a novel three-dimensional contact model for granulates...
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Computers and Geotechnics 65 (2015) 147–163
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Computers and Geotechnics
journal homepage: www.elsevier .com/locate /compgeo
Research Paper
A novel three-dimensional contact model for granulates incorporatingrolling and twisting resistances
http://dx.doi.org/10.1016/j.compgeo.2014.12.0110266-352X/� 2014 Elsevier Ltd. All rights reserved.
⇑ Corresponding author at: Department of Geotechnical Engineering, College ofCivil Engineering, Tongji University, Shanghai 200092, China. Tel.: +86 21 65980238; fax: +86 21 6598 5210.
E-mail address: [email protected] (M. Jiang).
Mingjing Jiang a,b,⇑, Zhifu Shen a,b, Jianfeng Wang c
a Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, Chinab State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, Chinac Department of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong
a r t i c l e i n f o a b s t r a c t
Article history:Received 7 May 2014Received in revised form 12 December 2014Accepted 15 December 2014
Keywords:Granular materialsDiscrete element methodContact modelRolling resistanceTwisting resistance
This paper presents a new three-dimensional (3D) contact model incorporating rolling and twisting resis-tances at inter-particle contact, which can be introduced into the discrete element method (DEM) to sim-ulate the mechanical behavior of particulates. In this model, two spheres were assumed to physicallyinteract over a circular contact area, where an infinite number of normal spring-dashpot-divider ele-ments and tangential spring-dashpot-slider elements were continuously distributed. The model consistsof four interactions, in normal/tangential/rolling/twisting direction, with physically-based stiffness, peakresistance and damping coefficient in each direction. The two main features of the model are that (1) thecontact behavior was physically derived and (2) only two additional parameters, shape parameter b (link-ing the contact area radius and particle size) and local crushing parameter fc (describing local contactcrushing resistance) were introduced when compared with the standard 3D DEM. The new model wasimplemented into the 3D DEM code and used to simulate conventional triaxial and plane-strain compres-sion (CTC and PSC) tests to examine its ability to capture the quasi-static behavior of sands. The numer-ical results show that the strength of the DEM material obtained in CTC tests increases with theparameter b and is within the experimentally observed typical strength range of sand (30–40�). Rollingand twisting resistances can lead to increased dilation in volume. Consistent with previous studies, thehigh level of out-of-plane confinement observed in the PSC tests can greatly increase the strength andproduce a strain-softening response. A unique critical state line (CSL) was identified in the CTC tests,where the intercept and slope on the e–lgp plane increased with b. Comparison with previous experimen-tal data confirms that the shape parameter b can be well correlated to a statistical measure of real particleshape. The new model is also compared with previous DEM models.
� 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Particle shape variation is important in the mechanical behaviorof compact granular materials. Sands are typical granular materialswhich develop varying degrees of particle sphericity and round-ness through chemical and mechanical processes [1,2]. In geome-chanics, sands are observed to exhibit high strength andapparent dilation when sheared due to angular particles. A typicalexample is the crushed lunar regolith (classified as silty sand) witha peak friction angle of up to 55� [3].
The discrete element method (DEM) can be effectively appliedto sands due to their particulate nature. This is a method ofnumerically investigating the mechanical behavior at the elementlevel [4–9] and in typical boundary value problems [10–14]. Forsimulation efficiency, the original DEM simplifies sand particlesas frictional discs or spheres, as large amounts of particles mustbe used to ensure a representative numerical assembly. This, how-ever, results in the loss of particle scale details such as angularityand the shear strength values of the disc/sphere particle assemblyare below those obtained experimentally.
To address this deficiency, non-disc/non-sphere grains havebeen modeled as particle agglomerates [12,15–24], polygonal/polyhedral particles [18,25–32], pill-like particles [33], and ellip-soids [34–36]. Other methods used to construct complex particleshapes include non-uniform rational basis-splines method [37],the potential particle method [38], the racetrack particle method
Fig. 1. Local Cartesian coordinate system and the transmitted forces and torque/moments.
148 M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163
[39] and the Fourier descriptor method [40], among others. Thesemethods make DEM modeling of complex particle shapes possiblewith variable accuracy. However, as pointed out in [9], particleshape acquisition, mathematical representation, contact detection,and particle shape evolution during loading are problems commonto all these methods.
Real sand particles primarily interact over areas that can trans-fer contact moments and reduce excess particle rotations. Incor-porating contact moments into the DEM contact model cantherefore be effective to obtain a more realistic bulk strengthand to identify the deformation behavior of the granular materi-als. Rolling resistance models have been extensively used in bothtwo-dimensional (2D) and three-dimensional (3D) analyses[9,19,20,24,41–47]. In [47], previously developed rolling resis-tance models in the literature were compared and it was foundthat the elastic–plastic spring-dashpot type models were generallythe most effective, although others could be appropriate in spe-cific instances. For either 2D or 3D analyses, it was found thatthe rolling resistance model was as effective as the particleagglomerate method regarding most aspects of the micro- andmacro-mechanical behavior of compact granular assemblies[19,24,27]. Only a combination of the rolling resistance modeland the particle agglomerate method has been found to effec-tively reproduce the flow behavior of rice grains [20]. Given thedifficulty in accurately modeling real particle shapes, the rollingresistance model must be applied to either sphere or non-spherebased DEM simulations, where the primary features to be mea-sured are the forms of bulk behavior resulting from the actualparticle characteristics of the medium.
Relative rotation of two 3D particles at a contact can bedecomposed into two components: one about the contact normaldirection and the other on the tangential contact plane. The firstleads to a twisting resistance torque, and the second induces arolling resistance moment, both of which hinder the relative rota-tion of two particles at contact. There are a few examples consid-ering both the rolling and twisting interactions in themicromechanics model (e.g. [48]) and in numerical simulations(e.g. [44,49,50]). However, most 3D DEM contact models ignorethe twisting resistance (e.g. [19,45,51]), leading to inaccuratekinematics of the 3D particles.
The primary goal of this study is to develop a complete and real-istic 3D contact model, incorporating both the rolling and twistingresistances together with the normal/tangential interactions, whichcan be used in both sphere and non-sphere based DEM simulations.This is because the few existing rolling and twisting resistance mod-els include arbitrary parameters with no explicit physical meanings,usually chosen by trial and error. Since Jiang et al. [9,43] proposed aphysically based rolling resistance model for 2D DEM analyses, inwhich only one additional shape parameter is required to incorpo-rate rolling resistance, this study will extend Jiang’s 2D rolling resis-tance model into a 3D physically based model.
In this paper, a novel physically based 3D elastic–plasticspring-dashpot type contact model incorporating rolling andtwisting resistances together with the standard normal/tangen-tial interactions was developed with only two additional param-eters. The different forms of contact behavior in the rolling andtwisting directions were first theoretically derived from idealdistributions of the normal and tangential stresses on anassumed circular contact area. The new 3D model was thenimplemented into a DEM software to simulate conventional tri-axial and plane-strain compression (CTC and PSC) tests with dif-ferent values of b. The CTC and PSC test results were thencompared and the effects of loading conditions on the strengthand deformation behavior of sands were examined. The resultswere also compared with those of previous experiments toassess the model’s ability to capture the quasi-static behavior
of sands. The shape parameter b can be linked with a statisticalmeasure of real particle shape.
2. Contact behavior over a circular contact area
It is assumed that two spheres microscopically interact at a con-tact over a circular flat contact area with a radius of R. Fig. 1 pre-sents the transmitted forces and torque/moments at the contactin a local Cartesian coordinate system where the z-axis is the nor-mal contact direction and the x–y plane is attached on the tangen-tial plane. Denote by Fn, Fsx (Fsy), Mrx (Mry) and Mt the normal forcedue to compression, the tangential force due to relative displace-ment on the tangential plane, the moment due to rolling resistanceand the torque due to twisting resistance, respectively. The resul-tant force of Fsx and Fsy is denoted by Fs and the resultant momentof Mrx and Mry is denoted by Mr. A contact model will be developedfrom the relative displacement and rotation at the contact to deter-mine the transmitted forces and torque/moments. The contactmodel has interactions in the normal, tangential, rolling and twist-ing directions. The contact behavior is theoretically derived and isbased on a physical model, where the circular contact area is con-tinuously distributed with an infinite number of normal spring-dashpot-divider and tangential spring-dashpot-slider elements.
2.1. Normal direction
Fig. 2(a) presents the physical model in the normal directionwhich includes a linear spring, a viscous damping dashpot and adivider. The normal contact force Fn and the viscous force Fv
n arecalculated as
Fn ¼ Knun ð1Þ
Fvn ¼ �cn _un ð2aÞ
where Kn is the normal spring stiffness, un is the overlap, cn is thecoefficient of the normal viscous damping and _un is the relativeapproach velocity.
According to the motion equation of a mass-spring-dashpotsystem [53], the normal viscous damping coefficient cn can berelated to another commonly used quantity, reconstitution coeffi-cient g, as
cn ¼2ffiffiffiffiffiffiffiffiffiffiKnmp
lnðgÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln2ðgÞ þ p2
q ð2bÞ
where m is the mass of a particle.
Fig. 2. Physical models in (a) normal direction and (b) tangential direction.
M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163 149
2.2. Tangential direction
Fig. 2(b) presents the physical model in the tangential directionwhich includes a linear spring, a viscous damping dashpot and aslider. The exchanged tangential contact force Fs and the viscousforce Fv
s are calculated as
Fs Fs þ KsDus ð3Þ
Fvs ¼ �cs _us ð4Þ
where Ks is the tangential spring stiffness, Dus is the incrementalrelative tangential displacement, cs is the coefficient of the tangen-tial viscous damping and _us is the relative tangential velocity.
The Mohr–Coulomb sliding criterion is used in the tangentialdirection,
Fmaxs ¼ lFn ð5Þ
where l is the inter-particle friction coefficient.Usually, the tangential viscous damping, if used, is activated in
the elastic state of tangential interaction. When the Coulomb fric-tion limit is reached, the tangential viscous damping will be deac-tivated because sliding is enough to dissipate energy. The sameprinciple applies to rolling/twisting viscous damping discussedbelow.
2.3. Rolling direction
It is assumed that the circular contact area is continuously dis-tributed with an infinite number of normal spring-dashpot-dividerelements and tangential spring-dashpot-slider elements. These arethe basic elements (BEs) used to formulate the contact behavior inboth the rolling and twisting directions. The unit system isassigned to normal and tangential BEs as
kn ¼Kn
pR2; ks ¼
Ks
pR2; gn ¼
cn
pR2; gs ¼
cs
pR2ð6Þ
where kn (ks) (N/m3) is the BE’s normal (tangential) stiffness and gn
(gs) (Ns/m3) is the BE’s coefficient of the normal (tangential) viscousdamping.
The rolling interaction law is derived under a given normalforce Fn. A moment Mr can be exchanged at the contact due tothe relative rolling rotation hr of the two spheres. The method ofcalculating Mr in the 3D model is similar to that of the 2D rollingresistance model in [43]. Fig. 3 schematically plots the rollinginteractions. The normal stress in the circular contact area isassumed to be linearly distributed and is related to the distanceto the rolling axis r–r0 passing through the contact center, shownin Fig. 3(a) and (b). Points 1 and 2 retain the maximum and mini-mum normal stresses in the contact area, expressed as
p1 ¼ �pþ knRhr; p2 ¼ �p� knRhr ð7a;bÞ
where
�p ¼ Fn
pR2ð8Þ
By integration, the linear distribution of the normal stress leadsto the following expression for Mr:
Mr ¼knpR4
4hr ¼ Krhr ð9Þ
where Kr is the rolling stiffness, which is not a free parameter butdepends on the normal stiffness Kn and the contact radius R, i.e.,
Kr ¼knpR4
4¼ KnR2
4ð10Þ
With viscous damping at the contact, _hr will lead to a linearlydistributed viscous normal stress in the contact area, as shown inFig. 3(c), i.e.,
pvl ¼ gnR _hr; pv
2 ¼ �gnR _hr ð11a;bÞ
Similarly, the viscous rolling moment produced by damping canbe integrated as
Mvr ¼ �
gnpR4
4_hr ¼ �cr
_hr ð12Þ
where cr is the coefficient of the rolling viscous damping, defined as
cr ¼gnpR4
4¼ cnR2
4ð13Þ
Again, cr is not a free parameter but depends on the coefficientof the normal viscous damping cn and the contact radius R.
In Eq. (7), the stress p1 (p2) increases (decreases) linearly withthe increase of hr. At a critical rolling rotation hr0, the normal BEon the far right is about to separate as no tension is permitted.Mr0 denotes the critical rolling moment at the instant of the sepa-ration of the normal BE at point 2. This critical state can be definedas
pl ¼ 2�p; p2 ¼ 0 ð14a;bÞ
and therefore
Mr0 ¼FnR4; hr0 ¼
�p
knR¼ Fn
KnRð15a;bÞ
Once hr exceeds hr0, the separation of normal BEs evolves con-tinuously toward the left as hr increases. Fig. 3(d) presents the dis-tribution of the normal contact stress in this stage, which isassumed to be linear. The normal stresses become
p1 ¼ Lknhr ; p20 ¼ 0 ð16a;bÞ
where L is the remaining contact size. Note that the normal overlapneeds to decrease as the contact rolls about the rolling axis throughthe contact center, maintaining the constant normal force condi-tion. The normal and rolling interactions then are coupled, althoughthis is ignored for simplicity as it has been in previous contact mod-els. Integrating the linearly distributed normal stress illustrated inFig. 3(d) leads to Mr and Fn,
Mr ¼8C1C2 þ C4
2C1 þ C2C3� FnR
4ð17aÞ
Fn ¼2C1 þ C2C3
p� KnRhr ð17bÞ
where C1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2a� a2Þ3
q=3, a ¼ L=R, C2 = a � 1,
C3 ¼ p=2� asinð1� aÞ � ð1� aÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a� a2p
, and C4
¼ p=2� asinð1� aÞ þ ð4a� 1� 2a2Þð1� aÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a� a2p
.Substituting Eq. (15) into Eq. (17), Mr and hr can be normalized
by Mr0 and hr0 as
Mr
Mr0¼ 8C1C2 þ C4
2C1 þ C2C3ð18aÞ
Fig. 3. Rolling interactions. (a) Rolling rotation of the contact area; (b) distribution of the normal contact stress without separate BE; (c) distribution of the normal viscousstress; (d) distribution of the normal contact stress with separate BEs.
150 M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163
hr
hr0¼ p
2C1 þ C2C3ð18bÞ
Fig. 4 presents the variation of Mr/Mr0 against hr/hr0 obtainedusing the intermediate variant a (the solid curve). In the first stage,Mr/Mr0 increases linearly with hr/hr0, without BE separation. Thefollowing non-linear stage is a result of the continuous separationof BEs as hr/hr0 increases. The elasticity of the springs in the normalBEs enables them to store all of the input energy and no plasticityis assumed during unloading. Plastic behavior has been widely dis-cussed in previous rotation resistance model studies but it is rarelyfully explained. The physical origin of this plasticity modifies theMr/Mr0–hr/hr0 relationship as to be discussed below. The solid curvein Fig. 4 can be seen as a reference curve, only applicable to infi-nitely strong particles. For real sand particles under a large normalforce or with a large rolling rotation, high local compressive stresscan develop at point 1 in Fig. 3, leading to local crushing of particleasperities, and any further increase in the rolling resistance
0 1 2 3 4 50.0
0.5
1.0
1.5
2.0
2.5
h g f
e
d
cb
r
r0
Μ /
Μ
r r0θ /θ
a
2.1
2.1
Fig. 4. The relationship between Mr/Mr0 and hr/hr0.
moment with the rotation is hindered. This crushing phenomenonresults in the deviation of the Mr/Mr0–hr/hr0 relationship from thereference curve. Consequently, a proportion of the total inputenergy is stored in the deforming springs of normal BEs, whilethe local crushing dissipates the remaining energy. The normalBEs used in this paper cannot accurately capture this local crushingeffect, which may be addressed by subjecting the normal BE to abearable threshold/peak stress level, although the contact modelwill then become extremely complex. Plastic behavior is insteadintroduced, as in previous 2D and 3D rolling resistance models,by modifying the rolling contact rule to account for the local crush-ing effects. To maintain the simplicity of the DEM, an elastic–plastictype rolling resistance model using a Mohr–Coulomb-like limit cri-terion is proposed, illustrated in Fig. 4 by the dashed line segments.Here, the total input energy in the non-linear stage (area of acdfgh)of the reference curve is divided into the elastic part (area of abgh)stored in the deforming normal springs and the plastic part (area ofbcefg) dissipated by local crushing. The energy conservation princi-ple, rationally assumed to be applicable here, demands that thearea of abc is equal to the area of cde, which yields Mr/Mr0 = 2.1and hr/hr0 = 2.1 at point b, as shown in Fig. 4. This modification con-cept is similar to the 2D bond rolling resistance model in [7]. Notethat the upper integration limit of hr/hr0 is 5 because an infinite rel-ative rolling rotation cannot be reached in practice. The introduc-tion of plasticity and the integration domain of hr/hr0 aresomewhat artificial as there is no rigorous limitation to the relativerolling rotation and local compressive stress. This issue deservesfurther investigation. The result of the modification is a simplifiedrolling resistance model that retains the efficiency of the DEM sim-ulation. The main difference between the original and the modifiedmodels is found in the distribution of energy storage/dissipation ata contact which should be very complicated. The modificationcompletes the 3D rolling resistance model as
Mr ¼Krhr hr < h�r
2:1Mr0 hr � h�r
�ð19aÞ
where h�r ¼ 2:1Mr0=Kr .
M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163 151
Furthermore, a local crushing parameter fc, describing theeffects of local asperity crushing and related to the hardness ofthe particle mineral material, is introduced here to control the peakrolling resistance. As such, the simplified rolling resistance modelcan be generally rewritten as
Mr ¼Krhr hr < h�rfcMr0 hr � h�r
�ð19bÞ
where h�r ¼ fcMr0=Kr . For the simplified model in Fig. 4, fc equals to2.1. In the limiting case of infinite rolling rotation for infinitelystrong material, the normal force tends to concentrate at point 1in Fig. 3(d), leading to fc = 4.0.
2.4. Twisting direction
Fig. 5 illustrates the twisting interactions with schematic dia-grams. In the twisting direction, a torque Mt is exchanged at thecontact due to the relative twisting rotation ht of two spheres.Assume that the shear stress imposed on the contact area fromtwisting increases linearly with the distance from the contact areacenter, as shown in Fig. 5(b). Then the shear stress in the peripheryof the circular area is
s0 ¼ ksRht ð20Þ
Through integration, the radially linear distribution of the shearstress leads to the following expression of Mt,
Mt ¼kspR4
2ht ¼ Ktht ð21Þ
where Kt is the twisting stiffness, which again is not a free param-eter but depends on the tangential stiffness Ks and the contactradius R, i.e.,
Kt ¼kspR4
2¼ KsR2
2ð22Þ
Fig. 5(c) shows that the viscous shear stress resulting fromdamping is also assumed to distribute in a radially linear way,and its value at the periphery of the contact area is
Fig. 5. Twisting interactions. (a) Twisting rotation of the contact area; (b) distributionstress; (d) distribution of the contact shear stress with sliding BEs.
sv0 ¼ gsR _ht ð23Þ
Similarly, the viscous twisting torque produced by the dampingcan be integrated as
Mvt ¼ �
gspR4
2_ht ¼ �ct
_ht ð24Þ
where ct is the coefficient of the twisting viscous damping, definedas
ct ¼gspR4
2¼ csR2
2ð25Þ
Again, ct is not a free parameter but depends on the coefficientof the tangential viscous damping cs and the contact radius R.
As ht constantly increases, the sliders of BEs in the periphery ofthe circular area will begin sliding once the shear stresses reach thelimit values defined by the Mohr–Coulomb criterion. At the instantwhen the outermost BEs begin sliding, the shear stress in theperiphery of the circular area is
so ¼ l�p ð26Þ
Similar to the rolling interaction, the critical twisting torque(Mt0) and twisting rotation (ht0) can be expressed as
Mt0 ¼lFnR
2; ht0 ¼
lFn
KsRð27a;bÞ
An annulus domain retains the maximum shear stress of lpwith an inner radius of R0 when ht exceeds ht0, and enlarges fromthe periphery of the contact area inward as ht increases, illustratedin Fig. 5(d). The shear stress distribution on the contact area canthen be described as
s ¼ KsRht r � R0
l�p R0 < r � R
(ð28Þ
It integrates the twisting torque directly as
Mt ¼2lFnR
3� ðlFnÞ4
6R2ðKshtÞ�3 ð29Þ
of the contact shear stress without sliding BE; (c) distribution of the viscous shear
152 M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163
Substituting Eq. (27) into Eq. (29) yields
Mt
Mt0¼ 4
3� 1
3ht
ht0
� ��3
ð30Þ
Fig. 6 presents the relationship between Mt/Mt0 and ht/ht0 (thesolid curve). Mt/Mt0 increases linearly with ht/ht0, then the increas-ing number of sliding BEs expanding inward from the periphery ofthe contact area leads to a non-linear relationship. In this stage, aproportion of the total input energy is stored in the deformingsprings of the non-sliding tangential BEs, while the remainingenergy is dissipated by the sliders. The tangential BEs capture theplastic behavior in the twisting interaction. The defined twistinginteraction law is however only applicable to monotonic loading.When the loading–unloading–reloading process is arbitrary, thebehavior becomes extremely complex. An elastic–plastic twistingresistance model is proposed to simplify this, illustrated by thedashed lines in Fig. 6. Previous DEM simulations have used similarsimplifications in the tangential contact behavior. By integration,the energy conservation principle (i.e., the area of abc being equalto the area of cde) demands Mt/Mt0 = 1.3 and ht/ht0 = 1.3 at point bin Fig. 6. Again, the arbitrary upper integration limit of ht/ht0 = 5 isused because infinite relative twisting rotation cannot be reachedin practice. Fig. 6 indicates that the simplified model is very closeto the theoretically derived model because the limit value of Mt/Mt0 = 1.3 is very close to the theoretical limit of Mt/Mt0 = 4/3 inEq. (30). The simplified model is greatly beneficial to the DEM sim-ulation efficiency while the basic physical meaning is retained. Thesimplified and efficient 3D twisting resistance model is nowcomplete,
Mt ¼Ktht ht < h�t
0:65lFnR ht � h�t
(ð31Þ
where h�t ¼ 1:3lFn= KsR� �
.The level of microscopic physical details captured in a contact
model depends on the purpose of the simulation and the interestof the research. This paper focuses on the macroscopic behavior,such as the strength, stress–strain relationship and critical statebehavior. As will be shown below, the novel contact model pro-posed is adequate for these purposes, although some simplifica-tions are introduced. This is because these simplifications retainthe basic physical meaning.
2.5. Model implementation and parameters
Fig. 7 presents the mechanical responses of the simplified con-tact model in each direction of interaction, which will be used in
0 1 2 3 4 50.0
0.5
1.0
1.5
1.3ed
cb
t
t0
Μ /
Μ
t t0θ /θ
a
1.3
Fig. 6. The relationship between Mt/Mt0 and ht/ht0.
the following simulations. The peak resistances in the shear, rollingand twisting directions are controlled by the normal force multi-plied by l, fcR=4 and 0:65lR, respectively. There are three signifi-cant differences between our new model in Fig. 7 and the classicHertz–Mindlin–Deresiewicz contact theory. (1) Hertz–Mindlin–Deresiewicz theory models contact behavior of two perfect sphereswith only normal and tangential interactions. Our new modelincorporates, in addition to normal and tangential interactions,rolling and twisting resistances to consider particle angularity.(2) In Hertz–Mindlin–Deresiewicz theory, the contact evolves froma point to a circular area due to normal force. In our new model, thecontact area is assumed independent of contact force since theevolution of particle angularity, which determines the contact area,is not considered at present. (3) Hertz–Mindlin–Deresiewicz modelusually produces a friction angle lower than the value for real sand.As to be analyzed below, our new model can reproduce frictionangle within the realistic range.
The derived formulations for the contact behavior are presentedin Table 1 by vector notations, for the convenience of DEM codeimplementation. Fig. 8 presents the vectors used in the modelimplementation. In Fig. 8 and Table 1, n is the unit vector in thecontact normal direction, and _un, _us, _xr and _xt are the relative nor-mal velocity, tangential velocity, rolling rate and twisting rate,respectively:
_un ¼ mi � mj� �
� n�
n ð32aÞ
_us ¼ mi � mj � _un þ r0in�xi þ r0jn�xj ð32bÞ
where ms and xs are the centroid velocity and rotation rate of parti-cle s, and r0s ¼ rs � un=2 (rs is the particle radius), for s = i, j. Using theincremental method [44], the relative rotation rate can be dividedinto the rolling and twisting rates as
_xi � _xj ¼ _xr þ _xt ð33aÞ
_xt ¼ _xi � _xj� �
� n�
n ð33bÞ
Eq. (33) enables simple DEM code implementation and is suffi-ciently accurate if the time step Dt is small enough in the DEM sim-ulation [49]. The total method, using quaternion algebra proposedin [49], is an alternative to this incremental method.
The contact model is also appropriate for non-spherical parti-cles, although it was designed to study the interactions of twospheres, because it is based on the relative velocity and rotationrate at a contact, irrespective of the shape of particles formingthe contact. For non-spherical particles, the contact area shouldbe non-circular. Therefore, it becomes necessary to consider thedependency of rolling resistance on the direction of relative rollingin the contact plane.
A dimensionless shape parameter b is introduced to link thecontact radius R and the radius of the contacting spheres:
R ¼ br ð34Þ
where r is the common radius of the two spheres defined as
r ¼ 2rirj
ri þ rjð35Þ
b can be regarded as a shape parameter, used to consider the effectsof particle shape on the overall mechanical behavior of granularmaterials.
As suggested in [52], the normal and tangential contact stiff-nesses of a sphere pair can be simply represented by
Kn ¼ 2rE; Ks ¼ Kn=n ð36a;bÞ
where E represents the modulus of the particle material and n is theratio of the normal to the tangential contact stiffness.
Fig. 7. Mechanical responses of the contact model: (a) normal direction, (b) tangential direction, (c) rolling direction, and (d) twisting direction.
Table 1Summary on the derived formulations for contact behavior.
Contact response Stiffness Peak resistance Damping response Damping coefficient
Normal Fn = Knunn Kn – Fvn ¼ �cn _un cn
Tangential Fs Fs þ Ks _usDt Ks uFn Fvs ¼ �cs _us cs
Rolling Mr Mr þ Kr _xrDt Kr ¼ 0:25KnR2 fcFnR=4 Mvr ¼ �cr _xr cr ¼ 0:25R2cn
Twisting Mt Mt þ Kt _xtDt Kt ¼ 0:5KsR2 0:65lFnR Mvt ¼ �ct _xt ct ¼ 0:5R2cs
Fig. 8. Relative velocities and rotation rates at a contact.
M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163 153
In summary, there are seven free parameters (E, n, b, fc, l, cn
and cs) in the new contact model: two new parameters (shapeparameter b and local crushing parameter fc) introduced to con-trol rolling and twisting resistances and five others from the stan-dard DEM.
2.6. Main comparisons with other similar contact models
Table 2 compares our model with three previous models of bothrolling and twisting resistances [44,48,50]. The models in [48,50]were not originally proposed for DEM simulations but are includedin Table 2 to provide a comprehensive comparison. The 3D contactmodel in [49], though complete and well derived, was proposed forbonded contact instead of unbonded contact in this study, and so itis not included here. Our model has several obvious advantages. (1)It is theoretically derived from a physical scenario whereby an infi-nite number of normal and tangential basic elements are continu-ously distributed over an assumed circular contact area. This is notthe case in other models. (2) The contact size is explicitly associ-ated with particle size so that the rolling and twisting resistancesand damping moments can be theoretically derived in a rigorousmanner. The mechanical responses are simply assumed in othermodels. (3) The clear physical meaning in our model requires onlytwo extra parameters (shape parameter b and local crushingparameter fc) to be added to control rolling and twisting resis-tances. Other models add up to six parameters and their valuesare usually chosen by trial and error. The proposed contact modelcan therefore be regarded as an improvement on or a refinement ofprevious models.
3. Quasi-static simulations of the mechanical behavior of sands
This new model was implemented through the widely usedDEM commercial software PFC3D [53]. Conventional triaxial com-pression (CTC) and plane-strain compression (PSC) tests were usedto examine the ability of the model in simulating the quasi-staticbehavior of sands. The loading condition effect on the strengthand deformation behavior of sands, a standard topic in soilmechanics, was preliminarily investigated.
3.1. Simulation procedures and input parameters
Fig. 9 illustrates the particle size distribution (PSD) used in thenumerical assemblies, and Fig. 10 provides a numerical assemblyfor analyses. Each numerical assembly consisted of 10,000 spheresof 3.5 mm � 3.5 mm � 7.1 mm, confined by six frictionless bound-ary walls. The assembly was colored by layers to enable the defor-mation to be observed.
Uniform assemblies with two distinct void ratios (eini = 0.84 and0.65) were generated using the Multi-layer Under-CompactionMethod (UCM) [54]. The UCM prepares a sample by compressingparticles one-dimensionally and by five layers, which mimics thenatural layer-by-layer depositing process of sands in hydrostaticcondition. For the assembly with a target initial void ratio ofeini = 0.65, the first layer of spheres was generated randomly andone-dimensionally compressed to a void ratio of 0.73. After eachof the second, third, fourth, and fifth layer of spheres were gener-ated, the assembly was compressed to a void ratio of 0.71, 0.69,0.67, and 0.65, respectively. Similarly, the sequence of void ratiofor the assembly with a target initial void ratio of eini = 0.84 was0.88, 0.87, 0.86, 0.85, and 0.84. The void ratios in 20 sub-volumes(also called measurement spheres in PFC3D, each containing about800 particles) within an assembly were measured. The spatial dis-tribution of void ratio exhibited an error margin of 10% and thusthe assembly was confirmed to be uniform. In the UCM, b wasset at 0, and l at 0 for assemblies with eini = 0.65 and at 1.0 forassemblies with eini = 0.84. Following the UCM, the top wall wasvertically moved until a vertical stress of 50 kPa was applied. Then,alternation was applied by assigning the specific value of b to allinter-particle contacts. Six different values of b (0, 0.1, 0.2, 0.3,0.4, and 0.5) were used in the CTC tests to investigate the effectsof the shape parameter b on the mechanical behavior of sands. Aconstant friction coefficient of l = 0.5 (a typical value of quartzsand particles) was used in all tests. After alternation, each assem-
Table 2Main comparisons with three similar contact models in the literature.
[48] [44] [50] This study
Application Micromechanicsmodel
DEM Contact dynamics method DEM
New parameters to incorporaterolling and twisting resistances
kRs, kRn kr, lr, cr, ko, lo, co lr, at b, fc
Number of new parameters 2 6 2 2Physical model Not given Not given Eccentricity of normal force
on one facet of a hexagonA contact area with continuously distributed infinitenumbers of normal/tangential basic elements
Contact area Not given Not given One facet of a hexagon Circular area, radius R ¼ brRolling
Stiffness kRs kr Not given 0.25Knb2r2
Peak resistance Not given lrFnR0(R0 = R/2) lrFnr (r: radius of a sphere) fcbFnr/4Damping coefficient Not given cr Not given 0.25cnb
2r2
TwistingStiffness kRn ko Not given 0.5Ksb
2r2
Peak resistance Not given loFnR0 atlFnr 0.65lbFnrDamping coefficient Not given co Not given 0.5csb
2r2
0.10 0.15 0.20 0.25 0.30 0.350
20
40
60
80
100
Perc
enta
ge sm
alle
r tha
n by
wei
ght (
%)
Particle diameter (mm)
Fig. 9. Particle size distribution of the numerical assemblies.
Fig. 10. Numerical assembly and applied stresses for the analyses.
154 M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163
bly, with its own specific value of b, was isotropically compressedto a confining pressure of 50 kPa, 100 kPa and 200 kPa. This wasfollowed in the CTC tests by axial loading, and the confining pres-sures (r2 = r3) remained constant. Every assembly was thereforetested with almost the same initial void ratio, fabric, and stressstate. Any variation in the mechanical behavior was solely theresult of the different values of b. The PSC tests were carried outwith b = 0.2 and l = 0.5, to investigate the effect of the loading con-dition on shear strength and the deformation of sands. After isotro-pic compression, plane-strain compression was applied by axial
loading in the z-direction, while fixing the two opposite boundarywalls in the y-direction and keeping the confining pressure in thex-direction (r3) constant.
Other contact model parameters were E = 100 MPa and n = 1.5. fc
was always equal to 2.1 (according to Fig. 4) unless stated other-wise. Note that viscous contact damping and local damping (PFC3Dmanual [53] has details on this form of damping) have no significanteffects in quasi-static behavior. Since local damping have severaladvantages identified in PFC3D [53], this study therefore used thelocal damping. A critical damping coefficient of 0.7 was chosen bytrial and error so that the particulate system is properly-damped,i.e., neither over-damped nor under-damped. The viscous contactdamping was turned off as cn = cs = 0. Viscous contact dampingcan be the proper damping in future simulations of dynamic prob-lems. The particle density was amplified to 2.655 � 1013 kg/m3 tofacilitate quasi-static test conditions. The axial loading rate was0.1%/min with a time step of 1.32 � 10�2 s.
3.2. Simulation results
3.2.1. Stress–strain relationship, void ratio and coordination numberFig. 11 presents the obtained stress–strain relationships and the
variations in the void ratio and mechanical coordination number(CN) in the CTC tests when b = 0, 0.2, and 0.5. The motion of theboundary walls and the forces acting on them were measured tocalculate the strains and stresses. The deviator stress was definedas r1 � r3. CN was calculated after [5], where rattler particles withzero or one contact were excluded.
In Fig. 11(a), when b = 0, the assemblies with eini = 0.65 exhibitstrain-softening and volumetric dilation behavior, and CN firstincreases slightly and then decreases abruptly to nearly constantvalues at large strain. The assemblies with eini = 0.84 show strain-hardening and volumetric contraction behavior, and CN firstincreases slightly and then tends to be constant at large strain. Acritical state is reached at an axial strain of approximately 35%,which is characterized by the same critical deviator stress, voidratio, and CN under each confining pressure, regardless of the ini-tial void ratio. These are typical mechanical responses of dense andloose sands.
Fig. 11(b) and (c) show that all assemblies exhibit strain-soften-ing and dilation behavior in the presence of rolling and twistingresistances. Under the same initial void ratio and confining pres-sure, both the peak deviator stress and the corresponding peak-state axial strain increase with b, indicating that more effort isrequired to collapse the force network with a larger b. The criti-cal-state deviator stress and void ratio also increase, but the criti-cal-state CN decreases as b increases, implying that the greater
0
100
200
300
400
500eini= 0.65 50kPa
100kPa200kPa
50kPa100kPa200kPa
Dev
iato
r stre
ss (k
Pa)
Axial strain (%)
eini= 0.84
0.0
0.1
0.2
0.3
0.4
0.5
(σ1−σ
3) /
(σ1+σ
3 )
Axial strain (%)
0.60
0.65
0.70
0.75
0.80
Voi
d ra
tio
Axial strain (%)
4.4
4.8
5.2
5.6
6.0
6.4
CN
Axial strain (%)(a)
0
200
400
600
800eini= 0.65 50kPa
100kPa200kPa
50kPa100kPa200kPa
Dev
iato
r stre
ss (k
Pa)
Axial strain (%)
eini= 0.84
0.00.10.20.30.40.50.60.7
(σ1− σ
3) /
( σ1+ σ
3 )
Axial strain (%)
0.60
0.65
0.70
0.75
0.80
0.85
0.90
Voi
d ra
tio
Axial strain (%)
3.64.04.44.85.25.66.06.4
CN
Axial strain (%)(b)
0
200
400
600
800
1000
1200eini= 0.65 50kPa
100kPa200kPa
50kPa100kPa200kPa
Dev
iato
r stre
ss (k
Pa)
Axial strain (%)
eini= 0.84
0.00.10.20.30.40.50.60.70.8
(σ1−σ
3) /
(σ1+σ
3 )
Axial strain (%)
0.600.650.700.750.800.850.900.95
Voi
d ra
tio
Axial strain (%)
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 353.23.64.04.44.85.25.66.06.4
CN
Axial strain (%)(c)
Fig. 11. Evolutions of deviator stress, mobilized internal friction angle, void ratio and CN versus axial strain for assemblies with different values of b in the CTC tests: (a) b = 0,(b) b = 0.2 and (c) b = 0.5.
M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163 155
rotational resistance at contact caused by the larger b can improvethe robustness of the force network, and is capable of maintainingoverall stability at an even looser state. Fig. 11 also presents themobilized internal friction angle defined as sin um
¼ r1 � r3ð Þ= r1 þ r3ð Þ. It is found that all curves with the samevalue of b finally tend to converge to the same critical state regard-less of the initial void ratio and the confining pressure.
When b increases up to 0.5, the void ratio at critical sate can beas high as 0.92, which seems unrealistically high but actually canbe observed in high-quality tests. ecr of Toyoura sand, for instance,is between 0.92 and 0.85 when the mean stress changes between 0and 500 kPa [58]. The effect of b on strength and critical state willbe discussed below by comparing it with experimental results.
Comparisons of the mechanical responses obtained in the PSCand CTC tests are shown in Fig. 12, where b = 0.2 and l = 0.5. Underthe same confining pressure, an assembly in the PSC test exhibitsgreater deviator stress and a larger strain-softening rate than inthe CTC test. The assemblies in the PSC tests reached a residualstate at an axial strain of approximately 10%, much earlier thanthe 35% for critical state in CTC tests. These observations are con-sistent with previous experimental results [55–57], confirmingthat the new contact model can adequately capture the mechanicalbehavior of sands in a quasi-static situation. Fig. 12 also presentsthe mobilized internal friction angle defined as sin um
¼ r1 � r3ð Þ= r1 þ r3ð Þ. All curves in PSC tests with the same value
of b finally tend to converge to the same residual state (0.45)regardless of the initial void ratio and the confining pressure.
It was found that the samples in PSC tests did not reach the crit-ical state over the whole sample because of localization. Fig. 13presents three typical assemblies in the CTC and PSC tests. All ofthe assemblies in the CTC tests deform uniformly with no identifi-able shear band. Obvious localized deformation, however, occurs inall of the PSC tests. The overall void ratio (calculated over thewhole sample) for PSC test in Fig. 12 is therefore lower than thereal critical value. Fig. 12 also presents the void ratio within theshear band, which is calculated locally using ‘‘measurementspheres’’ each containing about 500 particles. The void ratio withinthe shear band starts to deviate from the overall void ratio after thepeak state as the shear localization becomes important. It seemsthat a critical state void ratio is reached within the shear band,which is close to the value in axisymmetric triaxial compressiontest. Further study in this issue is needed using flexible boundariesso that the shear band can fully develop.
3.2.2. Strength indexesFig. 14 presents the peak and residual strength indexes
obtained in numerical simulations. In the CTC tests, both the peakand residual friction angles of the two groups of assemblies withdistinct initial void ratios increase with the shape parameter b.The peak friction angle ranges between 20� and 46� and the resid-
0 5 10 15 20 25 30 350
200
400
600
800
1000TC
50kPa100kPa200kPa
Dev
iato
r stre
ss (k
Pa)
Axial strain (%)
PSCσ
3=
0 5 10 15 20 25 30 350.00.10.20.30.40.50.60.70.8
(σ1−σ
3) /
(σ1+σ
3 )
Axial strain (%)
0.60
0.65
0.70
0.75
0.80
0.85
0.90
Voi
d ra
tio
Axial strain (%)
void ratio in the shear band (plane strain test)
overall void ratio
3.6
4.0
4.4
4.8
5.2
5.6
6.0
6.4
CN
Axial strain (%)(a)
0
200
400
600
800
Dev
iato
r stre
ss (k
Pa)
Axial strain (%)
TC50kPa100kPa200kPa
PSCσ
3=
0.00.10.20.30.40.50.60.70.8
(σ1−σ
3) /
(σ1+σ
3 )
Axial strain (%)
0.72
0.76
0.80
0.84
0.88
void ratio in the shear band (plane strain test)
Voi
d ra
tio
Axial strain (%)
overall void ratio
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 353.6
4.0
4.4
4.8
5.2
CN
Axial strain (%)(b) Fig. 12. Evolutions of deviator stress, mobilized internal friction angle, void ratio and CN versus axial strain in the PSC and CTC tests with b = 0.2: (a) assemblies witheini = 0.65 and (b) assemblies with eini = 0.84.
156 M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163
ual friction angle is between 20� and 34�, which is within the typ-ical range of sand strength (30–40�) observed in laboratory tests.Fig. 14 also shows that the peak and residual friction anglesobtained in the PSC tests are larger than the values obtained inCTC tests with the same shape parameter and initial void ratio.If, as in many previous DEM simulations, the rolling and twistingresistances are not considered (i.e. b = 0), the obtained strengthwould be apparently lower than the experimental results. It istherefore necessary to include rolling and twisting resistances toobtain satisfactory results in DEM simulations.
The overall macroscopic strength of an assembly is retained bymicroscopic sliding, rolling and twisting resistances at contacts.Their relative magnitudes determine the preference of particle-level rearrangement and the overall strength. Under fixed slidingresistance (fixed inter-particle friction coefficient), with the
increase of b, particles prefer to roll and twist as long as the slidingresistance is greater than the rolling (twisting) resistance. In thiscase, the strength increases with b. As the rolling (twisting) resis-tance approaches and finally exceeds the sliding resistance, contactsliding starts to prevail. As a result, the strength reaches a plateauand stops to increase with b. Similarly, if the rolling resistance (i.e.b) is fixed, with the increase of inter-particle friction, the overallstrength will first increase, then reach a plateau and finally stopto increase [59].
The difference in the mechanical behavior obtained from theCTC and PSC tests is because of the distinct loading conditions. Inthe PSC test, the intermediate principal stress (r2) changes withthe loading process, so the y-axis strain remains at zero throughoutthe test. Microscopically, the PSC test leads to a different inter-par-ticle contact failure configuration from that in the CTC test. Fig. 15
Fig. 13. Deformed assemblies at the steady state with b = 0.2 and r3 = 50 kPa: (a) assembly with eini = 0.65 in the CTC test, (b) assemblies with eini = 0.84 in the PSC test, and(c) assembly with eini = 0.65 in the PSC test.
0.0 0.1 0.2 0.3 0.4 0.515
20
25
30
35
40
45
50
CTC PSC Peak, e
ini= 0.65
Peak, eini
= 0.84 ResidualPe
ak a
nd re
sidu
al fr
ictio
n an
gle
(°)
Shape parameter
Range of friction angle for sands
Fig. 14. Peak and residual friction angles obtained in DEM simulations.
M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163 157
illustrates the different distributions of the contact normal vectorsprojected on the x-z and x-y planes in two assemblies witheini = 0.65 in the CTC and PSC tests at the peak state (b = 0.2 andr3 = 50 kPa). The contact percentage in each directional interval(5�) is calculated by Ni/Nc, where Nc is the total contact numberin all directions and Ni is the total contact number, or the slidingcontact number (i.e. Fs/(lFn) > 0.99), or the rolling contact number(i.e. Mr/(fcbFnr/4) > 0.99), or the twisting contact number (i.e. Mt/(0.65lbFnr) > 0.99) in each directional interval. These statisticsare applied to all contacts with a distance to the boundary wallsgreater than 2dmax (dmax is the maximum particle diameter), elim-inating possible boundary effects. Fig. 15 shows that the contactdistribution in the CTC test is cross-anisotropic and in the PSC testis orthotropic-symmetric. The percentage of sliding (or rolling ortwisting) contact in the PSC test is apparently smaller than in theCTC test, as the strong out-of-plane confinement in the PSC testcan effectively hinder the collapse of the force network. Conse-quently, greater strength is observed in the PSC than in the CTCtests.
The macroscopic plastic deformation of an assembly is due tomicroscopic sliding, rolling/twisting at contacts, which are basicmodes of micro-scale particle rearrangement. The proportion ofeach mode reflects the preference of particle-level deformation. Itis interesting to note that the percentage of rolling contact is largerthan that of sliding or twisting contact in the CTC test; however,the percentages of sliding, rolling and twisting contacts in thePSC test are very close. Fig. 15(a) implies that rolling rearrange-ment may be the preferred mode in the CTC tests possibly becausethe overall rolling resistance is less that the overall frictional resis-tance. In the PSC tests, it is inferred from Fig. 15(b) that rollingresistance is as strong as frictional and twisting resistances,
particles therefore tend to slide, roll and twist at contacts. The sig-nificance of sliding, rolling and twisting contacts in controlling themicrostructure and thus determining the deformation/strength ofgranular material will be of interest in further research.
3.2.3. Critical stateFig. 16 presents the loading loci of assemblies in the e–lgp plane
(p = (r1 + r2 + r3)/3 is the mean stress) obtained in CTC tests. Thesolid squares in Fig. 16 represent the states after isotropic com-pression, and the hollow circles represent the steady states. InFig. 16, assemblies with distinct initial void ratios, under the sameconfining pressure and shape parameter, are shown to ultimatelyreach the same critical state. The critical state points obtainedusing the same shape parameter b but with different confiningpressures are almost on the same critical state line (CSL). The fittedCSLs, by equation ecs ¼ C� k lg p=1 kPað Þ, are superimposed on theplots, where ecs is the void ratio at the critical state,p = (r1 + r2 + r3)/3 is the mean stress, while C and k are two fittingparameters. When b increases from 0 to 0.5, C increases from0.762 to 1.066, and k increases from 0.014 to 0.068.
3.2.4. Effect of local crushingThe local crushing parameter fc describes the resistance of local
asperity to crushing. Under the same shape parameter b, it is obvi-ous that the strength of a particle assembly increases with fc. Notethat with proper combination of b and fc, similar macroscopicmechanical behavior can be observed, as shown in Fig. 17. How-ever, the corresponding microscopic contact behavior can begreatly different. Therefore, when calibrating contact modelparameters in DEM simulations, it is crucial to identify the two dis-tinct microscopic mechanisms: particle shape (contact size) effectdescribed by b and local resistance to asperity crushing describedby fc.
3.3. Comparisons with experimental results
Because particles of real sand are not spherical, the void ratiosfrom experiments and numerical simulations are not directly com-parable. The intention is not to compare the DEM simulationresults with specific experimental data on sand, in terms of thepeak stress ratio, dilatancy rate, and stress–strain curves, as theseare void ratio-dependent. The ability of this contact model to cap-ture the general mechanical behavior of sands and to equate theshape parameter b with a statistical measure of real particle shapeis examined instead. The three critical state parameters (i.e., criti-cal state friction angle ucs, C and k) obtained from the DEM simu-lations with different shape parameters are compared with those
0.00.51.01.52.02.5
0
45
90
135
180
225
270
315
0.00.51.01.52.02.5 x
z
rolling
slidingtwisting
Con
tact
per
cent
age
(%)
total
0.0
0.5
1.0
1.5
2.0
2.50
45
90
135
180
225
270
315
0.0
0.5
1.0
1.5
2.0
2.5
y
x
rollingsliding
twisting
Con
tact
per
cent
age
(%) total
(a)
0.0
0.5
1.01.5
2.0
2.50
45
90
135
180
225
270
315
0.0
0.5
1.0
1.5
2.0
2.5 x
z
rolling
sliding
twisting
Con
tact
per
cent
age
(%) total
0.0
0.5
1.0
1.5
2.0
2.50
45
90
135
180
225
270
315
0.0
0.5
1.0
1.5
2.0
2.5
total
y
x
rolling slidingtwisting
Con
tact
per
cent
age
(%)
(b)
Fig. 15. Distributions of the contact normal vectors projected on the x–z and x–y planes in assembly with eini = 0.65 (at peak state) with b = 0.2 and r3 = 50 kPa in (a) CTC testand (b) PSC test.
100.6
0.7
0.8
0.9
1.0Γ = 0.762 λ = 0.014
Voi
d ra
tio e
Mean stress p (kPa)
β = 0
100.6
0.7
0.8
0.9
1.0Γ = 0.900 λ = 0.041
Voi
d ra
tio e
Mean stress p (kPa)
β = 0.1
100.6
0.7
0.8
0.9
1.0Γ = 0.963 λ = 0.049
Voi
d ra
tio e
Mean stress p (kPa)
β = 0.2
100.6
0.7
0.8
0.9
1.0Γ = 1.015 λ = 0.058
Voi
d ra
tio e
Mean stress p (kPa)
β = 0.3
100.6
0.7
0.8
0.9
1.0Γ = 1.045 λ = 0.064
Voi
d ra
tio e
Mean stress p (kPa)
β = 0.4
10
100 1000 100 1000 100 1000
100 1000 100 1000 100 10000.6
0.7
0.8
0.9
1.0Γ = 1.066 λ = 0.068
Voi
d ra
tio e
Mean stress p (kPa)
β = 0.5
Fig. 16. Loading loci of assemblies with different values of b (j: after isotropic compression, s: steady state).
158 M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163
values obtained from experiments on sands with various particleshapes.
For real sand particles, several dimensionless measures of parti-cle shape are defined in the literature, such as sphericity Sp, round-ness Ro, and regularity q [2]. Regularity q is a ‘‘comprehensive’’measure of the deviation of particle shape from spherical, and isdefined as q = (Sp + Ro)/2 after [2]. The smaller the q value, thegreater the deviation. Fig. 18(a) and (c) present variations of ucs,C and C � 2k regarding b in DEM simulations or 1 � q in experi-ments. The experimental data are adapted from [2], where the reg-
ularity q and critical state parameters (ucs, C and k) of many sandswere assessed using drained or undrained triaxial tests, or the‘‘simple CS tests’’ [60]. Given the small Cu (=1.3) of the assembliesused in our DEM simulations, only experimental results on sandswith a small Cu (<2.5) are plotted in Fig. 18. C � 2k is plottedinstead of k as it has a better correlation with 1 � q, as reportedin [2]. Fig. 18(a) and (c) show that the values of ucs, C andC � 2k obtained in the DEM simulations are within the typicalranges observed in experiments. These three parameters generallyincrease with b in DEM simulations, or with 1 � q in experiments.
0
100
200
300
400
500 β = 0.30, ζc = 1.0
β = 0.20, ζc = 2.1
β = 0.18, ζc = 4.0
Dev
iato
r stre
ss (k
Pa)
Axial strain (%)0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
0.60
0.65
0.70
0.75
0.80
0.85
0.90
Voi
d ra
tio
Axial strain (%)
Fig. 17. Evolutions of deviator stress and void ratio versus axial strain for assemblies with different combinations of b and fc (eini = 0.65, r3 = 100 kPa).
(a)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.710
20
30
40
50
Experimental dataDEM simulation results
ϕ cs (°
)
(1−ρ ) or β
DEM result fittingϕcs= 26.6β +21.2
(b)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.4
0.6
0.8
1.0
1.2
DEM result fittingΓ = 0.57β +0.82
(1−ρ ) or β
Experimental dataDEM simulation results
Γ
(c)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.4
0.6
0.8
1.0
1.2DEM result fittingΓ −2λ = 0.37β +0.77
(1−ρ ) or β
Experimental dataDEM simulation results
Γ −
2 λ
(d)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Γ -based(Γ−2λ)-basedϕcs-based
β =1−ρ
1−ρ
β
Fig. 18. Effects of particle shape on critical state parameters (experimental data adapted from [2]).
M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163 159
These comparisons confirm that the new contact model can accu-rately capture the mechanical behavior of sands in quasi-static sit-uations at a large strain range (i.e. the critical state).
Note that b controls the rolling and twisting resistances at acontact in the DEM contact model, while 1 � q represents thedegree of deviation of the particle shape from spherical, reflectingthe rotational frustration of a particle. Therefore, a correlationbetween b and 1 � q is expected. For each kind of sand, a represen-tative value of b producing the same ucs in the DEM simulations asin experiments can be determined by the linear fitting function ofthe DEM results (i.e. ucs = 26.6b + 21.2), shown in Fig. 18(a). Thestatistically obtained 1 � q of each type of sand now correspondsto a ucs-based contact model parameter b in the DEM simulations.The C-based and (C � 2k)-based relationships between 1 � q andb can be similarly established. Fig. 18(d) illustrates the obtainedpairs of (1 � q, b) where b generally increases with 1 � q. Largescatters from C-based and (C � 2k)-based correlations are how-ever observed in the dashed zone, which is possibly due to the dif-ference in the PSD between the DEM simulations and the
experiments. Because of the great variability of particle shapesand size distributions, Fig. 18(d) only provides a rough correlationbetween b and 1 � q, i.e., b = 1 � q (see the imposed line‘‘b = 1 � q’’ in Fig. 18(d)). Cho et al. [2] outlined a method to deter-mine the representative value of 1 � q for a particular sand. Intheir method, visual assessment with a handheld magnifying lensis sufficient to determine particle geometry (sphericity Sp, round-ness Ro, and regularity q) systematically in a standard geotechnicallaboratory. Therefore, it is practical to first obtain 1 � q experi-mentally, which then can be used as the first trial in parameter cal-ibration for DEM contact model. This is particularly useful in a DEMassembly with a large number of particles by reducing the timerequired to narrow the range of model parameters.
3.4. Comparisons with previous DEM simulations
The details and parameters in different DEM simulation meth-ods and contact models can be very different, which makes quan-titative comparison difficult. Therefore, the simulation results in
160 M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163
this study will be qualitatively compared with the numericalresults in the literature. Two typical types of DEM modeling to cap-ture particle shape effects will be compared with our new model.One is a sphere-based rolling resistance model in [45,51], whichuses spherical particles with rolling resistance at contact, the sametype of modeling technique as in our study. The other one in [22]uses particle clumps to represent complex particle shapes.
3.4.1. Effect of twisting resistanceDifferent from our new model, the contact model in [45,51]
only considers rolling resistance but ignores the twisting resis-tance. To investigate the effect of twisting resistance, Fig. 19 pre-sents the CTC test results obtained with (1) only rollingresistance, (2) only twisting resistance, and (3) both rolling andtwisting resistances. Sliding resistance in the tangential directionis always present in all the three cases. Fig. 19 shows that theabsence of either rolling resistance or twisting resistance leads tolower strength and less dilatancy when compared with the resultsobtained from a complete contact model incorporating both rollingand twisting resistances. Therefore, the contact model in [45,51] isnot complete and it is necessary to incorporate complete contactbehavior including normal, tangential, rolling and twisting resis-tances as in the model presented in this paper. Fig. 19 also showsthat the strength and dilatancy reduce more when the rolling resis-tance is removed than when the twisting resistance is ignored. Thisimplies that rolling resistance has more significant effect thantwisting resistance.
In the new model proposed here, the stiffness ratio in the rollingand twisting directions is Kr/Kt = 0.5n = 0.75 when n = 1.5. To probethe effect of twisting stiffness, the results with Kr ¼ Kt ¼ 0:25KnR2
(see Table 1) are presented in Fig. 19, which show that slight mod-ification of twisting stiffness brings negligible change to themechanical behavior.
3.4.2. Effect of particle angularityThis study and many previous works (e.g. [22,45,51]) contribute
to capturing the effects of particle shapes in DEM simulations. It iswell known that assembly with more angular particles physicallyshould exhibit higher peak and residual strengths and larger criti-cal state void ratio [2]. This will be the criterion to assess the per-formance of each model compared.
In [22], particle clumps were used to represent complex particleshapes, with which the major mechanical behavior of granularmaterial was successfully reproduced. In [45,51], a sphere-basedcontact model that only considers rolling resistance but ignorestwisting resistance was used. This model was largely based onthe model initially proposed in [41] for 2D simulations. From thesimulation results in [51], the peak strength increases with rollresistance, which is physically realistic; however, the residualstrength and critical state void ratio are nearly independent of roll-
0 5 10 15 20 25 30 350
100
200
300
400
500
600r tK K≠rolling and twisting resistances( )
rolling and twisting resistances( )only rolling resistanceonly twisting resistance
Dev
iato
r stre
ss (k
Pa)
Axial strain (%)
r t=K K
Fig. 19. Evolutions of deviator stress and void ratio versus axial strain obtained with onlyresistances (b = 0.3, eini = 0.65, r3 = 100 kPa. Sliding resistance in the tangential direction
ing resistance, which is not physically realistic. The contact modelin [45,51] therefore cannot completely reproduce the effects ofparticle angularity on the mechanical behavior of granular mate-rial, although it was claimed to be able to match experimentalresults well.
In our simulations, Fig. 11 shows that the increase of shapeparameter b leads to increase in peak and residual strengths, dilat-ancy rate and critical state void ratio, which is physically realistic.Therefore, in comparison with the rolling resistance model in[45,51], our new model represents a higher level of theoreticalcompleteness and rigor and can capture the effect of particle shapemuch better.
3.4.3. Energy storage and dissipationIt is important to track the evolutions of energy stored in elastic
deformation, energy dissipated in the plastic state at contacts andenergy dissipated by the local damping.
The elastic energy Ee is summed over all Nc contacts,
Ee ¼X
Nc
Fc2
2Kð37Þ
where Fc is the contact force or moment and K is the correspondingcontact stiffness.
The plastic dissipation Dp is summed over Ncp contacts that areat plastic state,
Dp ¼XNcp
FcDdp ð38Þ
where Ddp is the plastic displacement or rotation in the direction ofFc.
The local damping dissipation Dd is summed over all Np
particles,
Dd ¼XNp
FdDsp ð39Þ
where Fd is the local damping force or moment applied on a particleand Dsp is the translation or rotation of a particle in the direction ofFd.
The boundary work is summed over the six boundary walls,
W ¼X6
1
FwDuw ð40Þ
where Fw is the wall displacement and Duw is the correspondingwall displacement.
All the types of energy were calculated in each step and accu-mulated stepwise (except for the elastic energy). Because of thequasi-static condition, it is found that kinematic energy can beignored. The conservation of energy was checked in each stepand was satisfied throughout the simulation.
0 5 10 15 20 25 30 350.60
0.65
0.70
0.75
0.80
0.85
0.90
Voi
d ra
tio
Axial strain (%)
rolling resistance, or with only twisting resistance, or with both rolling and twistingis always present in all the three cases).
(a)
0
1
2
3 Normal interaction Tangential interaction Rolling interaction Twisting interaction
Elas
tic e
nerg
y (k
J⋅ m-3)
Axial strain (%) (b)
0
10
20
30
40
50
60Local damping dissipationTangential interactionRolling interactionTwisting interaction
Dis
sipa
ted
ener
gy (k
J⋅m-3)
Axial strain (%)
(c)
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 350
20
40
60
80
100
Total elastic energy change
Tangential dissipation
Axial strain (%)
Perc
enta
ge o
f ene
rgy
(%)
Rollingdissipation
Local damping dissipation
Twistingdissipation
Fig. 20. Evolutions of elastic energy and dissipated energy per unit volume versus axial strain (b = 0.3, eini = 0.65, r3 = 100 kPa): (a) elastic energy density, (b) dissipatedenergy density, and (c) percentage of each type of energy.
M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163 161
Fig. 20 presents the evolutions of different types of energy withthe axial strain. The energy is divided by the volume of the wholeassembly. Fig. 20(c) demonstrates the percentage of each type ofenergy in the boundary work and the total elastic energy changeis the change relative to the initial state before applying axialstrain. Fig. 20(a) shows that each component of the elastic internalenergy first increases with axial strain to a peak, then decreases ina softening stage, and finally reaches a constant value. The peakstate occurs around an axial strain of 5%, which is larger than anaxial strain of 3% corresponding to the peak strength in Fig. 19.The normal interaction stores the most elastic energy. Fig. 20(b)shows that the dissipated energy increases almost linearly withthe axial strain. The tangential interaction dissipates the mostenergy while the local damping dissipates the least. Fig. 20(c)shows that, at the very beginning of shearing, the boundary workis mainly used to increase elastic energy. With the increase of axialstrain, the dissipated energy starts to prevail. At the critical state,56% of the boundary work is dissipated by tangential interaction,35% by rolling interaction, 7% by twisting interaction, and theremaining is stored as elastic energy.
Note that most of these observations are similar to those in [22],where particle clumps were used to represent complex particleshapes. Again, this indicates that the new contact model proposedin this paper is capable of capturing the major behavior of granularmaterials.
Although simple particle shape (sphere) is used, the new con-tact model can effectively capture the effects of particle angularityon both peak and residual strengths and the critical state voidratio. The new model can also reproduce the energy storage anddissipation. The mechanical behavior which was produced withcomplex particle shapes before can now be easily reproduced withour new model incorporating both rolling and twisting resistances.Therefore, the proposed contact model can therefore be regardedas an improvement on or a refinement of previous models.
4. Conclusions
This paper presents a complete 3D contact model, incorporatingrolling and twisting resistances together with normal/tangentialinteraction. On a microscopic level, the main advantages of ourmodel are as follows. (1) Unlike previous DEM analyses, an areacontact instead of a point contact was assumed between particles,and the contact behavior was theoretically derived in a clear phys-ical scenario, in which the contact area was continuously distrib-uted with an infinite number of normal spring-dashpot-dividerand tangential spring-dashpot-slider elements. (2) Unlike previouscontact models, the interaction resulting from relative particlerotation was divided into rolling and twisting components. (3)The clear physical meaning allows complete rolling and twistingresistances to be incorporated in the new model using only twoadditional parameters (shape parameter b and local crushingparameter fc), while other models require up to 6 parameters(see Table 2). (4) The artificial shape parameter b achieves statisti-cal meaning in our model, as it is linked to a statistically obtainedmeasure of particle shape. The proposed contact model can there-fore be regarded as an improvement on or a refinement of previousmodels. Although simplified to some extent, the model maintainsphysical meaning and can be implemented in sphere or non-spherebased DEM code for highly efficient simulation.
The ability of the model to capture the quasi-static behavior ofsands was examined by implementing it in PFC3D, simulating con-ventional triaxial and plane-strain compression tests. Consistentwith previous studies, it reproduced several forms of mechanicalbehavior typical of sands: (1) our model can increase the peak(residual) strength to the typical value of real sand if the value ofb is properly selected, unlike the low strength sands obtained inprevious DEM simulations; (2) the strong out-of-plane confine-ment in the plane-strain compression tests increases the strengthof sands, leading to a high strain-softening rate; (3) conventional
162 M. Jiang et al. / Computers and Geotechnics 65 (2015) 147–163
triaxial tests identified a unique critical state in uniform deforma-tion, where the intercept and slope of the critical state line in e–lgpplot increased with b. The correlation of b to a statistical measureof real particle shape means the DEM can be regarded as an effec-tive tool in investigating the action of particle shape on themechanical behavior of sands.
Strain localization of granulates will be further analyzed by the3D DEM incorporating this new model, where particle rotation isbelieved to be critical [8,41,42,46]. This new model will be usefulin studying other fundamental theories of soil mechanics, such asanisotropy behaviors [23,37] and scale-crossing constitutive mod-els. The contact model can be further improved for problematicgeomaterials, by adding capillary force to investigate unsaturatedsands [6], and adding van der Waals force to investigate lunar soils[9], for example.
Acknowledgements
The funding of the research provided by the China NationalFunds for Distinguished Young Scientists, with Grant No.51025932, and by the National Natural Science Foundation ofChina, with Grant Nos. 51179128 and 51379180, is sincerelyappreciated.
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