Discrete-element model for dynamic fracture of a single particle

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Discrete-element model for dynamic fracture of a single

particle

Luisa Fernanda Orozco, Jean-Yves Delenne, Philippe Sornay, Farhang Radjai

To cite this version:

Luisa Fernanda Orozco, Jean-Yves Delenne, Philippe Sornay, Farhang Radjai. Discrete-element model

for dynamic fracture of a single particle. International Journal of Solids and Structures, Elsevier, 2019,

166, pp.47-56. �10.1016/j.ijsolstr.2019.01.033�. �hal-02414887�

Discrete-element

model

for

dynamic

fracture

of

a

single

particle

Luisa

Fernanda

Orozco

a ,b

_,

_Jean-Yves

_Delenne

c

_,

_Philippe

_Sornay

a

_,

_Farhang

_Radjai

b ,∗

a CEA, DEN, DEC, SA3E, LCU, Saint Paul les Durance 13108, France b LMGC, CNRS, University of Montpellier, France

c IATE, INRA, CIRAD, Montpellier SupAgro, University of Montpellier, France

Keywords: Granular materials DEM Contact dynamics Impact test Fracture energy Damage

a

b

s

t

r

a

c

t

We investigate the dynamic fracture of a single particle impacting a flat surface using 3D DEM simula- tions based on a fragmentation model involving both a stress threshold and a fracture energy. The particle is assumed to be perfectly rigid and discretized into polyhedral Voronoï cells with cohesive interfaces. A cell-cellinterface loses its cohesion when it is at a normal or tangential stress threshold and an amount of work equal to the fracture energy is absorbed as a result of the relative cell-cell displacements. Upon impact, the kinetic energy of the particle is partially consumed tofracture cell-cell contacts but also restituted to the fragments or dissipated by inelastic collisions. We analyze the damage and fragmenta- tion efficiency as a function of the impact energy and stress thresholds and their scaling with fracture energy and impact force. In particular, we find that the fragmentation efficiency, defined as the ratio of the consumed fracture energy to the impact energy, is unmonotonic as a function of the impact energy, the highest efficiency occurring for a specific value of the impact energy.

1. Introduction

Particlebreakagetakesplaceinnumerousnaturalandindustrial processes, and it has been addressed in multiple research fields and applications such as powder technology, construction engi-neering, mining, food industry and metallurgy (Tarasiewicz and Radziszewski, 1990; Vogel and Peukert, 2003; Leite et al., 2011; Xu et al., 2017 ).Theprocessesthatinvolveparticlebreakagerange from rock fracture in landslide events to the grinding of miner-alsinvariousapplications(Varnes, 1958; Touil et al., 2006; Bailon- Poujol et al., 2011 ). The initiationand evolution of particle frag-mentationaregovernedbyfracturemechanicsatthemateriallevel and by granulardynamics at the particle packing scale. For this reason, it is necessary to enhance our understanding of the ef-fectsofvariousparametersthatcontrolfracture-induced phenom-ena independently of the specific crushing or grinding machines employed indifferentapplications. Forexample,itiswell known that the comminution is an energy-intensive transformation and the size distribution of reduced particles is crucially dependent on various factors relatedto the material and loadingconditions (Tavares and King, 1998; Fuerstenau and Abouzeid, 2002; Stambo- liadis, 2007; Govender, 2016; Mayer-Laigle et al., 2018 ).Butmostof thepresentknowledgeontheenergyconsumptionandits

depen-∗ _{Corresponding author.}

E-mail address: franck.radjai@umontpellier.fr (F. Radjai).

denceonthe material propertiesisempirical innature.Different loadingmodessuchascompression,distortion,shear, andimpact cancauseparticlefracture, andtheir combinationleadstoa mul-titudeoflocalmechanismssuchasdamageandabrasionthattake place simultaneously during a comminution process (Tarasiewicz and Radziszewski, 1990; Govender, 2016; Kim and Santamarina, 2016 ).

Studying the fracture of a single particle is the first step to-wardsaquantitativedescriptionofthecomplexmulti-particle frac-turedynamics.Experimentally,thepreviousworkonthe fragmen-tation of one particle has been largely focused on compression testsunderquasi-staticconditions(Chau et al., 20 0 0; Brzesowsky et al., 2011 ).Dynamicfracturetestshavebeenperformedbymeans ofdropweight impact (Tavares and King, 1998; Wu et al., 2004; Ge et al., 2017 ) or air gun (Salman et al., 2002 ). Large-scale ex-periments were also performed on the fragmentation of rocks falling on a hard surface (Giacomini et al., 2009 ). The common goal ofsingle-particle testshas been to investigatethe fragment sizedistribution,crackpatterns,particlebreakageprobability,and failure modes (Potapov and Campbell, 1994; Kun and Herrmann, 1998; Tsoungui et al., 1999; Subero et al., 1999; Chau et al., 20 0 0; Salman et al., 2002; Wittel et al., 2008; Casini et al., 2013; Huang et al., 2014 ).Severalauthorshavealsoconsideredthefracture en-ergy consumed per unit mass (Stamboliadis, 2007; Tavares and King, 1998; Fuerstenau and Abouzeid, 2002; Casini et al., 2013 ). This is the energy consumed in producing new fractured sur-facesinsidetheparticle.Othersourcesofenergydissipationinthe

fragmentationprocess are the plastic deformations andfrictional orinelasticcollisions.Partofthetotalimpactenergyisalsotaken awayby thekineticenergyofthefragments,forwhichwefound noreported measurements in theavailable literature. Most mea-surementswerecarriedoutattheendofthetestsbecauseofthe difficultyofmeasuringinrealtimethestressdistributionsand par-ticlemotions(Wittel et al., 2008 ).

On the numerical modeling side, the simulation of parti-cle fragmentation began with Finite Element Method (FEM) ap-proacheswhereinthe dynamicmeshesmake itpossible to incor-poratecracks andtheir evolution intime (Hillerborg et al., 1976; Zubelewicz and Bažant, 1987 ).Later,thediscreteelement method (DEM)wasusedastheprivilegedtool forthesimulationof gran-ularmaterialswiththeadvantageofincorporatingvariousparticle interactionssuch asfriction,cohesionanddamage, andproviding accesstotheforcesandvelocitiesattheparticlescale(Cundall and Strack, 1979; Moreau and Solids, 1994; Radjai and Richefeu, 2009; Delenne et al., 2004; Taboada et al., 2006; Luding, 2008; Herrmann and Luding, 1998 ). Within the framework of the DEM, particle fragmentationhasoftenbeen modeled usingthe BondedParticle Method(BPM),in whichthe particles are modelledas agglomer-atesofglueddisks(Thornton et al., 1996 )orspheres(Subero et al., 1999; Mishra and Thornton, 2001; McDowell and Harireche, 2002; Moreno-Atanasio and Ghadiri, 2006 ). An issue with this method isthat the total volume of the material is not conserved during the fragmentationprocess. Another method consistsin replacing parent particles by smaller ones once a local failure criterion is achieved.Ithasbeenappliedtopackingsofdisks(Tsoungui et al., 1999; Lobo-Guerrero et al., 2006 ), spheres (Ciantia et al., 2015; Bruchmüller et al., 2011 ) andpolyhedral particles (Eliáš, 2014 ). A variantofthefirstmethodconsistsinreplacingsphericalprimary particlesbymultiplepolygonalcellsin2D(Potapov and Campbell, 1994; Kun and Herrmann, 1996; Nguyen et al., 2015 ) or polyhe-dralcellsin3D (Galindo-Torres et al., 2012; Cantor et al., 2017 ),a methodthatwascoinedBondedCellMethod(BCM).

In most discrete models of particle fracture using the BPM, the cohesive behavior at the inter-cell contacts is of brittle type and governed by a force or stress threshold. A pre-failure plas-tic force model was proposed by Luding (2008) by introducing two independent stiffness parameters for loading and unload-ing force-displacement relations, respectively.This plastic contact model,however,concernsonlythenormalcompressivepartofthe displacement, the failure in tension being simply governed by a force threshold. Another plastic contact model was proposed by Timár et al. (2010) whointroduced a healingtimeso that when-evertwo particles remain in contactfor a time longer than this time,anewcohesive linkis insertedbetweenthem.Thisprocess gradually modifies the referenceelastic configuration of the par-ticlesandleadstoplasticdissipationandpermanentdeformation. Thepower-lawdistributionoffragmentmassesfoundbyusingthis approachwasfoundtobeingoodagreementwithfracture exper-imentsperformedwithpolymericparticles.

Inthispaper,weintroduceafracturelawwithinadiscrete ele-mentapproachthatinvolvesbothastressthresholdandafracture energyin normalandtangential directions.This modelis imple-mented atthe inter-cell interfacesinside the particle in a three-dimensionalBCMapproach.Theintroductionofanenergycriterion allows for the simulation of dynamic fracture in impact tests. A cell-cellinterfacebreaksonlyifthestressthresholdisreachedand theworkabsorbedby theinterface dueto relativedisplacements along the normal and tangentialdirections is above the fracture energy.Usingthismethod,weinvestigatethebreakageofasingle particleimpactingarigidplanebymeansofextensivesimulations. Weanalyzethetotalfractureenergy,i.e.theenergyconsumeddue todebondingofcell-cell contacts,asafunction oftheimpact ve-locity.Inparticular,weshowthatthefragmentationefficiency,

de-finedastheratioofthetotalfractureenergytotheimpactenergy has a maximum value for a specific value of the impact energy. We introduce a fitting form that captures the observed behavior over the whole range of investigated energies. We also consider theeffectsofthenormalstrengthaswellastheratioofnormalto tangentialthresholds.

Inthefollowing,we firstintroduceinSection 2 thenumerical approach,withfocusonthefracturemodel,andgeneralconditions oftheimpacttest.Then,inSection 3 ,we analyzeparticledamage andfragmentation efficiencyasa function ofthe impact velocity andfractureenergy.InSection 4 ,weconsidertheeffective restitu-tioncoefficientandenergydissipationbyimpact.InSection 5 ,we focusontheeffectofthestressthresholdsandfrictioncoefficient. Finally,in Section 6 ,we briefly presentthesalient resultsofthis workanditspossibleextensions.

2. Numericalmethodandprocedures

Inthissection,we presentdifferentingredients ofour numer-ical approach for the simulation of particle fragmentation under impact. Weuse theBonded CellMethod (BCM) basedon the di-vision of the particle into polyhedral cells interacting withtheir neighboring cells via an interface characterized by a debonding stress thresholdand afracture energy (Nguyen et al., 2015; Can- tor et al., 2017 ). We describe the interface behavior, followed by thetessellationmethodandourDEM algorithmfordynamic sim-ulationoftheparticle.

2.1. Bonded-cellmethod

InBCM,thepolyhedralcellsofaparticleinteractonlythrough their interfaceareasandtheyareassumedtobehave as indepen-dent rigid particles so that their dynamic behavior can be sim-ulated by the DEM. During the fragmentation process, a subset of cell-cell interfaces break and the parent particle gives rise to fragments, each composed of several bonded cells. We used the Voronoï tessellation method for the division of the particle into randomcellsbymeansofthesoftwareNEPER(Quey et al., 2011 ). The cells are always convex and present adjacent faces. For the sakeofgeometricalconsistencybetweenaparticleandits consti-tutivepolyhedralcells,weuseparticlesoficosahedralshapeinthe simulations. The number of Voronoï mesh elements determines the maximumnumber of potential fragmentsthat can be gener-ated as a result of the fragmentation of a particle. Cantor etal. andNguyenetal.used theBCMforquasi-staticdiametrical com-pressionofparticles(Nguyen et al., 2015; Cantor et al., 2017 ).They found that the particle strength dependson the ordering of the cellular structure of particles. For this reason, we generate fully randomcellsbothintheirsizesandshapes.Fig. 1 displaysseveral examplesofparticleswithincreasingnumbersofcells.

The cell-meshedparticles in BCMare similar to aggregatesof primary particles orparticulate compounds in the Bonded Parti-cleMethod(BPM).Thereare,however,fundamentaldifferences be-tween aparticulatecompound anda particlein BCM.In thefirst place,aparticleinBCMhaszeroporosity,sothatthetotalvolume of theparticle is conserved duringits fracture whereas the total volumeofa particulatecompoundofsphericalparticlesofnearly the same size is reducedby 40% whenfully broken into its pri-maryparticles. Furthermore,the interfacesbetween cellsinBCM arewell-definedsurfaceareaswhereasinBPMtheyarepointwise contactsbetweenprimary sphericalparticles. Hence,the debond-ing stress thresholdof the cell-cell interfaces can be directly set equal to the tensile strength of the particle and the debonding force isgivenbytensilestrengthmultipliedby thecell-cell inter-facearea.

Fig. 1. Particles generated with different numbers of cells which are represented by different colors.

2.2. Internalcohesionandfracture

InBCM,thecellsshouldinteractthroughaninterface mechan-ical behaviorpertainingto thenatureofthe material.Inits most generalform,thisinterfacebehaviorischaracterizedbya relation-ship between thenormal andtangentialcomponents ofthe cell-cell stress,ononehand,andtherelativecell displacementor ve-locity,ontheotherhand.Wealsoneedacriterionfordebonding, i.e.thelossofinternalcohesionandthuscreationofacohesionless frictionalinterfacebetweentwocells.

In most discrete-element models applied to particulate com-pounds, the material parameters are the elastic moduli, and debondingisgovernedbyanormaland/ortangentialforce thresh-old. Hence, in these models the creation of cohesionless surface anditspropagationdonotexplicitlyobeythethermodynamic Grif-fithcriterion inwhichthepropagationofacrackrequiresthat an amountofworkperunit areaequaltoorlargerthanthefracture energyG_f tobe suppliedbythe actionofexternal forcesorfrom the variation of the elastic energy (Hillerborg et al., 1976; Rice, 1978 ). The original Griffith formulation isbased on a differential criterion, assumingthat thecrackgrowthiscontinuous. Hence,it can not be appliedassuch to acell-cell interface inBCM, which inthespritofDEM mustfailasawholebyreleasingafinitearea

s.Inother words,itisnotdesirableandcomputationallyefficient toconsidersub-cellscales andthetimeprocessofcrack propaga-tion inside the interface. Forthis reason, an incremental formof theenergycriterionshouldbeapplied(Leguillon, 2002 ):

−



Wp

s ≥ Gf, (1)

where



Wp isthevariationofthepotentialenergy.

InordertouseEq. (1) inDEM,wealsoneedtoexpressthe vari-ation



Wpofthe potentialenergyintermsofcell displacements

asthecellsareconsideredtoberigidsothatallinterfacevariables reflect those ofcells. We alsoneed to separate tensileandshear componentsun andut,respectively,oftherelativedisplacementof

thecellsattheinterface.Foraninterfaceattensileorshearstress threshold,theworkGnorGtperformedbythestressfromthetime

whenthestressthresholdisreachedt0tothetimetisgivenby

Gn

(

t

)

= t t0 Cnundt=Cn



n

(

t

)

, (2) Gt

(

t

)

= t t0 Ctut dt=Ct



t

(

t

)

, (3)

whereCnandCtarethetensileandshearstressthresholds,

respec-tively.EitherGnorGt (dependingonwhetherthenormalstressor

theshear stress is atits threshold),is the workabsorbed by the interface,andassuming that



Wpisfullyconsumedinthiswork,

accordingto(1) theinterfacefailsattimet1wheneither

Gn

(

t1

)

=Gf n=Cnn, (4)

or

Gt

(

t1

)

=Gf t=Ctt, (5)

whereGfnandGft are fractureenergies fornormalandtangential

rupture, respectively, n=



n

(

t1

)

and t=



t

(

t1

)

. In this way, a cell-cell interface fails when the stress threshold is reached and thecumulativeworkabsorbedbytheinterfaceduetotherelative displacementsalongthenormalortangentialdirectionisequalto thecorrespondingfractureenergy.

Notethat the formulation ofthe energy criterion in terms of finite increments is consistent with the Finite Fracture Mechan-icsbasedontheassumptionthat crackpropagationalwaysoccurs over a minimal length n≈ Gfn/Cn or t≈ Gft/Ct (Leguillon, 2002 ).

Asimilarlengthscaleisalsointroduced inCohesiveZone models inwhichitisassumedthatthematerialismicro-crackedandcan stilltransmitstressesinaregionoffinitelength behindthecrack tip(Hillerborg et al., 1976 ).Inourformulationofthework calcu-latedfromcell displacementsinequations(2) and(3) ,thelength scalesn andt representtheorders ofmagnitudeofthe relative

normalandtangentialdisplacementsbeforefailure.Hence,the ra-tios n/dc andt/dc,where dc isthe mean cell diameter, are the

inelasticdeformationsofaparticlebeforefragmentation. Aslarge inelasticdeformationsmustbeavoidedinDEMsimulations(aswe wantthecellstokeeptheircloseneighborsaslongasthefracture hasnotoccurred),itisimportanttomakesurethatthecellsizedc

islargecomparedtotheratiosGfn/CnandGft/Ct.

Inpractice,whenastress thresholdisreachedataninterface, thetwoconnectedcellsareallowedtoseparateorslidealongtheir interface.But,even ifthecorresponding relativecell velocitiesun

andutaredifferentfrom0,theinterfaceisallowedtocarrya

nor-malforce sCn ora tangentialforce sCt.Inother words, the

inter-faceremainsactiveandstress-transmittingalthoughthetwocells canmove withrespecttoeach other.Letfn andft be thenormal

andtangentialinterfaceforces,respectively, and

μ

the coefficient offriction.Fortheimplementation,ourinterfacemodelcanbe ex-pressedbythefollowingrelations:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

Gn≤ Gf n ∧



_

n=0 ⇒ fn≥ −Cns



n>0 ⇒ fn=−Cns Gn>Gf n ∧



_

n=0 ⇒ fn≥ 0



n>0 ⇒ fn=0 (6)

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

Gt≤ Gf t ∧

⎧

⎨

⎩



t>0 ⇒ ft=−Cts



t=0 ⇒ −Cts≤ ft≤ Cts



t<0 ⇒ ft=Cts Gt>Gf t ∧

⎧

⎨

⎩



t>0 ⇒ ft=−

μ

fn



t=0 ⇒ −

μ

fn≤ ft≤

μ

fn



t<0 ⇒ ft=

μ

fn (7)

Fig. 2 shows a graphical representation of these relations. Note that, theinterface becomesnoncohesiveonly inthe cases where

fn=0or

|

ft

|

=

μ

fn.

Oncean interfacelosesitscohesion,itturnsintoapurely fric-tionalcontact.Ifthegapcreatedasa resultofinterface deforma-tionisnonzero(



n>0),thenormalandtangentialforcesareboth

zeroandthecreatedcontactisopen.Otherwise(



n=0),the

con-tactremains active andthe relationbetweenthe normalforce fn

Fig. 2. Interface behavior along (a) normal direction and (b) tangential direction. The solution for each pair ( n , f n ) and ( t , f t ) lies on the thick line. See the text for

the definition of the variables.

Fig. 3. Frictional contact law defined at the contact framework in the (a) normal direction (b) tangential direction.

inequalities:

⎧

⎨

⎩



n= 0 ∧



_u n=0 ⇒ fn≥ 0 un>0 ⇒ fn=0



n>0 ⇒ fn=0 (8)

TheseinequalitiesareshowninFig. 3 asagraph(Jean et al., 2001 ). In the same way, the frictional component is governed by the Coulombdryfrictioninequalitiesbetweenthefrictionforce ft and

therelativetangentialvelocityut:

⎧

⎪

⎨

⎪

⎩



n=0 ∧

⎧

⎨

⎩

ut>0 ⇒ ft=−

μ

fn ut=0 ⇒ −

μ

fn≤ ft≤

μ

fn ut<0 ⇒ ft=

μ

fn



n>0 ⇒ ft=0 (9)

shown in Fig. 3 (b). The rigid-plastic contact model intro-duced in this work is similar in spirit to that introduced by Timár et al. (2010) inwhichtheelasticbeamsconnectingthe parti-clesarere-establishedafterahealingtime,allowingthusfor plas-tic(irreversible)deformation.In ourmodel,the particledoesnot breakaslongastheworkisbelowthefractureenergy.Thismeans thattheparticlecandeformasawholewithoutfailure.

2.3.Contactdynamics

For the simulation ofthe dynamics of undeformableparticles (includingrigidcellsinthiswork), weemployed theContact Dy-namics(CD)method(Moreau, 1993; Jean et al., 2001; Radjai and Richefeu, 2009 ).Asinmoleculardynamics(MD)orsimilarDEM al-gorithms,theequationsofmotionareintegratedintimebymeans ofa time-stepping scheme.However, in contrastto MD,the Sig-norini and Coulombinequalities are implemented in CD as con-straintsthat are takenintoaccount forthe calculation ofcontact forcesandvelocitiesinanimplicitscheme.Aniterativealgorithm isused to calculate simultaneously at all contacts and interfaces

the relativevelocities andforces atthe endofeach time step. It shouldbenotedthat,asintheCDmethodtheinterfaceand con-tactbehaviorsarenotbased onanelastic forcelawinvolvingthe overlapbetweenparticles(orcells),thetimestepcanbelarge,and thecalculatedforcerepresentsatime-averagedforceduringatime step.

TheimplicitnatureoftheCDmethodcanbe describedas fol-lows:

1. Anetworkofpotential contacts(orinterfaces)is definedfrom theparticlepositions.

2. Thecontactforcesandvelocitiesarecalculatedby aniterative process accountingfor equationsof motiontogether withthe CoulombandSignorinirelations.

3. Theparticlepositionsandrotationsareupdated.

Thesecondstepensuresthat,whentheparticlesaremoved ac-cordingto theircomputed velocities,they willnot overlapatthe endofthetimestep.Becauseofthisimplicitnatureofthemethod, the time-steppingscheme is unconditionallystable, so that large timestepscanbeused.

Itis alsoimportanttomentionherethemeaningof the resti-tutioncoefficientintheframeworkoftheCDmethod.Fora colli-sionbetweentwo particles,the normalandtangentialrestitution coefficients,en andetrespectively,areclassicallydefinedfromthe

relativenormalandtangentialvelocitiesafterandbeforethe colli-sion.Thisconceptcannotbeusedinadensegranularmaterialin which theparticle momentuminvolves a network ofparticles so that the momentapropagate through the whole contactnetwork andmayleavethesystemthroughtheboundaryconditions.Inthe CDmethod,aconceptuallydifferentapproachisused.Infact,the SignoriniandCoulombrelations(8) and(9) involvethevelocities

un andut, whichrepresentthe relativevelocities attheend ofa

timestepbecauseoftheimplicitformulationofthetime-stepping scheme.But a more generalapproach consistsin replacingthese velocitiesbyweightedmeans(Moreau, 1993; Radjai and Richefeu, 2009 ): un=un +_+enun− 1+en , (10) ut= ut++etut− 1+

|

et

|

, (11)

whereu−n andu−t are the velocitiesatthe beginning ofthe time

stepandu+n andu+t arethevelocitiesattheendofthetimestep.

According to (10) , a contact occurs between two particles when

un=0,implying u+n =−enu−n, whichcorresponds to thecommon

interpretation ofen.But,in contrastto thisclassical definitionof

en, the conditionu−n =0 (i.e. a persistent contact) does not

nec-essarilylead to u+_n =0.The lattercan only arise asa solutionof globaldetermination offorces andvelocitiesthrough an iterative process together withrelation(10) inwhich u−n =0. Forthe

tan-gential restitution coefficient, the condition u_t−=0 means that a contactis in the rollingstate (no sliding butone particle rolling on theother orsimply norelative motion).The above discussion regardingthe normal velocitiesapplies also to thetangential ve-locities.

Fortheinterfacebetweencells,aslongasthecohesionis effec-tive,the relativevelocity ofthecellsshouldbe interpretedas in-elasticdeformationslocalizedinsidetheinterface.Thismeansthat thecoefficientofrestitutionshouldbeconsistentlysettozero.This isalsotrueforcohesionlesscontactsbetweencellssincetheir con-figurationinsidetheparticleistoodense(withzeroporosity)fora normalrestitutioncoefficienttobe effective.Forthesereasons,in allsimulationresultsreportedinthispaperweseten=et=0.

In application to cells of polyhedral shape, the CD method should resolve also the types of contacts between the cells at

Fig. 4. Generic contact types between polyhedra.

eachtimestepbeforetheiterativedeterminationofdynamic vari-ables.Therecanbeseveraltypesofcontactsbetweentwo polyhe-dra: face-face, face-edge, edge-edge, vertex-face,.... Geometrically, a face-face contactisa plane anditcan be representedby three points(termed ‘triplecontact’).A face-edge contactisa line and can becharacterizedby twopoints(termed‘doublecontact’);see Fig. 4 .Avertex-facecontactisapointtermed‘singlecontact’.The edge-edge contacts are generally of single type whereas vertex-vertex and parallel edge-edge contacts are statistically rare and their occurrencesensitively dependson thegeometrical precision of the detection procedure. To determine the contact types, we usetheCundallCommonPlanmethod(Cundall, 1988 )andwe at-tributethreepointstoatriplecontact,twopointstoadouble con-tact and one point to a single contact. All pointsare treated as independent point contacts to which the iterative procedure de-scribedpreviously isapplied.Thecontactforceofatripleor dou-blecontactistheresultantforce ofthethreeortwoforcesacting atthecorrespondingpointswithitsapplicationpointdetermined astheircentroid.

The initial Voronoï tessellation ofa particle leadsto a config-uration of polyhedral cells that, by construction, have face-face, vertex-vertex and parallel edge-edge contacts. We only consider the face-face contacts that define the cohesive interfaces. The edge-edgeandvertex-vertexcontactsinthecellconfigurationare omittedastheinternalcohesionoftheparticleiscarriedbythe in-terfaces.However,asthecellsmoveorfractureattheir interfaces, contacts ofother types mayoccur, andthe consequent evolution needs todetectperiodicallythecontacts.Becauseofitstreatment ofcontactsasgeometrical constraints,theCDmethodprovidesin thiswayageneralframeworkforthesimulationofparticlesof ar-bitraryshape.Forthesimulations,weusedtheCDmethodas im-plementedinthesoftwareLMGC90(Dubois et al., 2011 ).

2.4. Impacttest

To investigate the fragmentationof particles, we perform im-pacttests,whichconsistinreleasingaparticlefromaheightequal to 2timesitsradius,measured fromthelowestpoint ofthe par-ticle,ontoarigidplane.Inordertostudytheeffectoftheimpact energyW_k−(kineticenergyoftheparticlebeforecollision),the im-pactvelocitywasvariedbyapplyinganinitialvelocitytothe par-ticle withthe gravity set to g=9.81 m/s2_{. The impact energy is} given byW_k−=m

v

2_{/2, wherev is theparticle velocity atimpact} time with theplane andm isthe particle mass.We used a par-ticle diameter equal to 1mm in all tests. The friction coefficient between the particle and the plane was set to 0.4. Each impact

Fig. 5. Snapshots of a particle impacting a rigid plane, and the evolution of particle breakage. This test was performed with an impact velocity of 6 m/s.

Fig. 6. Effect of the number of cells on the number of generated fragments for three different values of the impact velocity.

test was repeated10 times,each with a different tessellation of theparticleinto cells.The datapointspresentedinthe following areaveragevaluesoverthe10testswithanerrorbarrepresenting theirstandarddeviation. Itshouldalsobe notedthat thepointof impactwiththe plane israndom, sothat theparticle, whichhas apolyhedralexternal shape,canfall onaface,edge orvertex.To avoidsystematicerrorsduetothiseffect,werotatetheparticlein anarbitrarydirectionbeforeeachimpacttest.Asequenceof snap-shotsofaparticleduringanimpacttestisshowninFig. 5 .

In all the simulations described in this paper, we set G_{f n}=

Gf t≡ Gf withseveralvaluesintherange[0.2,2]J/m2,

correspond-ing to typical measured values of the fracture energy for glass beads.We performedaparametric studyby changingthe impact velocityupto 10m/s,theinterface stressthresholdsCn andCtup

to15MPa,andthe frictioncoefficient

μ

betweenfragmentsfrom 0.2to0.6.

Theobjectivityofafracturemodelrequiresthefractureprocess tobeindependentofnumericalparameters.InourBCMmodel,the numericalparameters are relatedto the Voronoï tessellationand the number of cells Ncells. For the tessellation, we use the most

random distributionof cellsbut their numbermay influence the fracture process. Fig. 6 shows the number of fragments Nf as a

functionof Ncells forthree differentvaluesofthe impactvelocity.

Wesee thatthe numberoffragmentsdeclines asNcells increases,

butasymptotically tendstoaconstantvalueindependentofNcells.

reduce the finite size effects that influence the number of frag-ments forlower numbersof cells. For thisreason, inall simula-tions ofimpact test we used 100 cellsto tessellatetheparticles. We analyze indetail below the effects of various parameters on theparticlefracture.

3. Damageandfragmentationefficiency

During an impact,partoftheinitial kinetic energyW_k− of the particleis transmittedto thefragments. LetW_k+ be thetotal en-ergyofthefragments.ThedifferenceWd=Wk−− Wk+isconsumed

infracture andpossiblydissipativeinteractions, includingfriction andinelasticcollisions, betweenfragments.Ifsis thetotal cohe-sionlesssurfaceareacreatedduringfracture,thetotalfracture en-ergyisgivenby

Wf=sGf. (12)

Intheframework oftheBCM, thisenergymaybecomparedwith thetotalfracture energyWT

f =s T_G

f requiredto breakallcell-cell

interfacesoftotalarea sT_{.Hence,the particledamage canbe}

de-finedas Dw= Wf WT f = s sT. (13)

Obviously, the value of sT _{dependson the size or number of}

cells.Thephysicalmeaningofthislimitinmaterialscanberelated tothescale ofheterogeneities.Forexample,ina porousmaterial of porosity

φ

with a distribution of pores of typical volume Vp,

themean-freepath =

(

Vp/

φ

)

1/3 maybe considered asthe

typi-calsizeofelementarycells(Laubie et al., 2017 ).Insuchmaterials, asa resultofstressconcentration, thevalue offracture energyis farbelowthe theoretical thresholdof thesame materialwithout pores.For thisreason, the fracture energyof fragmentsof size 

ismuchhigherthanthe initialporoussamples.Inthe sameway, inmaterials withagranulartexture, the grainsareusually much harderthan their assembly, andthus the grainsplay the role of buildingblocksasthecellsinourmodelmaterial.

A crucialaspect ofcomminution isits energetic efficiency,i.e. theamountofenergyconsumedforfractureasafunction of im-pactenergy.Wedefinethefragmentationefficiency

η

astheratio ofthetotalfractureenergytotheimpactenergy:

η

= Wf

W_k−. (14)

Thecomminutionisgenerallynotanefficientprocessinthesense that mostof thesupplied energyis not consumed infracture. It isthusinteresting tosee howthevalue of

η

fora singleparticle dependson theimpact parameters.This informationcanthen be usedtounderstandandpredictthefragmentationefficiencyforan assemblyofparticlesinarotatingdrumoranyothercrushing de-vice.

Fig. 7 shows particle damage Dw as a function ofthe impact

velocityvforseveralvaluesofthefractureenergyGf.Thedata

fol-lowanS-shapedcurveinwhichthedamagefirstincreasesrapidly withv andthen slowlytends to 1. Asexpected, particledamage for a given value of v declines as Gf increases. The asymptotic

valueDw=1correspondstothelimitcasewheretheparticlefully

breaksinto its buildingcells. Wealso seethat theerrorbars are small,indicatingthat the variabilityoffracture asa resultof the variationsofimpactpositionisnotaninfluentialfactor.

Fig. 8 showsthe fragmentationefficiency

η

as afunction ofv

for different values of Gf. We see that

η

is unmonotonic: it

in-creasesrapidlywithvup toavalue oftheorder of0.3andthen slowly declines towards zero. The velocity at which

η

takes its peak value increaseswithGf.Such an optimum value forenergy

utilizationasafunctionofthesuppliedenergyperunitmasswas

Fig. 7. Particle damage D w as a function of impact velocity v for different values

of fracture energy G f and C n = C t = 1 MPa. For each test, the error bar represents

standard deviation over 10 independent tests.

Fig. 8. Fragmentation efficiency ηas a function of impact velocity v for several val- ues of fracture energy, C n and C t were kept constant at 1 MPa.

alsoobserved in impactexperiments ofquartz beads ofdifferent sizes(Rumpf, 1973 ).Thisunmonotonicbehaviormeansthat there isa characteristic velocityat whichthe conversionof kinetic en-ergytofractureenergyisoptimal.Belowandabovethe character-isticvelocitythesuppliedenergyismostlyeitherdissipatedby in-elasticcollisionsortakenawaybythefragments.Theenergy con-sumedbyfragmentmotionsandinelasticcollisionsatthe charac-teristicvelocityisalmosttwotimeslargerthanthefractureenergy. Since the different plots of Dw and

η

differ according to the

value of the fracture energy Gf, we expect that they can be

col-lapsedonthesameplotwhenconsideredasafunctionofthe sup-pliedenergyW_k− (ratherthan theimpactvelocity)normalized by

WT

f =sTGf.Up to statistical fluctuations, this is indeed what we

observeinFigs. 9 and10 ,displayingDw and

η

asafunctionofthe

normalizedimpactenergydefinedby

ω

=Wk−

WT

f

. (15)

ThefittingformshowninFig. 10 isgivenby

η

(

ω

)

= a

ω

/

ω

∗

1+

(

ω

/

ω

∗

₎

2, (16) with a=0.55 and

ω

∗=1.81. In this approximation of the col-lapseddata,thepeakvalueof

η

is ࣃ0.27anditoccursfor

ω

=

ω

∗_. This value means that the amount of impact energy required to fracture the particleinto its buildingblocks (cells) is almost two

Fig. 9. Particle damage D w as a function of the normalized impact energy ω. The

dotted line is the fitting form (21) . The error bars represent standard deviation for 10 independent events. The inset shows the same plot in the range ω < 1.5 together with a quadratic fit.

Fig. 10. Fragmentation efficiency ηas a function of the normalized impact energy

ω. The dotted line is the fitting form (20) . The error bars represent standard devia- tion for 10 independent events.

timesthetotalfractureenergyoftheparticle,andthe fragmenta-tionefficiencyisonly27%forthisamountofthesuppliedenergy.

The fitting form (16) can be understood by first noting that Eqs. (13) –(15) yieldtherelation

η

=Dw

ω

. (17)

At low values of

ω

i.e. (

ω

<1.8), the data points in Fig. 9 sug-gest that Dw in this regime increases quadratically with

ω

; see

the inset to Fig. 9 . Hence, according to (17) ,

η

increases linearly with

ω

.On the other hand, atlarge valuesof

ω

, Dw tends to 1

so that

η

asymptotically declines as1/

ω

.The fittingform (16) is the simplest interpolation betweenthese two asymptotic behav-iors.Eqs. (16) and(17) ,yieldthefollowingfittingformforDwasa

functionof

ω

:

Dw

(

ω

)

=

ωη

=a

ω

∗

(

ω

/

ω

∗

₎

2

1+

(

ω

/

ω

∗

₎

2, (18) SinceDw tendsbydefinitionto1as

ω

→∞,wehave

a= 1

ω

∗. (19)

Thisrelationisconsistentwithournumericaldataanditreduces thenumberofparametersinEq. (16) toasingleparameter,sothat

wehave

η

(

ω

)

=

_ω

1_∗

ω

/

ω

∗ 1+

(

ω

/

ω

∗

₎

2, (20) Dw

(

ω

)

=

(

ω

/

ω

∗

₎

2 1+

(

ω

/

ω

∗

₎

2. (21) Both functions are in good agreement with the data shown in Figs. 9 and10 .

4. Restitutioncoefficient

Wenowconsidertwo moredimensionlessvariablesthat char-acterizethetransferofkineticenergyfromtheimpacting particle tothe fragments.We define an effectiverestitution coefficient ek

fromtheratioofthepre-impactandpost-impactkineticenergies:

e2_k=Wk+

W_k−. (22)

Thiscoefficient can take anonzero value even when the restitu-tioncoefficient ebetweentheparticle andthe impactedplane or betweenthe fragmentsiszero.Anothervariableofinterest isthe ratioofthepost-impactkineticenergytothefractureenergy:

χ

=Wk+

Wf

. (23)

Thisvariablesimplyreflectstherelativeimportanceoftheenergy transported by the fragments with respect to that consumed in particlefragmentation.

Fig. 12 shows

χ

as a function of

ω

onthe log-log andlinear scales. We see that

χ

is nearly constant and quite small (ࣃ0.3) in the range

ω

<

ω

∗_{ࣃ1.8 andthen increases approximately as a} power-lawfunction:

χ

(

ω

)

=c

 ω

ω

∗

α

for

ω

>

ω

∗_{, (24)} with

α

ࣃ1.12andcslightlyincreasingwithGf.ForGfvaryingfrom

0.2J/m2 _{to 2J/m}2_{, c variesfrom 0.3 to 0.5, as observed in inset} to Fig. 12 . The kinetic energy of the fragments being negligibly small as compared to the energy consumed for fracture in the low-energyregime, all the suppliedkinetic energy iseither used forfracture or dissipated by inelastic collisionsand friction. The amountofdissipationbycollisionsandfrictionisgivenby

Wc=W_k−− W_k+− Wf ≡



₁

η

−

χ

− 1

Wf. (25)

IthasitslowestvalueWc/Wfࣃ2.2at

ω

∗where

η

ࣃ0.27.

Fig. 11 showse2

k as a function of

ω

.It is nearly constant and

smallinthelow-energyregime (

ω

_<

ω

∗) andthen growswith

ω

. Itiseasytoseethat

e2

k=

χη

(26)

In the low-energy regime, we have ekࣃ0 as

χ

is negligible. At

higherenergies,wehave

e2 k=

χη

= c

ω

∗

(

ω

/

ω

∗

₎

α+1 1+

(

ω

/

ω

∗

₎

2. for

ω

>

ω

∗ (27)

Thisform fits well the data asshown inFig. 11 .We see that as

ω

→∞,e2

kvariesasymptoticallyas

ω

1/2.

5. Influenceofstressthresholdsonthefractureprocess

In the last section, we extensively analyzed the effect of im-pactenergyandfractureenergyontheparticlefragmentationfor aconstant valueofthestress thresholdsCn andCt aswell asthe

Fig. 11. Squared restitution coefficient e 2

k as a function of ω. The dotted lines rep-

resent the fitting form (27) with the corresponding values of c (see inset to Fig. 12 ). The error bars represent standard deviation for 10 independent events.

Fig. 12. Variable χ as a function of ω. The dashed line represents a power-law function ∝ ωα_{with αࣃ 1.12. The inset shows the same data on the linear scale.}

The dotted lines are different fits with the same value of αbut different values of the prefactor c ; see Eq. (24) . The error bars represent standard deviation for 10 independent events.

frictioncoefficientbetweenfragments.Inthissection,weconsider theeffectofstressthresholds.

Fig. 13 showsparticledamageDw asafunction ofCn forthree

valuesoftheimpactvelocityv.AsexpectedforsmallvaluesofCn,

theparticleis fullydamaged (Dw=1) butasCn increases,Dw

de-clinesandtends toa constant valuecloseto 0.Foragivenvalue of Cn, particle damage is higher for larger velocity. Actually, the

valuesofCn canbe compared to themean stress generated

dur-ingthecollisionoftheparticlewiththeplane.Themeanstressis givenbythe momentumexchange mv,wherem isparticle mass, divided by collision duration, which is equalto the time step

δ

t

inour CDsimulations, andby the total cell-cellinterface sT_.The

samedata are showninthe inset ofFig. 13 asa function of the normalizedstress CnsT

δ

t/(mv). Wesee thatwithin statistical

fluc-tuations thedata for thethree values ofv collapseon the same curve.Itisimportanttoremarkherethatthecollisiondurationis animportant parameterforthisscaling. Fora compliantparticle, thecollisiondurationdependsontheelasticmodulioftheparticle andmv/

δ

tshouldbereplacedbythelargestforceachievedduring collision.

Another parameter that mayinfluence particle fracture is the ratio

ψ

=Ct/Cn. Fig. 14 shows Dw as a function of

ω

for

differ-entvalues of

ψ

. Withinstatisticalfluctuations, all the data

coin-Fig. 13. Particle damage D w as a function of the normal stress threshold C n for

three values of impact velocity v for G f = 1 J/m 2 . The inset shows the same data

for the values of C n normalized by the impulsion mv /( s Tδt ), where mv is the change

of momentum during collision and δt is collision time. The error bars represent standard deviation for 10 independent events.

Fig. 14. Particle damage D w as a function of ω for different values of ψ = C t /C n .

The dotted line is the fitting form (21) . The error bars represent standard deviation for 10 independent events.

Fig. 15. Particle damage D w as a function of friction coefficient μbetween cohe-

sionless cells. The dotted line is the fitting form (21) . The error bars represent stan- dard deviation for 10 independent events.

cide.Interestingly,evenforCt=0,weobservethesamebehavior,

meaningthatthefracturebasicallyoccursintensilemode.Relative tangentialdisplacementsbetweencellsareobviouslyincompatible withthe kinematicconstraints.Butthis doesnotexclude the ac-tivation oftheshearmode inthecaseofanoblique collision be-tweentheparticleandtheimpactedsurface.Wealsoobservethat thefrictioncoefficient

μ

,activatedonlyatcell-cellinterfaces hav-inglosttheir cohesion,hasnoinfluenceonthefractureasshown inFig. 15 .

6. Conclusionsandperspectives

In this paper,we used 3D DEM numericalsimulations to an-alyze the fragmentation of a single particle impacting a rigid plane. Theparticleisdiscretizedbymeans ofVoronoï tessellation into polyhedral cells representing potential fragments. The cell-cell interfacesare governedby afracture lawcombiningmaterial strengthandfractureenergyasnecessaryandsufficientcriteriafor debonding.In thismodel,theratioofthe fractureenergy to ma-terialstrength correspondstothe minimuminelasticdeformation beforeparticlefracture.

We showed that particle damage, i.e. the proportion of frac-turedinterfaces,andtheamountofenergyconsumedfor fragmen-tation scale withthe initial kinetic energy ofthe impacting par-ticle normalizedby fracture energy. The fragmentationefficiency, definedasfragmentationenergynormalizedbytheimpactenergy, is unmonotonic as a function of impact energy withoptimal ef-ficiencyoccurringforawell definedvalue oftheimpact velocity. Weintroducedafunctionalformthatfitsthecollapseddatawitha singlefreeparameter. Similarfittingformswere proposedforthe damage andeffectiverestitutioncoefficient. Wealso showedthat particle damage scales with the normalstress threshold normal-izedbythemeanimpactstress.

All the resultspresented in thisnumerical investigation show consistently the ability of our numerical approach to handle dy-namicfragmentationofsingleparticles. Itcan beapplied to sim-ulatethe compactionandshearofan assembly ofcrushable par-ticles in which the behavior of a single particle can provide in-sightsintothecollectiveevolutionofparticlesandtheirfragments. The mostbasiclimitationofthisapproach,asinall DEM simula-tions, isthenumberofpotential fragmentsthat canbe produced inthecomminutionprocess.Nevertheless,itisapowerfultoolfor a detailedanalysis of the local events and re-distribution of en-ergy inside the granular material. Forexample, in thispaper we foundthat thefragmentationefficiencyisatmost30%inthe sin-gleparticleimpacttests.Thequeryhereiswhethersuchalevelof efficiencycan bereachedin acollective fragmentationprocessof particles.

Finally, inthe caseof a singleparticle,we did not analyzein thispaperthecrackgrowth,fragment shapesandsizesfor differ-ent levels of fracture energy, and fracture modes ofthe particle. Thesefeatureswillbepresentedelsewhere.

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