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APS/123-QED Irreversible Incremental Behavior in a Granular Material Luigi La Ragione1,, Vanessa Magnanimo1, James T. Jenkins2, and Hernan A. Makse3 1Dipartimento di Ingegneria Civile e Ambientale Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy 2 Department of Theoretical and Applied Mechanics Cornell University, Ithaca, NY 14853 U.S.A. 3Levich Institute and Physics Department CCNY, New York, NY 10031 U.S.A. (Dated: July 22, 2011) 0 We test the elasticity of granular aggregates using increments of shear and volume strain in a 1 numerical simulation. We find that the increment in volume strain is almost reversible, but the 0 incrementinshearstrainisnot. Thestrengthofthisirreversibilityincreasesastheaveragenumber 2 of contacts per particle (the coordination number) decreases. For increments of volume strain, n an elastic model that includes both average and fluctuating motions between contacting particles a reproduceswellthenumericalresultsovertheentirerangeofcoordinationnumbers. Forincrements J of shear strain, the theory and simulations agree quite well for high values of the coordination 9 number. 2 PACSnumbers: 81.05.Rm,81.40.Jj,83.80.Fg ] t f o Granularmaterialshavereceivedtheattentionofmany We carry out numerical simulations using a distinct s researchers in the last decade because of unsolved prob- element method and focus on the first incremental re- . t lems with direct relevance to chemistry, physics and en- sponse of an isotropically compressed random aggregate a m gineering. Thebehaviorofagranularmaterialcanrange that consists of identical, elastic, frictional spheres. We between that of a gas and that of a solid, depending on consider dense aggregates,with solid volume fractions φ d- the applied loading and the regime of deformation con- near 0.64, that have different Z¯. Previous work [10–12] n sidered. Significant progresses have been made with the haveconsideredpoorlycoordinatedaggregatesandfound o introductionof numericaltools (e.g. [1])that permit the thattheshearmodulusisproportionaltoZ¯ Z¯ ,where iso c detailed analysis of an aggregate of particles. Although Z¯ is equal to four for a packing of fricti−onal spheres. [ such simulations have provided information about inter- Hiesroe, we find that when Z¯ decreases, there is an irre- 1 particle interactions, such as their elasticity, sliding, and versiblebehaviorofthe aggregatethatinvolveslocal,co- v deletion,andstatisticalmeasuresoftheircooperativebe- ordinated, irreversible motions of the particles that are 6 havior,suchas induced anisotropyand force chains,it is not resisted by forces. These motions result in a reduc- 6 still unclear how to incorporate these informations in a tionof the apparentstiffness of the aggregate(e.g. [13]). 4 predictive theoretical model. That is, the initial configurationof a poorly coordinated 5 . Attempts to do this have been made in the context of aggregate can not sustain any incremental strain unless 1 the effectivemedium theory(EMT)(e.g. [2,3])inwhich a change in the geometry of the packing occurs. When 0 the contact displacements are given by the applied aver- such irreversible changes are present, simple elastic the- 0 1 age strain. However, the predicted shear and bulk mod- ory can not reproduce the response of the aggregateand : uli are far fromthose measuredin numericalsimulations the utility of the elastic moduli is questionable. v (e.g. [4, 5]). In order to improve the theoretical predic- We introduce measures of these irreversible deforma- i X tion, particle displacements are given by the sum of an tions which vary with the coordination number. For in- r average and the fluctuation components (e.g. [6, 7]). In crementsinvolumestrain,thenumberofirreversiblemo- a particular,betterpredictionsofthe shearandbulk mod- tions is so small that their effect is negligible and the ulihavebeenobtainedinarecentworkthatemployspair response of the aggregate can be assumed to be elastic. fluctuations[8,9]. However,AgnolinandRoux[10]point More importantly, for shear increments, the strength of out that such predictions fail when the the coordination the irreversibility persists even at high Z¯ and increases number, Z¯, is low and close to the isostatic limit Z¯ at asZ¯ decreasestowardsitsisostaticvalue,indicatingthat iso which the aggregateis statically determinate. They sug- elasticity does not describe the aggregate response. gestthatforlowcoordinationnumber,morecomplicated Our numerical simulations consider 10,000 particles, models are needed that account for collective deforma- each with diameter d = 0.2 mm, randomly generated tions among particles. That is, the assumption of pair in a periodic cubic cell. We employ material properties fluctuations is not sufficient to capture the response of typical of glass spheres: a shear modulus µ = 29 GPa poorly coordinate aggregates of particles. Here, we ad- and a Poisson’s ratio, ν = 0.2. The interaction between dressthis issue andprovideanalternativeinterpretation particles is a non-central contact force in which the nor- ofthefailureofanelasticdescriptionatlowcoordination mal component is the non-linear Hertz interaction and numbers. the tangential component is bilinear: an initial elastic 2 and the second is associated with contact forces. We apply a forwardincrement in strain followed by an identical backward strain in which the reference configura- tionshouldberecoveredifthedeformationwereperfectly elastic. We define the parameterζ as the averageoverallcon- tacts of the ratio of the absolute values of the contact displacements after and before the backward increment in strain. A parameter, χ, is similarly defined in terms ofthe contactforces. Bothparameterswouldbe zerofor perfectly elastic behavior. We measure ζ and χ in all of the packings for increments in both shear and volume strain. We apply incre- mentsofstrainwithmagnitudesthatdependonthecon- fining pressure; the ratio of the associatedvolume strain with the confining pressure is constant at about 10−2. FIG. 1: Measurement of irreversibility in terms of displacements when increments of volume and shear strain are ap- TheresultsareplottedinFigs. 1and2,displayingζ and plied. χ as functions of Z¯. Figs. 1 and 2 show small variations in ζ and χ associated with increments in volume strain. The strength of the irreversibility is almost negligible, with slight increases as Z¯ is approached. We conclude that an ap- iso proximate elastic response for the aggregate is obtained for increments of volume strain, independent of the coordination number. For increments in shear strain, both ζ and χ increase as Z¯ is approached. We believe that these irreversible iso motions are associated with the presence of local insta- bility (e.g. [13]). Moreover, both ζ and χ are non-zero forhighvaluesofthecoordinationnumber,incontrastto whatwefindforincrementsinvolumestrain. Theaggre- gate seems to experience rearrangements over the entire rangeofthecoordinationnumber,withtheirriversibility becoming stronger as isostaticity is approached. Theconditionofisostaticityhasbeenanobjectofgreat FIG. 2: Measurement of irreversibility in terms of contact interest for many researchers ([5, 11, 12, 15–17]). The forceswhenincrementsofvolumeandshearstrainareapplied. contact forces in an aggregate are uniquely determined in terms of the applied loads, independent of the contact stiffness, when the aggregate is both statically and displacement followed by Coulomb sliding (e.g.[5]). We kinematically determinate (e.g. [18]). The condition for create different initial isotropic states by varying the co- static determinacy, often referred to as Maxwells con- efficient of friction between particles during the prepa- dition [19], insures the equality between the number of ration (e.g. [14]); all initial states have a solid volume unknowns and the number of equilibrium equations. In fraction φ 0.64 and a confining pressure that varies a granular aggregate, this necessitates that Z¯ = Z¯ . ∼ iso from 50KPa to 10MPa. The condition for kinematic determinacy insures that For all packings, we evaluate the response of the ag- there are no inextensional mechanisms in the aggregate; gregateto homogeneous increments in volume and shear thenarigidaggregateisabletosustainanyexternalself- strain by setting the particle coefficient of friction high equilibrated perturbation without changing the relative enough to prevent sliding. Because of the random po- positions of its particle centers. sitions of the particles and their different initial contact Thekinematicconditionisnotoftentakenintoaccount stiffnesses, the subsequent particle motions are the sum in recent work on granular aggregates (e.g. [20]) and ofthehomogeneousappliedstrainandafluctuationthat sometimes has been emphasized in a different way. For relaxes the particles towards a new equilibrium state. example, Moukarzel [13], in his description of network When this relaxation involves a rearrangement of par- rigidity, defines an isostatic system to be one in which ticles, the incremental response is irreversible. therankKofthe rigiditymatrixthatrelatesthe contact Weattempttocharacterizetheparticlerearrangement forcestotheappliedforcesalwaysequalstothenumberof and introduce two measures of the strength of the irre- equations, rather than one in which the simple Maxwell versibility: the first is related to contact displacement, condition is satisfied. Then, when K is less than the 3 number of equations, the network is flexible; this can occur when Z¯ =Z¯ . iso The presence of irreversibility in all of our aggregates, even in the limit that Z¯ = 4 (see Figs. 1 and 2) indi- cates that the kinematic condition does not hold. Con- sequently, inextensional mechanisms and the associated softorfloppyvibrationalmodesarepossible. This situa- tionmayoccurwhateverthevalueZ¯,ifparticleslocation do not correspond to those of a rigid network. In particular, in packingcharacterizedas isostatic by Z¯ =Z¯ iso (e.g. [11, 15, 16]), soft modes can occur as long as the kinematic condition is not satisfied. However, we should emphasizethattheinitialstatesthatweemployarecon- structed in a much different way than those constructed to insure an initial, stable, elastic response (e.g. [21]). We now turn from irreversibility to elasticity and re- portresultsfromnumericalsimulationforthebulk mod- 3 FIG. 3: Comparison between the numerical data and fluctu- 1 ulus, Θ = ∆σ /∆V, and the shear modulus, G = ation theory for thenormalized bulk modulus. ii 3 Xi=1 Using (1) in (2), we obtain ∆σ12/∆ε12, where σij is the average stress tensor, εij the average strain tensor, and V is the volume of the aggregate. When a shear increment is applied to poorly 1/3 ∞ 3(1 ν) coordinatedsystems,the irreversibilityincreasesandthe δ1/2 = − P1/3w(P)dP material response deviates from elasticity. That is, an (cid:20) 2µd1/2 (cid:21) Z0 elastictheorycanonlybeconsideredasprovidinganup- 3(1 ν) 1/3 4 per bound on the shear modulus. = 2µd−1/2 P¯1/3Γ 3 , (3) To make predictions of the moduli, we adopt the fluc- (cid:20) (cid:21) (cid:18) (cid:19) tuation theory developed by [9]. This theory improves where Γ is the Gamma function. upon the simplest average strain models [2, 3], because So the average normal contact stiffness K = N contacting particles are assumed to move with both the µd1/2δ1/2/(1 ν) becomes average deformation and fluctuations and because the − sotraytiesmticpslooyfsthfoercaeggarnedgamteoamreenttakeqenuiilnibaricucmoufnotr. aThtyeptihcea-l 1 1/2 9√3πµ2 1/3 4 pair of particles to evaluate the fluctuations, and then KN =d(cid:18)3(cid:19) "2Z(1−ν)2φp0# Γ(cid:18)3(cid:19). (4) usesthemtodeterminethestressintheaggregate. How- ever,atlowvaluesofthecoordinationnumber,thesimple The average shear stiffness is K = T statisticalmodelintroducedtodescribethe variabilityof 2K (1 ν)/(2 ν). N theneighborhoodofcontactingpairsofparticlesisprob- Takin−g in acco−unt the initial distribution of forces in ably too simple, (e.g. [22]) and the initial distribution this way, we obtain a new solution for the fluctuations; contact forces (e.g.[23]) is not taken into account. with this, the resulting expressionsfor the bulk modulus Here we repair the second deficiency and assume that Θ and the shear modulus G are, respectively, the distribution, w(P), of the normal component of the contact force, P is exponential (e.g.[4]): φK 14.5 38 33.6 Θ= N 2.8Z¯ + 12.7 (5) 1 P 5πd − − Z¯3 Z¯2 − Z¯ − w(P)= exp , (1) (cid:20) (cid:21) P¯ −P¯ (cid:18) (cid:19) ∞ where, by definition, P¯ = Pw(P)dP. G = φ KN −KT 1.7Z¯+ 8.7 22.8 + 20.2 7.6 From the effective mediZu0m theory (e.g.[24]), the con- (cid:0) 5πd (cid:1)(cid:20) Z¯3 − Z¯2 Z¯ − (cid:21) fining pressure p0 can be expressed as function of P¯, +φKT 6.6Z¯+ 66.9 154.7 + 128.6 46 . (6) p0 =Z¯φP¯/πd2, 5πd (cid:20) Z¯3 − Z¯2 Z¯ − (cid:21) and the relation between the normal component of the where a relation between the rms value of the fluctua- contact displacement, δ, and P¯ is tion in the number of contacts per particle and its average value has been adopted [14]. At the upper limit 3(1 ν) 2/3 of validity of equations (5,6), Z = 22/3, we recover the δ = − P¯ . (2) 2µd1/2 average strain prediction [2, 3]. This limit results from (cid:20) (cid:21) 4 tolowervaluesofZ,thepredictionstronglydeviatefrom the simulations, as also seen by others [10]. Here, we conclude that this discrepancy is due to the observed irreversible motions and that an elastic theory is not able to capture the behavior of the system for low values of Z. For the bulk modulus, the theory works quite well over the entire range of Z, because the irreversibility, measured by χ and ζ, can be neglected. In conclusion, we have measured the irreversible motions in a granular aggregate subjected to incremental strainsanddeterminedtheirinfluence onthe mechanical response of the material. We found that particles may experiencerearrangementsintheirgeometryandcontact forces, even when small perturbations are applied and that these rearrangements are sensitive to the coordination number. The strength of this irreversibility is neg- FIG. 4: Comparison between the numerical data and fluctu- ligible for increments in volume strain, while it strongly ation theory for thenormalized shear modulus. increases for increments in shear strain as Z approaches Z . The presence of irreversible motions in the aggre- iso gatesindicatesadeviationfromelasticbehavior. Thatis, the modelingofthe statisticaldistributionofparticlesin an elastic theory is appropriate to describe the material the assembly. behavior only if mechanisms do not play an important Comparison between the predictions of the bulk and role in the displacements of the particles. shear moduli and measurements in the numerical simulation are shown in Figs. 3 and 4. There is agreement La Ragione, Magnanimo, and Jenkins are grateful for at high values of the coordination number for the shear support from Strategic Plan-119, Regione Puglia (Italy) modulus;theslightdifferencemaybeattributedtotheir- and Gruppo Nazionale della Fisica Matematica. Makse reversibility. However, when the comparison is extended acknowledges support from DOE and NSF. [1] P.A.CundallandO.D.L.Strack,Géotechnique29,47 [13] C. F. Moukarzel, Phys.Rev. Lett 81, 1634 (1998); ibid (1979). Gran. Matter, 3, 41 (2001). [2] P.J. Digby,J. Appl.Mech. 48, 803 (1981). [14] V. Magnanimo, L. La Ragione, J.T. Jenkins, P. Wang [3] K.Walton, J. Mech. Phys.Solids 35, 213 (1987). and H.A. Makse. Europhys. Lett. 81, 34006 (2008). [4] J.T.Jenkins,P.A.Cundall, andI.Ishibashi,inPowder [15] M. Wyart, L.E. Silbert, S.R. Nagel and T.A Witten, and Grains, Ed. Biarez and Gourvès, Balkema, Rotter- Phys. Rev.E. 72, 051306 (2005). dam (1989). [16] L.E. Silbert, A.J. Liu, S.R. Nagel, Phys. Rev. Lett. 95, [5] H.A.Makse,N.Gland,D.L.Johnson,andL.Schwartz, 098301 (2005). Phys. Rev. Lett. 83, 5070 (1999); ibid, Phys.Rev. E 70, [17] C. Brito and M. Wyart, Europhys.Lett. 76,149 (2006). 061302 (2004). [18] S. Pellegrino, C.R., Calladine, Inter. Jour. of Sol. and [6] M.A. Koenders ActaMech. 70, 31 (1987). Str. 22(4), 409-428 (1986) [7] A. Misra and C.S. Chang Int. J. Solids Struc. 30, 2547 [19] J.C., Maxwell, Philos. Mag.27,294-299 (1864) (1993). [20] M. Wyart, H. Liang, A. Kabla, and L., Mahadevan, [8] J.Jenkins,D.Johnson,L.LaRagione,andH.A.Makse, Phys.Rev. Lett 101, 215501 (2008) J. Mech. Phys. Solids 53, 197 (2005). [21] C.S.O’Hern,L.E.Silbert,A.J.LiuandS.R.Nagel,Phys. [9] L.LaRagioneandJ.T.Jenkins,Proc.Roy.Soc.London Rev. E. 68, 011306 (2003). A,463, 735 (2007). [22] I.Agnolin,N.KruytandJ.N.Roux,inEM0818-21May [10] I. Agnolin and J.N. Roux. I, II, III Phys. Rev. E 76, 2008 Minneapolis (USA). 061302, 061303, 061304 (2007). [23] C.ThorntonandS.J.Antony.Phil.Trans.Roy.Soc.Lon- [11] W. G. Ellenbroek, E. Somfai, M. van Hecke,and W. van don A,356 (1998). Saarloos, Phys.Rev.Lett 97, 258001 (2006). [24] J.T. Jenkins and O.D.L. Strack, Mech. Matls.16 , 25 [12] E.Somfai,M.vanHecke,W.G.Ellenbroek,K.Shundyak (1993). andW.vanSaarloos,Phys.Rev.E75,020301(R)(2007).