × 103 GPa) results in gen-tle increases of qu and εp but has little effect on E (Fig. 6a),while an increase of kn up to 1.3×104 GPa (with a fixed ks/knof 1.0) results in much greater increases of qu and εp, and asmaller increase of E (Fig. 6b). The latter result is particularlyinteresting because it shows the effect of the relative stiffnessratio of SJ contact to particle contact on the sample behavior.The shear strain distributions and bond crack distributions of3 selected samples after failure are shown in Fig. 7, provid-ing insights into themacroscopic strength behavior. Clearly, adistinct change of the failure mode from the shear band alongthe SJ plane at a low kn value (e.g., kn = ks = 1.3×103 GPa)to compaction bands near the two ends of the sample at ahigh kn value (e.g., kn = ks = 1.3 × 104 GPa) is observed,indicating the dominant role of the intact part of the samplerather than the smooth joint in the unconfined compressivestrength behavior in the latter case. Correspondingly, bondcracks occur predominantly in the vicinity around the SJplane in the former case but are found to concentrate withinthe two compaction bands near the top and the bottom endsof the sample. It should be mentioned that bond cracks intensile failure and shear failure are marked in red and black,respectively, in Fig. 7d–f. No significant change of the failure

Table 4 Tested values of SJM micro-parameters

Mechanical properties Values Geometric properties Values

ks/kn(kn = 1.3 × 103GPa) 0.1, 0.2, 0.5, 1.0 θp (deg) 0, 30, 45, 60, 90

kn(GPa) (ks/kn = 1.0) 1.3 × 101, 1.3 × 102, 1.3 × 103,1.3 × 104, 1.3 × 105

θd (deg) 0, 30, 45, 60, 90

μ 0, 0.3, 0.58, 0.8, 1.0 rj (mm) 5, 10, 15, 16, 20, 30

ψ (deg) 0, 7, 15, 30, 50 λ 1.0, 1.2, 1.5, 2.0, 3.0

σj /pbn, τj /pbs 0, 0.1, 0.3, 0.5, 0.75

The bold values of other parameters are fixed and used when the value of each parameter is varied separately

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Fig. 6 Effects of SJM mechanical micro-parameters on model-predicted macro-properties: a ks/kn ; b kn ; c μ; d ψ; e σj /pbn

mode, however, is found as the ks/kn ratio is varied. To avoidthe unrealistic large joint to matrix stiffness ratio that is notnormally found in naturally occurring rock masses, we usedthe same stiffness value of the bonded particle contact forthe SJ contact (i.e., kn = kn = 1.3 × 103 GPa) in the othersimulations.

Strong effects of μ on qu and εp are also found in Fig. 6cwhen μ ≤ 0.58 but no significant effect can be seen whenμ > 0.58, and E appears to be insensitive to μ within thefull range of variation. Again, the final shear bands and bondcrack distributions of three selected cases are shown in Fig. 8.The above results suggest that the joint displacement will

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Fig. 7 Effects of joint stiffness on the sample failure mode: a kn = ks = 1.3 × 103 GPa; b kn = ks = 1.3 × 104 GPa; c ks/kn = 0.5 (kn =1.3 × 103 GPa)

occur prior to the mobilization of the peak strength, whichdepends on the intact matrix beyond the two ends of the joint.The value of μ, however, will determine the amount of jointdisplacement mobilized and the subsequent failure patternof the intact matrix. It is seen in Fig. 8 that the shear bandsdeviate from the SJ plane beyond both ends of the joint for thecase of μ = 0.1, but are generally aligned with the SJ planefor the other two cases; an apparently thicker and roughershear band is also observed in the case of μ = 1.0. The bondcrack distribution patterns are found to correspond well totheir respective shear band patterns in Fig. 8a–c.

A surprising result from this parametric study is the almostnegligible effect of ψ, within a quite wide range of varia-tion between 0 and 50◦, on the sample behavior, as shown inFig. 6d. To better understand this result, we plot the evolution

of the normal and shear stresses acting on the SJ plane (i.e.,calculated by dividing the sum of all SJ contact normal andtangential forces, respectively, by the initial SJ plane area)against the applied axial strain in Fig. 9a, and compare themwith the macroscopic axial stress versus axial strain relationsin Fig. 9b. It is seen in Fig. 9a that the full mobilization of theshear stress on the SJ plane is achieved well before the peakmacroscopic axial stress is reached in all the cases, indicat-ing the activation of the dilatancy law via Eq. (5) from thatinstance. Indeed, the influence ofψ is reflected in the increas-ingnormal and shear stresseswith the increasingψ.However,this effect almost dies out as it propagates towards the upperand lower boundaries through the contact force chains, asshown in Fig. 9c, leading to the negligible change on themacroscopic stiffness and strength behavior. This result indi-

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Fig. 8 Effects of joint friction coefficient on the sample failure mode: a μ = 0.1; b μ = 0.5; c μ = 1.0

cates the incompetence of the current dilatancy model toachieve the joint roughness-associated strength effects of realrock masses. The model may be improved by enforcing adirect kinematic-type dilatancy relation or proposing a newdilatancy stiffness parameter to be used in Eq. (5), whichwill be the subject of a further study. As a matter of fact, thislimitation has recently been addressed by Chiu et al. [27] inan effort to develop a modified SJ model in which the jointroughness angle was made a function of the normal stressacting on the joint plane.

Figure 6e shows the effects of the bond strength para-meters σj and τj on the sample behavior. Clearly, qu and εpincreasewith σj (and τj)when σj /pbn ≤ 0.5 and remain con-stant afterwards; little effect of σj (and τj) on E is observed.Figures 10 and 11 show the final shear bands and the cor-responding distributions of SJ bond breakage from three

selected cases, respectively. Interestingly, a transition fromthe failure along the SJ plane at a low bond strength (e.g.,σj /pbn = τj /pbs = 0.1) to an inclined shear banding fail-ure within the intact matrix above the SJ plane at a high bondstrength (e.g., σj /pbn = τj /pbs = 0.75) is observed. Inthe latter case, the macroscopic strength is clearly governedby the inclined shear banding since no SJ bond breakage isfound (Fig. 11c) although a thin line of strain localization isstill seen along the joint plane (Fig. 10c), which is causedby the elastic deformation of SJ bonds. In fact, a trend ofdecreasing amount of SJ bond breakage along the joint planewith the increasing σj /pbn(and τj /pbs) is observed, whichis consistent with the above failure mode transition. Furtherevidences supporting the above analysis are given in Fig. 12,where the strain localization and SJ bond breakage distrib-utions from two more simulations in which kn (and ks) is

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Fig. 9 Effects of ψ on a normal and shear stresses acting on the joint plane; b macroscopic stress–strain relation

Fig. 10 Effects of joint bond strength on the sample failure mode: a σj /pbn = τj /pbs = 0.1; b σj /pbn = τj /pbs = 0.5; c σj /pbn = τj /pbs =0.75

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Fig. 11 Effects of joint bondstrength on the SJ bondbreakage distribution: aσj /pbn = τj /pbs = 0.1; bσj /pbn = τj /pbs = 0.5; cσj /pbn = τj /pbs = 0.75

Fig. 12 Effects of enhanced SJ contact stiffness on the sample failure mode (a, b) and bond breakage distribution (c, d): a and c σj /pbn =τj /pbs = 0.5; b and d σj /pbn = τj /pbs = 0.75

doubled are illustrated. It is now seen that for the case ofσj /pbn = τj /pbs = 0.5, SJ bond breakage completely dis-appears (Fig. 12c); but the strain localization patterns for bothcases (Fig. 12a, b) also differ fully from those in Fig. 10b andc since the enhanced SJ contact stiffness has largely changedthe stress distribution within the sample. The stress contourmaps within the sample with the two different sets of SJ con-tact stiffness are shown in Fig. 13. It is clear that the twostress distributions are drastically different: the case with theoriginal SJ contact stiffness having a band of stress localiza-tion along the joint plane, while the case with the enhancedSJ contact stiffness having the stress localization towards thetwo corners of the sample.

3.4 Effects of geometrical micro-parameters of SJM

Figure 14 shows the relationships between the geometricalmicro-parameters of SJM and the macro-properties of SRM

Fig. 13 Stress contours within the sample with different SJ contactstiffness: a σj /pbn = τj /pbs = 0.5 with original SJ contact stiffness;b σj /pbn = τj /pbs = 0.5 with enhanced SJ contact stiffness

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Fig. 14 Effects of SJM geometrical micro-parameters on model-predicted macro-properties: a θp; b θd; c rj; d λ

samples from UC simulations. It is seen that all the geo-metrical micro-parameters show distinct influences on theunconfined compression behavior of the samples except thejoint dip direction θd, which has very slight effect on the sam-ple behavior due to the relatively high homogeneity of thesample, as expected. Specifically, for an unbounded joint, E ,qu and εp all decrease roughly linearly with the increasingjoint radius rj when rj ≥ 15mm; and the minimum val-ues of qu and εp are obtained at the joint dip angle θp of45◦. However, it should be noted that there exists a couplingeffect between rj and θp, the variation of whose values in theparametric study may lead to different quantitative correla-tions but the underlying mechanism of the competition ofthe SJ plane and the intact matrix in controlling the failure ofthe sample remains unchanged, as demonstrated in Figs. 15and 16. As compared to rj and θp, the effect of λ appearsrelatively mild, showing a gentle and roughly linear increaseof each of the three macroscopic properties with increasing

λ, as expected (Fig. 14d). No notable change of the failuremode upon the variation of λ is observed.

It is seen in Fig. 15 that when θp varies and rj takes afixed value of 20 mm, the final shear band cuts through thesample and keeps aligned with the SJ plane for the casesof θp = 30◦, 45◦ and 60◦, but apparently takes a differentform which denies the major role of the joint in the sam-ple failure in the cases of θp = 0◦ and 90◦. We comparethe results in Fig. 15 with the simulation results given byManouchehrian et al. [30], who made a similar study onthe effect of pre-existing flaw orientations on crack propaga-tion in brittle materials using BPM. The crack propagationpatterns resulting from different initial flaw orientations pro-vided by Manouchehrian et al. [30] and the current study areshown in Fig. 15 for comparison. It needs to be mentionedthat the initial flaws in Manouchehrian et al. [30] were cre-ated by removing the particles in a narrow rectangular zonesurrounding the initial flaw. By comparing Fig. 15a–e with

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Fig. 15 Effects of joint dip angle on the sample failure mode (a–e) andcomparison of the failure mode with simulation results from ref. [30](f–k): a θp = 0◦; b θp = 30◦; c θp = 45◦; d θp = 60◦; e θp = 90◦; f

β = 0◦; g β = 15◦; h β = 30◦; i β = 45◦; j β = 60◦; k β = 90◦. f–kare reproduced from ref. [30])

Fig. 15f–k and, it can be seen that similar crack propagationpatterns featured with the alignment of the wind crack orien-tations with the initial flaw orientations for the middle rangeof orientation (or dip) angle (i.e., θp = 30◦–60◦) are foundin both studies, indicating the effectiveness of SJM in sim-ulating crack propagation in BPM materials. For the case ofan initially horizontal flaw (i.e., θp = 0◦), some differencesare found between the crack propagation patterns from thetwo studies, which is not surprising due to the minor roleof the initial flaw in the crack propagation in this case anddifferences in the simulation methods between the two stud-ies, such as 2D versus 3D simulations, initial flaw modelingtechnique, initial sample properties and so on.

In Fig. 16 where rj varies and θp takes a fixed value of30◦, a transition from a compaction band failure near thebottom of the sample at a low rj value (e.g., rj = 10mm) to ashear band failure aligned with the SJ plane at a high rj value(e.g., rj = 20mm) is observed again, providing a clear pictureof the failure mode evolution of a SRM sample containingfrom a non-persistent joint to a through-going joint. Herein,

it is both scientifically and practically important to look intothe role of a non-persistent joint with a critical length (withrespect to the sample size) in the sample failure behavior andwhether it could be appropriately modeled using SJM.

A very useful technique related to the modeling of a non-persistent joint mentioned above is to set any newly brokenparallel bondwhich is located on the extension of the originalSJ plane to be a new SJ contact. For convenience, this tech-nique is called “SJ-extension” technique. Figure 17 shows theeffect of use of “SJ-extension” technique on the sample con-taining a non-persistent joint with rj = 16mm and θp = 30◦.Interestingly, it is found that the sample failed along the SJplanewith the use of “SJ-extension” technique and otherwisealong an inclined shear band within the intact matrix abovethe SJ plane. The corresponding bond crack distributions inFig. 17c and d agree well with the shear banding patternsin Fig. 17a and b. This example clearly indicates whether ornot to treat any newly generated particle contact followingthe bond breakage event as the smooth joint contact makea significant difference in the simulated behavior of fracture

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Fig. 16 Effects of joint radius on the sample failure mode: a rj = 10mm; b rj = 16mm; c rj = 20mm

Fig. 17 Effects of the “SJextension” technique on thesample failure mode withrj = 16mm: a and c without thetechnique; b and d with thetechnique

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development and propagation and consequent failure patternof a jointed rock mass. However, a further study is neededto investigate the physical soundness of the SJ-extensiontechnique, probably with the aid of available experimentalresults.

4 Conclusions

A comprehensive study has been carried out on the influ-ences of microscopic parameters of the smooth joint model(SJM) on the unconfined compression behavior of syntheticrock masses (SRM) using 3D DEM simulation method. Themodel was constructed by incorporating a smooth joint planeinto a parallel-bonded particle assembly which was used torepresent an intact rockmass and validated against the uncon-fined compression test results of marble samples taken fromthe site of Jinping II hydropower station in Sichuan, China.

Comparing with the behavior of a pure bonded-particlemodel, it is clear that the presence of a pre-existing joint maysignificantly alter the macroscopic unconfined compressionbehavior and the associated failure mechanism of a SRMsample. However, the nature and degree of such a changeis a function of a series of micro-parameters of SJM andis essentially determined by the competition between thediscontinuity plane and the intact rock matrix in control-ling the failure of the sample. Simulation results indicatethat for a SRM containing a non-bonded, non-persistentjoint, mechanical micro-parameters including joint contactstiffness and friction coefficient, and geometrical micro-parameters including joint radius and dip angle all showsignificant effects on the unconfined compressive strengthbehavior. They are reflected in the development of a majorshear band along the joint plane or within the intact matrixnot involving the joint plane. However, strong couplingeffects exist among thesemechanical and geometrical micro-parameters so the eventual sample failure is a result of thecomplicated interaction among these micro-parameters. Fora bonded joint, the above competition mechanism remainsunchanged but is reflected in the interplay between the bondstrength of smooth joint contacts and parallel bond strengthof particle contacts. A noteworthy finding is that, for a non-bonded joint, the dilatancy angle has negligible influenceon the sample behavior, indicating the incompetence of thecurrent force-dilatancy relation in producing realistic jointroughness effects.

Although the SJM has not been validated against labo-ratory test results due to the unavailability of the data ofrock joint sets, the numerical results presented in this paperprovide the basis for the full application of SJM to the inves-tigation of the behavior of real rock masses at the laboratoryor field scales. They are valuable for the evaluation of the cur-rent model capability in simulating and predicting the shear

failure behavior of rockmasses, and the further improvementof the current model in terms of its numerical capability andphysical soundness. In a more general sense, these resultswill also be useful for the research on the damage mechanicsof geomaterials as discontinuous media on an array of finescales, such as single sand particle breakage, etc.

Acknowledgments This research was supported by the GeneralResearch Fund No. CityU 122813 from the Research Grant Councilof the Hong Kong SAR and the Research Grant No. 51379180 from theNational Science Foundation of China.

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