Numerical simulation of breakage of two-dimensional...

(2006) 100–112www.elsevier.com/locate/powtec

Powder Technology 166

Numerical simulation of breakage of two-dimensional polygon-shapedparticles using discrete element method

E. Seyedi Hosseininia ⁎, A.A. Mirghasemi

Department of Civil Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

Received 7 June 2005; received in revised form 6 January 2006; accepted 24 May 2006

Abstract

Using DEM (Discrete Element Method), a model is presented to simulate the breakage of two-dimensional polygon-shaped particles. In thismodel each uniform (uncracked) particle is replaced with smaller inter-connected sub-particles which are bondedwith each other. If the bond betweenthese sub-particles breaks, breakage will happen. With the help of this model, it is possible to study the influence of particle breakage on macro andmicromechanical parameters. In this simulation, the evolution ofmicrostructure in granular assemblies can be seen by tracing of coordination numberduring the shear process. Also variation of contact normal, normal force and tangential force anisotropy can be tracked. To do so, two series of biaxialtest simulations (breakage is enabled and disabled) are conducted on assemblies of two-dimensional polygon-shaped particles and the results arecompared. The results are presented in terms of macro and micro mechanical behavior for different confining pressures.© 2006 Elsevier B.V. All rights reserved.

Keywords: DEM; Particle breakage; Strength; Dilatancy; Micromechanics; Anisotropy; Granular material; Confining pressure

1. Introduction

Stability of soil structures such as breakwaters and rockfilldams is in concern of shear parameters and the behavior ofgranular media. In general, shear resistance and behavior ofgranular materials depends on different factors such as mineral-ogical composition, particle grading, size and shape of particles,fragmentations of particles and stress conditions.

Breakage of particles will happen in such high structuresespecially in the lower layers where there are levels of significantpressure caused by the upper layers. Crushing of large particlesinto smaller ones result in changes in the design gradation curve;therefore, the mechanical behavior of granular material alters.

Influence of particle breakage on internal friction angle anddeformability of granular materials can be studied using expe-rimental tests such as triaxial and unconfined compression tests[9,1,4,8,27]. Performing triaxial compression tests on coarsegranular materials, Marsal [9] found out that the most importantfactor affecting both shear strength and compressibility is the

⁎ Corresponding author. Tel.: +98 912 226 3498; fax: +98 21 6640 3808.E-mail address: [email protected] (E. Seyedi Hosseininia).

0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.powtec.2006.05.006

phenomenon of fragmentation undergone by a granular bodywhen subjected to change in its state of stress both during uniformcompression stage and during deviatoric load application. Also theresults showed that in granular media, the compressibility is aconsequence of complex phenomenon that takes place as a resultof displacements between particles combined with the particlebreakage. Varadarajan et al. [27] have investigated the behavior oftwo dam site rock materials (Ranjit Sagar and Purulia) in triaxialcompression tests which the former consisted of rounded and thelatter angular particles. It is interesting to note that the volumechange behavior of two rockfill materials was significantly dif-ferent from each other. During the shearing stage of the triaxialtest, compression, rearrangement and breakage of particles tookplace. The rounded material exhibited continuous volume com-pression, while the angular particles dilated and expanded afterinitial compression in volume; angular materials provide a highdegree of interlocking and cause dilation during shearing. Alsothey observed that a greater degree of particle breakage occurswith the larger particles because of the greater force per contact[7]. There are two factors governing on the shear resistance asinterlocking between particle and particle breakage. The effect ofincrease in interlocking is to increase the shearing resistance, whilethe effect of breakage of particles is vice versa. Also it is noted that

mailto:[email protected]

http://dx.doi.org/10.1016/j.powtec.2006.05.006

Fig. 1. Breakage modeling (a) Base particle P and its sub-particles (b) The bondpoints of two adjacent sub-particles.

101E. Seyedi Hosseininia, A.A. Mirghasemi / Powder Technology 166 (2006) 100–112

angular particles are more susceptible to break than roundedparticles.

Performing such tests on material with large particles in orderto study the behavior of materials such as rockfill would be costlydue to the required large size of specimen. At the present research,an investigation is made to study the influence of particlebreakage on behavior of granular media using a numericalmethod. In recent years, along with the progress of numericalmethods and computer technology, different methods have beenused to model breakage of brittle bodies with the help of DiscreteElementMethod (DEM).Among thesemethods, are the approachused by Cundall [3], Potapov and Campbell [17,18], the methodbased on simultaneous utilization of Molecular Dynamics (MD)(Kun and Herrmann [6]), and the 3D approach used by Robertsonand Bolton [19], McDowell and Harireche [11].

2. Particle breakage simulation in DEM

Prior to description of the method used at the presentresearch, a brief overview of the above methods is presented.

2.1. Brief review of different modelings

Cundall who is a pioneer in use of DEM in studying behaviorof granular media and stability of rock slopes prepared a codecalled RBMC in which breakage mechanism of rock blocks wassimulated similar to that of Brazilian test [3,28]. In this code, ineach cycle of simulation all of the point loads applied to eachblock are checked and then the application point andmagnitude of

Fig. 2. Replacement of the relative displacement of two sub-particles with threenormal, shear and rotational components.

the two maximum loads, which are applied in opposite directionof each other, are determined.

RBMC is based on the conception that particle breakagehappens instantly (during 1 cycle) and the block is divided intotwo pieces through the line connecting application point of theloads. In this method, after each breakage occurrence, the pri-mitive block is omitted and two blocks with new geometry aregenerated; thus it is necessary to calculate the geometry, mass andmoment of inertia of these new particles and save them in a bigamount of computer memory.

The other procedure for breakage simulation is the methodapplied by Potapov and Campbell [17,18]. They have studied thebreakage induced in a single circular particle that impacts on thesolid plates (1994) and brittle particle attrition in a shear cell(1997). They tried tomodel the breakage of square particles placedbetween two top and bottom plates with 90° saw tooth patterns. Inboth simulations, a breakable solid material is created by gluingtogether unbreakable and non-deformable solid triangular ele-ments. It is assumed that a glued joint can only withstand normal

Fig. 3. Three simulation stages for breakable of particles for confining pressure ofPcp=0.5 MPa; (a) Initial generated assembly of particles, (b) Isotropicallycompacted assembly by hydrolstatic pressure, (c) Sheared assembly in biaxial test.

Table 1Parameter used in test A (breakage disabled)

Normal and tangential stiffness (N/m) 2.0×107

Unit weight of particles (kg/m3) 2500Time step (s) 3.2E–4Strain rate 0.005

102 E. Seyedi Hosseininia, A.A. Mirghasemi / Powder Technology 166 (2006) 100–112

tensile stress up to some limit. If the tensile stress on any portion ofthe joint exceeds the limit, the glue along that portion breaks andcan no longer support tensile stress; crack forms but only alongthat portion of the joint for which the tensile strength is exceeded.

The other study in the area of breakage modeling and crackformation in brittle bodies are based on simultaneous use of DEMandMD (Kun and Herrmann [6]). Molecular Dynamics (MD) is acomputational technique that considers a macroscopic material asan assemblage of microscopic particles.

In order to study the process of fragmentation in two-dimensional brittle blocks and observing the relationship of thesize of broken parts with one another, Kun and Herrmann [6]considered each block as a mesh of inter-connected tiny cellslocated in a plane. This cellular mesh is generated by the use of arandomprocess (VoronoiConstruction). Each cell is a rigid convexpolygon that as the smallest component of the block neither breaksnor deforms and acts as a distinct element of other cells. Cells haveone rotational and two linear degrees of freedom in the block planeand their behavior in contact is simulated by DEM.

In order to keep the unity of the cells forming a block, thecenter ofmass (center of area) of each cell is connected to themasscenter of neighboring cells through an elastic beam. If at a specifictime during the simulation, the relative displacement between twocells enlarges so much that the stress formed in the beam con-necting them exceeds the beam bearing capacity, the bond will bebroken. This is the staring point in crack formation and the crackenlarges gradually as the beams connecting the consequentparticles break. When an assemblage of inter-connected cells isthoroughly disconnected from the primitive block, breakage hasoccurred. In the recent model, it is feasible to study the process ofcrack formation in a brittle body and not in an assembly ofparticles that each one behaves as an individual body and theyhave no connection to each other.

In order to simulate three-dimensional crushable soils, anapproach is produced usingDEM. In thismethod, agglomerates aremade by bonding elementary spheres in ‘crystallographic’ arrays.This approach is used in the program PFC3D (Itasca ConsultingGroup, 1999). This program uses the soft contact approach of thedistinct element methods, which assumes that each element has afinite normal stiffness and represents elastic flattening at contactsby allowing the bodies to overlap. A stiffness model, a bondingmodel and slipmodel are included in the constitutive representation

Fig. 4. Variation of bond breakage during simulation in two stages of hydrostaticcompaction and biaxial shearing for Pcp=0.5 MPa.

of contact points between the elementary spheres that are the basicbuilding block. In the linear contact model, it is assumed that eachsphere have a normal and a shear stiffness. The simple contact bondcan be envisaged as a pair of elastic springs at a point of glue. Itserves to limit the total normal and shear forces that the contact cancarry by enforcing bond-strength limits. The maximum tensileforce that the bond can sustain in tension and the maximum shearforce it can withstand before breaking are specified when the bondis created and may be modified at any time during the simulation.The bond breaks if either of these values is exceeded. As the simplecontact bond acts over a vanishing small area of contact point, itdoes not resist bending moment. This means that it has noresistance to rolling of a sphere bonded adjacent to it if no thirdbody exists to restraint the motion. This approach has been used byRobertson and Bolton [19] and McDowell and Harireche [11].

A slip model acts between unbonded objects in contact, orbetween bonded objects when their contact breaks. It limits theshear force between objects in contact and allows slip to occurat a limiting shear force, governed by Coulomb's equation.

This approach has been accomplished for simulation of silicasand grains and the results compared with the real data for silicasand. [2] Although this method can model the behavior of sandswell, it can not be used for the particles with sharp angles suchas rockfills, since the sand agglomerate in this method consistsof smaller rounded spheres.

2.2. Breakage modeling in this research

The aim of this research is to find the influence of breakage ofangular shaped particles on the behavior of materials such asrockfill. It has been tried to model the particle breakage in a waythat less number of particles and computational effort be needed[12,16]. To achieve this, the particles are studied in a two-dimensional space.

In the present research, simulation of biaxial test is performedon assemblies of 500 particles within 1500 sub-particles usingpersonal computer (PC). For this purpose, the program POLY[13,14] is developed to model assemblies of irregularly shapedparticles with the ability of breakage [15,16,23].

It is assumed that each rockfill particle can break throughstraight lines with certain direction and position. The lines aredetermined in a way that two commonly observed behavior infragmentation can be simulated. These two kinds of behavior arecrushing of particle vertexes and cracking across a particle thatdivides particle into pieces. The lines are predefined and arerandomly distributedwithin the particles. According to Fig. 1(a), itis assumed that particle P can only break through the lines d1, d2and d3. Thus in this method, each uncracked particle (P) consistsof smaller bonded particles like P1, P2,…and Pn. Particle P is calledBase Particle and particles P1 to Pn are called Sub-Particles. The

Table 2Parameters used test B (Breakage enabled)

Normal and tangential stiffness (N/m) 2.0×107

Unit weight of particles (kN/m3) 2500Time step (s) 1.52E–4Strain rate 0.005Module of elasticity (E) (MN/m2) 9.0×104

Rock strength parameters Compressive strength (MN/m2) 350Tensile strength (MN/m2) 35Intercept (cohesion) (MN/m2) 75Coefficient of static friction 1.60


sub-particles are considered to be rigid bodies. They are notbreakable and not deformable. But they can overlapwhen they arepressed against each other. The base particles are not deformableeither, but they are breakable since they are made up of severalsub-particles. The both base and sub-particles are arbitrarilyconvex polygon-shaped.

In order to ensure the rigidity and continuity of the bonded sub-particles to form a base particle, it is assumed that two adjacentsub-particles are connected with a fixed connection at the middleof their common edge (points m1 and m2 in Fig. 1(b)). This fixedconnection plays the role of the bond between two bonded sub-particles. During the simulation, when the existing bond stressexceeds its final bearing capacity, the connection will break andwith separation of the two bonded particles, breakage takes place.

For modeling the connection between two bonded particles,two transitional and one rotational springs are introduced. One ofthe transitional springs that are perpendicular to the common faceof particles is normal spring and the other one which is parallel tothe common edge is the shear spring. Moment and forces at theconnection between two bonded particles are transferred throughrotational and transitional spring, respectively. They can becalculated according to relative displacement of two sub-particlesat each simulation cycle.

Fig. 2 illustrates a base particle P in an assembly of particlessubjected to an arbitrary loading. Due to interaction betweenparticles, the forces and moments are induced at base particle'scontact points with adjacent particles. As a consequence, the sub-

Fig. 5. (a)Relationship between sin ϕmobilized and axial strain (b)Relati

particles P1 and P2 are relatively displaced against each other.Therefore, the points m1 and m2 are no longer coincident. Todetermine the force and moment applied on each contact edgebetween two sub-particles, the relative displacement of the twosub-particles P1 and P2 is replaced with three components, Δn

(normal displacement), Δs (shear displacement) and Δθ (rota-tional displacement) (Fig. 2). Hence, the normal and shear forcesand the moment at the contact point can be expressed as follows:

Fn�Bond ¼ Kn�BonddDn

Fs�Bond ¼ Ks�BonddDs

MBond ¼ Kh−BonddDh

ð1Þ

where Kθ−Bond is the stiffness of the rotational spring and Kn-Bond

and Ks-Bond are unit length stiffness of the normal and shearsprings, respectively. Values of these parameters are considered tobe proportional to the stiffness of the particles.

With the existence of one of the two following conditions,the bond between the two adjacent sub-particles is broken andseparation happens:

1. If the shear stress at the bond between the two particles is largerthan the allowable shear stress (the bond bearing capacity).

2. If the maximum compressive or tensile stresses caused bymoment and normal force of the bond exceeds the allowablecompressive and tensile stresses.

In other words, despite of the model presented by Potapov andCampbell [17,18], the bond will be failed due to three modes ofcompression, tension or shearing.Thebondbearing capacity obeysfrom the Coulomb failure criterion for rocks which is extended inboth compressive and tensile stresses, but they are limited bymagnitudes of stresses obtained from unconfined compressivestrength and Brazilian tensile strength tests respectively.

The same procedure of modeling would be suggested in 3-Dcomputation. The base particles could be 3-Dmulti-surface blocksthat are made up of several bounded 3-D blocks. The sub-particleswould be 3-D blocks that have connecting faces in touch. These

onship between volumetric strain and axial strain (Pcp=2.0 MPa).

Fig. 6. Relationship between bond breakage and axial strain (Pcp=2.0 MPa).


faces act actually as the planes of resistance against compression,tension and shearing.

2.3. Biaxial simulation

Simulation of two series of biaxial compression tests is ful-filled with four levels of confining pressures (Pc.p.) of 0.5, 1.0, 2.0and 4.0 MPa to investigate the particle breakage in a granularmedia. In the series test A particles are rigid with no ability infragmentation while in the series test B the particles are breakable.The particles are placed in a circular area.

Each test includes three stages. At first, the initial computerassembly of particles is generated, then subjected to a definedconfining pressure and finally the assembly is sheared biaxially ata constant deviatoric strain rate (along the 2–2 axis). The assemblyof particles at different stages (breakage enabled) is presented inFig. 3. Also, two different areas of the whole assembly are mag-nified to showmore clearly the breakage of particles. Fig. 4 showsthe growth of bond breakages during hydrostatic compaction andshearing processes, respectively. The inter-particle frictioncoefficient is set to 0.5 for all tests and the particles are assumedto be cohesionless at the contact. Also the particles are consideredweightless. The parameters used for tests A and B are summarizedin Tables 1 and 2, respectively.

Fig. 7. Relationship between sin

3. Results

The results obtained from the simulations can be studiedfrom two different points of view as macro and micromecha-nical considerations.

3.1. Macromechanical observations

The results of biaxial simulations in tests A and B for Pc.p=2.0 MPa are presented in the form of curves of sin ϕmobilized

(Eq. (2)) versus axial strain and volumetric strain versus axialstrain (Fig. 5).

As shown in both tests, the shear strength (sin ϕmobilized),increases and then reaches to a constant value. In test A, the sinϕmobilized grows rapidly and reaches at a peak of 0.6, but in test B,it gradually increases and it becomes constant just under 0.5. Itseems that particle breakage has a decreasing effect on shearingresistance of the assembly.

Sin/mobilized ¼r2−r1r2 þ r1

¼ r2=r1−1r2=r1 þ 1

ð2Þ

In general, the angular particles have dilative behavior (Marsal[9]). Fig. 5(b) shows that the assembly with no breakage has amore dilative behavior than that with the ability of fragmentation.As investigated before, the more the assembly dilates, the larger isthe shear resistance. In test B, particles can not undergo the forcesimposed on them and breakage happens, therefore smallerparticles fill the voids and let the other particles move freely.This causes the assembly to show a compressive behavior inlarger axial strains followed by increasing volumetric strain. Thistrend can justify the reduction of sinϕmobilized in test B. The sameresult has been obtained in experimental test results [9,4].

Fig. 6 illustrates the variation of bond breakage in three modesof failure which have been tracked during biaxial shear test withconfining pressure of 2.0 MPa. The breakage percentage showsthe ratio of broken bonds to total number of bonds. It should benoted that initial amounts of breakage shown in the figure,pertains to the breakage of particles resulted in the isotropiccompression stage. The possible causes of fragmentations are due

ϕmobilized and axial strain.

Fig. 8. Volumetric strain versus axial strain during the tests.


to compression, tension and shear breakage. It shows that noparticle breakage has happened due to compression. Although inhydrostatic compression stage, most particles have been brokenbecause of shear failure (35%), in biaxial test the degree of tensionbreakage is higher which started from about 13% at the beginningand it reached at just below 40%. Also the number of shearbreakage is to some extent constant at large axial strain whereasthe tension breakage grows gradually.

Having performed triaxial tests on rockfill, Marsal [10]showed that at the beginning of the test, larger particles thatcontain more flaws and defects, break and it is why the breakagerate at the beginning of the test is high. At the primitive stages ofthe test, the smaller particles, produced by larger particlesbreakage, are located in the voids between the other intact largeparticles and consequently have no role in transferring the force totheir neighboring particles. After compaction of assembly duringnext stages, the gaps between particles become smaller and thesmall particles can play their role in transferring the force to theadjacent particles. Thus the mean contact stresses decrease owingto the increase of particles surrounding each grain; therefore, thebreakage quantity will reduce afterwards. Considering the totalnumber of breakage in Fig. 6, the rate of particle breakage is highat the beginning of simulation and then it slows down. Thereforevariation of breakage rate versus axial strain (and consequentlyaxial stress) during the simulated biaxial test is in agreement withthe trend observed byMarsal [10]. The same results can be foundout in DEM simulation [24].

The effect of confining pressure on shear strength is in reversewhere the higher pressure results in the lower mobilized frictionangle in both tests (Fig. 7). Also the axial strain corresponding to

Table 3Comparison of internal friction angle in simulated tests

Confining pressure With no breakage With breakage

0.5 MPa 37.6° 35.0°1.0 MPa 36.2° 27.4°2.0 MPa 34.8° 26.7°4.0 MPa 33.4° 26.1°

the maximum shear strength values increases with increasingconfining pressure.

In Fig. 8, it can be seen that assembly with no breakage has amore dilative behavior than that with the ability of fragmentation.The higher confining pressure on the specimen causes to compressit more and does not let the sample dilate. On the other hand, underhigher pressures particles havemore tendencies to be broken. Thiscauses the assembly to show a more compressive behavior underlarger confining pressures. This trend can describe the reason forthe reduction of sin ϕmobilized. On overall, for both series ofsimulations, the more the assembly dilates, the larger the shearresistance is. The same result has been obtained in experimentaltest results [9,4,27].

Table 3 shows obtained frictional angle of the assembly indifferent confining pressures. Also Fig. 9 illustrates the variationof bond breakage degree (in percent) which has been trackedduring different biaxial shear tests. This diagram confirms thathigher degree of breakage is achieved when the larger value ofconfining pressure is used in the simulations.

All the simulations presented here have been carried out in 2-D state. Thus, it is difficult to compare quantitatively theobtained results with the 3-D experimental tests such as triaxialcompression. However it is possible to compare the trend ofresults influenced by different parameters. In order to compare

Fig. 9. Variation of bond breakagewith axial strain in different confining pressures.

Fig. 10. Variation of principal stress ratio versus degree of breakage (%). Forresults obtained from simulation biaxial test and experimental tests.


qualitatively the obtained results from these simulations with theexperimental tests results, the values of the maximum principalstress ratio (σ2 /σ1)max in the biaxial simulation tests and theexperimental tests performed by Gupta [5], Venkatachalam [26],Marachi et al. [8] are shown in Fig. 10 versus the degree ofbreakage (Bg). The value of breakage is calculated from sieveanalysis of rockfill sample as follows. Before testing, the sampleis sieved using a set of standard sieves and the percentage of

Fig. 11. Histograms of different anisotropy coefficients for a test with breakable

particles retained in each sieve is calculated. Due to the breakageof particles, the percentage of particles retained in large sizesieves will decrease and the percentage of particles retained insmall size sieves will increase. The sum of decreases inpercentage retained will be equal to the sum of increases inpercentage retained. The sum of decreases (or increases) is thevalue of the breakage factor (Bg) [9]. It is observed that thesimulation results fall inside the lower bound shown in thefigure. As shown in Fig. 10, the degree of breakage increaseswith decrease of ratio (σ2 /σ1)max in both simulation andexperimental tests. If the ratio (σ2 /σ1)max can be interpreted asthe strength of the assembly, the observed trend is logical and thesimulation and experimental results are qualitatively inagreement.

3.2. Microscopic behavior

While it is obvious that forces in granular media must becarried bymeans of contacts between particles, it is only recentlythat a means of quantifying the arrangement of contacts has beendeveloped. For any angle θ, the portion of the total number ofcontacts in the system that are oriented at angle θ is E(θ). Thedistribution of contact normal orientations is described by afunction such that the fraction of all assembly contact normals

particles at Pcp=2.0 MPa: (a) at axial strain 2.0%; (b) at axial strain 16.0%.

Fig. 12. Displacement trajectories of all particles during the biaxial test in (a) unbreakable test; (b) breakable test.


falls within the orientation intervalΔθ. Rothenburg et al. (1989)[20] showed that the distribution of such contacts takes the form:

EðhÞ ¼ 12k

½1þ a cos2ðh−h0Þ� ð3Þ

where a is referred to as the parameter of anisotropy, and θ0 isthe major principal direction of anisotropy. The meaning of abecomes clear if it is noted that the number of contacts orientedalong the direction of anisotropy, i.e. when θ=θ0 is proportionalto 1+ a while the number of contacts oriented in theperpendicular direction is proportional to 1−a. The parametera, therefore, is proportional to the difference in the number ofcontacts oriented along the direction of anisotropy and inperpendicular direction. A similar expression was introduced byThornton and Barnes [25].

The magnitudes of the contact forces in an assembly withirregular geometry vary from contact to contact. Despite theapparent randomness in the variation of contact forces, regulartrends emerge when they are averaged over groups of contactswith similar orientations. The average contact force acting atcontacts with an orientation can be decomposed into an averagenormal force component, f̄ n

c(θ), and an average tangential forcecomponent, f̄ n

c(θ). By averaging the contact forces of the contacts

Fig. 13. Coordination number's variatio

falling within the group of similar orientation and following thesame logic as for the contact normals, symmetrical second-ordertensors may be introduced to describe average normal contactforces and average tangential contact forces. The average normalcontact force tensor can be defined as [20]:

f̄ nðhÞ ¼ f̄0

n½1þ ancos2ðh−hf Þ� ð4Þwhere an is the coefficient of normal force anisotropy, and θf isthe major principal direction of force anisotropy; f̄ sub n0(θ) is theaverage normal contact force from all assembly contacts.

The average tangential contact force tensor can be defined as:

f̄ tðhÞ ¼¼ f 0t ½atsin2ðh−h0Þ� ð5Þwhere at is the coefficient of tangential force anisotropy and θ0the direction of anisotropy.

The general expression for the average stress tensor can nowbe written as:

rij ¼ mvl0

Z 2p

0

f̄c

nðhÞnci ncj þ f̄c

t ðhÞtci ncjh i

EðhÞdh ð6Þ

where mv is the average number of contacts per unit area(volume), l0 is the assembly average contact vector length

n with axial strain (Pcp=2.0 MPa).

Fig. 14. Comparison of coordination number's variation with axial strain (a) with unbreakable particles; (b) with breakable particles.


(average distance from the particle centers to the contact point),nic is the contact normal vector, and ti

c is the contact tangentvector. Rothenburg and Bathurst [20–22] derived a relationshipbetween the measure of shear stress and the parameters a, an, atinvolved in the characterization of anisotropies in contactorientations and contact forces according to Eqs. (2)–(5). Forthe case when the directions of anisotropy in contact forces andcontact orientations coincide, as in a biaxial test, the relationshipis as follows:

r2−r1r2 þ r1

� �¼ aþ an þ at

2þ an � at

� �ð7Þ

The simplified expression suggests that the capacity of acohesionless granular assembly is directly attributable to its abilityto develop anisotropy in contact orientations or to withstanddirectional variations of average contact forces. Eq. (7) wasevaluated for assemblies of disc-shaped and elliptical particles[20]. The application of this theory is also evaluated for polygon-shaped particles [13]. So the same model can be applicable tostudy the microscopic behavior of breakable particles. For thispurpose, each base particle that has not been yet broken intopieces is considered as one particle, but the sub-particles are

Fig. 15. Evolution of contact normal anisotropy coefficient (Pcp=2.0 MPa).

considered as individual new particles as soon as they are se-parated from their base particles. In this condition, during thesimulation the number of particles is increased and accordingly allmicro parameters are being averaged in each step for the currentnumber of individual particles.

One can find out the evolution of the anisotropy coefficientsalong the increment of axial strain by drawing them against eachother. Also at each axial strain, one can sketch the anisotropy stateof the assembly with a histogram which shows a polar distributionof the anisotropy. If there is no anisotropy in an assembly, thehistogram shape should be close to a circle, which implies that theparameter magnitude is the same in all direction, but with thegrowth of anisotropy, the shape grows to be deformed such as apeanut. The histogram shape of anisotropy coefficient of tangentialforce (at) is different from the histogram shape of contact normal(a) and normal force (an) coefficients which are like a peanut.Apart from measuring the anisotropy of the parameters andsketching the histograms, fitted histograms can be obtained fromthe tensor functions stated above by having the coefficient and theprincipal angle of anisotropy. Fig. 11 shows three histograms ofdifferent anisotropy coefficients in the beginning (axialstrain=2.0%) and at the end (axial strain=16.0%) of a biaxialtest with breakable particles at Pc.p.=2.0 MPa.

One way of investigating how a microstructure of granularassembly evolves during the shearing process is to trace eachparticle displacements along the test. This is possible to see withnumerical simulation in which the locations of all particles can beunder controlled. Fig. 12 represents the movement of all theparticles in the unbroken test and all the sub particle displace-ments in the test with breakable particles. In this sketch, the initialand final locations of all particles are connected to each other. Itshows that all particles are trying to move towards the impliedmajor and minor stresses during the test whereas the particlessituated in the center of the assembly have the minimum move-ment during the test.

The other way of investigating the evolution of microstruc-ture of granular assembly is to study the change in the numberof contacts in the assembly or the average coordination numberof the system.

Fig. 16. Evolution of contact normal anisotropy coefficient.


Fig. 13 presents the evolution of the average coordinationnumber during shear deformation at Pc.p. =2.0 MPa. At thebeginning, the coordination numbers of two assemblies aredifferent owing to the hydrostatic compression stage carried outpreviously. But the way of creating contacts is different in thesetwo series of tests. In tests B, the particles can be broken intopieces and then each new particle makes new contacts with itsneighbor particles, while this phenomenon does not happen forthe particles in tests A. Therefore, both the number of particlesand the number of contacts during the hydrostatic loading areincreasing. Consequently, the coordination number at the finalstage of compaction will differ from that of assembly with non-broken particles. As can be seen, the growth of the numbers ofparticles and contacts due to bond breakages in test series B(with breakage), is in a way that at the beginning of shear stage,the coordination number is less than that of the tests A (with nobreakage). Also the trends of coordination number variation aredifferent during shearing process. As shown in Fig. 14, in testseries A, it decreases rapidly along axial strain and comes to beconstant, while in test B, it grows gradually towards a constantvalue in high confining pressures. During each test of both seriesof tests, contacts in the assembly began to degrade as the axialstress increased, mainly in the horizontal direction. But in testseries B, particle breakage is also happening at the same time and

Fig. 17. Evolution of normal force anisotropy coefficient (Pcp=2.0 MPa).

this phenomenon results more development of contacts betweenparticles. That is why the number of contacts grows. The effectof increasing the confining pressure on coordination number canbe observed from the figures. The higher confining pressure, themore contacts are induced in the assembly [24]. As mentionedbefore, the initial values of the coordination number are becauseof the isotropic compaction stage before the assembly is shearedbiaxially.

The variation of the contact normal anisotropy (parameter a)as a function of axial strain is illustrated in Fig. 15. It describes thedegree of anisotropy in contact orientations. This coefficient isessentially proportional to the difference in the number of contactsin each direction, and describes the degree of anisotropy in contactorientations.

The coefficient of fabric anisotropy evolves to the maximumvalues as contacts are lost, mostly oriented along the direction oftensile strain (horizontal direction). But this growth is more rapidin tests A than in test B where particles can break. Also in test A,the coefficient reduces to the lower ultimate values at large axialstrain, but in test B, it continuously and slowly grows to a peakvalue of half of that in test A. In Fig. 16, the effect of confiningpressure on this parameter can be investigated. The confiningpressure has a reverse effect on this coefficient, but this rule is notalways correct in the test series B.

Fig. 18. Evolution of tangentialforce anisotropy coefficient (Pcp=2.0 MPa).

Fig. 19. Evolution of force anisotropy coefficient.

Fig. 20. Verification of relationship between stress and fabric for assembly withand without breakable particles (Pcp=2.0 MPa).


Fig. 17 presents the development of anisotropy in normalcontact forces for Pc.p=2.0 MPa by plotting the variation of thecoefficient of normal contact force, an, with axial strain during thesimulations on assemblies with rigid and breakable particles. Inboth tests, by applying deviatoric axial strain in the verticaldirection, the normal forces carried by chains of particles in thevertical direction are increased, while the magnitude of averagenormal forces in the horizontal direction remains almost constant.The parameter an indicates the difference in the values of averageforces on vertical and horizontal contacts. A greater difference inthe magnitude of the average forces in the horizontal and verticaldirections provides a higher value for an. In test A, as the axialstrain increases, an shows a rapid growth at lower axial strain,followed by a reduction after the maximum value. This is becauseof loss of contacts and also the loss of the capacity of chains ofparticles to sustain high forces. In contrast to test A, when theparticles can be divided into smaller ones, the parameter an showsa gradual increasewhich reaches at a constant value. This behaviorof an is sensitive while the particles cannot tolerate the imposedforces and breakage happens, therefore, particles can not make achain to show a peak in an. Although the increasing of an are indifferent manners in these two tests, it might be anticipated that anreaches to the same ultimate value at large axial strain.

The coefficient of tangential contact force anisotropy, at,shows a rapid rise at small axial strain. In test A, the coefficient of

tangential force anisotropy reaches to the maximum value,followed by a slow reduction in magnitude, as illustrated inFig. 18. In spite of test A, at in test B continues to grow veryslowly to a constant value. The initial increase in at is due to thedevelopment of frictional resistance as a result of potential relativemovement between adjacent particles. As the number of contactsreduces, the particles gain more opportunity to rotate; therefore intest A, tangential forces are slowly released. In test B, particles

Table 4Strength parameters for three rockfill materials in the tests with breakableparticles

Strength parameters Sample #1 Sample #2 Sample #3

Rock strengthparameters

Compressive strength(MN/m2)

190 350 650

Tensile strength (MN/m2) 19 35 65Intercept (MN/m2) 40 75 140Coefficient of staticfriction (tan(ϕ))

1.6 1.6 1.6


cannot bear the tangential forces and get into pieces so they havemore opportunity to move freely in the voids between otherparticles. Particle breakage causes not to mobilize the shear forcescompletely; therefore shear force does not reach to a peak.

Also, it can be observed that assemblies with lower confiningpressure can provide more anisotropy in the media withsubsequent higher shear strength. Fig. 19 shows the variation ofanisotropy coefficients with different confining pressures [24].The higher the assembly is confined, the particles can not movemuch freely and the particles shall tolerate their imposed forces intheir obliged direction. So the anisotropy decreases both fornormal and tangential parameters.

Fig. 21. Variation of (a) sin ϕmobilized; (b)volumetric strain; (c) total breakage degree

The coefficients of anisotropy a, an and at were substitutedinto the stress–force–fabric relationship (Eq. (7)). Fig. 20 com-pares the computed shear resistance based on theory (right-handside of Eq. (7)) and the value of sin ϕ mobilized (left-hand side ofEq. (7)) provided by biaxial simulations on assemblies withangular particles for both tests A and B. As can be observed, thestress–force–fabric expression is in agreement with the measuredshear resistance in the test simulations.

4. Effect of rock strength on the behavior of the assembly

Three biaxial compression tests have been simulated withdifferent rockfill materials to investigate the effect of strength onthe behavior of the assembly. The strength parameters are showninTable 4. All these tests are performed in the same conditionwithconfining pressure of 2.0 MPa. The inter particle frictionalcoefficient is held constant and equal to 0.5. Fig. 21 shows thevariation of sin ϕ mobilized, volumetric strain and degree of totalbond breakage (%) along the axial strain during the tests.

As illustrated in Fig. 21, although the amount of breakage isreduced with increasing rock strength, there is little sense ofdifference in their corresponding shear strength (sin ϕ mobilized).Perhaps this is because of high amount of breakage in all tests and

; with axial strain in confining pressure of 2.0 MPa for different rock strengths.


also the same level (75%–92% at the end) of the degrees of bondbreakage. The differences in behavior would be probablyobserved more clearly if the variation of breakage degree werein a larger range.

The other point which can be found out in this series of tests isthe relation between the shear strength of the assembly and thedilation; the higher the assembly dilates, the higher the shearstrength becomes. This point was also gained when the assemblywas studied in different confining pressures.

5. Conclusions

The results of two simulated series of biaxial tests withseveral confining pressures indicated that higher confining pres-sure leads to decrease in shear strength and increases granularmaterial compressibility. In return, the dilatancy falls down. Thisshows that the higher the dilatancy is, the higher the shearstrength becomes. The same result was gained with a series testswith different rock strengths. Also the rate of particle breakage indifferent modes was investigated. The higher the confiningpressure, the more the degree of breakage is. The results aresimilar to data obtained from experimental tests on real rockfillmaterials.

The influence of confining pressure on the variation ofmicromechanical parameters was also studied. The assembly inwhich breakage is enabled, the coordination number remainsalmost constant during the shearing test, but in the other group, itdecreases along with axial strain. As observed, the magnitude ofnormal contact force and tangential force anisotropy coefficientsare smaller in the case of breakable particles than those in rigidparticles. But the confining pressure has a reverse effect on theanisotropy coefficients. The shear strength of granular assembly isdirectly attributable to the ability to develop anisotropy.

The stress–force–fabric expression developed by Rothenburgand Bathurst [20,21] was verified for the assemblies of angularparticles in which the particles can break.

Also, comparisons between simulations results and observa-tions obtained from experimental tests show that the methodpresented for modeling breakage, can help us to have a qualitativeview about the effect of breakage phenomenon on behavior ofgranular materials.

References

[1] P. Bertacchi, R. Bellotti, Experimental research on materials forrockfill dams, 10th Int. Congress on Large Dams, ICOLD, vol. Q36,1970, pp. 511–529, Montreal.

[2] Y.P. Cheng, Y. Nakata, M.D. Bolton, Discrete element simulation ofcrushable soil, Geotechnique 53 (7) (2003) 633–641.

[3] P.A. Cundall, Ball – a computer program to model granular media usingdistinct element method, Technical Note TN-LN-13, Advanced Technol-ogy Group, Dames and Moore, London, 1978.

[4] E. Fumagali, B. Moscini, P.P. Rossi, Laboratory tests on materials and staticmodels for rockfill dams, 10th Congress on Large Dams, ICOLD, vol. Q36,1970, pp. 531–551, Montreal.

[5] Gupta, A. K. (2000). “Constitutive modeling of rockfill materials”, Ph.Dthesis, Indian Institute of Technology, Delhi, India.

[6] F. Kun, H.J. Herrmann, A study of fragmentation process using a discreteelement method, Comput. Methods Appl. Mech. Eng. 138 (1996) 3–18.

[7] T. William Lambe, Robert V. Whitman, Soil Mechanics, SI version, JohnWiley and Sons Inc.0 85226 502 6, 1979.

[8] N.D. Marachi, C.K. Chan, H.B. Bolton, Evaluation of properties of rockfillmaterials, J. Soil Mech. Found. Div. 98 (SM1) (1972).

[9] R.J. Marsal, Large scale testing of rockfill materials, J. Soil Mech. Found.Div. 93 (SM2) (1967).

[10] R.J. Marsal, Mechanical Properties of Rockfill, in: R.C. Hirshfield, S.J.Poulos (Eds.), Embankment-Dam Engineering, Casagrandeh Volume,John Wiley and Sons Inc, New York, 1973, pp. 109–200.

[11] G.R. McDowell, O. Harireche, Discrete element modeling of soil particlefracture, Geotechnique 52 (2) (2002) 131–135.

[12] A.A. Mirghasemi, R.Mousavi Nik, Influence of particle breakage onmecha-nical behavior of assemblies of two-dimensional polygon-shaped particles,Proc. of 54th Canadian Geotechnical Conference, Calgary, Alberta, 2001, pp.614–620.

[13] A.A. Mirghasemi, L. Rothenburg, E.L. Matyas, Numerical simulations ofassemblies of two-dimensional polygon-shaped particles and effects ofconfining pressure on shear strength, Soil Found. Jpn. Geotech. Soc. 37 (3)(1997) 43–52.

[14] A.A. Mirghasemi, L. Rothenburg, E.L. Matyas, Influence of particle shapeon engineering properties of assemblies of two-dimensional polygon-shaped particles, Geotechnique 52 (3) (2002) 209–217.

[15] A.A. Mirghasemi, E. Seyedi Hosseininia, Micromechanical considerationsof particle breakage using discrete element method, Proc. of 57th CanadianGeotechnical Conference, Quebec City, Canada, 2004.

[16] Mousavi Nik, S., R., (2004). Modeling Particle Breakage of GranularMaterials in Biaxial Test Using Discrete Element Method, M.Sc Thesis,University of Tehran, Tehran, Iran.

[17] A.V. Potapov, C. Campbell, Computer simulation of impact-inducedparticle breakage, Powder Technol. 81 (1994) 207–216.

[18] A.V. Potapov, C.S. Campbell, Computer simulation of shear-inducedparticle attrition, Powder Technol. 94 (1997) 109–122.

[19] D. Robertson, M.D. Bolton, DEM simulations of crushable grains andsoils, Proc. Powders and Grains, Sendai, , 2001, pp. 623–626.

[20] L. Rothenburg, R..J. Bathurst, M.B. Dusseault, Micromechanical ideas inconstitutive modeling of granular materials, in: Biarez, Gourves (Eds.),Powders and Grains, Balkema, Rotterdam, ISBN: 906191 984 3, 1989.

[21] L. Rothenburg, R.J. Bathurst, Analytical study of induced anisotropy inidealized granular materials, Geotechnique 39 (4) (1992) 601–614.

[22] L. Rothenburg, R.J. Bathurst, Micromechanical features of granularassemblies with planar elliptical particles, Geotechnique 42 (1) (1992)79–95.

[23] Seyedi Hosseininia, E. (2004), Micromechanical considerations of particlebreakage using discrete element method, M.Sc Thesis, University of Tehran,Tehran, Iran.

[24] E. Seyedi Hosseininia, A.A. Mirghasemi, Effect of confining pressure onparticle breakage of assemblies of simulated angular particles using dis-crete element method, Proc. of 11th International Conference of IACMAG,Turin, Italy, 2005.

[25] C. Thornton, D.J. Barnes, Computer simulated deformation of compactgranular assemblies, Acta Mech. 64 (1986) 45–61.

[26] Venkatachalam, K. (1993), Prediction of mechanical behavior of rockfillmaterials, PhD thesis, Indian Institute of Technology, Delhi, India.

[27] A. Varadarajan, K.G. Sharma, K. Venkatachalam, A.K. Gupta, Testing andmodeling two rockfill materials, J. Geotech. Geoenviron. Eng., ASCE(March 2003).

[28] P.A. Cundall, R. Hart, Development of generalized 2-D and 3-D distinctelement programs for modeling jointed rock, Itasca Consulting GroupMisc. U.S. Army Corps of Engineers, 1985, Paper SL-85-1.

Numerical simulation of breakage of two-dimensional...

Documents

Transcript of Numerical simulation of breakage of two-dimensional...