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VOLUME 74, NUMBER 14 PHYSICAL REVIEW LETTERS 3 APRIL 1995Nonlinear Dynamics in Granular Columns

Robert S. SinkovitsCode 6420, Naval Research Laboratory, Washington, D.C. 20375-5344

Surajit SenPhysics Department, State University of New York, Buffalo, New York 14260(Received 16 May 1994)Atomistic simulations of the vertical propagation of perturbations in deep gravitationally compactedgranular columns characterized by the intergrain potential V(6) ~ 6", with n ~ 2 and 6 the grainoverlap, are shown to recover the results of elasticity theory for weak perturbations. For strongperturbations, the sound velocity est,~ c k as z ~ with the deviations from c k best expressedvia a certain recursion relation in g. We predict that voids in real granular columns lead to c ~ 1 e,when the void fraction e is small, and show that the velocity power spectrum of a grain resembles thatof a harmonic oscillator chain as z

PACS numbers: 46.10.+z, 03.20.+i, 62.30.+dNonlinear elasticity and sound propagation in dry granu-lar columns at small strains [15] has received significantattention in recent years. The characteristic feature of ver-tical sound propagation in granular columns as predicted

by elasticity theory, referred to in the literature as Hertziancontact theory [3,4,6], is that the sound velocity c scalesas z''/, z being the depth. This power law behavior is ob-tained from the assumption that the grains interact via thewell accepted potential for contact between noncohesivespheres V(6) ~ Bs12 for r ( rwith r, the cutoff radius,0 otherwise, where 6 denotes the normal displacement ofone grain against another [7].This prediction is consistent with experimental resultsfor c at large depths or pressures. Discrepancies betweenHertzian theory and experiments, however, persist at shal-low depths [5,6], and one must invoke altered contactmechanisms to explain shallower depth sound propagation[5]. In general, however, the elasticity theoretic treatmentof sound propagation, while simple and appealing, is pos-sibly inadequate for describing the propagation in realgranular columns at shallow depths, in the presence ofvoids, and when perturbations of large magnitude areimportant.In this Letter, we use molecular dynamics simulationsto first recover and generalize the elasticity theory basedpredictions mentioned above for very deep 1D granularcolumns of -104 grains. We then extend our studyand consider the problems of real granular systems byaddressing the deviations from the scaling law for largeamplitude perturbations at shallow depths followed by anextension of the study to 2D columns with voids and therole of voids in modifying the scaling law for c. Weclose with calculations of the velocity power spectra of thegrains at large depths which reveal that the grain dynamicsat various depths can be related to the simpler problem ofthe dynamics of particles in a harmonic oscillator chainsubjected to a weak linear field. Thus, the present Letterdiscusses a simple way to study the propagation of strong

perturbations in real granular media in a highly nonlinearregime.We model the granular material (GM) as a collection ofspheres interacting as follows (for studies on the propertiesof GM with other potentials, see [8,9]):V(&) =lJ 0 r )where r;, = ~r; r, ~ is the separation between grainsi and j, 6;~ r, r;, is the grain overlap, and r,. isthe cutoff distance for the potential. V(6;,) leads to arepulsive force between grains in intimate contact [7].For noncohesive spheres, it can be shown that n = 5/2[5]. For grains with conical imperfections, on the otherhand, n = 3 [5]. We study the system for arbitrary n torecover and extend upon the scaling law for c at largez. In addition, we subject the grains to the gravitationalforce F = mgz, z being the unit vector in the verticallyupward direction and m the mass of the grain. In all of thesimulations units are employed in which m and 2r,. are setequal to 1 and g is set equal to 0.01. The system dynamicsis obtained by time integration of the coupled Newtonianequations of motion for a N [O(10 )] grain system viathe third-order Gear algorithm [10] using a time step inthe range 1.0 X 10 to 5.0 X 10We first focus on pristine systems. Given that cdepends sensitively on 6, care is taken to insure that thecolumn is relaxed (to the extent possible in a numerical

study) and is hence in its "ground state" thus possessingzero effective granular temperature (i.e., total kineticenergy

0). This step is critical for the study ofthe perturbation that we initiate in the system (via avery low energy impact) to probe the nature of verticalsound propagation in granular columns. The method ofdetermining the ground state is as follows. For a 1Dsystem consisting of a single gravitationally compactedcolumn, the location of the bottom grain is first fixedand the positions of the remaining grains are set suchthat the repulsive forces due to the overlap between

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VOLUME 74, NUMBER 14 PHYSICAL REVIEW LETTERS 3 APRIL 1995the adjacent grains exactly equal the forces required tosupport the grain column. For a system of W grainsin which the bottom grain is labeled 1, the overlapbetween grains i and i + 1 is determined via the 1D sumrule gg, +, mj = an6,';+'~. For the 2D case, the initialconfiguration is taken as the gravitationally compactedperfect triangular lattice. The coordinates of the bottomrow of grains are fixed and periodic boundary conditionsare imposed in the horizontal direction. In most ofthe 2D simulations, a large height to width aspect ratio1000 was chosen. In complete analogy with the1D systems, the overlap between the grains in adjacentrows is determined via the 2D sum rule gg, , +, m, =an6,",+',Ql 1/(2 26;;+~/r, )~ for 6;;+~. The 2D sumrule is obtained by allowing the separation between grainsin adjacent rows to be reduced from I., to r, 6 sothat the z components of the intergrain forces balancethe weight of the supported column, while the distancesbetween grains in the same row are kept constant. Theseparation between adjacent rows i and i + 1 is thenreduced from the uncompacted triangular lattice rowseparation by the amount

Az = 2 r, 1 1 3(2r,6 62)cFor the weakly disordered system, starting from theequilibrium configuration for the gravitationally com-pacted 2D triangular lattice, we remove the grains in a"semirandom fashion, " i.e., a grain is removed from arandomly chosen site for every i rows. The configu-ration thus obtained, though significantly ordered in thesense that each grain still very nearly resides at a per-fect triangular lattice site, possesses considerable disorder

in the force network. The removal of grains using theabove mentioned procedure allows one to tune the poros-ity and the degree of disorder in the system. The simu-lations were limited to cases with up to 12.5% of thegrains removed (i.e., up to void concentrations of 12.5%).Upon removal of the grains from the lattice, the system isno longer in its ground state. Obtaining the new globalground state would require relaxing the system until all ofthe voids were filled and the compacted triangular latticewas recovered. Instead, we wanted to obtain a metastablestate in which the energy is at a local minimum and thevoids are trapped in the lattice. We have tried several ap-proaches to find the metastable configuration. Althoughit is not the only viable approach, we found that inte-grating the Newtonian equations of motion with an addi-tional time-dependent viscous damping term of the formF = b(t)v is an efficientway to relax the system intoa metastable configuration in which the positions of thevoids are preserved and the effective granular tempera-ture does not rise significantly after the viscous dampingis turned off.In all of the simulations, the sound speed was deter-mined by monitoring the position of the weak perturbationin the column as a function of time. The perturbation was

initiated at time t = 0 by imparting an initial downwardvelocity to the top particle or row of particles. For thepristine 1D and 2D systems, initiating the weak perturba-tion in this manner results in a spatially well-defined pulsethat travels downward through the column. Althoughthere was some tendency for the pulse to broaden slightlyover time, the shape of the pulse remains approximatelyinvariant over the course of the simulation. The locationof the pulse was defined by the position of the particleor row of particles with the highest velocity, and its localspeed was determined from the time derivative of pulselocation. To simplify the analysis of the results, the sys-tem parameters were chosen so that the density of the col-umn as a function of depth does not change significantlydue to compaction. This requires that 6&2 r, /2, or .interms of the system parameters (mgN/an)'~~" '~ && r, /2Since sound speed scales as c Qp, /p [7j, p, being thebulk modulus, and p being the density, variations in cas a function of depth in the column are due entirely tochanges in the stiffness of the system.We have performed the simulations for a family of po-tentials with a set of values for n in Eq. (1), and for a set ofmagnitudes of the initial perturbations, v; p,. Althoughexperimental and theoretical works indicate that the po-tentials that best describe real systems are V(6) 65~2 forcontact between perfect spherical grains and V(6) 63 forcontact between grains with conical imperfections, calcu-lations were carried out using a range of exponents from

n = 5/2 to n = 10 in order to study the power law de-pendence of sound speed as a function of depth and de-viations from the predictions of the theory. The resultsof these calculations are plotted in Fig. 1 for n = 5/2 andn = 6. For small v' p 1 c k determined from the 1Dsimulations scaled with depth as c~kz ' ' ~" '~~ . Athigher v' p 1 and small z, we saw large deviations fromthe above law, with the deviation increasing with v; p,.In these studies, c~does not show a simple power lawbehavior (i.e. , z'~6 or like) until greater depths are reachedwhere it asymptotically approaches that found in the lowv p 1 studies. c~increases more slowly with z thanz'~6 for small z (see Figs. 1 and 3). The relative differ-ence between c for large v; p, and v; p, 0 can be de-fined as g(z) = (csc k)/ck. A functional formfor g(z) is expected to decrease monotonically as z in-creases and show the limiting behaviors lim, o $(z)and lim, g(z) ~ 0.Having attempted many different functional formsto fit s(z) we conclude that $(z) does not follow asimple power law behavior. The results of nonlinearcurve fitting show that the functional form of g(z) iswell approximated by a function in z which is bestexpressed as gbk exp( diaz) where thecoefficientsbk and dk follow the recursion relations bI, +~ = bj, n anddq+i = dqP with n and P constants that depend on thechoice of system parameters. As an example, a best fitof g(z) over the range 100 ( z ( 10000 for v; ~, = 1.0and n = 3 to a triple exponential function gives

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VoLUME 74, NUMBER 14 PH YS ICAL REVIEW LETTERS 3 APRIL 1995

100

L

yggglfA

vimp act=1.0vimpact=0. 1o vim pact=0. 01

41010210~ 1O'I0 1O'CL 10'

-810-1010

HOz=1600

z=100

I100 I I I I I I I1000Z 10000~ ~ ~ ~ ~ I I)0.1 ~ E ~ ~ ~ ~ ~ I I I I I ~ ~ ~ ~ l)10 ~ ~ ~ ~ ~ ~ I I]100

FIG. 2. Velocity power spectra for grains at five differentdepths, z = 100, 200, 400, 800, and 1600, in a 1D columndue to passage of vertical disturbance. Shown for comparisonis the velocity power spectrum for a particle in an infiniteharmonic oscillator chain. The system parameters are n = 5/2and v; p f 0.05.100

I100 I I I ~ I I I1000z

pact=1.0pact=0. 1pact=0. 01

10000

position. The leading term in a power series expansionof the force about dz = 0 is 2a(n 1)6" 'dz, hence fora strongly confined grain in the limit of small amplitudeoscillations the behavior approaches that of a harmonicoscillator.FIG. 1. Speed of vertical disturbances in 1D columns for(a) n = 5/2 and (b) n = 6. Solid lines show scaling behaviorpredicted from generalized Hertzian contact theory. Slopes oflines in (a) and (b) are 1/6 and 2/5, respectively. 100the result g(z) = 0.04107 + 0.2519exp(0.0003244z) +0.4527 exp(0.001 911z) + 0.8019exp(0.010 19z), whichhas four parameters. For this case a = 1.8 and P = 5.8.n and P are both greater than unity, insuring that $(z)converges for all g ) 0 and diverges at g = 0. Althoughn and P depend on v; ~, and n, in all cases studied bothquantities are found to be greater than 1. It is reasonableto argue that at each value of g, as the strong perturbationpropagates progressively downward, some energy goesto promote local excitations. It follows then that asz ~ this energy loss becomes vanishingly small. Ourcalculations suggest that the origin of the above recursionrelation lies in this iterative process. At this time weare unable to provide a simple explanation for the abovefunctional form for $(z) for this highly nonlinear process.Typical velocity power spectra for grains at five differ-ent depths in the 1D column during the passage of a ver-tical disturbance are shown in Fig. 2. As one goes deeperand deeper into the column, one finds that the grains aremore strongly confined to their equilibrium positions andthat their velocity power spectra approach that of a massin an infinite harmonic oscillator chain [11,12]. For adeep lying grain the total force on the grain is approxi-mately F = a(B dz)" ' a(6 + dz)" ' mg, wheredz is the displacement of the grain from its equilibrium

I100 I I I I I1000

b, 5 ~ phdggb,b ~lpga100

t=1.0t=0.1t=0.01

100 1 I I I I I I1000FIG. 3, Speed of vertical disturbances in 2D columns for(a) n = 5/2 and (b) n = 6. Solid lines show scaling behaviorpredicted from generalized Hertzian contact theory. Slopes oflines in (a) and (b) are 1/6 and 2/5, respectively.

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VOLUME 74, NUMBER 14 PHYSICAL REVIEW LETTERS 3 APRIL 1995

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CI I I I100 I I ~ I ~ ~ I )1000

dsCISICISICOS

FIG. 4. Speed of vertical disturbances in 2D columns withvoids for n = 5/2 in the limit of weak impact. Solid lines arethe best fits of the data to the functional form c = az". Insetshows the prefactor a plotted against 1 e illustrating lineardependence of c on void fraction.

To understand the behavior of estg let us return to theHertzian contact theory in which c k~k, k = df/d 6,f Ir'" ' for V(B)Ir'", n ~ 2. In this theory, oneassumes that although k is a function of z or P it isessentially constant during the passage of a perturbation.This assumption fails for large perturbations and hencecs,g does not scale with z as c k does. To studycs(z)one must therefore solve for 6(r) in d~6/dr~ =an6" ' subjected to appropriate initial conditions on 6.For fixed total energy, knowledge of 6(t) and hence ofV(t) will yield information about v;(t) of the individualgrains, which in turn will yield the behavior of estg as afunction of z.The results of calculations on 2D systems are consis-tent with those from the 1D simulations. The same scal-ing of sound speed on depth and deviations from thegeneralized Hertzian theory for strong perturbations areobserved in the 2D calculations. The results of the 2Dsimulations on the systems without voids are shown inFig. 3. We find that in the limit of weak perturbationck~ P ' 'I(" ')]I~ in pristine 2D systems (see Fig. 3),while c ~ (1 e)P(' 'I(' ')]I~ for weakly disordered 2Dsystems with void fractions of up to 12.5% (see Fig. 4).The main difference between the perfect and perturbedsystems is that in the 2D systems, with voids, inhomo-geneities in the force network lead to a significant transferof energy from the coherent elastic disturbance into therandom motion of the particles. Figure 5 shows the av-erage grain kinetic energy as a function of z for a systemwith 3.3% void fraction. With the exception of the strongdissipation of the pulse in the topmost, loosely packedlayers of the system, the details of the sound propagationin the 2D systems with and without voids are unchanged.On the basis of our calculations for 2D columns, we findit reasonable to expect similar behavior for close-packed(hcp) 3D systems. The issue of sound propagation in bothclose-packed and randomly packed 3D columns will beaddressed in the future to definitively answer this point.

1.41.2

L. .1.0-UJA 08V 0.6 t=25.0

t=37.5

0.2 t=12.5I I ~ I / ~ I I I I I ~ I I $ I l ~ I ( ~ I I I t l I I I l I I I I $ ~ I I I l ~ 1 '4 I 1 I I ~ 10 1000 2000 3000 4000 5000

FIG. 5. Average kinetic energy as a function of depth forsound propagation in 2D column with 3.3% void fraction,v' p t:0 1 and n = 2.5.In conclusion, we have performed molecular dynamicssimulations to recover the elasticity theory based resultc kI''~ at large depths from microscopic considera-tions. Our approach allows us to probe new issues withregard to the propagation of perturbations in real granu-lar columns. These issues are (i) propagation of weakperturbations in shallow depths, (ii) propagation of strongperturbations at all depths, and (iii) propagation of weakperturbations in granular columns with voids. In addition,we argue that the velocity power spectra of grains at largedepths can be understood by studying the velocity powerspectrum of particles in a harmonic oscillator chain.We thank Professor J.T. Jenkins, Dr. Elaine Oran,

and Mark Wedloe for valuable conversations. R.S.S.was supported by an ONR postdoctoral fellowship andby DARPA. S.S. acknowledges support of the PhysicsDepartment at SUNY, Buffalo.

[1] H.M. Jaeger and S.R. Nagel, Science 255, 1523 (1992).[2] C-h. Liu and S.R. Nagel, Phys. Rev. Lett. 68, 2301(1992); Phys. Rev. B 48, 15 646 (1993).[3] P. J. Digby, J. Appl. Mech. (ASME) 48, 803 (1981).[4] K. Walton, J. Mech. Phys. Solids 35, 213 (1987).[5] J.D. Goddard, Proc. R. Soc. London A 430, 105 (1990).[6] J. Duffy and R.D. Mindlin, J. Appl. Mech. (ASME) 24,585 (1957).[7] A. E.H. Love, A Treatise on the Mathematical Theoryof Elasticity (Dover, New York, 1944), 4th ed. ; K.L.Johnson, Contact Mechanics (Cambridge Univ. Press,Cambridge, 1985).[8] P.A. Thompson and G. S. Grest, Phys. Rev. Lett. 67, 1751(1991).[9] S. Sen, N. Mousseau, and G. Overney, Phys. Rev. E 49,4712 (1994).[10] C.W. Gear, Numerical Initial Value Problems in OrdinaryDifferential Equations (Prentice-Hall, Englewood Cliffs,1971).[11] M. H. Lee, J. Florencio, Jr. , and J. Hong, J. Phys. A 22,L331 (1989).[12] S. Sen and R. Sinkovits (to be published).2689