Contact disorder and force distribution in granular materials

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Contact disorder and force distribution in granular

materials

Jean-Noël Roux

To cite this version:

MATERIALS.

DESORDRE DE CONTACT ET DISTRIBUTION DES FORCES DANS LES MATERIAUX GRANULAIRES.

Jean-Noël ROUX

LaboratoireCentral des Pontset Chaussées,

58boulevard Lefèbvre, 75732 Paris edex 15,Fran e

ABSTRACT: Thanks to extensive numeri alsimulationsof a simple two-dimensional model ofa granular pa king,we identify a hara teristi lengthfor stress homogeneity thatis signif-i antlylargerthan the grainsize, and a urately ompute the onta tfor e distribution. The ee tsofloadorientation(biaxialtest)arestudied, andtheinuen eofsomebasi simplifying assumptions (small displa ements, no fri tion...) is dis ussed, whi h allows us to shed some light on the possible mi ros opi origins of some generi me hani al behaviours of granular materials.

RESUME: Une étude systématique, par simulations numériques, d'un modèle simple d'assemblage granulaire bidimensionnel,nous onduità l'identi ationd'une longueur ara -téristiquedel'homogénéitédes ontraintesnettementplusgrandequelatailledugrain,etàun al ulpré isdeladistributiondesfor esde onta t. L'évaluationdel'inuen edel'orientation (testbiaxial)du hargement,etladis ussion des onséquen esde quelques hypothèses simpli- atri es(petitsdépla ements,absen ede frottement...) fournissentun é lairagedespossibles originesmi ros opiques de ertains omportements mé aniques génériques des granulats.

1 INTRODUCTION.

One spe i feature of dense granular me-dia, as distinguished from other disordered me hani al systems, is the unilaterality and extremelyshortrangeoftheparti le intera -tion. Twoneighbouringgrains,ifthey tou h, may stronglyrepell ea h other, but on ethe onta t opens, whi h requires but an arbi-trarily small motion, no for e is transmit-ted any longer. In aseeminglyhomogeneous well ompa ted medium, opening and los-ing of onta ts leads to a hara teristi het-erogeneity (1-4) of stress transport: for es appeartobelo alizedonpreferredpaths(or `for e hains'), while some grains arry no load (`ar hing ee t'). Dening a

Represen-tativeVolumeElement(RVE),a ru ialstep (5) in the derivation of ma ros opi onsti-tutive laws, is not straightforward.

limitwithgooda ura y. Se tion7 presents theresultsofabiaxialtest(ma ros opi on-stitutivelaw, distribution of onta t for es), and the on lusive se tion (part 8)dis usses the general relevan e of the model and the role of itsbasi assumptions.

2 THE MODEL

Consider a lose-pa ked 2D assembly of n dis s of diameter a, on a regular triangu-lar latti e. To any dis i, assign a num-ber Æ

i

, that is randomlypi ked up with uni-form probability in the interval [0; ℄, with 0   1, and redu e its diameter to a

i = a(1 Æ

i

). This(g. 1)isthereferen e ong-uration of the system, from whi h displa e-ments are evaluated. Then, submittingit to

Figure 1: A hexagonal sample. Intergranular gaps arewidenedonthegure.

some given load, sear h for equilibrium dis-pla ementsand onta tfor es, underthe fol-lowing three assumptions.

1. dis s are rigid

2. fri tion isignored

3. onemayusetheapproximationofsmall displa ements(ASD).

Assumption 2 means that the for e F ij

ex-erted by any dis i onto its neighbour j is arried by the unit ve tor n

ij

pointing from the entre of i to the entre of j : one has F ij = f ij n ij , with f ij > 0. A ording to as-sumption 1, the onta t law relating f

ij to the interstitial thi kness h

ij (initially equal to h ij =(Æ i +Æ j

)a=2) is the Signorini ondi-tion: ( f ij =0 if h ij >0 f ij 0 if h ij =0 (1)

The ASD (assumption 3) amounts to evalu-ate displa ements and deformations to rst order in  (regarded as innitesimal), while positionsandfor es areevaluatedonthe ref-eren e onguration. In parti ular, any ve -torn

ij

stays paralleltoone of the three ve -tors n 1 (1;0), n 2 ( 1 2 ; p 3 2 ) and n 3 ( 1 2 ; p 3 2 ). h ij is linearlyrelated tothe displa ements :

h ij =h 0 ij Æu ij ; with Æu ij =n ij :(u i u j ): (2)

In the following, any su h pair i;j of neigh-bours on the latti e is alled a onta t. N denoting the number of onta ts, those are labelled by index l, 1l N, and one may write h l , Æu l , n l

, et ... Conta t l is losed if h

l

=0. It isa tive if f l

>0.

3 STRESSES AND STRAINS

In order to impose a state of uniform stress, several boundary onditions (BC) are used. One may spe ify the motion of the walls of a ontainer, orexert some externalfor es on the grains near the boundary. If the sam-ple shape paves the plane, periodi bound-ary onditions might also be implemented. ForwhateverBC,anoverall,generalized dis-pla ement(or, respe tively, for e) is dire tly dened, and the onjugate generalized for e (resp.,displa ement)identiedonwritingthe work of external eorts. Those global stati and kinemati parameters may respe tively beinterpretedasstress,,andstrain,, ten-sors. We implement the dierent BC's in su hawaythatthe lassi alrelationship(15)

 = a A N X l =1 f l n l n l (3) holdsexa tly,A= Na 2 2 p 3

beingthesample sur-fa e area. (Compressions are ounted posi-tively). It is onvenient to part onta ts in three subsets C

k

k l k notingthenthe average ofthe onta tfor es inthe set C k , eqn.3 be omes : 8 > > < > > :  11 = 1 a p 3  2F 1 + 1 2 (F 2 +F 3 )   12 = 1 2a (F 2 F 3 )  22 = 1 2a (F 2 +F 3 ) (4)

whi h, onversely, may bewritten as 8 > > > < > > > : F 1 = a p 3 2   11 1 3  22  F 2 = a   22 p 3 + 12 )  F 3 = a  22 p 3  12 )  : (5)

Su h alinearrelationgivingthe angular dis-tribution of normal for es on e  is known

was onje tured in the general ase (15). It is automati ally satised here be ause there are onlythree onta t dire tions.

(Æu l

) 1l N

is anadmissible set of normal relativedispla ementsifitis ompatiblewith some a tual dis displa ements and the BC. It is then asso iated with a value of .

Sim-ilarly, (f l

) 1l N

is an admissible set of on-ta t for es if dis s are in equilibrium, with somevalueof. Foranysu hpair ofrelative

displa ements (supers ript (1)) and onta t for es (supers ipt (2)), on has the following formof Hill's lemma(5):

N X l =1 f (2) l Æu (1) l =A (2) : (1) : (6)

Now, due to latti e regularity, the repla e-ment of any f (2) l by the average F (2) k when-ever l 2 C k

yields another admissible set of onta tfor es,withthe same

(2)

(this prop-erty,infa t,guidesthe hoi eofa onvenient implementation of the BC's). Dening then

 k

, for 1 k  3, asthe average of Æu l a over alll 2C k , eqn.6 entails A (2) : (1) = Na 3 3 X k=1 F (2) k  (1) k ;

whi h, using eqn. 5, leads to the following kinemati analogueof eqn. 4:

8 > < > :  11 =  1  12 = 1 p 3 ( 2  3 )  22 = 1 3 (  1 +2 2 +2 3 ) (7)

The following properties stem from lassi al results in linear programming (a parti ular ase of the Kuhn-Tu ker theorem in onvex optimization). Let us submit the system to a given external load. If m is the number of degrees of freedom, m-dimensional (general-ized)displa ementandexternalfor eve tors

~ U and

~

F may be dened. The value of ~ F is imposed. The impenetrability onstraint may be written (see eqn. 2) as (summation overrepeated indi es impliedin the sequel)

G l j U j h 0 l ; (8)

involving anN m matrix G the transpose of whi h appears in the equilibrium require-ment G l j f l =F j : (9)

Let us onsider problems P 1 (unknown ~ U) and P 2 (unknowns f l ) P 1 ( Maximize F j U j with onstraints: ( 8) P 2 ( Minimize f l h 0 l

with onstraints (9)and f l 0 P 1 and P 2

are dual linear optimization prob-lems. Toanysolution

~ U  toP 1 orrespondsa solution ~ f  toP 2

,and re ipro ally,su hthat

f  l  h 0 l G l j U  j  =0: (10)

Conversely,displa ementsand onta t for es satisfying the onstraints of P

1

and P 2

and relation 10are solutionsto P

1

and P 2

. But, on e onstraints are satised, eqn. 10 is ex-a tly equivalent to the Signorini ondition, eqn. 1. Thus, sear hing for an equilibrium state amounts to looking for solutions to P

1 and P 2 . Whenever P 1 or P 2 possess a solu-tion, there exists a `basi ' solution, i.e., lo- atedina ornerofthesimplexofadmissible variables,where amaximumset ainequality onstraintsare satisfedasequalities. The set C



of a tive onta ts orresponding toa ba-si solutiontoP

2

isminimal,sothatrea tion values, f

 l

, are found on solving an isostati problem : they are determined, on e C

pliesequalityof the riteriainP 2

,and hen e requiresoneofanitenumberoflinear om-binations, with xed oe ients (asso iated with some parti ular subsets of the latti e), of the random numbers Æ

i

, to be equal to zero. Su h an event has a zero probability. Therefore, Almost surely, the set C



of a -tive onta ts and the for es f

 l

they arry, are uniquely determined, under a givenload, by the initial hoi e of the random diame-ters. Astodispla ements,la kofuniqueness of the solution to P

1

is only due to motions su hthat Æu

l

=0 for any l2C 

.

5 NUMERICAL METHOD

When the number n of grains ex eeds a few hundreds,thesimplexmethodproves impra -ti able,and the solutionstoP

1

and P 2

,i.e., the displa ements and for es have to be de-terminedbydierentmeans. Inpra ti e,any granular dynami s might be used, provided onvergen e to the unique equilibrium state is ensured. We found it e ient (7,8) to re-sorttolubri atingvis ousfor es, oftheform

f l =(h l ) dÆu l dt ;

withade reasingfun tion that possessesa non-integrabledivergen eash!0. Negle t-ing inertia, vis ous for es balan e external for es, and velo ities, at ea h time step, are the solution to a system of linear equations. Ast!+1,iftheload anbesupported, ve-lo itiesvanish,f l !f  l andh l !0ifl 2C  , f l ! 0 and h l

tends to a nite limit oth-erwise (this might be proved). Equilibrium is thusasymptoti allyapproa hed. Even for the largest samples (n = 12600) that were studied, the set C



weobtainalwaysexa tly satisesthe isostati ity requirement (8).

6 LARGE SYSTEM LIMIT

Studyingmanysamplesofdierentsizeswith various BC's, we he k the existen e of an RVE, and evaluate its size. The pra ti al pro edures are illustrated here in the ase

(8,9).

Intensive quantities des ribing the inter-nal state of the system should approa h a BC-independent limit as n ! 1. When a systemati dependen e on the system di-ameter L is apparent, one might attribute it to a boundary ee t. Near a rigid wall, for instan e, fewer onta ts are a tive than in the bulk. Near a free boundary where somestressisimposed,many onta tsare a -tive, sin e all perimeter grains must arry a load. Su hboundaryinuen esyield system-ati linear variations with 1=L (the relative weightof someperipheralzone). Fig.2isan illustrationofthispoint,withthe proportion of a tive onta ts, N

 =N.

Figure 2: Proportion N 

=N of a tive bonds, aver-agedoverhexagonalsampleswith pdis s peredge, versus1=p, for 3 BC's : rigid rough wall ( rosses), periodi BC (open squares), and uniform pressure (bla ksquares). Errorbarsextendtoonesampleto sample standard deviation. Forperiodi BC's, the system has no wall and the limit is rea hed mu h sooner.OtherBC'sareae tedbywallee ts,hen e the linear ts. All data extrapolate to the limit 0:3930:003. 32systemswith p=60(n>10000)

werestudied.

pre-avoid thinking in terms of ` typi al onta t for es'.

Figure 3: log 10

(p(f)) versus f in units of aP. p(f) de reasesfromanitevalueforf !0. Forf 2,it mightbetted(dottedline)as0:92f

1:5

exp[ 1:28f℄

For the isotropi strain, of the form  =

1, we nd  

!0:3440:003 as L ! 1. Equivalently, the maximum pa king fra tion 



of slightly polydisperse dis s, when the diameter distribution is uniform, is, to rst order in   =  2 p 3 (1 k ); with k = 0:3120:006.   is well dened here,withintheASD,be auseofthe unique-ness property.

A linearsize l mightbeattributedtothe RVE as orrelation length for spatial distri-butionofstress, lengths ale forlo al distur-ban estoattenuate,ordepthofawallee t. All thesepro edures give l 10a.

7 BIAXIAL TEST

If some admissible set of non-negative on-ta tfor esexist,then(fromse tion4)P

1 and P

2

bothpossessasolution,and(asremarked in se tion 3) another admissible set is then obtained on equating ea h f

l

with the aver-age F

k

of all for es arried by the onta ts that share the same orientation n

k

. From

tain the load  if, and only if, the following onditions hold: 8 > < > : 0  22  3 11  22 p 3  12   22 p 3 : (11) Keeping  12 =0 and  11 + 22 onstant, wehavesubmittedsquare systems ofvarious sizes(upto12600grains,4samples)toloads ofdierentdire tions,withthefollowing val-ues of the ratio r = 

11 =

22

: 0:361, 1=2, 2=3, and,1 being alreadystudied,

p

3,3, 10. As g. 4 shows, the anisotropy of the for e hains ree ts that of the load: as r grows, the ontribution of the set C

1

to the trans-mission of stress in reases fromnegligible to dominant. Asthesystem isanisotropi ,only a part of the me hani al behaviour is ex-plored, orresponding,bysymmetry,to

12 = 0. Due touniqueness, weobtainstrains that fun tionnallydepend onload dire tion, and, thankstotheoptimizationproperties,results anbeunderstoodinthefollowingway, illus-tratedong.5. Inthe

11 ;

22

plane,the

sim-Figure5: Strainstates orrespondingtothe7 dier-entloaddire tions that were simulated. Error bars indi ate un ertainties in theL ! 1 extrapolation. The urve(

11 ;

22

)=0istheenveloppeunderthe tangentsthataredrawnforea hpoint. The

asymp-totesarethethi kstraightlines.

pro-from 1=3to 1, taking thevaluesgiven in thetext. A tive onta tsare drawn aslines joining dis entres, withathi knessproportionaltothefor eintensity.

in reases,thenumberoflinearse tions limit-ingthisplanesimplextendstoinnity,while their lengths tend to zero. Consequently, in the limit L ! 1 of ma ros opi systems, the set of strains 

11 ;

22

that are permitted by grain impenetrability is limited by some smooth urve (

11 ;

22

) = 0. For any sus-tainableload (with 

12 =0), 11  11 + 22  22 , the riterionof problemP

1

,is maximizedon that urve, where its tangent is orthogonal to the ve tor of oordinates (

11 ;

22

). The existen e of this point is equivalent to load sustainability, and the urve has therefore

marginallysupported loads given by eqn. 11 (r = 1=3 and r = +1). The BC's require all onta ts of C 1 to lose for r ! +1, hen e  11 =  1

=  =2, and all onta ts of C

2

and C 3

to lose for r = 1=3, hen e  2 =  3 = =2,and  22 =(2=3) (1=3) 11 . Note that, as sample surfa e area is only mini-mized for an isotropi pressure, the system naturally exhibits dilatan y. Our numeri al results strongly suggestthat an RVE always exists, ex ept right on the limit of the do-main of supported loads, for 

22 = 0 or for  11 = 22

given value of  , the ASD should eventually break down asthe urve approa hes anyone of itsasymptotes (the later the smaller ).

Fig. 6 is a plot, versus 1=r, of the pro-portionof a tive onta ts inthe dierent di-re tions. The typi al aspe t of for e

prob-Figure 6: Proportion of onta ts in dire tion 1 ( rosses)and2and3(squares)thatarea tive,versus

1=r. Thedottedlines areguidestotheeye.

ability densities p k

, depending on the dire -tion k = 1;2;3 of the onta t, in the ase of ananisotropi load, is displayed ong. 7,in the ase r = 1=2. In this ase, F

3 = F 2 , p 3 = p 2 and F 1 = F 2 =4. The anisotropy inuen e on the for e distribution de reases with the for e intensity. Most for e hains arrying large eorts are oriented along the prin ipalaxisofstresswith thehigher eigen-value (p

1

(f)  p 2

(f) at large f) while the smallest onta t for es remain isotropi ally distributed (p

1

(f) ' p 2

(f) for small f). A similartenden y wasnoted by otherauthors (12).

8 DISCUSSION

Thanks to the simpli ity of the model and the e ien y of the numeri al method, we derive,fromgrain-levelsimulations,a ma ro-s opi onstitutive law for biaxial ompres-sion. The existen e of an RVE (of typi al

Figure 7: log 10

(p 1

(f)) ( rosses) and log 10

(p 2

(f)) (squares)when r =1=2,versusf, the unit of for e

beinga( 11

+ 22

)=2.

and the for edistributionis a urately om-puted. Load anisotropy mainly ae ts the orientationof the largest onta t for es.

For e hains,wide for edistributions, re-distribution of onta ts on altering the load dire tion, are now re ognized as hara teris-ti featuresofgranularmaterials. Thesimple modelwe study should providea onvenient test for theories attempting a quantitative predi tion of su hphenomena.

dila-as a model for usual biaxial experiments is likely to be limited to the reversible part of the stress-strain urve. It shows that, if the geometry of the pa king is su h that the set of a tive onta ts might hange signi antly with small grain displa ements, the elasti " partoftheelasti -plasti behaviourisnot ne -essarily due to the elasti ity of the grains.

Another remarkable property is the iso-stati ity inthe equilibriumstate. In (9)itis shown that, if one dispenses with the ASD, though the uniqueness is lost, one still nds anisostati for e- arrying stru ture at equi-librium. As spe ial latti ealignments disap-pear ontaking real, nitedispla ementsinto a ount, it is isostati ity in a generi sense, with a oordination number, ounting only load- arrying dis s (n



) and a tive onta ts (N



), of 4. This seems to hold quite gener-ally, and adding some Coulomb fri tion, the inequality N

 < 2n



should persist. There-fore, the oordinan e of a generi pa king of rigid dis s in 2 dimensions(or rigid spheres in 3 dimensions) should never ex eed 4 (re-spe tively: 6).

Theisotropi ompa tionofthesame sys-tem made of elasti dis s is studied in (10). The in rease, as pressure grows, of the den-sity of a tive onta ts, results in a gradual lossofthe hara teristi stress heterogeneity of granular materials.

In future work, with more sophisti ated models, we intend to introdu e (separately, at rst) three features that are possible mi- ros opi sour es of plasti ity and dissipa-tion: solidfri tion,largedispla ementsofthe grains, non- onvex attra tive potential be-tweenthegrains( ohesion). Anyoneofthose ingredientsdestroystheuniquenessproperty, andisthuslikelytoyieldanin remental ma ro-s opi behaviour(asopposed toafun tional relationship between stress and strain). In parti ular, it is expe ted that, provided sig-ni antgranularrearrangementso uronsmall s ale, theplasti " part of the elasti -plasti behaviourisnotne essarilydueto intergran-ular fri tion.

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