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i) Ph.D Candidate, Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Korea (asaca83＠maver.com).

ii) Master, Department of Civil and Environmental System Engineering, Konkuk University, Korea (031608＠hanmail.net).iii) Associate Professor, Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Korea

(gyechun＠kaist.edu).iv) Associate Professor, Department of Civil and Environmental System Engineering, Konkuk University, Korea (swlee＠konkuk.ac.kr).

The manuscript for this paper was received for review on July 7, 2010; approved on March 15, 2011.Written discussions on this paper should be submitted before May 1, 2012 to the Japanese Geotechnical Society, 4-38-2, Sengoku, Bunkyo-ku,Tokyo 112-0011, Japan. Upon request the closing date may be extended one month.

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SOILS AND FOUNDATIONS Vol. 51, No. 4, 701–712, Aug. 2011Japanese Geotechnical Society

PARTICLE SHAPE AND CRUSHING EFFECTSON DIRECT SHEAR BEHAVIOR USING DEM

SEON-AH JOi), EUN-KYUNG KIMii), GYE-CHUN CHOiii) and SEOK-WON LEEiv)

ABSTRACT

Numerical analyses using the PFC2D are conducted to study the relative changes of particle crushing and the shearbehavior of granular materials according to the quantiˆed particle shapes. A total of seven particle shapes are stan-dardized and quantiˆed. Three diŠerent particle models, including a circular particle model, a non-crushing particlemodel, and a crushing particle model, are developed and analyzed. The results show that shear strength is mobilized insize order: 3 ballÀ6 ball_TÀ2 ballÀ6 ball_RÀ4 ballÀ9 ballÀ1 ball model, corresponding to triangleÀrectan-gleÀsquareÀcircle shape in both the non-crushing particle model and the crushing particle model. Within the sameshape but with a diŠerent number of sub-particles, it is found that an increase in the number of sub-particles within aparticle coincides with smaller shear strength per model. The non-crushing particle model shows the increase of porosi-ty not only in the shear band, but also in other layers. However, in the case of the crushing particle model, the increaseof porosity is mainly focused within the shear band. It is found that with a larger circularity and convexity, that is, as aparticle becomes more circular in shape, the shear strength decreases, regardless of particle crushing. It can be conclud-ed that the standardized particle shape model suggested in this research has broader applications for future studies.

Key words: direct shear test, particle bonding, particle crushing, particle shape, PFC, roughness (IGC: D3)

INTRODUCTION

Granular materials with a large particle size are widelybeing used in large scale civil engineering constructionssuch as dams, harbors, railroads, foundations, and em-bankments. It is important to understand the shear be-havior of granular materials for the safe and economicaldesign or construction of such structures. Granularmaterials such as crushed aggregates or sand can be par-tially or fully crushed by external loads, and such crush-ing can cause the instability of the structure by changingthe soil properties of the foundation. Even under thesame loads, the degree of particle crushing can diŠer ac-cording to particle size, shape, mineral components, dis-tribution, and relative density. The gradual developmentof particle crushing derived from the external loads,and/or the chemical or physical characteristics of parti-cles, pulverizes the particle structure, and the volumecompression produced by crushed ˆne grained particles,ˆlling up gaps between particles, makes the soil structuredenser. The ground condition, particularly its vulnerabil-ity to particle crushing, can cause the issue of instabilityof the structure due to large scale compressive deforma-tion, and can result in an engineering problem.

In general, laboratory experiments such as unconˆnedcompression tests, triaxial compression tests, and directshear tests, or ˆeld experiments, have been used to under-stand the characteristics of shear behavior. Though theseexperiments have produced reliable results, their draw-back is that it is impossible to identify the particle behav-ior visually. Consequently, numerical analysis techniquesare being adapted to compensate for such shortcomings.Especially, the model studies for the prediction of shearbehavior of granular soils using DEM (discrete elementmethod), which adapts discontinuum mechanics, are con-tinuously increasing. DEM can consider the interlockingeŠect, the inherent quality of the particle, and the in-‰uence of porosity, and has the advantage of being ableto simulate the shear behavior visually in the process ofanalysis.

For this reason, the PFC (particle ‰ow code), a numeri-cal analysis program based on DEM, was used in thisresearch to study the particle crushing and shear behaviorof granular materials according to particle shape andbonding state. That is, each shear behavior was analyzedwith direct shear test modeling by means of generating sixparticle shapes using the clump model within the PFC interms of particle shape, and the impact of particle shape

702702 JO ET AL.

on shear behavior was then studied by comparing it withthe direct shear behavior of the circular particle. In addi-tion, the shear behavior where particle crushing had beentaken into consideration was analyzed for six particleshapes, using the crushing particle models developedfrom the cluster model within the PFC. Consequently,the impact of particle crushing on shear behavior hasbeen studied by comparing the result with that of themodel in which crushing was not considered.

The six particle shapes deˆned in this researchrepresent particle surfaces with a range of roughnessfrom the circular particles of Ottawa sand to the angularparticles of blasting sand, and an attempt was made tocompare the relative characteristics of the shear behaviorand particle crushing according to particle shape. In thatrespect, it should be noted that the intention of thisresearch is not to numerically analyze the laboratory ex-periment by determining the micro-properties throughthe calibration process based on the result of a speciˆclaboratory shear test. In other words, the main purposeof this study is to consider the relative changes of particlecrushing and shear behavior of granular materials ac-cording to the quantiˆed various particle shapes deˆnedin this research.

LITERATURE REVIEW

Direct shear test modeling of granular soils has beenconducted not only in two dimensions, but also in threedimensions; and the stress-displacement behavior and/ordilatancy have been the constant research subjects untilnow (Hainb äuchner et al., 2003; O'Sullivan et al., 2004;Lobo-Guerrero and Vallejo, 2005; Liu, 2006; Zhang andThornton, 2007). At the same time, the study on theshear behavior, considering the particle shape and parti-cle crushing of granular soils, has progressed considera-bly. Many researchers have selected the modeling tech-nique of binding several circular particles with a constantbonding strength to simulate the geometrical shape ofparticles through numerical analysis using DEM (Jensenet al., 2001; McDowell and Harireche, 2002; Cheng et al.,2003).

Jensen et al. (1999) conducted a DEM-based numericalanalysis to study the in‰uence of particle shape on shearbehavior by using clustered particles and non-clusteredparticles. More interlocking occurred with a rougher par-ticle shape; as a result, high shear strength was proved.Moreover, the distinctive shear band derived from parti-cle crushing was also proved by means of the modeling,considering particle crushing. However, it has beenproven that the decrease of shear stress caused by particlecrushing is not signiˆcantly large. Masson and Martinez(2001) tried to simulate a direct shear test using a twodimensional DEM. They created 1,050 cylinder-type par-ticles and performed analysis in both dense and looseconditions, and showed the typical shear behavior ap-peared in granular soils. In addition, they observed thatthe shear zone occurred mainly in the middle of the shearbox, formulating a shear band showing a range of 5 to 6

times larger than the maximum aggregate size in a densestate by studying the displacement and rotation of parti-cles. On the other hand, they showed that no such shearband was veriˆed in a loose state, and displacement andparticle rotation occurred over the total range relativelyuniformly. Matsushima et al. (2003) performed modelingof the angular shape of an actual particle with threedimensional DEM, and analyzed its impact using thedirect shear model. In this case, LAT (Laser-AidedTomography) was applied to the numerical model afteranalyzing the actual three dimensional shape. Lobo-Guerrero and Vallejo (2005) also simulated particlecrushing with a two dimensional direct shear model. Intheir research, in the case of particle crushing, if a circu-lar particle meets the crushing criteria, it can be trans-formed into eight circular particles with three diŠerentsizes. This study showed that particle crushing was fo-cused on the shear zone. From the literature review,however, it is considered that existing references are notsu‹cient to draw an integrated conclusion, since theycontain fragmentary studies of particle shape and crush-ing suited only for the purpose of the speciˆc research.

In spite of vigorous adaptations of numerical analysis,the numerical modeling of a complicated natural particleshape is not easy, and the deˆnition of the relationshipwith particle crushing via the quantitative deˆnition ofparticle shape is still a di‹cult task. Despite the limitationof the actual representation of particle shape, we stan-dardized particle shapes by simplifying them into a trian-gle, a rectangle, and a square, and quantiˆed the surfaceroughness of the six particle shapes hypothesized in ourstudy by selecting two parameters among various rough-ness parameters deˆned in previous research. Further-more, even if a model had represented the same particleshape, we changed the numbers of sub-particles constit-uting the certain particle shape to consider the sizes ofcrushed ˆne grained particles. In summary, we suggestthe possibility of the new particle model which combinesthe three standardized particle shapes suggested in thisstudy, by presenting the result of shear behavior of eachof the three shapes with surface roughness measurement.Also, we suggest a possible approach to consider the sizeof crushed ˆne grained particles, by presenting the resultof shear behavior modeled from the alteration of thenumbers of sub-particles which constitute the sameshape.

NUMERICAL MODELING USING PFC

Modeling of Particle Shape and CrushingBasically, PFC only supports a single circular particle,

and particle crushing is not considered. Consequently, aclump model was applied to create the various shapes ofparticles intended to be implemented in this study. Aclump model, combining more than two circular sub-par-ticles, allows various shapes of particles to maintain theiroriginal shapes in the process of shearing by preventingthe destruction of contact points. The various particleshapes formulated in this research are constructed, as

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Fig. 1. Particle shapes

Table 1. Particle properties

Model 1 ball 2 ball 3 ball 4 ball 6 ball_R 6 ball_T 9 ball

Radius [mm] 0.360 0.255 0.208 0.180 0.147 0.147 0.120

emax 0.267 0.279 0.492 0.747 —

emin 0.183 0.180 0.119 0.155 —

Dr (z) 92 89 81 94 —

Fig. 2. Schematic diagram of direct shear test model

703PARTICLE SHAPE AND CRUSHING EFFECTS

shown in Fig. 1 by binding with 2, 3, 4, 6, and 9 small cir-cular sub-particles. Figure 1(a) is regarded as a single cir-cular particle, where (b) and (e), (c) and (f), and (d) and(g) are regarded as a rectangle, triangle, and square, re-spectively. Each sub-particle which formulates one parti-cle shape is assumed to be a cylinder of a unit thickness.The sizes of respective sub-particles forming 6 shapes(Figs. 1(b)¿(g)) were determined given that the sum ofthe areas of respective sub-particles is equal to the area ofthe 1 ball model (Fig. 1(a)). The radii of sub-particles ofeach model are summarized in Table 1. As a result, atotal of seven particle shapes including clump models(Figs. 1(b)¿(g)), a non-crushing particle model, of whichthe binding is permanently not destructible, and the 1 ballmodel made of a single circular particle (Fig. 1(a)), havebeen constructed in order to study the impact of particleshape on shear behavior.

A cluster model, which is a crushing particle modelwhich binds more than two circular sub-particles withˆnite bonding strength, was applied to study the impactof bonding state on particle crushing and shear behavior.A cluster model is a model in which the binding of sub-particles is destructed when an external force over thelimit of bonding strength is applied. That is, the bindingof sub-particles for six particle shapes (Figs. 1(b)¿(g)), isset to be destroyed in the case where a certain amount ofexternal force is applied. Each numerical analysis for thebonding strengths of 200 kPa and 800 kPa was carriedout in this research in order to verify the impact of bond-ing strength between sub-particles on particle crushing.More details on the determination of bonding strengthwill be described later in this paper.

In summary, we classiˆed particle models into circularparticle model, non-crushing particle model, and crush-ing particle model according to the shape and bindingcondition of sub-particles. The numerical analysis of thedirect shear test on each particle model was then per-formed and its result in terms of particle crushing andshear behavior was analyzed.

Modeling of Direct Shear TestThe direct shear test was modeled by using PFC2D to in-

vestigate the shear behavior of granular materials. Thesize of the direct shear box used for the model was 6 cm indiameter, and 2 cm in height, and the modeling was in-tended to transform a three dimensional cubic box into atwo dimensional plane. Figure 2 shows a schematic dia-gram of the direct shear test model. The modeling of thedirect shear test consisted of the upper box and the lowerbox. The upper box consists of four walls of EF, FG,GH, and HI, and the bilateral walls (HI and GF) are ˆx-ed. The HG wall is free to move up and down and its roleis to maintain the normal stress by the servo-controlledsystem. The lower box is composed of four walls (AB,BC, CD, JK) performing shear displacement, moving leftto right with constant velocity. EF and JK walls were in-stalled to prevent the ball from falling out into the gapcreated during shearing progress between FG and CD,and HI and AB. The friction coe‹cient of the wall whichconstitutes the shear box was set to 0, while 1.0×109

N/m, which was larger than the stiŠness of the balls, wasapplied for wall stiŠness to prevent balls from penetrat-ing out of the wall.

Numerical Analysis Method and ScopeThe particle being supported in the PFC is circular and

is a cylinder with unit thickness under two dimensions,and its three dimensional modeling is implemented into asphere. In this case, material properties are attributed toeach generated particle. The input variables regarding theparticles in the PFC2D are as follows: particle size, stiŠ-ness, friction coe‹cient, and bonding strength betweensub-particles. The same particle size and consequently thesame number of the generated particles of 2,474 were ap-plied to all numerical analysis models conducted in thisstudy. The random generation principle for the genera-

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Table 2. Input values in PFC2D

Micro properties Value

Density (kg/m3) 2,650

Normal stiŠness (N/m) 1×108

Shear stiŠness (N/m) 1×108

Normal/shear stiŠness ratio 1

Friction coe‹cient 0.75

Gravity (m/s2) 9.81

R (mm) 0.72

Initial porosity 0.160

Ball number 2,474

Parallel bondNormal (Pa) 2×105 or 8×105

Shear (Pa) 2×105 or 8×105

Fig. 3. Stress-displacement curve of circular particle model

Fig. 4. Failure envelope of circular particle model

704 JO ET AL.

tion of particles, adjusting porosity using the ˆsh func-tion, and applying su‹cient stiŠness to avoid overlaps ofparticles, were applied for modeling. Moreover, frictionbetween particles was set to 0 to reduce the strengthgenerated between particles in the process of particlegeneration. The friction coe‹cient of 0.75 was then ap-plied after reaching complete equilibrium. Other inputvariables were evaluated, and the ˆnal input data isshown in Table 2. In this study, to simulate the densestate of the specimen, the initial porosity was set to be0.16 and the number of particles constituting a specimenwas identically 2,474 for all specimens. The relative den-sity can diŠer under the same porosity according to thediŠerence of the particle shape. For comparison, theoret-ical emax and emin of respective models were calculated andthe relative density on the porosity of 0.16 applied to allmodels was calculated under two dimensional model con-ditions, as summarized in Table 1. In the case of the 2ball and 4 ball models, the diŠerence between relativedensities is minor and the 3 ball model also shows a densestate with the relative density of 81z. In other words, therelative densities of the models with a porosity of 0.16show slight diŠerences, but in general they are all denselypacked. Therefore, it is considered that the porosity tun-ing method, which has the identical number of producedparticles, can be applicable for the relative comparisonamong the models. As mentioned above, the numericalanalysis of the direct shear test model was implemented,altering the particle shape, bonding strength, and normalstress concerning the non-crushing particle model andcrushing particle model.

CHARACTERISTICS OF SHEAR BEHAVIORACCORDING TO PARTICLE SHAPE

Circular Particle ModelFigure 3 shows the relationship between shear stress

and horizontal displacement resulting from the directshear modeling of the circular particle model. An initialporosity of 0.16 was applied and normal stresses of 50

kPa, 100 kPa, 200 kPa, and 300 kPa were performed.Figure 4 shows the failure envelope and the internal fric-tion angle was 27.89. The reliability of this model was es-tablished by conˆrming that the peak shear stress valueand the horizontal displacement point where that peakshear stress occurs, increased as normal stress increasedstepwise.

Non-crushing Particle ModelThe numerical analysis of the direct shear model, using

six shaped non-crushing particle models as shown in Fig.1, was carried out in order to study the shear behavior ac-cording to the changes of particle shape. Figure 5(a) is thestress-displacement curve of the non-crushing particlemodel per each shape when normal stress is 50 kPa, anddemonstrates that the shear stress of the non-crushingparticle model composed of more than two sub-particlesis larger than the shear stress of the circular particlemodel. Therefore, the fact that particle shape has a sig-niˆcant eŠect on the changes of shear behavior was con-ˆrmed. The comparison result of peak shear stress be-tween the non-crushing models showed that the 3 ballmodel has the largest value, while the 9 ball model has thesmallest value. In terms of particle shape, the peak shear

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Fig. 5. Stress-displacement curve of non-crushing particle model

Fig. 6. Failure envelope of non-crushing particle model

Table 3. Internal friction angle of non-crushing particle model

Particle shape Internal friction angle (9)

2 ball 50.0

3 ball 54.7

4 ball 46.9

6 ball_R 49.1

6 ball_T 52.6

9 ball 46.5

705PARTICLE SHAPE AND CRUSHING EFFECTS

stress appeared in the following order of size: 3 ball, 6ball_T, 2 ball, 6 ball_R, 4 ball, and 9 ball models. Con-sidering this result, the diŠerence of shear strength ac-cording to particle shape is assumed to be in connectionwith the degree of angularity of particle, and this will bereviewed later in this paper. Figures 5(b) and (c) show thestress-displacement curves of six shapes when normal

stress is increased to 100 kPa and 300 kPa, respectively.The trend of peak shear stress was the same as the case of50 kPa, and the increase of peak shear stress was propor-tional to the increase of normal stress. The residual shearstrength was achieved in the lower normal stress, howevera distinct peak shear stress was found in the higher nor-mal stress. In this study, it was assumed that the stiŠnessof sub-particles constituting the particle shapes is identi-cal and consequently the stiŠness of the particles is alsoidentical regardless of the particle shapes. Although theinitial stiŠness may change according to the types of par-ticles, the results shown in Fig. 5 are with no considera-tion of stiŠness change.

Figure 6 and Table 3 show the failure envelope and in-ternal friction angle of non-crushing particle models, re-spectively. The 3 ball model showed the largest value(54.79), and the 9 ball model showed the smallest value(46.59). In addition, having standardized the particleshape into a triangle (3 ball and 6 ball_T), rectangle (2ball and 6 ball_R), and square (4 ball and 9 ball), thetriangle was the largest, followed by the rectangle andthen the square in size order.

When the particles with the same shape but a diŠerentnumber of sub-particles (for example, 2 ball and 6 ball_Rfor rectangle), are relatively compared, the internal fric-tion angle decreases in size as the number of sub-particlesin a particle increases. This can be explained in terms ofthe roughness of the particle surface. In other words, the

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Fig. 7. Relation between bonding strength and peak shear stress

706 JO ET AL.

particle surface becomes rougher as the number of sub-particles in a particle decreases. A greater roughness in-creases the interlocking eŠect between particles andresistance against shearing. As mentioned previously,since the total area of the sub-particles which constituteparticle shape now remains the same, in the case wherethe number of sub-particles increases, the size of the sub-particle decreases proportionally, hence the surface of theparticle becomes smoother than in the case where parti-cles have fewer sub-particles. Therefore, it can be inter-preted that the models with smoother surfaces (6 ball_R,6 ball_T, and 9 ball) have less shear resistance than themodels with less sub-particle numbers (2 ball, 3 ball, and4 ball).

CHARACTERISTICS OF PARTICLE CRUSHINGACCORDING TO PARTICLE SHAPE ANDBONDING STATE

The accurate prediction of the crushing strength ofparticles or the degree of the actual particle crushing is anextremely di‹cult task due to various internal or externalfactors such as anisotropy of stress, physical or chemicalproperties of the particle, geometrical shape of the parti-cle, etc. The study on particle crushing, adapting a clustermodel in the PFC, has been conducted in this research inorder to obtain reasonable results by simplifying theseproblems.

A crushing particle model can be created by attributinga proper bonding strength between the sub-particleswhich constitute particle shape. Therefore, if an externalforce beyond the bonding strength is applied to the con-tact between sub-particles, the contact will be separated;hence simulating particle crushing. The parallel bondmodel was used in this research. The parallel bond modelconsists of a set of elastic springs with a normal and shearstiŠness and acts on the contact plane centered at the con-tact point. This model can transmit both the force andmoment at the contact point which is related to maximumnormal and shear stresses acting within the bond materi-al. When either of these stresses exceeds its correspondingbond strength, the parallel bond breaks (Itasca, 2004).

First, the relation between bonding strength and peakshear stress was investigated as shown in Fig. 7, in orderto determine the proper bonding strength for modeling.At that time, the 2 ball model was used as a representativemodel, and peak shear stress according to bondingstrength was obtained under the normal stresses of 50kPa, 100 kPa, and 300 kPa. Peak shear stress increasedsharply within the bonding strength range of 100 kPa to1,000 kPa, but showed a convergent tendency to a certainvalue if the range was higher or lower than this bondingstrength range. In PFC, when the bonding strength be-tween the sub-particles constituting particle shapes is verysmall, the particle bonds in the entire region of a speci-men break, resulting in a decrease in shear strength. Thisis related to the low threshold of bonding strength, whichcorresponds to the lower boundary of bonding strength.On the other hand, as the bonding strength increases, it

shows a reverse tendency and eventually the highthreshold of bonding strength where the particle bondingdoes not break at all. It corresponds to the upper bound-ary of bonding strength. Consequently, this shows that ifthe bonding strength is under 100 kPa, most of the con-tacts are separated by the external force applied to theparticle, displaying similar shear behavior as that of thecircular particle model; conversely, if the bondingstrength exceeds 1,000 kPa, the bonding strengthbecomes relatively larger than the applied external force,and the contacts are not signiˆcantly separated and main-tained, showing a similar behavior to that of the non-crushing particle model. Based on the results, the bond-ing strength values of 200 kPa and 800 kPa, corre-sponding to 20z and 80z values, respectively, between100 kPa to 1,000 kPa, were selected as the bondingstrengths of the crushing particle model in this study.

Of the 7 particle shapes shown in Fig. 1, the 1 ballmodel was excluded from the crushing particle model be-cause it was a single particle model and not crushable.Figure 8(a) shows the stress-displacement curve under thenormal stress of 100 kPa and bonding strength of 200kPa. Peak shear stress was contained within the range of20¿60 kPa, showing a value of almost the same as, orsmaller than, 55 kPa, which was the peak shear stress ofthe circular particle model as shown in Fig. 3. The reasonwhy peak shear stress was smaller than for the circularparticle model could be explained as follows: each sub-particle which constituted the particle shape was separat-ed and became smaller than the particle size of the circu-lar particle model which formulated the specimen, and italso had better gradation distribution. Figure 8(b), ob-tained by increasing the bonding strength to 800 kPa,demonstrates that the increase of shear stress is propor-tional to the increase of bonding strength, since a greaterbonding strength causes less particle crushing. The stress-displacement curve of the non-crushing particle modelconˆrmed that all the peak shear stresses appeared dis-tinctively; however, in the case of the crushing particlemodel, the curve had a more gradual shape due to particle

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Fig. 8. Stress-displacement curve according to bonding strength

Fig. 9. Degree and tendency of particle crushing according to normalstress; cluster model, 6 ball_T model, bonding strength＝800 kPa

707PARTICLE SHAPE AND CRUSHING EFFECTS

crushing; hence, it showed a relative decrease of peakshear stress. In summary, in order to determine theproper bonding strength for particle shaping, the follow-ing steps are suggested: experimentally ˆnding the degreeof particle crushing of the actual particle for research;performing the numerical analysis as shown in Fig. 7 us-ing the shape of the actual particle; and matching the ex-perimental result with the numerical result.

Since the visual discrimination of the degree of particlecrushing was di‹cult owing to frequent occurrences ofparticle crushing in most models in the case of bondingstrength of 200 kPa, the bonding strength was set to 800kPa hereafter and the analysis of the crushing particlemodel was conducted. The degree and tendency of parti-cle crushing, occurring in the process of shearing at thelevel of each normal stress, was studied, and the resultsfor the representative 6 ball_T model are shown in Fig. 9.In other words, Fig. 9 shows a visual representation ofthe degree of particle crushing after the shear test accord-ing to the normal stress. Figures 9(a), (b), and (c) showthe degree and tendency of particle crushing at the nor-mal stress of 50 kPa, 100 kPa, and 300 kPa, respectively.If normal stress was 50 kPa, particle crushing did not oc-

cur in certain speciˆc areas, occurring rather less over theentire area; however, particle crushing gradually expand-ed as normal stress increased. In particular, particlecrushing, mainly occurring on the shear plane, formulat-ed the shear band as normal stress increased as shown inFig. 9(c). Statistically, the degree of particle crushing(i.e., the number of broken particles in percentage) ap-pears to be 1.31z at 50 kPa, 2.24z at 100 kPa, and6.62z at 300 kPa. In particular, it is obvious that as thenormal stress increases, the particle crushing occursmostly near the shear plane forming the shear band. Thisphenomenon has already been veriˆed in previous studieson the direct shear test of granular materials. More de-tails on the shear band will be discussed later in thispaper. In terms of particle shape, the degree of particlecrushing ranged from largest to smallest in the followingsize order: triangle, rectangle, and square; even within thesame shape, more particle crushing occurred as the num-ber of sub-particles decreased.

The stress-displacement curve of the crushing particlemodel according to the normal stress is shown in Fig. 10;in this case, the bonding strength was 800 kPa. Thecrushing particle model showed gradual curve shapedespite the increase of normal stress, unlike the non-crushing particle model. The ratio of peak shear stressagainst normal stress decreased. In short, due to particlecrushing, peak shear stress decreased compared with thatof the non-crushing particle model. In respect of particleshape, the peak shear stress appeared in the following sizeorder: triangle, rectangle, and square; even in the sameshape, as the number of sub-particles decreased, a higher

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Fig. 10. Stress-displacement curve of crushing particle model

Fig. 11. Failure envelope of crushing particle model

708 JO ET AL.

peak shear stress was induced. If normal stress was small,the increase of porosity on the shear plane was observedwhere the contacts between sub-particles were not brokenand dilation was induced. Conversely, if normal stresswas increased, dilation was thought to be restricted bynormal stress, and porosity on the shear plane no longerincreased. According to the previous studies, when shear

tests are conducted on dense specimens, the shear behav-ior moves from the brittle behavior, in which usually aclear peak is observed, to a ductile behavior, which showsno noticeable peak, with an increase in the normal stress.Namely, as the rearrangement and crushing of particlesoccur by the high normal stress, even dense specimensshow similar shear behavior to loose specimens. Conse-quently, in summary, in the high level of normal stress,the result of the previous studies, which suggests that thespecimen of granular soils in a dense condition under ahigh conˆning pressure showed a ``loose state shear be-havior'', is conˆrmed by numerical analysis conducted inthis study.

Figure 11 shows the failure envelope that originatedfrom the crushing particle model of the bonding strengthof 800 kPa. According to the results of the previous stu-dies, the envelope changed to non-linear behavior in thecase where the particle crushing occurred (Bolton, 1986;Feda, 2002). This study also showed that the failure enve-lope of the models which allowed particle crushingshowed non-linearity compared with the non-crushingparticle models, and consequently showed a considerabledecrease of the internal friction angle. Appearances ofnon-linearity in the failure envelope were derived fromthe acceleration of particle crushing and particle rearran-gement due to the increase of normal stress, and the rela-tive decrease of shear resistance resulted from particlecrushing. This situation can also cause the decrease of di-lation. Eventually, the crushing particle model provedcapable of expressing the shear behavior of granular soilsmore or less closer to the case of actual particle crushingthan the non-crushing particle model. Table 4 shows theinternal friction angle of the crushing particle model ofeach particle shape derived from the linearization of thefailure envelope. Here, for linearization, the adhesion in-tercept was assumed as 0 and the linear regression wasconducted. In most models, R2 is 0.96 which is a quitereliable value. The crushing particle model also showed alarge internal friction angle in the triangle models, anddecreased in size in the following order: rectangle,square.

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Table 4. Internal friction angle of crushing particle model

Particle shape Internal friction angle (9)

2 ball 38.3

3 ball 42.6

4 ball 36.5

6 ball_R 34.4

6 ball_T 39.2

9 ball 31.9

Fig. 12. Comparison of non-crushing particle model with crushingparticle model

Fig. 13. Position of each layers of specimen divided into 10 layers

709PARTICLE SHAPE AND CRUSHING EFFECTS

COMPARISON BETWEEN NON-CRUSHINGPARTICLE MODEL AND CRUSHING PARTICLEMODEL

Figure 12 shows the stress-displacement curve of thenon-crushing particle model and the crushing particlemodel, concerning the 3 ball triangle model under thenormal stress of 100 kPa. The non-crushing particlemodel showed a larger peak shear stress than that of thecrushing particle model. Also, the crushing particlemodel having a bonding strength of 800 kPa generatedless particle crushing and consequently larger peak shearstress than the crushing particle model having a bondingstrength of 200 kPa.

If particle crushing does not occur, the interlockingphenomenon arises, increasing the porosity of the speci-men. The changes of porosity per each layer during theshear test were compared by dividing the specimen into 10layers at the same height of 2 mm from the bottom in avertical plane as shown in Fig. 13. In Fig. 13, the 1st layerand the 10th layer, both of which contact with the bound-ary, were excluded from the analysis due to the possibleboundary eŠect. The result is shown in Fig. 14. Shearingof specimens was terminated at the target shear displace-ment stages and porosity proˆles for each stage were eval-uated: (i) before a shearing stage (d＝0 mm), (ii) at ashear displacement stage when a peak shear stress isgenerated in the stress-displacement curve (d＝1.15, 1.88or 0.85 mm), (iii) at a shear displacement stage when aresidual shear stress is beginning to appear in most of themodels (d＝3 mm), and (iv) at the completion stage ofshearing (d＝6 mm). While three models have slightdiŠerences in the amount of an increase in porosity, theporosity in the shear band obviously increased comparedwith the initial porosity in most cases. The shear band ex-isted between the 5th layer and the 6th layer; and the 5thlayer had slightly larger porosity; hence implying moredilation occurrences in the lower box which was movinghorizontally. The non-crushing particle model, whichhad the largest shear stress among three models, showedthe largest changes in porosity during shearing, and thecrushing particle model with the bonding strength of 200kPa, which had the smallest peak shear stress resultingfrom many occurrences of particle crushing, showed thesmallest changes in porosity. This can be judged as theresult of the rearrangement of particles having gaps that

were ˆlled with broken sub-particles originating fromparticle crushing rather than from the occurrence of ex-pansion behavior due to interlocking. Moreover, the non-crushing particle model showed an increase of porositynot only in the shear band, but also in other layers.However, in the case of the crushing particle model, theincrease of porosity was mainly focused within the shearband, and the changes in other layers were not as large.The diameter of the particle is about 0.72 mm (in Table1) and the height of each layer in Fig. 13 is 2 mm, andtherefore each layer corresponds to 2.8 times the diameterof the particle. Since the region where the porosity greatlyincreases can be considered as the shear zone, it can be in-ferred from Fig. 14 that the zones equivalent approxi-mately to the height of 8 times the diameter of the particleat 800 kPa and to the height of 5 times the diameter of theparticle at 200 kPa are involved in shear behavior in thecase of the crushing particle model.

RELATION BETWEEN PARTICLE ROUGHNESSAND SHEAR STRENGTH

It is necessary to quantify the particle shape preferably

710

Fig. 14. Porosity changes in the 3 ball triangle model

Table 5. Evaluation of particle roughness by circularity and convexity

Triangle Rectangle Square

3 ball 6 ball_R 2 ball 6 ball_T 4 ball 9 ball

Circularity 0.777 0.836 0.886

Convexity 0.881 0.883 0.880 0.889 0.887 0.902

710 JO ET AL.

before studying the relation between particle roughnessand shear strength. According to the previous studies,particle shape has been evaluated quantitatively usingvarious roughness parameters such as sphericity, round-ness, smoothness, etc (Wadell, 1932; Krumbein, 1941;Power, 1953; Krumbein and Sloss, 1963; Barret, 1980).Based on the literature review, the concepts of circularity

and convexity were selected in this study and the particleroughness was evaluated and expressed with the values inthe range of 0¿1.0. Circularity and convexity can be ex-pressed as Eq. (1) and Eq. (2), respectively. Circularityand convexity means that as the values become smaller,the shape of a particle becomes sharper and more irregu-lar.

Circularity＝pD

Perimeter(1)

Convexity＝Area of actual particle

Area created by connecting each edge of particle(2)

Table 5 shows the evaluated values of circularity andconvexity concerning the particle shapes shown in Fig. 1.Circularity is a main index for particle shape, and thesmallest value was 0.777 for the triangle, while the rectan-gle and square had the values of 0.836 and 0.886, respec-tively. Convexity is a main index for roughness of particlesurface. In the case of the triangle shape, the 3 ball whichhas smaller numbers of sub-particles, showed smallerconvexity than the 6 ball_T. In the same manner, in thecase of the rectangle and the square, the 2 ball and the 4ball (which have smaller numbers of sub-particles),showed smaller convexity than the 6 ball_R and the 9ball, respectively. Therefore, it can be inferred that thetriangle is the most irregular and angular shape withrespect to particle shape, whereas the square is the shapeclosest to a circle.

Figures 15 and 16 show the relations between the inter-nal friction angles and the quantiˆed particle shapes.Here, for the cluster model, the bonding strength of 800kPa was applied to each particle shape model. Figure 15shows the relation between the internal friction angle andcircularity concerning the 3 ball, 2 ball, and 4 ball, whichrepresent the triangle, rectangle, and square, respectively.Regardless of particle crushing, it was conˆrmed that thelarger circularity (meaning the closest in shape to the cir-cle), resulted in the smaller internal friction angle. Thelargest internal friction angle was obtained in the triangleshape, and became smaller in the following order: rectan-gle, square. Figure 16 shows the relation between the in-ternal friction angle and convexity according to particleshape. Similar to circularity, the larger convexity resultedin the smaller internal friction angle within the same par-ticle shape. In summary, the triangle, which has thesmallest circularity, is proved to be the most angular andthe roughest, while the rectangle is more angular androugher than the square. Circularity and convexityshowed an inversely proportional relation with the inter-

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Fig. 15. Relation between circularity and internal friction angle

Fig. 16. Relation between convexity and internal friction angle

711PARTICLE SHAPE AND CRUSHING EFFECTS

nal friction angle; and this was exactly consistent with theresults of previous studies (Lee and Farhoomand, 1967;Hagerty et al., 1993; Lade et al., 1996).

CONCLUSIONS

Numerical analyses using the PFC, a numerical analy-sis program based on DEM, are conducted to study therelative changes of particle crushing and shear behaviorof granular materials according to the quantiˆed particleshapes. A total of seven particle shapes are standardizedand quantiˆed. Three diŠerent particle models includinga circular particle model, a non-crushing particle model,and a crushing particle model are developed and ana-lyzed. The main conclusions drawn from this research areas follows:(1) With respect to particle shape, the shear strength is

mobilized in the following size order: 3 ballÀ6 ball_TÀ2 ballÀ6 ball_RÀ4 ballÀ9 ballÀ1 ball model,corresponding to triangleÀrectangleÀsquareÀcircleshapes in both the non-crushing particle model andthe crushing particle model. Within the same shape

but where there are diŠerent numbers of sub-parti-cles, it is found that as the number of sub-particles ina particle increases the shear strength of a modeldecreases. This phenomenon is veriˆed with respectto particle surface roughness.

(2) In the PFC, if the bonding strength is under or overthe certain range, most of the contacts are almostseparated or not separated, respectively, by the exter-nal force applied to a particle. Therefore, the properbonding strength between sub-particles for the crush-ing particle model should be determined in advance,corresponding to the speciˆc purpose of the research.The guideline for the determination of appropriatebonding strength is suggested in this study.

(3) In the crushing particle model, particle crushing inlow normal stress does not occur in a certain speciˆcarea, occurring somewhat less over the entire area,however particle crushing is gradually expanded asnormal stress increases. This phenomenon induces agradual stress-displacement curve shape dissimilar tothat of the non-crushing particle model, and thedecrease of the ratio of peak shear stress against nor-mal stress. This phenomenon also induces a non-linear failure envelope and consequently the consider-able decrease of the shear strength compared with thenon-crushing particle model, especially in high nor-mal stress.

(4) With respect to the shear band, the non-crushing par-ticle model shows the increase of porosity not only inthe shear band, but also in other layers. However, inthe case of the crushing particle model, the increaseof porosity is mainly focused within the shear band.It can be concluded that the zones approximately cor-responding to the height of 8 times the diameter ofthe particle and to the height of 5 times the diameterof the particle, are involved in shear behavior in thecase of the crushing particle model with 800 kPabonding strength and with 200 kPa bonding strength,respectively.

(5) It is found that the larger circularity, that is, as a par-ticle becomes more circular in shape, the shearstrength reduces, regardless of particle crushing.Similarly to circularity, the larger convexity, that is,as the particle surface becomes less rough, the shearstrength decreases. It can be concluded that the stan-dardized particle shape model suggested in thisresearch has laid the groundwork for future applica-tions such as studies of shear behavior of granularsoils including particle crushing.

(6) This study has a limit of not taking the modeling ofreal soils into consideration. However, the particleshapes of real soils are roughly classiˆed into sevenshapes and thus it is considered that if a real soil has asingle particle shape, the micro shear behavior of thereal soil according to the change of particle shape andcrushing can be qualitatively predictable using theˆndings in this study. Therefore, this study can beconsidered as a fundamental research to implementthe engineering behavior assessment of real soils with

712712 JO ET AL.

having complicated shapes.

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