tavarez_et_al-2007-international_journal_for_numerical_methods_in_engineering.pdf

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2007; 70:379–404Published online 17 October 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1881

Discrete element method for modelling solidand particulate materials

Federico A. Tavarez‡ and Michael E. Plesha∗,†

Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI, U.S.A.

SUMMARY

The discrete element method (DEM) is developed in this study as a general and robust technique forunified two-dimensional modelling of the mechanical behaviour of solid and particulate materials, includingthe transition from solid phase to particulate phase. Inter-element parameters (contact stiffnesses andfailure criteria) are theoretically established as functions of element size and commonly accepted materialparameters including Young’s modulus, Poisson’s ratio, ultimate tensile strength, and fracture toughness.A main feature of such an approach is that it promises to provide convergence with refinement of aDEM discretization. Regarding contact failure, an energy criterion based on the material’s ultimate tensilestrength and fracture toughness is developed to limit the maximum contact forces and inter-elementrelative displacement. This paper also addresses the issue of numerical stability in DEM computationsand provides a theoretical method for the determination of a stable time-step. The method developedherein is validated by modelling several test problems having analytic solutions and results show thatindeed convergence is obtained. Moreover, a very good agreement with the theoretical results is obtainedin both elastic behaviour and fracture. An example application of the method to high-speed penetrationof a concrete beam is also given. Copyright q 2006 John Wiley & Sons, Ltd.

Received 11 April 2006; Revised 25 July 2006; Accepted 2 August 2006

KEY WORDS: discrete element method; particulate media; solid media; clusters; convergence; fractureenergy

∗Correspondence to: Michael E. Plesha, Department of Engineering Physics, University of Wisconsin-Madison, 1500Engineering Drive, Madison, WI 53706, U.S.A.

†E-mail: [email protected]‡Present address: ExxonMobil Upstream Research Company, P.O. Box 2189, Houston, TX 77252, U.S.A.

Contract/grant sponsor: National Highway InstituteContract/grant sponsor: U.S. Air Force Office of Scientific Research (AFOSR); contract/grant number: F49620-03-0216

Copyright q 2006 John Wiley & Sons, Ltd.

380 F. A. TAVAREZ AND M. E. PLESHA

1. INTRODUCTION

The fracture behaviour of solids such as concrete, rock, ceramics, and other brittle materialsunder severe loading has become an area of increased research in recent years. These materialsare complex and extremely heterogeneous, especially after they degrade from solid to particulate.Reproducing the behaviour exhibited by these materials with continuum methods requires complexconstitutive models containing a large number of parameters and/or internal variables, such asyield surfaces and equation of state descriptors. The discrete element method (DEM) originallydeveloped by Cundall and Strack [1] has proven to be a powerful and versatile numerical toolfor modelling the behaviour of granular and particulate systems [2–11], and also for studying themicromechanics of materials such as soil at the particle level. However, the method also has thepotential to be an effective tool to model continuum problems (i.e. solids), especially those thatare characterized by a transformation from a continuum to a discontinuum. Such problems includefailure of concrete structures, fragmentation of rock due to blasting, and fracture of ceramics andother quasi-brittle materials under high velocity impact.

For problems with severe damage, DEM offers a number of attractive features over continuumbased numerical methods, with the primary feature being a seamless transition from solid phaseto particulate phase. Continuum based methods such as the finite element method are challengingto apply to these problems and are plagued by the need for continuum constitutive models, severeelement distortion, and frequent re-meshing. In contrast, DEM has the capability to capture thecomplicated behaviour of actual materials by using a discretization scheme that is simple in conceptand implementation, and with simple assumptions and parameters that govern the micro levelelement–element interactions. Complex macroscopic behaviour such as strain softening, dilation,and fracture arise automatically from extensive microcracking in the DEM medium.

The present study develops the DEM as a general, robust, and scalable computer technique forunified modelling of the mechanical behaviour of solid and particulate materials, including thetransition from solid phase to particulate phase. Most of the past research in DEM for modellingcontinuum behaviour and fracture rely on calibration processes to determine the correct inter-element parameters for a specific problem [12–15]. Consequently, the results that are producedare likely to be DEM element size dependent. In this work, we attempt to theoretically determinesuch parameters as a function of known material properties and DEM element size. A weaknessof DEM is that its convergence properties are not understood. The crucial question is whetherconvergence (in both elastic behaviour and material failure) is obtained as DEM element sizevanishes in the limit of model refinement. Therefore, another goal of our investigation is to studyDEM convergence for modelling elastic behaviour and fracture of solids.

2. THE DISCRETE ELEMENT METHOD (DEM)

The DEM discretizes a material using rigid elements of simple shape that interact with neighbouringelements according to interaction laws that are applied at points of contact. The rigid elements canhave a variety of shapes, although most often they are circular or spherical due to the simplicityand speed of the contact detection algorithm. The analysis procedure consists of three majorcomputational steps: internal force evaluation, in which contact forces are calculated; integrationof equations of motion, in which element displacements are computed; and contact detection, wherenew contacts are identified and broken contacts are removed. In a DEM analysis, the interaction of

Copyright q 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 70:379–404DOI: 10.1002/nme

ELEMENT METHOD FOR MODELLING SOLID AND PARTICULATE MATERIALS 381

the elements is treated as a dynamic process that alternates between the application of Newton’ssecond law and the evaluation of a force–displacement law at the contacts. Newton’s second lawgives the acceleration of an element resulting from the forces acting on it, including gravitationalforces, external forces prescribed by boundary conditions, and internal forces developed at inter-element contacts. The acceleration is then integrated to obtain the velocity and displacement. Theforce–displacement law is used to find contact forces from known displacements. The equationsof motion are integrated in time using the central difference method. Details of this process aregiven in [1]. The method can be computationally very demanding and thus, efficient algorithms,especially for the internal force evaluations and contact detection, must be used. Computationaleffectiveness will be particularly important for three-dimensional discretizations, the use of whichis inevitable for obtaining fully realistic and accurate models for many applications.

3. DEM FOR MODELLING PARTICULATE MATERIALS

It is well known that the mechanical behaviour of sands and other granular cohesionless geoma-terials is different in important respects from that of other materials, and that this difference isdue to the particulate nature of the medium. As such, the DEM has become accepted and widelyused to model the mechanical behaviour and flow of particulate geomaterials. An extension toindustrial geomaterial applications has occurred over the past several years with modelling of ballmills, dragline excavators, and material transport using conveyor belts. A largely separate researchactivity to model powders and materials processing is also ongoing.

The present work uses a modified version of the program TRUBAL [16, 17]. Since 1978, manyresearchers have adapted this program to solve specific problems with emphasis on applicationsto granular materials. Soil is inherently discontinuous and can be effectively modelled by theDEM. In terms of computational convenience, two-dimensional circular disk elements and three-dimensional spherical elements are ideal choices for DEM element shape. However, in someapplications, such as flow problems, the medium may not be adequately represented by usingsuch ‘smooth’ elements due to excessive element rotations that occur, and often the agreementbetween the shear strength of the DEM medium and the natural material being modelled ispoor. In order to improve agreement, some researchers have employed the ad hoc practice ofartificially constraining the rotations of elements periodically throughout the course of a simulation.A better approach is to impart some shape complexity to the particles being modelled in a DEMdiscretization, and to this end, several investigators have developed DEM elements having morecomplex shape than circular or spherical. For instance, Ghaboussi and Barbosa [2] developed three-dimensional DEM elements consisting of rigid six-sided polygons that can interact with each otherthrough different types of contacts, and Ting [11] developed DEM elements having elliptical shape.A contact algorithm for DEM elements of arbitrary geometries was developed by Williams et al.[18]. Another technique to improve the accuracy of DEM models while retaining the simplicity andspeed of the contact detection algorithms for simple-shape elements was introduced by Jensen et al.[6, 7], where particles are modelled by combining several smaller DEM elements of simple shapeinto clusters that act as a single, larger rough particle. In their research, DEM clusters were formedby bonding a number of circular-shaped elements into a semi-rigid configuration. In simulationsof media-structure interface shear tests, their results indicate that clustering gives rise to lowerelement rotation and increased shear resistance of the medium that is in better agreement withnatural materials. Another form of clustering has been reported in which displacement constraints



are used to render a cluster to be truly rigid [8]. While this approach can be more efficient, primarilybecause internal force computations for the intra-cluster contacts is not performed, knowledge ofthese intra-cluster forces is desirable for purposes of implementing cluster failure mechanisms, asis discussed next.

A very significant feature of the clustering developed in [6, 7] is the opportunity to easilyincorporate particle damage such as grain crushing, wear of roughness, etc. Indeed, it appearsthat grain crushing is an important result, or initiator, of shear zone formation in sand [9]. Jensenet al. [10] used an energy density criterion to determine if enough work has been accumulated by aDEM element of a cluster to warrant its separation (breakage) from the cluster. During the courseof computation, the increment of sliding work done on a particular DEM element i is computedat each time step by summing the work increments for all extra-cluster contacts for the element

dWi = ∑number ofextra-clustercontacts

Ftd�st (1)

where Ft is the tangential force at the contact and d�st is the increment of relative tangential plastic(sliding) displacement between the DEM element and a contacting neighbour. The total work isthen obtained by integrating (summing) the work increments

Wi =∫

dWi (2)

Therefore, an element separates from its cluster once the total work Wi exceeds a maximumWmax

i , computed by the product of a user-specified critical energy density W0 and the DEMelement volume Vi .

An important conclusion to be drawn here is that it appears that accurate modelling of aparticulate material, such as sand, requires that individual particles be modelled as solids, with duerespect for the their ability to undergo damage due to wear, and damage due to fracturing. Thework of Jensen et al. [10] presents reasonable criteria for modelling damage due to wear, which isa more or less gradual process that depends on frictional sliding. The work reported in subsequentsections of this paper will allow damage due to grain fracture, which should enhance the abilityof DEM to model particulate materials.

4. DEM FOR MODELLING SOLID MATERIALS

In our DEM approach to modelling solids, we expand on the clustering concept introduced byJensen et al. [6, 7, 10] to use megaclusters, wherein the entire volume of a solid is modelled bybonding individual DEM elements, or clusters of DEM elements, together. For a solid with noinitial cracks (i.e. a nominally homogeneous material), all contacts in the DEM discretizationare initially bonded with the behaviour that is schematically illustrated in Figure 1. During thecourse of a simulation, if a bonded contact fails, according to criteria to be described, the contactbecomes frictional as also shown in Figure 1, if indeed the contact still exists (i.e. if the twoDEM elements are still being pressed into one another), or the contact is destroyed if the twoelements have separated. Thus, for a bonded contact, the normal direction spring acts in tension andcompression, with a relationship between normal force and normal relative displacement given by



nK

nC

tK tC

bonded contact frictional contact

nK

nC

tK tC

µ

Figure 1. Discrete element interaction models for bonded and frictional contacts.

Figure 2. Generation of a cantilever beam model by consolidating three-element DEM clusters.

the elastic stiffness Kn, and in the tangential direction, changes of shear force are linearly relatedto changes of shear relative displacement by a stiffness Kt. For frictional contact, the normaldirection spring acts only in compression and the shear force is limited by a Coulomb frictionlaw. Consequently, when the computed shear force reaches this maximum value, the inter-elementcontact undergoes sliding. The computer implementation uses Rayleigh or proportional damping[19], and therefore damping parameters are chosen as fractions of critical damping at two desiredfrequencies.

Figure 2 shows an example of the creation of a DEM model of a cantilever beam. In thegeneration of this model, a domain is initially populated by a large number of clusters (three-element clusters are used in Figure 2), which are then compressed or consolidated into the shapeof the beam. After consolidation, the three-element clusters that make contact are bonded yieldinga single megacluster that has the desired shape. After this step, all contacts (intra-cluster and inter-cluster) are treated as inter-cluster contacts. The main purpose of using clusters in the consolidationprocess is to add irregularity and randomness to the discretization, and to avoid having large regionsbeing consolidated into a close-packed arrangement. For circular or spherical elements, as shownin Figure 3(a), a frictional contact is detected when

d�RA + RB (3)

Due to the consolidation process, adjacent elements might slightly overlap, or may be slightlyseparated, as shown in Figure 3(b). In our algorithm to determine which contacts should be bonded,we prescribe a search radius rs around each element, so that if the surface of any other element is



element overlap

initial element separation

1x

2x

aR

bRd

sr

(a) (b)

Figure 3. Frictional contact detection and initial bonded contacts.

found within this radius, the element will be initially bonded to that neighbour. For this study, thevalue of rs was chosen rather arbitrarily (rs ≈ Ri + Ri/10, where Ri is the element radius), sincethe main purpose of this parameter is to ensure a dense bond network in the medium. In orderto have zero initial contact forces throughout the megacluster after consolidation is completed,element displacement interactions are measured relative to the initial distance between bondedelements. As such, the packing scheme and wall forces used for the consolidation process are notlikely to significantly influence the bulk macroscopic properties of the model.

Once the model’s geometry is established, boundary conditions are prescribed, such as thebuilt-in support at the left-hand side shown in Figure 2. Note that the medium that is produced hassome irregularity and porosity that is typical of a material such as concrete. As such, this methoddoes not rely upon any assumptions about a distribution of initial flaws in the medium. Also, thisgeneration scheme allows for ‘molding’ of structures or structural components of arbitrary shape.Further, for modelling higher porosity solids, DEM elements may be randomly removed until thedesired porosity is obtained. It should be noted that although the true microstructural propertiesof a material such as concrete could in principle be represented using this method, this was notattempted in the present study, and therefore the DEM medium generation is not being done in acontrolled fashion to model the actual microstructure of a material such as concrete.

5. DETERMINATION OF INTER-ELEMENT NORMAL AND SHEAR STIFFNESSES

The macroscopic linear elastic behaviour of isotropic materials can be characterized by two elasticconstants: Young’s modulus E and Poisson’s ratio �. The model parameters governing DEM elementinteractions (i.e. Kn and Kt) are typically determined by the ad hoc process of validating a numericalsimulation of a standard laboratory test with an actual experimental result [12, 13]. However, suchresults are highly likely to depend on DEM element size. That is, if a new discretization for aparticular problem is produced but with different DEM element size (such as making the elementsize smaller to improve accuracy of the simulation), the calibration process would need to berepeated in its entirety to obtain the new DEM parameters.



h R32

4R

1 2

34

5

6 7

σ elementmaterialisotropic

l

σ

(thickness = t)

1 2

34

5

6 7

(a) (b) (c)

Figure 4. Close-packed DEM unit cell for determination of inter-element spring constants.

In the present study we use a new approach wherein we attempt to theoretically establish the inter-element normal and tangential stiffnesses (and failure parameters, to be discussed later) as functionsof element size and commonly accepted material parameters including Young’s modulus andPoisson’s ratio. This is done by developing a DEMmodel of a unit cell of material. Figure 4(a) showsan isotropic solid material element (with known elastic modulus and Poisson’s ratio) subjected touniaxial stress. The volume of material is then modelled using the DEM close-packed unit cellshown in Figure 4(b). The unit cell contains seven elements having three degrees of freedom (d.o.f.)per element (two translations and one rotation). Due to the symmetry of loading, all rotations inthe unit cell are zero. Therefore, the matrix equation for the 14 translational d.o.f. can be expressedas

[K]{d} = {f} (4)

where the contents of the matrix [K] are given in Reference [20]. When subjected to loading shownin Figure 4(a), the material element will undergo rigid body motion and uniform straining. The rigidbody motion is irrelevant for our purposes, and thus we impose displacement constraints on theunit cell of material shown in Figure 4(b) that describes a uniform straining deformation, includingthe Poisson effect. For a uniform straining deformation, such constraints can be implemented byexpressing the 14 d.o.f. unit cell displacement vector as a function of only 2 d.o.f. (horizontaldisplacement of element 1 and vertical displacement of element 3, shown in Figure 4). That is,

{d} = [T]{d∗} (5)

where [T] is a transformation matrix and {d∗} is the displacement vector containing the tworetained d.o.f. (displacements are measured relative to element 5). Substituting Equation (5) intoEquation (4) and premultiplying by [T]T provides

[T]T[K][T]{d∗} = [T]T{f} (6)

Therefore, the reduced system can be expressed as

[K∗]{d∗} = {f∗} (7)



where [K∗] = [T]T[K][T] and {f∗} = [T]T{f}. Assuming small deformations, the reduced systemin Equation (7) is⎡

⎣2Knc2 + 2Kts

2 + 4Kn 4Kncs − 4Ktcs

4Kncs − 4Ktcs 8Kns2 + 8Ktc

2

⎤⎦

{dx1

dy3

}=

{8Rs�xx t

0

}(8)

where R is the DEM element radius, t is the thickness (out-of-plane depth) of the unit cell, ands and c are the sine and cosine of �/3, respectively. The in-plane macroscopic strains for the unitcell can be expressed as a function of the displacements in Equation (8) as

�xx = 2dx1l

= dx1R

(9)

�yy = 2dy3h

= dy32Rs

(10)

Equating the results obtained by Equations (9) and (10) to the expressions for in-plane strains givenby classical 2-D plane elasticity in terms of Young’s modulus and Poisson’s ratio, the displacementsdx1 and dy3 are found to be

dx1 = 2R�xx

E∗ (11)

dy3 = −2�∗Rs�xx

E∗ (12)

where

plane stress plane strain

E∗ = EE

(1 − �2)(13)

�∗ = ��

(1 − �)(14)

Substituting Equations (11) and (12) into Equation (8), the normal and tangential stiffness in theunit cell simplify to

Kn = 1√3(1 − �∗)

E∗t (15)

Kt = (1 − 3�∗)(1 + �∗)

Kn = (1 − 3�∗)√3(1 − �∗2)

E∗t (16)

which are independent of the DEM element radius R. Complete details for this process can be foundin Reference [20]. To reiterate, Equations (15) and (16) are the normal and tangential stiffnessesneeded so that a close-packed DEM discretization (Figure 4(b)) gives exact displacement results,



including the Poisson effect, for a linear elastic material under plane stress/strain conditionssubjected to x-direction uniaxial loading. As will be shown later, Equations (15) and (16) arealso shown to be adequate for irregular discretizations created using the procedure described inSection 4. For systems with a prescribed element size distribution, a new representative DEMunit cell should be constructed, and the procedure described above should be repeated in order todetermine a potentially different set of normal and shear stiffnesses.

Other researchers have derived a different expression for the normal stiffness where Kn = Et[15, 21]. It is important to mention that this expression can only be theoretically obtained byassuming a simple cubic arrangement of DEM elements. The theory provides no basis for thedetermination of the shear stiffness, and therefore this parameter is usually arbitrarily chosen as afraction of Kn [15].

Due to the location and orientation of the contacts in the DEM unit cell, the loading shownin Figure 4(c) was also considered in order to investigate possible anisotropy in the unit cell.Surprisingly, the normal and tangential stiffness produced by this exercise agree exactly withEquations (15) and (16). Assuming the shear stiffness must be non-negative, it is interesting tonote that these equations limit the maximum value of Poisson’s ratio to � = 1/3 for plane stress and� = 1/4 for plane strain. Such values correspond to a tangential stiffness of zero. Equations (15)and (16) were also obtained by Griffiths and Mustoe [22]. However, in their work they used adifferent approach in which the strain energy density of three DEM elements in a close-packedhexagonal arrangement was computed as a function of the normal and tangential stiffnesses andthe in-plane strains of the contact elements.

The DEM unit cell used in this study also proved to be useful to theoretically determine atime-step bound that ensures stability in explicit time integration, and this topic is discussed indetail in the following section.

6. STABILITY FOR EXPLICIT TIME INTEGRATION

The subject of DEM numerical stability in explicit time integration is a topic that is usually notclearly explained in the DEM literature. Explicit integration of the second order equation of motionby the central difference method is conditionally stable, and the time step �t should satisfy

�t<2

�max

(√1 − �2 − �

)(17)

for linear viscous damping, where � is the fraction of critical damping at �max, which is thehighest natural frequency of the mesh. Mass proportional damping can be implemented implicitlyif the mass matrix is diagonal, although this improves the time step only marginally since massproportional damping decreases at higher frequencies. Nonetheless, if mass proportional dampingis used and is treated implicitly, then Equation (17) applies where � is the fraction of criticaldamping at �max due to stiffness proportional damping only.

6.1. Estimates of �max

For practical problems, �max is rarely known and it is customary to use an upper bound for thisfrequency for purposes of determining a stable time step size. In finite element analysis, it is usual tobound �max for the mesh by the maximum frequency among all unconstrained individual elements,



DEM cell unit

nK

K

Figure 5. DEM discretization and unit cell used for computing the maximum frequency.

(�max)e [23]. However, in the DEM, the contact springs are massless, so that (�max)e = ∞, andhence the bound is of little practical usefulness, and other methods must be used.

6.1.1. Traditional estimation of �max in DEM computations. Most of the literature on time stepselection for DEM discretizations state that the critical time step is given by

�tc = 2

�max≈ �

2√kmax/mmin

(18)

where kmax is the largest inter-element spring stiffness, mmin is the mass of the smallest element,and hence

√kmax/mmin is a crude estimate of the highest natural frequency of vibration for the

model. Moreover, this estimate does not explicitly account for damping. For these reasons, auser-selected parameter � is included in Equation (18), and computational experience shows thatvalues of � near 0.1 are typically satisfactory to provide for stable computation [6]. However, thead hoc scheme given by Equation (18) is far from ideal, for several reasons. For one, too small atime step requires unnecessarily long execution time, and for problems with strong nonlinearities,the stable time step size may change throughout a simulation, and furthermore, if instability doesoccur, it may be difficult to detect because of the numerous energy-dissipative mechanisms. In thefollowing subsection, we discuss a new method for determining the stable time step size.

6.1.2. DEM unit cell approach for estimating (�max)e. While it is possible to use the Rayleighquotient or Gerschgorin bound to accurately bound �max for a DEM discretization [19], thefollowing approach using a unit cell is effective. The idea rests on the fact that for any discretization,the imposition of constraints raises the frequency spectrum. Thus, we consider the arrangementof elements shown in Figure 5, where the seven-element cluster of DEM elements is seen to beidentical to the unit cell used earlier in Figure 4(b). In Figure 5, the frequencies of a constrainedseven element cluster can be computed if all d.o.f. in the full model are constrained, except thoseassociated with the seven elements. Thus, the highest natural frequency of the seven elementcluster, (�max)e, will bound the highest natural frequency of the entire, unconstrained DEMdiscretization [24].

Mass and stiffness matrices were computed for the DEM unit cell, which consists of 21 d.o.f.(14 translational d.o.f. and seven rotational d.o.f.). Values for the stiffness and mass matrices canbe found in [20]. The stiffness matrix for the unit cell was computed using a value for Kt equalto Kn, which is the maximum value that Kt has for values of Poisson’s ratio ��0. Table I liststhe eigenvalues for the DEM unit cell shown in Figure 5. As shown in the table, the maximum



Table I. Eigenvalues of DEM unit cell shown in Figure 5.

1 2 3 4 5 6 7 8 9 10 11

2.3542 2.3542 4.7085 5.0 5.0 5.0 5.0 7.0 7.0 7.0 7.0

12 13 14 15 16 17 18 19 20 21

7.6458 7.6458 8.0 8.0 10.0 10.0 14.0 14.0 15.2915 16.0

natural frequency is (�max)2e = 16.0Kn/m, which gives (�max)e = 4.0

√Kn/m, where m is the

mass of each DEM element and Kn is the normal contact stiffness. Substituting this frequencyinto Equation (17), we obtain an upper bound estimate for the critical time step �tcrit given by

�tcrit = 12

√m/Kn

(√1 − �2 − �

)(19)

It is important to mention that our critical time step bound is a function of the unit cellconfiguration used to discretize the medium. For instance, a DEM unit cell having a simple cubicassembly will produce a different bound for the critical time step than a close-packed assembly.Note however that a close packed assembly involves the largest number of contacts possible for anarrangement of elements (assuming constant element diameter), and hence the highest frequenciesare expected for this situation, and Equation (19) should thus apply to other arrangements of DEMelements. For assemblies with a distribution of particle sizes (hence with varying mass and contactstiffness values), a DEM element could potentially have a larger number of contacts, and hencethe critical time step should be computed using Equation (18), where an estimate of �max wouldneed to be obtained by other methods.

Related to our approach here is the work of O’Sullivan and Bray [25] for determining a criticaltime step size. In their approach, they consider one contact between two DEM elements, and useonly the normal and tangential direction springs in this contact to determine a stiffness matrix withof six d.o.f. Because the contact springs have no mass associated with them, simple assumptionswere made to assign portions of the DEM element masses to the d.o.f. based on the number ofcontacts for each DEM element. For a close packed assembly, they report a critical time stepof �tcrit ≈ 0.408

√m/Kn (with no damping), which is lower than that given by Equation (19).

However, it is unlikely that this estimate is an upper bound.

7. ELASTIC RESPONSE OF A CANTILEVER BEAM

To test the accuracy of the expressions for contact stiffnesses obtained in Section 5, a cantileverbeam subjected to a slow sinusoidal tip load with frequency � was considered. The accuracy of theDEM simulations for this problem was studied by comparing the DEM results with simulationsperformed using finite element analysis (FEA) with the program ANSYS. In the FEA simulations,the cantilever beam was discretized using 1440 four-node plane quadrilateral elements.

Figure 6 shows three DEM discretizations for the cantilever beam using close-packed assemblieswith 81, 729, and 6561 elements, respectively, and for each refinement, the element radius was



tFF ωsin0=

tFF ωsin0=

tFF ωsin0=

Figure 6. DEM models of cantilever beam using a close-packed assembly.

6561 elements

FEA solution

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.005 0.01 0.015 0.02 0.025

81 elements729 elements6561 elements

tip

dis

pla

cem

ent,

m

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

tip

dis

pla

cem

ent,

m

time, sec

0 0.005 0.01 0.015 0.02 0.025

time, sec(a) (b)

Figure 7. DEM tip displacement results using a close-packed assembly and comparison with FEA solution.

reduced by a factor of 3. All DEM simulations were performed in plane stress, using the valuesfor normal and tangential spring stiffness given by Equations (15) and (16) with E = 1.2 GPaand � = 0.3. The forcing frequency was chosen to be 0.2 times the value of the fundamentalnatural frequency of the cantilever beam, and mass proportional damping with a fraction of criticaldamping of � = 10% was used at this frequency. Figure 7(a) shows the tip displacement as afunction of time for the three models. Figure 7(b) shows the tip displacement obtained by thefinest mesh along with the FEA solution. As shown in Figure 7(b), the agreement between theDEM solution and the FEA solution is excellent, and results also appear to converge to a definitevalue in the limit of mesh refinement. Applying Richardson extrapolation [19] to the results of



tFF ωsin0=

tFF ωsin0=

tFF ωsin0=

Figure 8. DEM models of cantilever beam using an irregular element assembly.

these three DEM models provides an estimate of �max = 0.4754 m as the steady-state maximumamplitude of vibration that the DEM models will converge to. The finite element solution provides�FEM = 0.4749 m, which is within 0.1% of the DEM extrapolated solution.

Figure 8 shows three DEM discretizations for the cantilever beam using an irregular assemblyof DEM elements, consisting of 600, 1200 and 2400 elements, respectively. The models wereconstructed using the procedure described in Section 4, and for each model refinement, the DEMelement radius was reduced by a factor of

√2. For these simulations, the forcing frequency was

chosen to be 0.2 times the value of the fundamental natural frequency of the beam, and massproportional damping with a fraction of critical damping of � = 10% was used at this frequency.Values for normal and tangential spring stiffness were obtained from Equations (15) and (16) usingE = 1.2 GPa and � = 0.3. Figure 9(a) shows the tip displacement as a function of time for thethree DEM models. Figure 9(b) shows the tip displacement obtained by the finest mesh along withthe FEA solution. Even though the results appear to show convergence, it is interesting to notethat contrary to the case using a close-packed arrangement, these models become more flexible asthe mesh is refined. Moreover, results appear to converge to a solution that is considerably lower(about 6.5%, using Richardson extrapolation) than the FEA solution.

One possible explanation for why the models for the close-packed arrangement become stifferas the DEM mesh is refined could be that for each refinement, there are contact springs that arecloser to the top and bottom surfaces of the beam, increasing the flexural stiffness of the model.This appears to have a greater effect on the behaviour of the model than adding more d.o.f., whichusually results in making a model more flexible. On the other hand, the addition of d.o.f. appears tohave a greater effect on the behaviour of the model when using an irregular element arrangement,resulting in a numerical solution that becomes ‘softer’ as the mesh is refined.



-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 0.001 0.002 0.003 0.004 0.005 0.006

2400 elements

FEA solution

tip

dis

pla

cem

ent,

m

time, sec

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 0.001 0.002 0.003 0.004 0.005 0.006

600 elements1200 elements2400 elements

tip

dis

pla

cem

ent,

m

time, sec(a) (b)

Figure 9. DEM tip displacement results using an irregular element assemblyand comparison with FEA solution.

Table II. Summary of convergence results for the element arrangements used.

Number of elements usedDEM elementarrangement Mesh 1 Mesh 2 Mesh 3 � p

Close-packed 81 729 7290 3 1.52Irregular 600 1200 2400

√2 3.00

7.1. Rate of convergence

Using the results of three meshes, the rate of convergence (if uniform) can be estimated using [19]

p= log

(1 − 2

2 − 3

)1

log �(20)

where i are, in this case, the values of the maximum steady-state displacement for the threedifferent discretizations, and � is the element size reduction factor for each mesh refinement,which must be the same for each refinement. As discussed in [19], Equation (20) is accurate onlyunder fairly strict requirements, including all meshes must be sufficiently fine to yield a constantrate of convergence. Table II shows the rate of convergence computed for the two differentDEM element packing arrangements used for the cantilever beam. The irregular element-packingarrangement shows a surprisingly fast rate of convergence with a value of p= 3. However, thisvalue may be artificially high because the 600 element mesh is too coarse. The rate of convergencefor the close-packed arrangement was p= 1.52, which is a more reasonable value. While theseresults are the first of their kind that we are aware of, considerable additional work is still requiredto adequately understand the convergence properties of DEM simulations of elastic responsein solids.



8. GENERAL CRITERIA FOR DEM CONTACT FAILURE

Most of the work reported in the literature for modelling fracture in DEM discretizations use asimple tensile force limit for contact failure [14, 21, 26–30]. That is, if the tensile force in a certaincontact exceeds a limit value, the contact breaks and can no longer support tensile forces. This forcelimit is usually determined by calibrating DEM simulation results to available experimental datafrom tensile and compressive tests for a particular material. Several researchers have imposed limitsfor both the tensile and shear strengths in contacts [12, 13, 15]. However, since force transmissionthrough a DEM medium can only occur via the inter-element contacts, the number of contactswithin a DEM medium will most likely affect the fracture behaviour. Therefore, this contact forcelimit most likely depends on element size.

Several researchers have used the maximum normal stress failure model to develop a simplefailure criterion for DEM models that is element size dependent, as follows. Considering a DEMmodel consisting of two elements, as shown in Figure 10, failure occurs when the contact stressreaches the tensile strength of the material [30]. Hence, the tensile force required to break thecontact is given by

fult = �ult(2R)t (21)

where R is the radius of the DEM elements, t is the thickness (out-of-plane depth), and �ultis the tensile strength of the material. Despite the linear dependence on R in Equation (21),if one considers the contact springs immediately in front of a crack tip, it is clear that use ofEquation (21) will always under-predict the far-field stress that would yield crack growth in alinear elastic material for sufficiently refined discretizations. In the remainder of this paper, wedevelop failure criteria that are objective to mesh refinement, using criteria based on a material’sfracture toughness, KIc, and critical energy release rate, G f .

σ

2R

nK

f

σ

σ

t

Figure 10. Tensile strength criterion for failure.



σ

θ

σ

στ

+

=

θθ

θ

θ

θπσ

σσ

θ

θθ

Figure 11. Crack tip stress field for pure Mode I cracking.

The results to be described here begin with the original work of Potyondy and Cundall [15]. Intheir work, they postulate a breaking strength between the bonds of contacting DEM elements, andthen address the issue of determining the fracture toughness of the material that is being modelledby the DEM discretization. The approach described herein differs from theirs in that we view thefracture toughness as a fundamental material parameter, and then attempt to use theory to developexpressions for the behaviour of bonds between contacting DEM elements. A main feature of suchan approach is that it promises to provide convergence. That is, it provides a precise specificationof how the contact behaviour between DEM elements must change as a function of element size sothat convergence to the exact solution is achieved as the DEM element size becomes smaller duringmodel refinement. To model the progressive failure of a solid due to crack growth, a local rupturecriterion is applied to the bonded contacts between interacting elements. The plane stress/strainnear crack tip stress field for pure Mode I loading in a homogeneous, isotropic, linear elasticmaterial is given by [31]

�i j (r, �) = KI√2�r

Fi j (�) +∞∑n=0

Anrn/2F (n)

i j (�) (22)

Since crack growth is controlled by the stresses near the crack tip, i.e. at small r , the first term inEquation (22) gives the highest contribution to the stress field at the crack tip. Therefore, keepingonly the first term in Equation (22), the well-known expression for the stresses close to the cracktip, shown in Figure 11, is obtained. Restricting attention to � = 0, the stress �� is then

�� = KI√2�r

(23)

The force required to advance the crack by breaking the contact at the crack tip of a DEM model isdetermined by integrating the stress field over the DEM element diameter and thickness as shownin Figure 11. Therefore,

f =∫ t

0

∫ 2R

0�� dr dt =

∫ t

0

∫ 2R

0

KI√2�

r−1/2 dr dt (24)



Figure 12. Mixed mode fracture in DEM contacts for a close-packed assembly.

f = 2KI t

√R

�(25)

where f is defined in Figure 10. Assuming linear elastic fracture mechanics (LEFM) conditions,cracking will occur when the value of KI reaches the material’s fracture toughness KIc. Therefore,the required force to break the contact is given by

fult = 2KIct

√R

�(26)

As shown in Equation (26), material failure parameters in this approach are dependent onDEM element size and the fracture toughness of the material. As described above, this result issignificant because it prescribes exactly how the DEM inter-element strength must change as afunction of element size so that convergence to the LEFM behaviour is obtained as element radiusR vanishes in the limit of mesh refinement. However, it is important to mention that this result wasobtained for a DEM model consisting of a simple cubic element arrangement, and considers onlyMode I cracking, where there is no force being carried by the tangential spring at the crack tipcontact. Since in a two-dimensional simulation both Modes I and II can be present simultaneously,especially for a close-packed discretization, a fracture criterion should be developed to account forthis type of loading where both the normal and tangential springs at the crack tip contact are beingdeformed. For instance, Figure 12 shows a DEM model of a plate with an edge crack subjectedto a uniaxial stress. Even though macroscopically this behaviour would be pure Mode I, since thefar-field stress is perpendicular to the crack plane, both the normal and tangential springs at thecrack tip are neither parallel nor perpendicular to the crack plane. As such, both springs contributeto resisting the crack advance.

Another important issue to consider is that failure according to Equation (26) assumes thatfracture mechanics controls failure over other phenomena such as the yield strength or maximumtensile strength. Due to the nature of the DEM medium that is proposed in this work (irregular ele-ment arrangement with voids), clearly there should be a correlation between the fracture toughnessof the material as a criterion for fracture, and the material’s ultimate tensile strength. Figure 13shows a general DEM model consisting of an irregular assembly of elements of different size. Forsuch a model, if pre-existing flaws of width a0 populate the medium, the fracture stress should



Figure 13. Irregular assembly of DEM elements of different size.

follow from the fracture toughness KIc of the material as

�ult = KIc√�a0

(27)

In other words, there is a transition crack length given by

at = 1

�

(KIc

�ult

)2

(28)

below which there will be little or no strength reduction due to the initial flaws in the material [32].In a DEM model with no initial flaws, where the fracture toughness is a constant material property,the failure criterion given by Equation (26), which is based on the material’s fracture toughness,will be inappropriate. For this case, the ‘crack length’ will be given by the lattice contact spacing,and therefore the fracture toughness will be element size dependent. Since there are no guaranteesthat, in the limit of mesh refinement, the models developed in this work will have flaws greater insize than that given by Equation (28), an alternate criterion should be developed in order to accountfor a condition in which failure will be dictated by the ultimate tensile strength of the material. Inwhat follows, we develop a contact failure criterion that incorporates both the material’s tensilestrength �ult and fracture toughness KIc.

The energy release rate in a homogeneous, isotropic linear elastic material is determined bycalculating the total work balance per unit thickness for extending a crack a distance �a [31].Therefore,

G = 1

�a

∫ �a

0

∫ � f

0�yy d� dx (29)

where � f is the crack opening displacement at position x behind the crack tip after crack extension�a, and �yy is a function of both position x and the crack opening displacement � behind the



crack tip. By solving Equation (29) and taking the limit as �a → 0, we obtain the well-knownrelationship between energy release rate and stress intensity factor KI

G = K 2I

E∗ (30)

where E∗ is given by Equation (13).In the present work, it is assumed that the energy required to fracture an isotropic material is

independent of the mode of fracture. Therefore, the fracture energy G f or critical energy releaserate is then given by Equation (30) when the value of KI reaches the material’s fracture toughnessKIc. That is,

G f = K 2I c

E∗ (31)

For application in the DEM, �a in Equation (29) is replaced by the element diameter andtherefore �yy is averaged by dividing the contact force by the diameter and thickness of theDEM element. Moreover, the crack opening displacement is also an average computation over thediameter of the element and consists of the stretch of the normal spring in the crack tip contact.Therefore, in the present approach, the critical energy release rate is obtained by computing thearea under the curve of �yy vs � at contact failure.

Returning to the DEM unit cell described earlier and shown in Figures 4 and 5, we can obtainboth the normal and tangential contact forces as a function of the applied uniaxial tensile stresssupported by the unit cell. When the applied stress approaches the material’s tensile strength �ult,using the unit cell results for normal and tangential contact stiffness, these forces are

f critn = (Rt)�ult2(1 − �∗)

(√3 − �∗

√3

)(32)

f critt = (Rt)�ult2(1 − �∗)

(1 − 3�∗) (33)

where �∗ is given by Equation (14). The derivation of Equations (32) and (33) can be foundin Reference [20]. These critical forces for a close-packed assembly are analogous to the cri-terion leading to Equation (21), where a simple cubic arrangement is implied. For a flaw-freemedium, contact normal and tangential forces will reach the critical forces given by Equations (32)and (33) simultaneously and then the model will reach the ultimate tensile strength of the material.Regardless of the material’s fracture toughness, the contact forces should not exceed those givenby Equations (32) and (33). Otherwise, the DEM model’s tensile strength will be greater than thematerial’s tensile strength.

When cracks are present, the contact forces at the crack tip reach the critical forces given byEquations (32) and (33) relatively fast during loading since stresses at a DEM crack tip becomeinfinite in the limit of mesh refinement. However, it is proposed that complete failure shouldnot occur before the contact achieves the material’s fracture energy G f . In order to satisfy thiscondition, the stress–displacement curve shown in Figure 14 is then adopted for both the normaland tangential contact at the crack tip. As shown in this figure, the contact springs (both normal andtangential) behave linearly up to the point where the forces satisfy Equation (32) or Equation (33)



fG

nδ

tσ

fG

critnσ

nσ

nδcrit

nδ

crittσ

tσ

tδcrit

tδ t

maxδnmaxδ

contact tensilecritical stress

contact shearcritical stress

(a) (b)

Figure 14. DEM contact stress–displacement behaviour and fracture energy.

for the normal and tangential springs, respectively. Once these forces are reached, the behaviouris proposed to be plastic until the area under the stress–displacement curve for the normal spring,shown in Figure 14(a), reaches the fracture energy of the material, given by Equation (31). Itshould be noted that the area under the curve in Figure 14(a) is the minimum energy lost, asenergy may also be stored in the tangential direction of the contact. However, this approach provedto be accurate enough since the critical energy release rate is computed considering pure Mode Iloading only.

The maximum normal displacement �nmax is obtained by computing the area under the curvein Figure 14(a) and equating this to the material’s fracture energy G f . Therefore, the maximumnormal displacement is given by

�nmax = K 2I c

�critn E∗ + �critn

2(34)

where

�critn = f critn

2Rt= �ult

4(1 − �∗)

(√3 − �∗

√3

)(35)

�critn = f critn

Kn= R�ult

2E∗ (3 − �∗) (36)

Equation (34) gives a displacement-based failure criterion, where the maximum displacement is afunction of the material’s tensile strength, elastic modulus, Poisson’s ratio, and fracture toughness(each of which could be specified independently). Since contact stresses are computed using theradius of the elements in contact, the failure criterion is also element size dependent. It shouldbe noted that if unloading occurs in the contact after it reaches the plastic range, the slope of theunloading force–displacement curve will be equal to its initial slope (Kn and Kt for the normaland tangential stiffnesses, respectively).

In order to test the failure criterion given in Equation (34), three DEM models representing aplate with an edge crack subjected to pure Mode I loading were developed. The models, shown in



σ

σ

σ

σ

σ

σ

-0.5

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6

stre

ss, M

Pa

time, ms

radius = 7.812500 mmradius = 3.906250 mmradius = 1.953125 mm

Figure 15. Convergence for crack growth in a DEM model of a cracked plate subjected topure Mode I loading; hollow data points in the plot represent the value of far-field stress

that cause unstable crack growth.

Figure 15, consist of 1172, 4713, and 18 899 elements, respectively, arranged in a close-packed as-sembly. The material modelled by these DEM meshes has elastic modulus E = 29.7GPa, Poisson’sratio �= 0.3, ultimate tensile strength �ult = 3.44MPa, and fracture toughness KIc = 1.0MPa

√m

(i.e. concrete). The edge crack (shown by the solid line at the centre of each model) was cre-ated by giving contacts along the crack the standard behaviour described by the frictional contactmechanism shown in Figure 1. All simulations were performed in plane strain, applying a uniformstress to the top and bottom boundaries, which was slowly increased until failure in the model.The results of the DEM simulations are also shown in Figure 15, where the hollow data pointsrepresent values of the far-field stress that cause the crack to propagate unstably across the model.

An exact solution for this problem (finite width plate with an edge crack with pure ModeI loading) does not exist due to the difficulty of finding a solution that satisfies the traction-free boundary conditions on the free edges. However, several numerical techniques have beendeveloped in order to obtain very accurate approximate solutions, and a commonly used solution isgiven by [33]

�F = KIc√�a

[1.12 − 0.23(a/c) + 10.6(a/c)2 − 21.7(a/c)3 + 30.4(a/c)4]−1 (37)

where �F is the value of the far-field stress that causes the crack to propagate unstably across theplate, a is the crack length (a = 0.03125 m) and c is the width of the plate (a/c= 1/8 for all themodels of Figure 15). Equation (37) was obtained by a least-squares polynomial fit to data obtainedby the Boundary Collocation Method, which is accurate to within 0.5% for a/c�0.6. Using theparameters cited earlier, this equation provides �F = 2.61MPa. An extrapolated solution based onthe results of the three DEM models, assuming regular mesh refinements, can be computed by



fG fG

critnσ

nσ

nδcritnδ n

maxδ

contact tensilecritical stess

Figure 16. Alternate DEM contact normal stress–displacement behaviour.

Richardson extrapolation [19] using

e = 13 − 22

1 − 22 + 3(38)

where in this case, i represents the value of the far-field stress that causes the crack to propagateunstably for model i , and e is the value to be extrapolated. Applying Equation (38) to the resultsof the three DEM models in Figure 15 provides an estimate of �ult = 2.595 MPa as the value ofthe far-field stress that the DEM models will converge to. This value is within 0.5% of the valueprovided by Equation (37), which as cited earlier could be in error by as much as 0.5%. Therefore,the proposed criterion for contact failure appears very promising.

Applying Equation (20) to the results of the three models, a convergence rate p= 1.8 (almostquadratic) was obtained, which is a reasonable value and is encouraging. The models shown inFigure 15 were also analysed with no initial crack, and all three models failed at essentially thesame maximum stress of �= 3.44 MPa (independent of model discretization), which agrees withthe value of the ultimate tensile strength of the material �ult. Therefore, the proposed methodprovides excellent results regardless of the type of failure (fracture toughness criterion or ultimatetensile strength criterion).

Rather than the plastic behaviour proposed in this work, as shown in Figure 14, severalresearchers [34–36] have proposed the DEM contact stress–displacement behaviour shown inFigure 16, where the contact normal stress decreases linearly to zero after reaching a criticaltensile stress. Analogous to the approach presented in this work, the slope of the curve in thesoftening stage is determined such that at failure, the area under the contact stress–displacementcurve is equal to the material’s fracture energy. However, when this contact behaviour was usedin the DEM models shown in Figure 15 with the contact parameters developed in this work(i.e. contact stiffness Kn, and critical tensile stress �critn , and fracture energy G f ), the DEM resultsunder-predicted the far-field stress that causes the crack to propagate unstably. That is, even thoughthe approach could be used in some cases to model material softening phenomena, it does notappear to guarantee that the DEM model results will converge to the exact solution (assumingLEFM conditions) in the limit of mesh refinement.



9. DEM FOR MODELLING PENETRATION

To illustrate the application of the DEM method and to demonstrate its utility for modelling chal-lenging problems, we will consider the example of projectile penetration of a concrete structuralmember. While problems of this type have been successfully analysed using FEA (e.g. [37, 38]),there are numerous difficulties confronting such analyses, including the need for continuum con-stitutive models which are not well established for many materials of engineering importance, overthe range of conditions and states pertinent to extreme loading.

Figure 17 shows a projectile penetrating a concrete beam at several stages of a DEM sim-ulation. The concrete beam consists of 12 000 DEM elements and was constructed using theprocedure described in Section 4. The projectile is steel and consists of 351 DEM elements ar-ranged in a close-packed assembly. While it is possible to allow for damage (e.g. erosion) ofthe projectile using the methods discussed in this paper, in this example the projectile undergoesno damage. As shown in Figure 17, nonlinear phenomena including widespread cracking andthe progressive gross failure and fragmentation of the solid beam into particulates is simulatedautomatically.

It is important to mention that projectile impact can generate very high loading rates which canproduce strain rate effects including increased tensile strength, compressive strength, and fractureenergy, in addition to other modifications of material behaviour. Some of these effects have beenincluded in the results shown in Figure 17. However, because a detailed discussion of these strainrate enhancements are not central to the main theme of this paper, we refer to Reference [20]for details.

Stage 1 (t = 0 ms) Stage 2 (t = 1.0 ms)

Stage 3 (t = 2.67 ms) Stage 4 (t = 6.0 ms)

Figure 17. DEM simulation of projectile penetration into concrete beam.



10. SUMMARY AND CONCLUSIONS

A DEM was developed as a general computer technique for unified modelling of the mechanicalbehaviour of solid and particulate materials, including the transition from solid phase to particulatephase. Megaclustering was used to model solids, and the method by which a problem domain isconstructed proved to be effective for introducing irregularity in the model, with no anisotropiesor preferential directions for fracture. An energy-based criterion was developed for contact failureand this criterion was shown to be more appropriate than a criterion based purely in a material’sultimate tensile strength and fracture toughness. For purposes of selecting a time step size forintegrating the equations of motion by the central difference method, a method was developedto estimate the maximum frequency of vibration in a DEM model, which gives a more accurateprediction than the ad hoc procedure that has historically been used.

Using a DEM model of a unit cell of homogeneous material, a theoretical method was developedto determine the DEM contact normal and tangential stiffnesses as functions of known materialproperties so that convergence is obtained with model refinement and the correct elastic behaviouris produced. To summarize, the normal and tangential stiffnesses are given by

Kn = 1√3(1 − �∗)

E∗t (39)

Kt = (1 − 3�∗)(1 + �∗)

Kn = (1 − 3�∗)√3(1 − �∗2)

E∗t (40)

Regarding contact failure for close-packed and irregular DEM assemblies, a novel DEM contactbehaviour was developed that respects failure governed by the material’s fracture toughness andthe material’s ultimate tensile strength. The contact stress–displacement curve is proposed to belinear until the contact normal and shear stresses reach critical values given by

�critn = f critn

2Rtwhere f critn = (Rt)�ult

2(1 − �∗)

(√3 − �∗

√3

)(41)

�critt = f critt

2Rtwhere f critt = (Rt)�ult

2(1 − �∗)(1 − 3�∗) (42)

After the contact forces reach the values given by Equations (41) and (42), the contact behaviour isproposed to be plastic until the area under the stress–displacement curve for the normal directionspring reaches the material’s fracture energy, given by

G f = K 2I c

E∗ (43)

and the maximum normal displacement is then given by

�nmax = K 2I c

�critn E∗ + �critn

2where �critn = f critn

Kn(44)



Even though the simulations showed very good agreement with theoretical results for bothelastic behaviour and fracture, the rate of convergence was not clear, and it appears to dependon the element arrangement. While these results are the first of their kind that we are aware of,considerable additional work is still required to adequately understand the convergence propertiesof DEM models of solids. It should also be noted that all expressions and criteria presented inthis paper are for two-dimensional DEM models. However, the theory used could in principle beapplied to develop analogous expressions and criteria for three-dimensional systems.

ACKNOWLEDGEMENTS

The financial support of the National Highway Institute in the form of a Dwight D. Eisenhower Trans-portation Fellowship for Federico Tavarez, and the Computational Mathematics Program of the U.S. AirForce Office of Scientific Research (AFOSR), Grant No. F49620-03-0216, is gratefully acknowledged.We are also grateful for the valuable assistance and collaboration of Dr Richard P. Jensen, Sandia NationalLaboratory, and numerous conversations with Dr David Potyondy, Itasca Consulting Group, Inc., and Prof.Walter Drugan, Department of Engineering Physics, University of Wisconsin-Madison.

REFERENCES

1. Cundall PA, Strack ODL. A discrete element model for granular assemblies. Geotechnique 1979; 29(1):47–65.2. Ghaboussi J, Barbosa R. Three-dimensional discrete element method for granular materials. International Journal

for Numerical and Analytical Methods in Geomechanics 1990; 14:451–472.3. Dobri R, Ng T. Discrete modelling of stress-strain behaviour of granular media at small and large strains.

Engineering Computations 1992; 9:129–143.4. Morgan JK, Boettcher MS. Numerical simulations of granular shear zones using the distinct element method—

1. Shear zone kinematics and the micromechanics of localization. Journal of Geophysical Research 1999;104(B2):2703–2719.

5. Cleary P. Modeling comminution devices using DEM. International Journal for Numerical Methods inGeomechanics 2001; 25:83–105.

6. Jensen RP, Bosscher PJ, Plesha ME, Edil TB. DEM simulation of granular media—structure interface: effects ofsurface roughness and particle shape. International Journal for Numerical and Analytical Methods in Geomechanics1999; 23:531–547.

7. Jensen RP, Edil TB, Bosscher PJ, Plesha ME, Kahla NB. Effect of particle shape on interface behaviour ofDEM-simulated granular materials. The International Journal of Geomechanics 2001; 1(1):1–19.

8. Thomas PA, Bray JD. Capturing nonspherical shape of granular materials with disk clusters. Journal ofGeotechnical and Geoenvironmental Engineering 1999; 125(3):169–178.

9. Boulon M, Nova R. Modeling of soil structure interface behaviour: a comparison between elastoplastic and ratetype laws. Computers and Geotechnics 1990; 9:21–46.

10. Jensen RP, Plesha ME, Edil TE, Bosscher PJ, Kahla NB. DEM simulation of particle damage in granularmedia—structure interfaces. The International Journal of Geomechanics 2001; 1(1):21–39.

11. Ting JM. A robust algorithm for ellipse-based discrete element modelling of granular materials. Computers andGeotechnics 1992; 13(3):175–186.

12. Potyondy DO, Cundall PA, Lee CA. Modeling rock using bonded assemblies of circular particles. Proceedingsof the 2nd N. American Rock Mechanics Symposium, Montreal, 1996; 1937–1944.

13. Magnier SA, Donze FV. Numerical simulations of impacts using a discrete element method. Mechanics ofCohesive-Frictional Materials 1998; 3:257–276.

14. Donze FV, Bouchez J, Magnier SA. Modeling fractures in rock blasting. International Journal of Rock Mechanicsand Mining Sciences 1997; 34(8):1153–1163.

15. Potyondy DO, Cundall P. A bonded-particle model for rock. International Journal of Rock Mechanics and MiningSciences 2004; 41:1329–1364.

16. Cundall PA, Strack ODL. The distinct element method as a tool for research in granular media—Part I. NSFReport Grant ENG76-20711, 1978.



17. Cundall PA, Strack ODL. The distinct element method as a tool for research in granular media—Part II. NSFReport Grant ENG76-20711, 1979.

18. Williams JR, O’Connor R. A linear complexity intersection algorithm for discrete element simulation of arbitrarygeometries. Engineering Computations 1995; 12(2):185–208.

19. Cook RD, Malkus DS, Plesha ME, Witt RJ. Concepts and Applications of Finite Element Analysis (4th edn).Wiley: New York, 2002.

20. Tavarez FA. Discrete element method for modelling solid and particulate materials. Doctoral Thesis, Universityof Wisconsin-Madison, 2005.

21. Masuya H, Kajukawa Y, Nakata Y. Application of the distinct element method to the analysis of concretemembers under impact. Nuclear Engineering and Design 1994; 6(2):283–294.

22. Griffiths DV, Mustoe GG. Modelling of elastic continua using a grillage of structural elements based ondiscrete element concepts. International Journal for Numerical and Analytical Methods in Engineering 2001;50:1759–1775.

23. Belytschko T, Liu WK, Moran B. Nonlinear Finite Elements for Continua and Structures. Wiley: New York,2000.

24. Fried I. Influence of Poisson’s ratio on the condition of the finite element stiffness matrix. International Journalof Solids and Structures 1973; 9:323–329.

25. O’Sullivan C, Bray JD. Selecting a suitable time step for discrete element simulations that use the centraldifference time integration scheme. Engineering Computations 2004; 21(2/3/4):278–303.

26. Zubelewicz A, Bazant ZP. Interface element modelling of fracture in aggregate composites. Journal of EngineeringMechanics 1987; 113(11):1619–1630.

27. Bolander JE, Saito S. Discrete modelling of short-fiber reinforcement in cementitious composites. AdvancedCement Based Materials 1997; 6:76–86.

28. Brara A, Camborde F, Klepaczko JR, Mariotti C. Experimental and numerical study of concrete at high strainrates in tension. Mechanics of Materials 2001; 33:33–45.

29. Mishra BK, Thornton C. Impact breakage of particle agglomerates. International Journal of Minerals Processing2001; 61:225–239.

30. Sawamoto Y, Tsubota H, Kasai Y, Koshika H, Morokawa H. Analytical studies on local damage to reinforcedconcrete structures under impact loading by the discrete element method. Nuclear Engineering and Design 1998;179:157–177.

31. Anderson TL. Fracture Mechanics—Fundamentals and Applications (2nd edn). CRC Press: Boca Raton, 1994.32. Dowling NE. Mechanical Behavior of Materials—Engineering Methods for Deformation, Fracture and Fatigue.

Prentice-Hall: Englewood Cliffs, NJ, 1993.33. Cook RD, Young WC. Advanced Mechanics of Materials (2nd edn). Prentice-Hall: Englewood Cliffs, NJ, 1999.34. Bazant ZP, Planas J. Fracture and Size Effect in Concrete and other Quasibrittle Materials. CRC Press:

Boca Raton, 1993.35. Mohammadi S, Forouzan-Sepehr S, Asadollahi A. Contact based delamination and fracture analysis in composites.

Thin-Walled Structures 2002; 40:595–609.36. Davie CT, Bicanic N. Failure criteria for quasi-brittle materials in lattice-type models. Communications in

Numerical Methods in Engineering 2003; 19:703–713.37. Schwer LE, Day J. Computational techniques for penetration of concrete and steel targets by oblique impact of

deformable projectiles. Nuclear Engineering and Design 1991; 125:215–238.38. Yadav S, Repetto EA, Ravichandran G, Ortiz M. A computational study of the influence of thermal softening

on ballistic penetration in metals. International Journal of Impact Engineering 2001; 25:787–803.


tavarez_et_al-2007-international_journal_for_numerical_methods_in_engineering.pdf

Documents