A discrete particle model andnumerical modeling of the failure

modes of granular materialsXikui Li and Xihua Chu

The State key Laboratory for Structural Analysis of Industrial Equipment,Dalian University of Technology, Dalian, Republic of China, and

Y.T. FengCivil & Computational Engineering Centre, School of Engineering,

University of Wales Swansea, Swansea, UK

Abstract

Purpose – To present a discrete particle model for granular materials.

Design/methodology/approach – Starting with kinematical analysis of relative movements of twotypical circular grains with different radii in contact, both the relative rolling and the relative slidingmotion measurements at contact, including translational and angular velocities (displacements) aredefined. Both the rolling and sliding friction tangential forces, and the rolling friction resistancemoment, which are constitutively related to corresponding relative motion measurements defined, areformulated and integrated into the framework of dynamic model of the discrete element method.

Findings – Numerical results demonstrate that the importance of rolling friction resistance,including both rolling friction tangential force and rolling friction resistance moment, in correctsimulations of physical behavior in particulate systems; and the capability of the proposed model insimulating the different types of failure modes, such as the landslide (shear bands), the compressioncracking and the mud avalanching, in granular materials.

Research limitations/implications – Each grain in the particulate system under consideration isassumed to be rigid and circular. Do not account for the effects of plastic deformation at the contact points.

Practical implications – To model the failure phenomena of granular materials in geo-mechanicsand geo-technical engineering problems; and to be a component model in a combineddiscrete-continuum macroscopic approach or a two-scale discrete-continuum micro- macro-scopicapproach to granular media.

Originality/value – This paper develops a new discrete particle model to describe granular media inseveral branches of engineering such as soil mechanics, power technologies or sintering processes.

Keywords Particle physics, Mathematical modelling, Numerical analysis,Physical properties of materials

Paper type Research paper

1. IntroductionThe discrete element method (DEM) combined with the use of different discrete particlemodels has become an increasing popular approach for studying the behavior of

The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at

www.emeraldinsight.com/researchregister www.emeraldinsight.com/0264-4401.htm

The authors are pleased to acknowledge the support of this work by the National Key BasicResearch and Development Program (973 Program) through contract number 2002CB412709 andthe National Natural Science Foundation of China through contract/grant numbers 50278012,10272027.

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Received July 2004Revised April 2005Accepted April 2005

Engineering Computations:International Journal forComputer-Aided Engineering andSoftwareVol. 22 No. 8, 2005pp. 894-920q Emerald Group Publishing Limited0264-4401DOI 10.1108/02644400510626479

granular materials such as sand and clay, which consist of packed assemblages ofparticles with voids at the microscopic level. In addition, DEM, as a numericaltechnique, can be used to simulate the flow phenomena of granular materials such asmud avalanches, while the continuum model, that is efficient to numerically simulatemacroscopic motion of granular assembly as a whole, fails to do so. Continuum modelsperform exceptionally well at a laboratory level but experience severe difficulties atindustrial scale due to complex geometric configurations and time scales. For a typicalLagrangian analysis mesh distortion is near intractable and there are a unique set ofchallenging problems regarding the evolution of damage and subsequent elementdeactivation. The DEM approach is based on modeling of the interaction betweenindividual grains. Therefore it is crucial to correctly present quantitative descriptionsof constitutive relations between interacting (normal and frictional) forces and relativemotions of two particles in contact.

There exist many different models that describe the normal and tangential contactforces in particulate systems. Among them are the pioneering work of Cundall andStrack (1979), with subsequent significant publications including the book of Oda andIwashita (1999), the models introduced and discussed in Elperin and Golshtein (1997),the papers of Iwashita and Oda (1998), Zhou et al. (1999), Zhang and Whiten (1999) andFeng et al. (2002). Thornton (1991), Vu-Quoc and Zhang (1999) and Vu-Quoc et al. (2001)discussed the mechanisms governing the generation and evolution of contactingforces, particularly the quantitative relations of the reduction in the contact radius tothe contacting forces. Vu-Quoc and Zhang (1999) and Vu-Quoc et al. (2001) furtherdeveloped their models for normal and tangential force – displacement relations forcontacting spherical particles, accounting for the effects of plastic deformation at thecontact points on variation of the radii of the particles at the contacting points due toplastic flow.

Oda et al. (1982) and Bardet (1994) observed significant effects of particle rolling onthe shear strength and consequently on the occurrence and evolution of shear bands inparticulate system. Iwashita and Oda (1998) developed a modified distinct elementmodel on the basis of the classical DEM proposed by Cundall by introducing a rollingfriction resistance moment at each contact as an additional component mechanism fortaking into account the effects of particle rolling, but the model did not distinguishbetween the rolling and the sliding frictional tangential forces at contact, which shouldbe constitutively related to the tangential components of the relative rolling and therelative sliding motion measurements respectively at the contact.

Feng et al. (2002) pointed out that although the study for developing a modelcapable of capturing the nature of the friction between two typical grains in contactand therefore capable of simulating the friction properly has been in progress for manyyears, a universally accepted (rolling) friction model has still not been achieved. Theyalso discussed and stressed the importance of rolling friction resistance in the correctsimulation of physical behaviors in particulate systems and took into account both therolling and the sliding friction forces in their model, but not constitutively related tocorresponding rolling and sliding motion quantities and not fully incorporated it intothe framework of discrete particle models.

It is remarked that the models proposed by Vu-Quoc and Zhang (1999) and Vu-Quocet al. (2001) are rather advanced as they were developed on the basis of the continuumtheory of elastoplasticity, but it seems to the authors that the practical use of the

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models in the framework of DEM may be problematic. Suppose that two particles Aand B keep contacting each other over a portion of the load history. As the contactingpoint at particle A at a particular time instant, as a material point, will no longercontact with particle B in general at the next time instant, it will be difficult to keep theinformation, i.e. the corrected radii due to plastic deformations and the positions, of allcontacting points at particle A, over the loading history.

It should be pointed out that a rational model, which is established on the basis of anadequate description of the mechanisms governing the generation and evolution of thetangential friction forces between two particles at contact, especially an adequatedescription of the mechanism relating to the rolling, has not been achieved as thephysical nature of the friction forces between two moving particles in contact are stillnot fully understood, particularly from a microscopic view point.

The objective of the present paper is to propose a discrete particle model for granularmaterials. Each grain in the particulate system under consideration is assumed to berigid and circular. Starting with kinematical analysis of relative movements of twotypical grains in contact, both the relative rolling and sliding motion measurements,including the translational and angular velocities (displacements) are defined. Theformulae to calculate both the rolling and sliding friction tangential forces, and therolling friction resistance moment, which are constitutively related to the definedrelative motion measurements respectively, are given. In addition, the viscous normal,sliding and rolling dashpots are respectively introduced into each of componentconstitutive relations to calculate the normal and the tangential (the sliding and rolling)forces and the rolling friction resistance moment. In addition, the normal compressionstiffness coefficient is defined to be a function of the normal “overlap” between twoparticles in contact. The constitutive relations that take into account both the sliding androlling resistance forces (resistance moment) are integrated into the framework of theDEM. The numerical results are presented to verify the performance of the proposedmodel in the simulation of different types of failure modes, such as the landslide (shearbands), the compression cracking and the mud avalanching, in granular materials.

2. Kinematical analysis of moving grains in contact2.1 Relative sliding and rolling between two particles in contactTo analyze interacting forces attributing to relative motions between two particles incontact, we consider two typical disks A and B with radii rA and rB respectively, in atwo-dimensional assemblage of particles. Assume that these two particles remain incontact at a forward neighborhood I n ¼ ½tn; tn þ Dt� of time tn in the time domain asshown in Figure 1. Let X(On) denote the coordinates of the contacting point On, as aspatial point at tn, in the global Cartesian coordinate system X.

A local Cartesian coordinate system xn with its origin at X(On) is defined in such away that the y n axis is chosen to be along the line connecting centers Ao, Bo of disks Aand B with its positive orientation from Bo to Ao, while the x n axis is selected to bealong the tangent of the two disks with its positive orientation determined to form aright-hand coordinate system.

Let X(On) denote the coordinates of the contacting point Ot between the two disks Aand B, as a spatial point at t [ I n; referred to the global Cartesian coordinate systemX. The translational and rotational velocities of the coordinate system xn at tn are thendefined as

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Vo ¼t!tnlim

XðOtÞ2XðOnÞ

t 2 tn; Vo ¼

t!tnlim

at 2 an

t 2 tnð1Þ

where an, at are the inclined angles of the x n axis and the x t axis to the X axis,respectively, which are assumed positive counter-clockwise as shown in Figure 1.

Denoting VoA and Vo

B; VA and VB as the translational and angular velocities ofcenters Ao and Bo at tn respectively. The translational velocities Vc

A and VcB of the

contacting points Ac and Bc at time tn, as the material points, at disks A and B referredto the coordinate system X can be expressed as

VcA ¼ Vo

A þVA £ rc;nA ; Vc

B ¼ VoB þVB £ rc;n

B ð2Þ

where

rc;nA ¼ XðOnÞ2XðA0Þ; rc;n

B ¼ XðOnÞ2XðB0Þ ð3Þ

VA ¼ VAeZ; VB ¼ VBeZ ð4Þ

The directions of pseudo-vectors VA and VB are expressed in terms of the unit vectoreZ normal to the x-y plane of the coordinate system X with the right-hand screw rule.For clarity in expressions, the superscript n of rc;n

A and rc;nB will be omitted hereafter.

Obviously, for the two disks in contact we have

rcB ¼ 2

rB

rArc

A:

It is noted that as a repulsive force between the two disks A and B in contactis considered in the model of forces certain amount of “overlap” of the two disks isassumed to exist. Nevertheless in kinematical analysis of two moving disks in contactthe point contact (in the 2D case) between the two disks is assumed as anapproximation for simplifying the model of kinematics.

The relative sliding translational velocity DV and relative rolling angular velocityDV of two particles A and B at the contacting point On at tn, can be defined, with theuse of equations (2)-(4), as

Figure 1.Kinematical analysis of

two moving grains incontact from tn to tn þ Dt

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DV ¼ VcA 2 Vc

B ¼ VoA 2 Vo

B þ VA þVBrB

rA

� �eZ £ rc

A; ð5Þ

DV ¼ VA 2VB ¼ ðVA 2VBÞeZ ð6Þ

If DV ¼ 0; DV – 0; pure rolling occurs at the contacting point, whereas if DV ¼ 0;DV – 0 pure sliding occurs even if VA;VB – 0; provided VA ¼ VB: In the case ofpure sliding, equations (5)-(6) lead to

VA ¼ VB; DV ¼ VoA 2 Vo

B þVA 1 þrB

rA

� �eZ £ rc

A ð7Þ

while in the case of pure rolling it has

VA – VB; VoB ¼ Vo

A þ VA þVBrB

rA

� �eZ £ rc

A ð8Þ

It is remarked from equations (5) and (6) that if only the translational motion of the twoparticles exists, i.e. Vo

A – 0 and/or VoB – 0; and VA ¼ VB ¼ 0; a possible relative

movement between the two particles in contact is only pure sliding; however, when the twoparticles only rotate, i.e. VA – 0 and/or VB – 0; and Vo

A ¼ 0; VoB ¼ 0; possible relative

movements, in general, may include not only pure rolling, but also both sliding and rolling.It is noticed that Vo

A, VoB are defined as the time derivatives of the material points

(i.e. the Lagrangian derivatives) while Vo is defined as the time derivative of the spatialpoint (i.e. the Eulerian derivative). As shown in Figure 1 we have rc;n

A ¼ rcA

� �rc

A þ VoDt ¼ VoADt þ rc;t

A ð9Þ

for particle A in which Vo and VoA are approximately assumed to be constant within the

time interval [tn,t ]. There is a similar equation for particle B. It is noted that On, Ot arenot the same material point, and rc;n

A ; rc;tA shown in Figure 1 are also not the same

material line segment vector. As the velocity quantities at time tn are considered fromEquation (9) and the similar equation

rcB þ VoDt ¼ Vo

BDt þ rc;tB

for particle B with

t!tnlim rc;t

A 2 rcA

� �=Dt ¼ V0 £ rc

A andt!tnlim rc;t

B 2 rcB

� �=Dt ¼ V0 £ rc

B;

VoA;V

oB can be expressed in terms of Vo, Vo of the local coordinate system xn as

VoA ¼ Vo 2Vo £ rc

A; VoB ¼ Vo 2Vo £ rc

B ð10Þ

Substitution of equation (10) into equation (2) gives

VcA 2 Vo ¼ ðVA 2VoÞ £ rc

A; VcB 2 Vo ¼ ðVB 2VoÞ £ rc

B ð11Þ

where VcA 2 Vo and Vc

B 2 Vo represent respectively the translational velocities of thecontacting points at particles A and B that contact each other at spatial point X(On) attn, relating to the local coordinate system xn.

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2.2 Relative movement measurements between two particles in contactAs the relative movement measurements within an incremental time step ½tn; tnþ1� areconcerned, the forward difference scheme in time is assumed to be employed. With the useof equations (5) and (11), the relative sliding displacement increment vector DUs betweentwo particles A and B in contact that occurs within the time interval can be defined as

DUs ¼ DVdt ¼ VcA 2 Vc

B

� �Dt ¼ Vc

A 2 Vo

� �Dt 2 Vc

B 2 Vo

� �Dt ð12Þ

By using equation (11) and noting

rcB ¼ 2

rB

rArc

A;

the projection of DUs to the x n axis of the local coordinate system xn, i.e. the relativetangential sliding displacement increment Dus between two particles A and B at thecontacting point can be expressed by

Dus ¼ rAðVA 2VoÞDt þ rBðVB 2VoÞDt ð13Þ

By denoting DuA ¼ VADt; DuB ¼ VBDt; Duo ¼ VoDt and

Da ¼ rAðDuA 2 DuoÞ; Db ¼ rBðDuB 2 DuoÞ ð14Þ

we can re-write Equation (13) in the form

Dus ¼ Daþ Db ð15Þ

According to Equation (6) the relative rolling angular displacement increment betweentwo particles A and B in the time interval from tn to tnþ1 is given by

Dur ¼ DVDt ¼ ðVA 2VBÞDt ¼ DuA 2 DuB ð16Þ

It is noticed that the two terms in the last equality of equation (12) represent the relativesliding displacement increments of two particles A and B relating to the contacting pointOn atX(On). The relative rolling displacement increments of two particles A and B relatingto On at X(On) can be similarly defined by

DUAor ¼ DtðVA 2VoÞ £ rc

A DUBor ¼ DtðVB 2VoÞ £ rc

B ð17Þ

The vector DUr of the relative rolling displacement increments between two particles Aand B at On can be expressed in the form

DUr ¼ DUAor þ DUBo

r ð18Þ

The projection of DUr to the x n axis of the local coordinate system xn, i.e. the relativetangential rolling displacement increment Dur between two particles A and B at thecontacting point is then expressed by

Dur ¼ Da2 Db ð19Þ

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3. A computational model for friction resistances between particles incontactEven though the importance of the rotations of soil particles in simulation ofmechanical behaviors of granular materials such as soils and clays is well recognized(Iwashita and Oda, 1998; Oda et al., 1982; Bardet, 1994), the effects of rolling friction arestill largely neglected in existing discrete particle models (Elperin and Golshtein, 1997;Zhang and Whiten, 1999; Tanaka et al., 2000; Takafumi et al., 1998; Zhang and Whiten,1998). Feng et al. (2002) pointed out that correct modeling of rolling friction is still anunder-developed area and many issues remain unanswered and they developed arolling resistance model and incorporated it within the sliding friction model.

It should be remarked that as two particles in contact move against each otherthrough sliding and rolling, rolling resistances may be neglected in the model forcalculating the tangential force as the rolling friction coefficient is, in general, muchsmaller than the sliding friction coefficient. However, as the relative motion that occursat the contacting point is only or almost only the relative rolling, a model for tangentialforces with omission of rolling resistance may lead to unrealistic results in numericalcalculations.

Based on the previous work (Cundall and Strack, 1979; Iwashita and Oda, 1998;Feng et al., 2002), a model for calculation of tangential forces between two particles atcontact, which takes into account both rolling resistances (rolling friction tangentialforce and rolling resistance moment) and sliding frictional force, is proposed. Themodel is incorporated into the framework of the DEM and applied to simulate differentfailure modes of granular materials. Each of the friction resistances applied to twoparticles at contact, i.e. the rolling friction tangential force Fr, the rolling resistancemoment Mr, the sliding friction tangential force Fs, are related to correspondingrelative movement measurements Dur, Dur, Dus and their variation rates with respectto time.

3.1 Rolling and sliding friction tangential forcesHereafter within this section the subscript t ¼ r; s represents in turn the rolling and thesliding friction tangential force, respectively.

The predictor Fnþ1t;tr of rolling/sliding friction tangential force Fr at tnþ1 due to the

relative tangential rolling/sliding displacement increment DutðdutÞ; which occurswithin the typical time sub-interval ½tn; tnþ1�; can be calculated as

Fnþ1t;tr ¼ f nþ1

t þ dnþ1t ð20Þ

in which

f nþ1t ¼ f nt þ Df t; Df t ¼ 2ktDut; dnþ1

t ¼ 2ctdutdt

ð21Þ

where kt, ct stand for the stiffness coefficient and the coefficient of viscous damping ofthe rolling/sliding tangential friction, dnþ1

t reflects the damping effect on rolling/slidingfriction tangential force in the dynamic model. Fnþ1

t;tr has to satisfy Coulomb law offriction and rolling/sliding friction tangential force is determined by

Fnþ1t ¼ Fnþ1

t;tr ; if Fnþ1t;tr

��� ��� # mt Fnþ1N

�� �� ð22aÞ

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Fnþ1t ¼ sign Fnþ1

t;tr

� �mt F

nþ1N

�� ��; if Fnþ1t;tr

��� ��� . mt Fnþ1N

�� �� ð22bÞ

where Fnþ1N is the normal contact force at tnþ1; mt is the (maximum) static

rolling/sliding tangential friction coefficient.

3.2 Rolling friction resistance momentThe predictor Mnþ1

r;tr of rolling friction resistance moment Mnþ1r at tnþ1 due to the

relative rolling angular displacement increment Dur(dur), which occurs within the timesub-interval ½tn; tnþ1�; can be calculated as

Mnþ1r;tr ¼ Mnþ1

rs þMnþ1rv ð23Þ

in which

Mnþ1rs ¼ Mn

rs þ DM rs; DM rs ¼ 2kuDur; Mnþ1rv ¼ 2cu

dur

dtð24Þ

where ku, cu stand for the stiffness coefficient and the coefficient of viscous damping ofthe rolling friction moment, Mnþ1

rv reflects the damping effect on rolling frictionresistance moment in the dynamic model. Mnþ1

r;tr has to satisfy Coulomb law of frictionand rolling friction resistance moment is determined by

Mnþ1r ¼ Mnþ1

r;tr if Mnþ1r;tr

��� ��� # mur Fnþ1N

�� �� ð25aÞ

Mnþ1r ¼ sign Mnþ1

r;tr

� �mur Fnþ1

N

�� �� if Mnþ1r;tr

��� ��� . mur Fnþ1N

�� �� ð25bÞ

where mu and r are the (maximum) static rolling friction moment coefficient and theradius of the particle in consideration respectively. In fact e ¼ rmu represents aneccentricity of the normal contact traction Fnþ1

N toward the rolling direction withregard to its stationary position (Feng et al., 2002) in view of the line contact (in the 2Dcase) between the two disks A and B in contact, which occurs in practice and weconsider in the model of forces. Hence, the limit mur Fnþ1

N

�� �� of rolling friction resistancemoment defined in equation (25a) and (25b) for each of two disks A and B in contactpossesses the same value, i.e.e ¼ rAmuA ¼ rBmuB and it will be reached for the twoparticles simultaneously. Indeed e ¼ rAmuA ¼ rBmuB is assumed for each two particlesin contact in numerical examples of present work.

3.3 The tangential friction resistance force due to both rolling and slidingIn general, both the sliding and the rolling coexist between two particles in contact.The predictor Fnþ1

T;tr of the tangential friction force Fnþ1T at tnþ1 due to the relative

tangential sliding and rolling displacement increments Dus(dus) and Dur(dur), whichoccur within a typical time sub-interval ½tn; tnþ1�; can be calculated as

Fnþ1T;tr ¼ Fnþ1

s þ Fnþ1r ð26Þ

Fnþ1T;tr has to satisfy Coulomb law of friction and the tangential friction force is

determined by

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FT ¼ FT;tr if jFT;trj # msjFN j ð27aÞ

FT ¼ signðFT;trÞmsjFN j if jFT;trj . msjFN j ð27bÞ

It is noted that since ms $ mr in general, we have from equations (22a) and (22b)

maxðjFTjÞ ¼ maxðjFsjÞ ¼ msjFN j ð28Þ

For simplifying the discussion below in this section, we assume particle B is fixed andthe subscript and the superscript for identifying particle A is omitted for clarity inexpressions.

It is emphasized that in a pure rolling case, a rolling frictional force should beapplied in the tangential direction (Feng et al., 2002). To clarify this point, consider thecase of pure rolling between two particles A and B at contact, i.e. the relative slidingvelocity at the contacting point vc

sð¼ _usÞ ¼ 0 due to DV ¼ 0 and therefore, VcA ¼ 0:

For particle A we have

vcs ¼ vo þ rV ¼ 0 ð29Þ

where v o is the tangential translation velocity of the center of the particle, r,V are theradius and the rolling angular velocity of the particle. It is obtained from equation (29)that

_vo þ r _V ¼ 0 ð30Þ

On the other hand, Newton’s second law governing the tangential translation and therolling movements of particle A subjected to an tangential external force Fe at theparticle center can be expressed as

m_vo ¼ Fe þ FT ð31Þ

Im_V ¼ FTr þM r ð32Þ

where m and Im are the mass and the mass moment of inertia of particle A. As thesteady rolling ð _V ¼ 0Þ under the condition Fe – 0 is considered, we have _vo ¼ 0 fromequation (30). Since the relative movement between the two particles is pure rolling, i.e.Fs ¼ 0ðus ¼ 0Þ; if the rolling friction resistance is omitted, then FT ¼ Fs ¼ 0: Thisobviously violates the motion equation (31). Hence, it is concluded that even thoughms $ mr in general and jFsj $ jFrj in most cases when the relative sliding and therelative rolling coexist, to ensure the reliability of the numerical simulations it isnecessary to take into account the effects of both the sliding and the rolling frictionresistances in the model for calculation of tangential friction resistances. Furthermore,as steady state pure rolling is concerned ð _V ¼ 0Þ and an external force Fe – 0is assumed to apply at the center of particle A, we must have both a resultingFr ¼ FT ¼ 2Fe and M r ¼ 2FTr ¼ 2Frr ¼ Fer at the contact point from equations(29)-(32). This conclusion justifies the inclusion of both Fr and Mr simultaneously in theproposed model, which provides a mechanism for energy dissipation during steadystate rolling.

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3.4 Repulsive normal contact forceA repulsive normal force between two particles A and B in contact arises due to thestiffness of the particles. The model for calculating the normal force is based on theassumption that a small amount of “overlap” of the two particles in contact is allowedand compressibility of the particle is finite. In the proposed model, the normal contactforce is related to the relative normal movement measurement, i.e. the “overlap” unþ1

N atcurrent time instant tnþ1 and its variation rate with respect to time. unþ1

N is defined asthe difference between the sum of the radii of the two particles and the distancebetween the centers of the two particles, i.e.

unþ1N ¼ rA þ rB 2 X Anþ1

o

� �2X Bnþ1

o

� ���� ��� ð33Þ

The normal contact force between the two particles in contact can be calculated by

Fnþ1N ¼ 2kNu

nþ1N 2 cN

unþ1N 2 unN

Dtif unþ1

N . 0 ð34aÞ

Fnþ1N ¼ 0 if unþ1

N # 0 ð34bÞ

where kN, cN are the compression stiffness coefficient and the coefficient of viscousdamping of the normal contact deformation for the granular material. In the presentmodel both the non-linear normal compression stiffness coefficients and the linear (i.e.kN is taken as a constant) at contact are employed in the first and the second numericalexamples respectively. The non-linear normal compression stiffness coefficient isproposed on the basis of combination of the power law model (Han et al., 2000) and thelinear model for the coefficient, i.e.

kN ¼ kN unþ1N

� �¼ kN0 exp

unþ1N

rA þ rB

� �ð35Þ

in which kN0 ¼ kNð0Þ is the initial normal compression stiffness coefficient.

4. The governing equations for dynamic equilibrium of discrete particulatesystem – a discrete element approachLet JA and nA denote the set of neighboring particles and the number of these particlesfor a given typical particle A at time t [ ½tn; tnþ1�: The neighboring particles ofparticle A are defined as those particles, which have the possibility of keeping contactwith particle A at time t [ ½tn; tnþ1�: A subset J

cA [ JA is formed after checking each

of the particles in the set JA: Each of the particles in the subset JcA keeps contacting

with particle A at t [ ½tn; tnþ1�: Using the DEM, the motion equations of particle A inthe two-dimensional case can be written as

mA €UA

X ¼i[J

cA

XFiX þ Fe;A

X ð36aÞ

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903

mA €UA

Y ¼i[J

cA

XFiY þ Fe;A

Y ð36bÞ

IAm€uA ¼

i[JcA

XFi

TrA þMir

� �þM e;A ð36cÞ

where mA; IAm are, respectively, the mass and the mass moment of inertia of particle A,

€UA

X ;€U

A

Y ;€uA stand for the translational accelerations in X,Y axes and the angular

acceleration in the X-Y plane of particle A. Fe;AX ;Fe;A

Y ;M e;A are the external loadsapplied to particle A corresponding to the degrees of freedom UA

X ;UAY ; u

A: The rollingfriction resistance moment Mi

r exerted by particle i on particle A can be calculated by

using equation (25a) and (25b). FTX ;i ¼ Fi

XFiY

h iis the contacting force produced by

particle i on particle A referred to the global coordinate system X. Let ai denote theangle between the X axis of global coordinate system X to the x axis of the localcoordinate system with its origin at the contacting point between particle A andparticle i as shown in Figure 1, in which the local coordinate system is denoted as xn.

The contacting force exerted by particle i on particle A in the local coordinate

system xn is denoted by FTx;i ¼ Fi

TFiN

h i: The tangential force Fi

T is defined by

equation (27a)and (27b) and can be calculated by using equations (20)-(22a), (22b). Thetransformation between Fx,i and Fx,i can be expressed in the form

Fx;i ¼ TTi Fx;i; Ti ¼

cosai sinai

2sinai cosai

" #ð37Þ

It is noted that the right hand side terms of equation (36a), (36b) and (36c) are not onlydetermined by the external loads locally applied to particle A, but also related to therelative movements of particle A with its neighboring points in contact,i.e.UA

X ;UAY ; u

A; UiX ;U

iY ; u

i i [ JcA

� �and their derivatives with respect to time.

An explicit time integration algorithm is used to solve the motion equations (36a),(36b) and (36c) for each particle in the particulate system under consideration. Theconditional stable nature of explicit algorithms in time domain imposes a limitation onthe maximum time step that can be employed in the solution procedure. A formula forcalculating critical time step size is given below (Tanaka et al., 2000)

Dtcr ¼ lcr

ffiffiffiffiffiffiffiffiffiffiffim=kn

pwhere kn and m are the (equivalent) normal contact stiffness coefficient and the(equivalent) mass of a typical particle, lcr is the coefficient to take into account theviscous damping effect and lcr ¼ 0:75 is chosen in (Tanaka et al., 2000). In the presentwork, the critical time step size for the whole of the particulate system is calculatedaccording to the following formula

Dtcr ¼ lcri

minffiffiffiffiffiffiffiffiffiffiffiffiffiffimi=kn;i

qð38Þ

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5. Nominal strains of granular materialsTo measure the change of the position of a particle in consideration relating to itsneighboring particles two nominal strains, termed the effective strain and thevolumetric strain, respectively, are defined at the center of the particle.

Consider the change of the position of particle A relating to one of its neighboringparticles denoted particle B, as shown in Figure 2. Let Xn

A XnB and Xnþ1

A ; Xnþ1B denote

the coordinates of the centers of particles A and B, referred to the global coordinatesystem X at two time instants tn and tnþ1: The relative positions of particles A and Breferred to the global coordinate system at tn and tnþ1 can be expressed as DXn

BA ¼Xn

B 2XnA and DXnþ1

BA ¼ Xnþ1B 2Xnþ1

A ; respectively. Referred to the local coordinatesystem xnl ; y

nl

� �defined in terms of the positions of particles A and B at time tn as

shown in Figure 2, the relative positions of particles A and B at tn and tnþ1 can also beexpressed as Dxn

BA ¼ xnB 2 xn

A and Dxnþ1BA ¼ xnþ1

B 2 xnþ1A ; where xn

A;xnB and

xnþ1A ;xnþ1

B are the local coordinates of the centers of particles A and B at tn andtnþ1: According to the coordinate transformation, we have

DxnBA ¼ TDXn

BA; Dxnþ1BA ¼ TDXnþ1

BA ; T ¼cosa1 sina1

2sina1 cosa1

" #ð39Þ

The change in the relative position of the pair of material particles A and B from time tnto tnþ1 can be described by the following deformation gradient fn referred to the localcoordinates

fn ¼Dxnþ1

BA

DxnBA

¼ RnUn ð40Þ

where

Rn ¼cosða2 2 a1Þ 2sinða2 2 a1Þ

sinða2 2 a1Þ cosða2 2 a1Þ

" #; Un ¼

lAB 0

0 1

" #ð41Þ

Figure 2.Change of the position of

particle A relating aneighboring particle

A discreteparticle model

905

lAB ¼lnþ1AB

lnAB

; lnAB ¼ DxnBA

�� ��; lnþ1AB ¼ Dxnþ1

BA

�� �� ð42Þ

in which a1 and a2 denote the angles between the X-axis of the global coordinatesystem X to the x axis of the local coordinate systems at the two time instants, asshown in Figure 2. Substitution of equation (39) into equation (40) gives

DXnþ1BA ¼ FDXn

BA ð43Þ

in which

F ¼ TTfnT ð44Þ

Further we denote the “displacement derivative” matrix

D ¼ F2 I ð45Þ

where I is the identity matrix, and

gAB ¼2

3DijDij

1=2

ð46Þ

with componentsDij of matrix D. The effective strain at the center of particle A measuredin terms of the changes in the relative positions of particle A with its nA neighboringparticles can be defined, according to the theory of continuum mechanics, as

gA ¼1

nA

XnA

B¼1

gAB ð47Þ

The volumetric strain at the center of particle A is then defined as

gvA ¼

1

nA

XnA

B¼1

gvAB; gv

AB ¼ lAB 2 1 ð48Þ

6. Numerical example6.1 Uni-axial compression of a rectangular panelThe first example concerns a granular assembly with 86:7 £ 50 cm rectangular profile.The assembly is generated by 4,950 homogeneous particles with radius of 5 mmcollocated in a regular manner and subjected to a uniaxial compression between tworigid plates applied by a vertical displacement control, as shown in Figure 3. Thisgranular assembly with radius of 5 mm is characterized by “smaller grains” andtermed “S.G..” for short. The gravitational forces are neglected. The particles on the leftand the right boundaries are free in the vertical and the horizontal directions. Thecontact between the particles on the top and the bottom boundaries with the plates ismodeled as a vertical sticking, i.e. only the vertical displacements of the particles on thetop and the bottom boundaries are specifies as uniformly prescribed values under thevertical displacement control. The horizontal movements of the particles on the topand the bottom boundaries relating to the plates are allowed with the sliding coefficient0.5 being used. The material parameters of the granular assembly used in the exampleare listed in Table I.

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To demonstrate the importance of the rolling resistance, particularly the rolling frictiontangential force, on the shear strength and consequently on the occurrence andevolution of shear bands in particulate system, the numerical results obtained by thecases with and without the rolling resistance accounted into the model are compared.The four cases are particularly considered, i.e.

(1) without rolling ðkr ¼ 0; ku ¼ 0Þ;

(2) with rolling A ðkr ¼ 0; ku ¼ 2:5 Nm=radÞ;

(3) with rolling B ðkr ¼ 106 N=m; ku ¼ 2:5 Nm=radÞ;

(4) with rolling C ðkr ¼ 108 N=m; ku ¼ 2:5 Nm=radÞ:

Figure 3.A granular assembly with

86:7 £ 50 cm rectangularprofile generated by 4,950

homogeneous particleswith radius of 5 mm

collocated in a regularmanner

Parameter Selected value

Particle density (r) 2000 (kg/m3)Stiffness coefficient of normal force (kn) 2.5 £ 108 (N/m)Stiffness coefficient of sliding force (ks) 1.0 £ 108 (N/m)Stiffness coefficient of rolling force (kr) 1.0 £ 106 (N/m)Stiffness coefficient of rolling moment (kq) 2.5 (Nm/rad)Damping coefficient of normal force (cn) 0.4 (N s/m)Damping coefficient of sliding force (cs) 0.4 (N s/m)Damping coefficient of rolling force (cr) 0.4 (N s/m)Damping coefficient of rolling moment (cq) 0.4 (Nm s/rad)Sliding friction force coefficient (ms) 0.5Rolling friction force coefficient (mr) 0.05Rolling friction moment coefficient (mq) 0.02

Table I.The material parametersof the granular assembly

used in the rectangularpanel example

A discreteparticle model

907

Here effects of the rolling friction tangential force are particularly symbolized by thematerial datum kr, which does not appear in existing discrete particle models as therolling and the sliding friction tangential forces at contact were not distinguished inthese models.

The load-displacement curves to show the load history applied on the top of thegranular assembly with the increasing vertical displacements of top surface of theplate are shown in Figure 4(a). The results illustrate the reduction of the load-carryingcapability, i.e. the softening behaviour, of the assembly. Figures 5-7 show evolutions ofthe effective strain, the volumetric strain and the particle rotation distributions,respectively, with increasing vertical displacements 0.6, 1.2, 1.8 and 3.6 cm prescribedto the particles at the top boundary of the assembly, which show that the effective andthe volumetric strains and the particle rotation occur and develop sharply into anarrow band of intense straining characterized strain localization phenomena. It is

Figure 4.The load-displacementcurves for the rectangulargranular assembliesregularly or randomlygenerated by particleswith different radiisubjected to uniaxialuniform compression;(a) for regularly generatedparticulate assembly; and(b) for randomly generatedparticulate assembly

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observed from Figures 4-6 that the present discrete particle model is capable ofmodeling the failure mode of shear bands characterized by the softening behavior andstrain localization phenomena.

To show the dependence of the width of the shear band and the load-carryingcapability on the particle radius (the particle size), the granular assembly with the sameprofile is re-generated by 1,225 homogeneous particles with radius of 10 mm collocatedin a regular manner. The abbreviated name of the assembly is termed “L.G..” (i.e. largergrains) for short. The material parameters used in this case are chosen as the same asthose used for the case with the finer size particles except for mq ¼ 0:01: Figure 8shows the effective and the volumetric strain distributions as a vertical displacement of1.8 cm is prescribed to the top boundary of the coarser granular assembly. It isobviously observed in the figure that the shear bands obtained by using the coarsergranular assembly are wider than those shown in Figures 5-6 by using the finergranular assembly. The particle size can be regarded to act as the internal length scalein the discrete particle model and is related to the width of the shear band. On the otherhand, Figure 4(a) shows that the limit of load-carrying capability of the coarser

Figure 5.The effective straindistributions in the

rectangular granularassembly with increasing

vertical displacementsprescribed to the top

boundary of the assemblyðks ¼ 1:0 £ 108 N=mÞ;(a) 0.6 cm; (b) 1.2 cm;

(c) 1.8 cm; and (d) 3.6 cm

A discreteparticle model

909

granular assembly is lower than half of that of the finer granular assembly. It may beattributed to the less total contacting points, therefore, the smaller total internal frictionresisting the external force causing material failure.

It is remarked that the shear band mode and its width depend not only the particlesize as shown in Figures 5-6 and Figure 8, but also the material property data,particularly the stiffness coefficient ks of the sliding tangential friction. Figures 9-10shows evolutions of the effective strain and the volumetric strain distributions, whichoccur and develop in the “S.G..” assembly as the material parameters are chosen thesame as those listed above except ks ¼ 1:0 £ 106 N=m: It is shown in Figures 9-10 thatthe shear band mode in this case is rather different from that occurs as shown inFigures 5-6, where the four shear bands appear at the final stage of the simulationwhile only two shear bands develop until the end of the simulation in Figures 9-10, thatis similar to the shear band mode developed in the rectangular panel example shown in(Li et al., 2003), in which the example is modeled by using the continuum theorycombined with the finite element discretization.

Figure 6.The volumetric straindistributions in therectangular granularassembly with increasingvertical displacementsprescribed to the topboundary of the assemblyðks ¼ 1:0 £ 108 N=mÞ;(a) 0.6 cm; (b) 1.2 cm;(c) 1.8 cm; and (d) 3.6 cm

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Figure 7.The particle rotation

distributions in therectangular granular

assembly with increasingvertical displacements

prescribed to the topboundary of the assembly

ðks ¼ 1:0 £ 108 N=mÞ;(a) 0.6 cm; (b) 1.2 cm;

(c) 1.8 cm; and (d) 3.6 cm

Figure 8.The effective and the

volumetric straindistributions as a verticaldisplacement of 1.8 cm is

prescribed to the topboundary of the coarser

granular assemblyðks ¼ 1:0 £ 108 N=mÞ;

(a) effective straindistribution; and

(b) volumetric straindistribution

A discreteparticle model

911

The mechanical properties of granular materials not only depend on the physicalproperties and the geometrical sizes of individual particle in the granular assembly, butare also related to the collocation manner of all particles in the assembly. To illustratethis, we consider a granular assembly with 80 £ 60 cm rectangular profile. Theassembly is randomly generated by 3,679 particles with different radii of 4.35, 5.8 and7.25 mm and termed “R.G..” for short. The material parameters used in this case arechosen as the same as those used for the above assemblies generated by homogeneousparticles except mu;iri ¼ 1 £ 1024 m ði ¼ 1; 2; 3Þ; where ri;mu;i are the radius andcorresponding value of mu for each of the three types of particles. The gravitationalforces are neglected. It is assumed that there exist neither initial overlaps (initialcontacting forces) between each two particles in contact nor initial microforcesundertaken at any particles in the assembly. Figure 11 shows evolutions of theeffective strain distribution with increasing vertical displacements 1.2, 12, 30 and43.2 cm prescribed to the particles at the top boundary of the randomly generatedparticle assembly.

Figure 9.The effective straindistributions in therectangular granularassembly with increasingvertical displacementsprescribed to the topboundary of the assemblyðks ¼ 1:0 £ 106 N=mÞ;(a) 0.6 cm; (b) 1.2 cm;(c) 1.8 cm; and (d) 3.6 cm

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It should be noted that no obvious shear band characterizing failure mode of strainlocalization, but severe vertical settlement with dramatic dilation in horizontaldirection are observed in the assembly randomly generated by particles with differentradii. It can be attributed to the stochastic distribution of voids with different sizes inthe assembly, therefore, the particles with smaller radii tend to fill into the poresenclosed by the particles with larger radii under vertical compression; and thestochastic anisotropy due to random collocation manner of particles with differentsizes around each local point, i.e. the deformation structure and the mechanicalbehavior of the assembly randomly generated by the particles with different sizes arerelated to microscopic structures of the granular assembly.

It is remarked that much more apparent influence of the rolling friction tangentialforce on the load-carrying capability is observed in the randomly generated granularassembly as shown in Figure 4(b) than that displayed in the assembly generated withhomogeneous particles shown in Figure 4(a). It is also shown in Figure 4 that therolling friction resistance increasing with the rolling stiffness coefficient kr enhancessomewhat the load- carrying capability of the assembly.

Figure 10.The volumetric strain

distributions in therectangular granular

assembly with increasingvertical displacements

prescribed to the topboundary of the assembly

ðks ¼ 1:0 £ 106 N=mÞ;(a) 0.6 cm; (b) 1.2 cm;

(c) 1.8 cm; and (d) 3.6 cm

A discreteparticle model

913

6.2 A slop stability exampleThe second example concerned is a slope stability problem. The geometry of the slope isshown in Figure 12. The particles on the right boundary are free and fixed in the verticaland horizontal directions respectively. The particles on the bottom boundary are fixed inthe vertical direction while their movements tangent to the bottom surface and theirrolling movements are allowed with the same sliding, rolling friction coefficients and thesame rolling friction moment coefficient, which are used for two typical particles in

Figure 11.The effective straindistributions in theassembly randomlygenerated by particles ofdifferent radii withincreasing verticaldisplacements prescribedto the top boundary of theassembly; (a)1.2 cm;(b)12 cm; (c)30 cm; and(d) 43.2 cm

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contact, between the particles and the surface. The slope is loaded by a footing modeledas a rigid plate resting on its crest as shown in Figure 12. An increasing load is applied tothe slope via the increasing vertical displacement prescribed to point A at the plate, sothat the footing is also allowed to rotate around it. The material parameters used forcalculation of the external forces applied to the particles contacted with the plate are thesame as those used for two contacted particles. One of the main objectives of thisexample is to show the capability of the proposed discrete particle model in reproducingthe different failure modes, i.e. the landslide, the compression cracking and the mudavalanching, which occur in the slope made of granular materials.

First, we consider the slope with geometrical sizes of b ¼ 50 cm; H ¼ 50 cm; L ¼25 cm; a ¼ 15 cm: The slope is modeled by an assemblage of 3,710 homogeneousparticles with the radius of 5 mm collocated in a regular manner. The mechanicalproperties of the granular material used in this case are listed in Table II. Gravitationalforces are taken into account. The rate of the increasing vertical displacement prescribedto point A is 69 cm/s and the incremental time step sizeDt < 1:441856 £ 1025 s equals to

Figure 12.The slope problem

modeled by granularassembly: geometry and

external load pattern

Parameter Selected value

Particle density (r) 2000 (kg/m3)Stiffness coefficient of normal force (kn) 5.0 £ 106 (N/m)Stiffness coefficient of sliding force (ks) 2.0 £ 106 (N/m)Stiffness coefficient of rolling force (kr) 2.0 £ 102 (N/m)Stiffness coefficient of rolling moment (kq) 1.0 (Nm/rad)Damping coefficient of normal force (cn) 0.4 (N s/m)Damping coefficient of sliding force (cs) 0.4 (N s/m)Damping coefficient of rolling force (cr) 0.4 (N s/m)Damping coefficient of rolling moment (cq) 0.4 (Nm s/rad)Sliding friction force coefficient (ms) 0.5Rolling friction force coefficient (mr) 0.05Rolling friction moment coefficient (mq) rmu ¼ 5:0 £ 1025 m

Table II.The material parametersof the granular assemblyused in the slope example

with the failure mode ofshear band

A discreteparticle model

915

the critical time step size Dtcr calculated according to equation (38). Figure 13 shows thedevelopment of effective strains in the deformed slope with the increase of the prescribedvertical displacement of point A at 3, 6 and 9 cm. The failure mode of the landslide, i.e.shear bands developed at first and then expanded to the surrounding zone can beobserved. The latter phenomenon observed is different from the failure mode of shearbands occurring in the continuum model. It can be explained by the tendency ofinstability of the particles at the surrounding zone from their original positions due tothe severe movements of the particles at the shear bands and the gravitational forceeffect.

Secondly, we consider the slope with the same geometry and modeling parametersof the granular assembly as in the first case. The material properties and theincremental time step size are also as same as those used in the first case exceptcn ¼ cs ¼ cu ¼ 0; i.e. viscous damping effects are neglected. Gravitational forces arealso neglected. The slope is loaded by the enforced increasing vertical displacementprescribed to point A with the rate 69 cm/s and the body force accelerationwith prescribed increasing rate 226.4 m/s3 until the time when the acceleration

Figure 13.The failure mode: thelandslide. Evolution ofeffective strains in thedeformed slope withincreasing prescribedvertical displacement ofpoint A equals to (a) 3 cm;(b)6 cm; and (c)9 cm

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10 m=s2ð< 1 gÞ is reached and then the value of the acceleration is kept unchanged.Figure 14 shows the evolution of the deformation of the slope. It is remarked that thegranular assembly is likely to deform as granular flow, i.e. the failure mode of the slopein this case is rather characterized by the mud avalanching while no intensestrain localization concentrated into narrow shear bands is observed as shown inFigure 15.

Finally, the slope with geometrical sizes of b ¼ 60:5 cm; H ¼ 35:5 cm; L ¼ 30 cm;a ¼ 20 cm is analyzed. The slope is modeled by 3,710 homogeneous particles withthe radius of 5 mm collocated in a regular manner. The mechanical properties of the

Figure 14.The failure mode: the mud

avalanching evolution ofthe deformation of theslope with increasing

prescribed verticaldisplacement of point A

equals to (a) 3.3 cm;(b)4.8 cm; (c)7.8 cm; and

(d)10.8 cm

A discreteparticle model

917

granular material used in this case are listed in Table III. Gravitational forces aretaken into account. The slope is loaded by the enforced increasing verticaldisplacement prescribed to point A with the rate 690 cm/s. The incremental timestep size Dt < 1:441856 £ 1026 s is used. The failure mode of the compression

Figure 15.Evolution of effectivestrains in the deformedslope with increasingprescribed verticaldisplacement of point Aequals to (a) 2.4 cm;(b)3.3 cm; and (c)4.8 cm

Parameter Selected value

Particle density (r) 2000 (kg/m3)Stiffness coefficient of normal force (kn) 5.0 £ 108 (N/m)Stiffness coefficient of sliding force (ks) 2.0 £ 108 (N/m)Stiffness coefficient of rolling force (kr) 2.0 £ 102 (N/m)Stiffness coefficient of rolling moment (kq) 1.0 (Nm/rad)Damping coefficient of normal force (cn) 0.4 (N s/m)Damping coefficient of sliding force (cs) 0.4 (N s/m)Damping coefficient of rolling force (cr) 0.4 (N s/m)Damping coefficient of rolling moment (cq) 0.4 (Nm s/rad)Sliding friction force coefficient (ms) 0.5Rolling friction force coefficient (ur) 0.05Rolling friction moment coefficient (mq) rmu ¼ 5:0 £ 1025 m

Table III.The material parametersof the granular assemblyused in the slope examplewith the cracking failuremode

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cracking characterized by a number of macroscopic cracks running through theslope, including a vertical crack due to the compression, can be observed inFigure 16.

7. Concluding remarksA systematic analysis of the kinematics of two moving circular grains withdifferent radii in contact is carried out. Accordingly, the relative motionmeasurements, i.e. the relative rolling and sliding motion measurements,including the translational and angular velocities (displacements) are defined.Both the rolling and sliding frictional tangential forces, the rolling frictionresistance moment, which are constitutively related to corresponding relativemotion measurements defined, are formulated and integrated into the framework ofdynamic model of DEM.

A discrete particle model for granular materials is proposed. In view of the factthat the rolling resistance mechanism described in currently available models inthe literature is inadequate in both physical and numerical aspects, the proposedmodel is thus focused on an adequate description of the rolling resistancemechanism in the model. However the present work may be regarded as afurther attempt to highlight the issue rather than to provide a comprehensivesolution.

The definitions of nominal effective and volumetric strains are proposed on theformalism of the continuum theory to illustrate the movements of each particle relatingto its surrounding particles in a particulate system.

The three types of failure modes, i.e. the landslide (shear bands), the compressioncracking and the mud avalanching, in granular materials are reproduced todemonstrate the capability and the performance of the proposed discrete particle modeland corresponding numerical method in the simulation of failure phenomena observedin granular materials.

Figure 16.The failure mode: the

compression cracking.Evolution of effective

strains in the deformedslope with increasing

prescribed verticaldisplacement of point Aequals to (a) 2.1 cm; and

(b)2.4 cm

A discreteparticle model

919

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