Solitary Waves Under Hertz Contact

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Solitary waves in a chain of beads under Hertz contact

C. Coste, E. Falcon, and S. FauveLaboratoire de Physique, Ecole Normale Superieure de Lyon, URA CNRS 1325, 46 Allee dItalie,

69364 Lyon Cedex 07, France

Received 7 April 1997; revised manuscript received 11 June 1997

We study experimentally the propagation of high-amplitude compressional waves in a chain of beads in

contact, submitted or not to a small static force. In such a system, solitary waves have been theoreticallypredicted by Nesterenko J. Appl. Mech. Tech. Phys. USSR 5, 733 1984. We have built an impact

generator in order to create high-amplitude waves in the chain. We observe the propagation of isolated

nonlinear pulses, measure their velocity as a function of their maximum amplitude, for different applied static

forces, and record their shape. In all experiments, we find good agreement between our observations and the

theoretical predictions of the above reference, without using any adjustable parameter in the data analysis. We

also show that the velocity measurements taken at three different nonzero applied static forces all lie on a

single curve, when expressed in rescaled variables. The size of the pulses is typically one-tenth the total length

of the chain. All the measurements support the identification of these isolated nonlinear pulses with the solitary

waves predicted by Nesterenko. S1063-651X9704211-6

PACS numbers: 03.40.Kf, 46.10.z, 43.25.y

I. INTRODUCTION

Granular materials are widely spread in both geophysicaland industrial contexts. Acoustic waves are frequently usedas nondestructive testing tools in the laboratory and aresometimes the only accessible way to get information in geo-physical investigations. Thus the acoustic behavior of granu-lar materials has been widely investigated for a long time1,2.

The shape of the grains, the details of the contact lawbetween adjacent grains, the geometry of the contact lattice,and hence the dimensionality of the grain piling all affect thepropagation of sound in granular material. This variety of

phenomena makes the problem difficult and many openquestions subsist see3 for a recent review. Many experi-mental investigations were concerned with the propagationof seismic waves in materials submitted to very high pres-sures 4 6, where all those problems are somewhat mixed.Recent works on sound propagation in sand focus on geo-metrical effects due to the disorder of the piling 7,8. In thispaper we are concerned with the effect of the contact lawbetween adjacent grains and the resulting nonlinear and dis-persive behaviors.

The interaction law between two adjacent elastic spheresis an exact solution of linear elasticity, known as Hertzs law9. Because of purely geometrical effects, the relation be-tween the force F0exerted on the spheres and the distance of

approach of their centers 0 is nonlinear, F003/2 . A s a

consequence of Hertzs interaction law, the velocity of linear

sound waves in one-dimensional systems scales as F01/6 ; this

classical result, experimentally demonstrated long ago 1,2for more recent results see 10, does not seem to be veri-fied in higher dimensions because of geometrical effects 3.To get rid of such problems, a one-dimensional ordered sys-tem such as a chain of identical elastic beads is a good can-didate to study the nonlinear regime.

The nonlinear behavior of a chain of beads was originallyinvestigated with the help of numerical simulations, in order

to study the propagation of shock waves in the chain, forseveral nonlinear interaction laws 1113. Nesterenko

14,15 gave an analytical solution to the problem for Hertzs

law. He showed that strong compressional waves, that is,

with an amplitude much greater than the applied static force,

may propagate as isolated solitary wavesin the chain. Later,experimental evidence of the existence of such waves wasgiven by Nesterenko and Lazaridi 1618. A strongly re-lated problem, if not exactly identical, is the propagation ofwaves in a vertical column of grains subjected to gravity,which is crucial to the understanding of the details of thewhole column dynamics after an impact 19,20.

In this paper we report quantitative experiments on non-

linear wave propagation in a chain of identical elastic beads,allowing a comparison of the shape and the velocity of thewaves with the theoretical predictions of Ref. 14. We ob-serve the propagation of nonlinear isolated pulses and a de-tailed analysis, in which no free adjustable parameter is used,strongly supports their identification with the solitary wavesdescribed by Nesterenko 14. Our experimental setup allowsa systematic study of the waves in a large range of ampli-tudes for several applied static forces. This is in contrast withthe work of Refs. 16, 17, in which the excitation of thewaves cannot be varied and only indirect velocity measure-ments were performed.

The paper is organized as follows. In Sec. II we review

the calculations of Nesterenko for convenience and furtherreference. Section III A is devoted to the presentation of ourexperimental apparatus and Sec. III B to the analysis of thedata and a comparison with the theoretical predictions. Theexperimental results are presented in Sec. IV. We display theexperimental shape of the nonlinear pulses in Sec. IV A for achain submitted to moderate static forces 29.4 and 167 N.In Sec. IV B we report on measurements of the velocity ofpropagation of the nonlinear pulses for three different ap-plied static forces 9.8, 29.4, and 167 N. We show that whenthe velocity is rescaled by the linearsound velocity and themaximum amplitude of the pulses is rescaled by the applied

PHYSICAL REVIEW E NOVEMBER 1997VOLUME 56, NUMBER 5

561063-651X/97/565/610414/$10.00 6104 1997 The American Physical Society


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static force, the previous measurements all lie on a singlecurve. Section IV C is devoted to the behavior of a chain inthe absence of any static force because the theoretical statusof this case is somewhat singular. In Sec. IV D we discussour results in the light of previous works 1618. Our con-clusions are given in Sec. V.

II. THEORETICAL ANALYSIS

When two identical elastic spheres of radius a are in con-tact and submitted to a static force F0 , the distance of ap-proach 0 of their centers reads

02F0

2/3

a 1/3 with

3 12

4E , 1

where E is Youngs modulus and Poissons ratio of thebead material. This is an exact result of linearelasticity,known as Hertzs solution9, and the nonlinear relationshipbetween 0 and F0 is a purely geometrical effect.

This solution is expected to remain valid when the forceF and hence the distance of approach are both slowlyvarying functions of time. The variations may be consideredas slow if every typical time scale involved in the motion ismuch greater than the time needed by a bulk longitudinalacoustic wave to travel across the diameter of a bead. For thestainless-steel beads used in our experiments, which are 8mm in diameter, this condition is fulfilled when all timescales are much greater than 2 s or all frequencies muchless than 500 kHz. Figures 68 and 1315 below show thatit is indeed the case in all our experiments. Moreover, thedeformation of the beads in contact is strongly confined to avery small region near the contact point; it is thus possible tomodel a chain of identical beads in contact as a chain ofpoint masses m4a3/3, where is the density of each

bead, linked by nonlinear springs governed by Eq. 1. Thedynamics of the chain, at sufficiently low frequencies, is thusdescribed by the system of coupled nonlinear differentialequations

una/2

2m 0 u nu n1

3/20 un1u n

3/2,

2

where u n is the displacement of the nth bead from its equi-librium position and un its second time derivative.

Two other approximations are hidden in Eq. 2. First,plastic deformation of the beads is neglected. The nonlinear

waves are generated by impacting the first bead of the chainwith a bead of maximum speed roughly 0.5 m/s see Sec.III A. For steel beads, plastic deformation is negligible if therelative speed of the impacting bodies is less than 1.3 m/s21, so that this effect is actually not relevant to our experi-mental situation. Dissipation of the waves may also occurbecause of the viscoelastic behavior of the bead material22, but this effect is presumably very small for steel beads,and we neglect it too. The validity of those approximations ischecked experimentally, a posteriori. In the limit of verysmall amplitude waves, for which the plastic effects are notrelevant, the behavior of the chain is very well accounted forby the dissipationless model 2 10. In the strongly nonlin-ear regime, there is very good agreement between the theo-

retical predictions, which do not take dissipative phenomenainto account, and the observations see Sec. IV.

The linear approximation of Eq. 2, obtained in the limitunu n10 , gives for the linearized spring constant

k0F0

1

3

4

aF0 1/3

2/3 3

and we recover the well-known results for a chain of identi-cal point masses linked by identical linear springs. The dis-persion relation between the wave number q and the pulsa-tion is

2 km

sin qa , 4

the cutoff frequency fc of the chain reads

fc1

k

m

3

43/2F0

1/6

1/3a 4/31/2, 5

and the sound velocity cs of the chain, not to be confusedwith the velocities of acoustic waves in the bulk material ofthe beads, is given by

c s limq0

q2ak

m

3

2

F01/6

1/3a 1/31/2. 6

The simple results 5 and 6 are of course well knownsee, e.g.,3and were recently tested for an actual chain ofidentical elastic beads 10. They both have been confirmedexperimentally with a great accuracy; this constitutes a clearexperimental proof of the validity of the approximationsleading to Eq. 2, which are otherwise rather uncontrolled.We will return to the same tests in the nonlinear regime inSec. IV.

An interesting feature displayed by Eq.6 is that withoutany static force F00, no linear acoustic waves propagate inthe chain: c s0. Nesterenko 14 called this situation thesonic vacuum. He also showed that strongly nonlinear wavesdo propagate, even when the beads are just in contact, but inthe absence of any static force. For convenience and furtherreference, we reproduce below his analysis, which may befound in 14.

The strongly nonlinear limit corresponds to unu n10 . In the long-wavelength approximation, that is, when

the characteristic size L of the perturbation is much greaterthan the radius of a bead, we may write u n(t)u(x, t),where x represent the abscissa along the chain, and proceedto the development

u n1 tux2a ,tu2aux2a2uxx

4a3

3 uxx x

2a4

3 uxx xx , 7

where uxu/x. Inserting this development in Eq. 2, weget

56 6105SOLITARY WAVES IN A CHAIN OF BEADS UNDER . . .


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u ttC23

2 ux

1/2uxxa 2

2 ux

1/2uxx xxa2uxx uxx x

2ux1/2

a 2

16

uxx 3

ux3/2 , ux0, C2 2a

3

m, 8

where u is now the total displacement of the bead centerfrom its position when no static force is applied. Equation 8is valid up to order ( a/L) 2. Looking for progressive wavesu(xVt), where the wave velocity V remains to be de-termined, and setting u , we transform Eq. 8 into

V2

C2

3

2 1/2

a2

2

1/2

a 2

16

3

3/2. 9

This equation may be integrated and put in dimensionlessform with the help of successive changes of variables

z4/5, zVC

5

y , 25

a, 10

so that we finally get a particularly simple expression

yd

dy Wy

with

Wy 58y

8/5

12y

2

54 y

4/5y

6/5y 4/5, 11

where y is the value taken by y when . The inte-gration constant is such that y0, which means thatthe solution will have the form of a localized excitation. ThefunctionW(y ) has a maximum for yy and a minimum for

y12 1y

2/5 1y

2/5 13y 2/55/2, 12

defined for 0y 1; moreover, yy for 0y

( 23 )5/2. Thus, when y belongs to this last range, the curve

W(y ) has the shape displayed in Fig. 1a.Equation 11, in an obvious mechanical analogy, de-

scribes the motion of a particle of unit mass at position y , inthe potential W(y ), during the time . If the particle is ini-tially at the position y , with energy E , it leaves its un-stable equilibrium position up to position y m, defined by

Wy mWy 0, 13

FIG. 1. aGraph of the function W(y ), illustrating the mechanical analogy between the solutions of Eq.11and the motion of a point

particle of unit mass in a potentialW(y ); the parameter of the graph is y 0.128, which means y m1.007. The motion of the particle with

energy E0.011 corresponds to the solitary wave solution of Eq. 11. Also shown is a particle of energy E0.005, corresponding to a

nonlinear periodic wave solution of Eq. 11. b Same as a, but the respective motions are shown in the phase plane (y ,y); the homoclinic

orbitsolid line corresponds to the solitary wave, the periodic orbit dashed line to the nonlinear periodic wave. c Shape of the solitary

wave corresponding to the motion of the particle with energy E. dShape of the nonlinear periodic wave corresponding to the motion of

the particle with energy E.

6106 56C. COSTE, E. FALCON, AND S. FAUVE


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in an infinite time because y is the location of a maximumof the potential, and then returns to position y again in aninfinite time; the corresponding trajectory in the phase plane(y ,y) is a homoclinic orbitsee Fig. 1b. Returning to ourwave problem, this describes the propagation of a solitarywave of amplitude y at and of maximum ampli-tude y m at 0, say; the shape of a typical wave profile isdisplayed in Fig. 1c. Also shown in Fig. 1d is a solutionof energy EE, represented by the dashed closed orbit inthe phase plane, and corresponding to a nonlinear periodicwave.

Those conclusions are valid for y 0 because the changeof variables 10 is singular for y0. The solution for y 0, which corresponds physically to F00, is a nonlinearperiodic wave train 14 with an analytic expression thatreads, in the original variables and ,

54

V2

C2

2

cos4

a10. 14

This solution is mathematically singular because it does notfulfill the inequality u0 see 8. On the otherhand, those nonlinear periodic waves are modulationallystable, as shown in Ref. 23, so that they may be relevant to

the actual behavior of the chain. Experimentally, it seemsthat solitary waves exist also when no static force is appliedto the chain see 16,17 and Sec. IV C. It is very unlikelythat a qualitative change suddenly appears at zero static forcebecause solitary waves exist at any infinitesimal appliedstatic force. The shape of a solitary wave with a small staticforce is extremely similar to one arch of the nonlinear wave14, as shown by Fig. 2. This behavior is confirmed bynumerical simulations of Nesterenko and Lazaridi 17,18,who found the same properties for a wave of maximum am-plitude 200 N propagating either under a static force of 2 Nor in an uncompressed chain. Moreover, the wave velocityalso tends continuously toward its zero static force limit, asshown in Sec. III B. We return to the theoretical status of

those nonlinear periodic wave in Sec. III B and discuss theirexperimental relevance in Sec. IV C.

III. MEASUREMENT METHODS

A. Experimental apparatus

The experimental apparatus is sketched in Fig. 3. Thesystem under study is a chain of 51 identical stainless-steelbeads Association Francaise de Normalisation AFNORnorm Z 100 C 17, each 8 mm in diameter, with a toleranceof 4 m on the diameter and 2 m on the sphericity, and amaximal roughness of 0.06 m. The physical properties ofthe beads are summarized in Table I. The beads are sur-rounded by a framework of polytetrafluoroethylene PTFE;this material is chosen because of its high density and lowrigidity, leading to a velocity of 400 m/s for bulk acousticwaves, smaller than the velocity of the nonlinear waves ob-served in our experiments, which is greater than

FIG. 2. Evolution of the force in N as a function of time in

s for a maximum amplitude of 100 N and, respectively, a static

force of 1 N solid line the wave profile is determined by Eq. 23

and no static forcedashed line the wave profile is determined by

Eq. 28. The similarity of the two curves is striking; the choosen

value of 1 N roughly corresponds to the available resolution on the

static force see Sec. III A.

FIG. 3. Sketch of the experimental apparatus not to scale. The

framework of PTFE consists of two parts, each one 30 mm high, 40

mm wide and 400 mm long, with a straight channel of squared

cross section, 8.02-mm sides, milled in the lower part, that contains

the beads. A very small clearance of 2/100 mm is managed in the

channel, so that the beads move freely along the axis but not in theperpendicular direction. Only two of the sensors perpendicular to

the chain axis are shown as well as the one that is parallel to the

chain axis; they are all piezoelectric quartz transducers with charge

amplifiers included hence the need of a stabilized electrical alimen-

tation. The impact generator is described in Fig. 4. Two mechanical

devices, not represented here, ensure longitudinal i.e., parallel to

the chain axis displacement of both the dynamometer and the vi-

bration exciter.

TABLE I. Relevant physical properties of the beads used in the

chain; they are made of stainless steel, corresponding to the

AFNOR norm Z 100 C 17.

Symbol Signification Value

a bead radius 4 mm2m

E Youngs modulus 2.261011 N/m2

Poissons ratio 0.3

density 7650 kg/m3

see Eq.1 3.021012 m2/N

C see Eq.8 4.55103 m/s



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500 m/s see Secs. IV B and IV C. The observation of thewave is thus not perturbed by a quicker wave propagating inthe surrounding medium. The static force is controlled by adynamometer, and ranges from 0 to 170 N, with a precisionof2 N.

Three force sensors Dytran, sensitivity 50 mV/LbF

are held with their axis perpendicular to the chain axis, overbeads 6, 26, and 46, allowing the measurement of the waveflight time, but not of its shape. Indeed, those sensors aresensible to the transversedeformation of a bead, perpendicu-lar to the direction of propagation of the wave, which isrelated in a rather complicated manner to the force exertedon the bead by its two neighbors, in the direction of propa-gationsee24. Moreover, the total force is the algebraicsum of the forces exerted on each side of the bead and thusrelated to the gradient of the deformation u in thecontinuous limit. The wave profile is obtained with the helpof a fourth force sensor Dytran, sensitivity 10 mV/LbFheld at the end of the chain, with its own axis parallel to the

chain axis. In this configuration, the force measured by thissensor is related in a simple way to the deformation wavethat propagates along the chain see Eq. 20. As shown inFig. 3, the sensors are linked to a digital scope, suitable forthe observation of nonrepetitive pulses; the signal may betransferred to an Apple computer for further analysis, tocompare the experimental wave profile to the theoretical onesee Secs. IV A and IV C.

The three sensors perpendicular to the chain axis aremounted on brass supports and we monitor the mountingtorque with a torque wrench. The last sensor is centered onthe chain axis, with its contact surface normal to this axis,and is mounted on a brass cylinder in the same manner as the

other three. The cylinder is guided in order to allow thelongitudinal displacement of the sensor along the chain axis,required by the settlement of the static force. The dynamom-eter may press on the brass cylinder, to exert a longitudinalforce on the chain, but the contact is not a permanent one andmay be broken when no static force has to be applied on thechain. The relevant device, together with the one that allowsadjustment of the impact generator, is not shown in Fig. 3.

In order to send high-amplitude compressional waves inthe chain, we have built an impact generator sketched in Fig.4. A bead, contained in a carefully adjusted bore drilled in aduralumin cylinder an inlet for air evacuation is milled inthe cylinder in order to reduce air friction on the bead,moves freely between the first bead of the chain and a dur-

alumin piston held on a Bruel&Kjaer vibration exciter.

When the piston oscillates, the moving bead successively

impacts the piston and the first bead of the chain; for piston

oscillation frequencies ranging from 70 to 110 Hz and a free

path of several millimeters for the moving bead, the succes-

sive impacts occur in a regular fashion. Since the duration of

an impact is very small typically 50 s; this duration is

nevertheless sufficient for the quasistatic approximation to bevalid; cf. the beginning of Sec. II, a great amount of mo-mentum may be transferred, generating a maximum ampli-tude of the wave up to 1200 N see Sec. IV. In the stationaryregime, the period of successive impacts is the same as theone of the piston and is much greater than both the durationof an impact typically 50 s and the time taken by thewave to travel along the chain, which is at least 1 ms in theworst case. The wave is thus completely damped betweentwo successive impacts, mostly because of the very smallreflection coefficient at the beginning of the chain, so thattwo successive pulses do not interact. The amount of impul-

sion transferred during the impact increases with the ampli-tude of the piston oscillations, their frequency, the length ofthe moving bead free path, the restitution coefficient for bothimpacts experienced by the moving bead, and the density ofthe moving bead material. Although in a strictly empiricalfashion, a proper choice of the parameters listed above al-lows us to set the amplitude of the traveling pulse to a con-venient value. In most cases, the moving bead was made oftungsten carbide and 8 mm in diameter mass 4 g and a veryhigh restitution coefficient, but we also used other impactingbeads listed in Table II.

FIG. 4. a General sketch of the impact generator. The impacting bead see Table II for details moves freely in a hole drilled in a

cylinder of duralumin, between the first bead of the chain and a piston mounted on a vibration exciter.b Cross section of the cylinder along

the plane A A normal to its axis; this sketch emphasizes the inlet ensuring quick air evacuation, thus a great reduction of air friction on the

bead. In the stationary regime, this apparatus send periodically, with the period of the piston motion, almost identical nonlinear pulses in

the chain.

TABLE II. Characteristics of the different impacting beads used

in our experiments. The materials are sorted in decreasing only ina qualitative senserestitution coefficient order. The 8-mm tungsten

carbide bead was the most frequently used one. This variety of

impacting beads allows a large-amplitude range for the pulses sent

in the chain.

Impacting bead material Diametermm Mass g

tungsten carbide 8 3.91

tungsten carbide 12 13.54

bronze 7.95 2.33

stainless steel 8 2.05

duralumin 8 0.71



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B. Data analysis

In our experiments we have access to the velocity of thewave, by performing time of flight measurements, and to thetime evolution of the force experienced by the sensor held atthe end of the chain. The static force applied on the chain isset to a known value with the help of the dynamometer. Theactual value of the static force gives the distance of approach

0between two adjacent beads, with the help of Eq. 1, so

that

0

2a

2/3

a 4/3F0

2/3 . 15

Using Eq. 10 we derive a relationship between the un-known quantities y and V,

VC

4

y 4/5 , 16

where the reader is reminded that the constant C depends

only on the physical properties of the bead material. Themaximum amplitude of the solitary wave y m is a solution ofEq.13. The definition ofW(y) is given by Eq. 11and wesee that Eq. 13 is an algebraic equation of fifth order in the

unknown y m2/5 ; y my is of course one of the roots and

sinced W/dy y 0 it is a double root. We thus have to solve

only a third-order algebraic equation, and we obtain explic-itly25the function y m(y ). The unknown y is thus givenby

y my y

4/5

m

, 17

where m is related to y m by the change of variables 10.The measurement of the maximum force experienced by theforce sensor at the end of the chain gives the experimentalvalue of m. The quantity is the gradient of the totaldisplacement of the bead center and is the sum of the con-stant value at equilibrium0/2a and a time varying part

(t), which is the only part measured by the force sensor.We have to take into account that the contact between thesensor and the last bead of the chain is between a plane anda sphere, so that 9

plane-spheresphere-sphere

2 1/3 . 18

Moreover, the contact surface is at a distance a from thecenter of the bead. Let x w be the position of the sensor andw the distance of approach between the sensor and the lastbead; we thus have

wuxwa ux wa ux xw

a. 19

From this equation we deduce the relationship between thesignal given by the sensor F(t), of maximum value Fm, and

(t), which reads

t2/3

a 4/3 2F t2F0

2/3 2F0

2/3. 20

We thus get

m

2Fm2F02/3 2F0

2/3F0

2/3

F02/3 . 21

The knowledge of the applied static force F0 , togetherwith the measurement of the maximum forceFmexperiencedby the sensor, gives the experimental value of the ratiom/ from Eq. 21. This value is inserted in Eq. 17,which is solved numerically to obtain the experimental valueofy ; the numerical root finding is particularly simple herebecause we know the existence of one and only one root in

the range y 0,(23 )

5/2 see in Sec. II the discussion fol-

lowing Eq.12. We then deduce from Eq. 16 the value Vof the velocity of the solitary wave predicted by the theory,which may be compared to the observed one see Sec. IV B.

Introducing the linearsound velocity c s, given by Eq.

6, we may rewrite Eq. 16 as

V

cs2

3 y Fm/F0

1/5, 22

where we have emphasized the fact that through Eqs. 21and17, y is a function of the ratio Fm/F0 only. Thus Eq.22 means that all the velocity measurements, taken at dif-ferent applied static forces, may be displayed on a singlecurve in properly rescaled variables. This is experimentallydemonstrated below in Fig. 12.

The theoretical shape of the wave, as measured by theforce sensor, may be computed with the help of Eq. 21,

which also gives the relationship between ( t)

( t)and F( t), and Eq. 10, and reads

F tF0

2 y Vt2/5a

y

4/5

12 2/3

3/2

2 ,23

where the function y () is obtained by a numerical integra-tion of Eq. 11. The wave shape predicted by Eq. 23 maythen be compared to the experimental observations; this isdone in Sec. IV A. Note that once F0 and Fm are known, thetheoretical shape of the wave and the theoretical value of its

velocity are given without any adjustable parameter.The case without static force is formally somewhat sim-

pler, in the sense that there is an analytical solution 14 toEq. 11. However, the physical situation is less clear, andthis case deserves particular discussion because the theorypredicts propagation of nonlinear periodicwaves rather thansolitarywaves like in the previous case. The analytic solu-tion 14, together with the relation 20 for the particularcaseF00, gives an explicit relation between the velocity ofthe wave and its maximum amplitude,

VF00C 45 1/2

2Fma 2

1/6

. 24



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The scaling of V with Fm may be understood in a verysimple fashion. Let us drop in Eq. 8 the highest derivatedterms; we obtain

u tt3

2 C

2ux 1/2uxx, 25

which is a wave equation with an amplitude-dependent wave

velocityVC(ux) 1/4; using Hertzs law 1, we may write(ux)(Fm)

2/3/a4/3, so that we get

V

C

a 2

1/6

Fm1/6 . 26

In the general case, the velocity depends on both the staticforceF0 and the maximal amplitude of the wave Fm, so thatthis too simple expression has to be corrected to

V

C

a 2

1/6

Fm1/6fF0

Fm , 27

where the function f is only implicitly known, through Eqs.

21, 17, and 16. Formula 24 states that f(0 )

( 45 )1/221/6, and the opposite limit is given by Eq. 6, which

states, using the definition of C in Eq. 8, that

limFm0Fm1/6

f(F0/Fm)3/2F01/6 .

When no static force is applied on the chain, formula 26is exact up to a numerical constant. It expresses the fact thatthe Hertz interaction between adjacent beads is responsiblefor the propagation of the wave and is a common feature ofall waves propagating in that type of medium. For example,the same scaling exists for step waves 19 propagating up-

ward in a chain of beads in contact. The Fm1/6 scaling is

clearly consistent with experimental observations in Fig. 16,

which emphasize that the basic approximations leading tothe nonlinear spring-point masses model of Eq. 8 are alsovalid in the nonlinear regime.

We also stress that although the limit F00 is singularfor the shape of the wave, because the solitary wave solutiondisappears, the velocity continuously tends toward expres-

sion24. Indeed, Eq. 11 implies y m(y 0)(54 )

5/2, from

which we deduce the velocity as a function ofm with Eq.10 and finally as a function ofFmwith the help of Eq. 20,thus recovering Eq. 24. Most probably Eq. 24 is a verygood approximation of the solitary wave velocity in the ab-sence of a static force. The shape of the wave, as predictedby this theory, is accurately given by the shape of one arch of

the nonlinear periodic wave 14 see Fig. 2 and the discus-sion at the end of Sec. II and reads

F tFm cos6VF00t

10a. 28

IV. EXPERIMENTAL OBSERVATIONS

A. Shape of the wave for nonzero static force

Figure 5 shows a long time recording of the force expe-rienced by the sensor. The pulse corresponds to a compres-sional wave and exerts only a positive force on the sensor;the oscillations appearing after the arrival of the pulse are

due to the resonant response of the sensor its resonance

period is 13 s, not too far from the duration of the pulse,

which is about 50 s and multiple reflections in the differ-

ent parts of the apparatus at the end of the chain.

In Figs. 68 we show the experimental shape of the soli-

tary wave recorded by the force sensor at the end of the chainand compare it to the theoretical prediction derived from Eq.23. The static force is 29.4 N in the case of Figs. 6 and 7and 167 N for Fig. 8; the available values of the nonlinearparameterm/are thus much smaller in this case. Indeed,

the nonlinear parameter ranges from 3.7 in Fig. 6a to 17.0in Fig. 7d, whereas it ranges from 2.5 in Fig. 8ato 5.2 inFig. 8d, for a range of maximal force experienced by thesensor that is essentially the same for both applied staticforces. All the figures exhibit very good agreement with thetheory, although the arrival of the pulse seems sometimesslightly different from the theoretical expectations. It is im-portant to note that in this test of the theory no adjustableparameter is involved.

The typical duration of a pulse is about 3040 s and itsvelocity is about 1000 m/s; the typical length scale of a pulseis thus 34 cm or 45 beads. The long-wavelength approxi-mation 7 is thus fulfilled and the pulse propagates on a

range much greater than its spatial extension, so that we canindeed call it a solitary wave. Another general conclusionthat may be drawn is that in all cases the duration of thepulse is much greater than the time taken by a bulk acousticwave to travel across a bead diameter, which is a necessarycondition to apply Hertzs theory to a nonstatic situation.

B. Velocity of the wave for nonzero static force

We give below experimental results on pulse propagationin the chain for three different values of the applied staticforce: 9.81, 29.41, and 1671 N. An important pa-rameter to consider is the ratio m/, given by Eq.21interm of the static and dynamic forces, which characterize the

FIG. 5. The dotted curve displays a time recording of the force

experienced by the sensor at the end of the chain during a long time,

typically 10 times the duration of the first pulse. The abscissa is the

time in s, the ordinate the force in N; the experimental condition

are a static force of 167 N and a nonlinear parameter m/2.74. The first positive pulse is identical below with Nesterenkos

14 solitary wave; the solid line shows the theoretical shape of a

solitary wave of the same amplitude, derived from Eq. 23. Apart

from resonant oscillations of the sensor, we observe an isolatedpulse.



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nonlinearity of the wave. This ratio must be much greaterthan 1 in order that Eq. 8 be valid; in our experiments, wehave

m/1.9,17.0 for F09.8 N,

m/ 1.4,16.5 for F029.4 N, 29

m/ 1.5,5.2 for F0167 N.

We obtain the wave velocity very simply from time-of-flight measurements between the four sensors available along

the chain see the details on the experimental apparatus in

Sec. III A. As we stressed before, the three force sensorsthat are held perpendicularly to the chain are well suited forsuch types of measurements, although they give no informa-tion about the actual shape of the wave. The theoretical valueof the wave velocity is derived as explained in Sec. III B,with no free parameter once the applied static force and themaximal amplitude of the wave are both known.

We show in Figs. 911 the results of wave velocity mea-surements for static forces of, respectively, 9.8, 29.7, and167 N. For the two smallest static forces, there is good agree-

FIG. 6. Shape of four different solitary waves recorded at the end of the chain, for a static force of 29.7 N. Each graph displays the

evolution of the force in N with time in s. The dots are experimental points, whereas the solid lines are the theoretical predictions

derived from Eq. 23. We give in each case the maximum amplitudeFm, the velocity V, and the nonlinear parameter m/of each wave:

a Fm100 N, V895 m/s, m/3.7; b Fm138 N, V921 m/s, m/4.5; c Fm209 N, V960 m/s, m/5.8; andd

Fm290 N, V994 m/s, m/7.2.

FIG. 7. Same as Fig. 6, with the same notations, but for greater values ofm/. The respective characteristics of each pulse are a

Fm503 N, V1060 m/s, m/10.4; b Fm608 N, V1086 m/s, m/11.8; c Fm705 N, V1106 m/s, m/13.0; and

d Fm1060 N, V1168 m/s, m/17.0.



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ment between the theoretical predictions and the measure-ments. In each case the nonlinear parameter ranges betweenthe same limits, as shown by Eq. 29. A small discrepancyis shown by the measurements at 167 N; a possible explana-tion is an increase of energy transfer between the wave andthe framework containing the beads, due to the high staticforce. A wave arriving at the end of the chain has lost energyand has thus propagated quicker than what is predicted fromits final amplitude; this interpretation is supported by thegood agreement of the experimental and theoretical shape of

the wave at the same static force shown in Fig. 8. If thediscrepancy was due to the smallness of the nonlinear param-eter see Eq. 29, it should have been exhibited in Fig. 8too. The successive time-of-flight measurements along thechain do display some scattering of the data, hence the error

bars of the graphs. The accuracy of the data is not sufficientto observe a systematic evolution of the velocity, whichseems rather constant during the pulse propagation, even fora static force of 167 N.

As we explained above, in the paragraph following Eq.22, the measurements of Figs. 911 may be displayed on asingle curve when expressed in rescaled variables V/c s andFm/F0 . This is demonstrated in Figs. 12a and 12b. Fig-ure 12a shows that all the previous data lie on a singlecurve, when expressed in the variables V/cs and Fm/F0 .

Figure 12b shows that, as predicted by Eq. 22, V/c s isindeed a linear function of y (Fm/F0)

1/5. A fit of theproportionality constant gives 0.84, which compares well

with the expected value2/30.8165. Note also that Fig. 12clearly shows that the nonlinear waves are supersonic, a re-sult predicted by Nesterenko 14.

C. The case of zero static force

In order to be experimentally as close as possible to thecase of zero static force, we proceed in the following man-

FIG. 8. Same as Fig. 6, with the same notations, but for a static force of 167 N. The respective characteristics of each pulse are a

Fm274 N, V1131 m/s, m/2.5; bFm474 N, V1178 m/s, m/3.3; cFm842 N, V1239 m/s, m/4.7; anddFm1005 N, V1260 m/s, m/5.2.

FIG. 9. Evolution of the wave velocityin m/s with the maxi-

mum amplitude of the wave in N for an applied force of 9.8 N.

The solid line is the theoretical prediction, derived from Eqs.21,

17, and 16.

FIG. 10. Same as Fig. 9, but for an applied static force of 29.7

N.



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ner. We exert a static force on the chain in order to set thebeads into contact and then we relax this force until the con-tact between the dynamometer and the brass cylinder thatsupport the longitudinal force sensor is broken. There is al-most no adhesion between the steel beads, but there may be

some uncontrolled friction with the walls of the channel con-taining the beads. We expect this friction to be small forsmooth steel beads in contact with PTFE. Experimentally,the static force between two adjacent beads is certainly verysmall, but we cannot be sure that it is strictly zero. In thatsense, the singularity of the zero static force case is rather amathematical oddity than an actual physical effect.

When no static force is applied to the chain, the theorypredicts the propagation of nonlinear periodic waves ratherthan solitary waves as in the previous case. We have re-corded the shape of the pulses arriving at the end of the chainin a large range of amplitudes, from 40 to 700 N. Experi-mentally, the comparison of the Figs. 68 with Figs. 1315

proves that there is no qualitative difference between thecases of zero and nonzero static force. Concerning the shapeof the wave, as it is clearly demonstrated in Fig. 2, it is notpossible to distinguish a solitary wave in the limit of verysmall static force from an arch of the nonlinear periodicwave. In Figs. 1315 we compare the experimental shape ofa pulse to one arch of a solution28with the same maximalamplitude for different values of the maximal amplitude ofthe pulse. The agreement between the observed shape of thepulse and the theoretical prediction is striking, except for thesmallest amplitude pulses displayed in Fig. 13. The duration26of the pulse is about 30 s and its velocity is about 900m/s. The spatial extension of the pulse is then about 30 mm,

roughly four beads, which is enough to ensure the validitythe long-wavelength approximation 7. Indeed, one knowsfrom numerical simulations 27 that in discrete nonlinearchains, the excitations are perfectly described by the continu-ous approximation if their size is greater than or equal to fiveparticles. This length is also much less than the total lengthof the chain, which means that the pulse has kept its shapeover a great distance, as a solitary wave should. The differ-ence from the previous case of Sec. IV A is that we are in thefully nonlinear regime, so that the development 8 is cer-tainly correct.

As shown in Fig. 13, the agreement between the shape ofthe pulse and the theoretical prediction becomes poor for anamplitude roughly less than 70 N. A possible interpretation

for the discrepancy between the observed shape of the pulses

and the predicted one could be that the friction of the beads

on the walls causes a residual static force F0

*0. But any

order of magnitude derived from Fig. 8, e.g., will give for F0*

a value much too high to be accepted. Another possibility isthat, at low impact amplitude, it takes too much time for thepulse to reach its asymptotic shape i.e., the solitary waveprofile from the initial wave profile. Unfortunately, there isno theoretical information about the times taken by a givenprofile to reach its asymptotic shape and we are unable to testthis interpretation. We stress that the amplitudes of thewaves in those experiments are much smaller than in theexperiments reported in Sec. IV A, in which this effect wasnot observed.

From the pulse velocity measurements, we can first verify

thatVF00 scales as Fm

1/6 ; Fig. 16 shows that this is indeed

FIG. 11. Same as Figs. 9 and 10, but for an applied static force

of 167 N.

FIG. 12. a Plot of the dimensionless ratio V/c s versus the

dimensionless quantity Fm

/F0

. The data are represented by circles

for a static force of 29.7 N, by triangles for a static force of 9.8 N,

and by squares for a static force of 167 N. All the experimental

points lie on a single curve. The solid line is the curve const

y(Fm/F0)1/5, where the experimental value of the constant is

0.84. This is to be compared with the theoretical prediction 22 in

which const2/30.8165. b Plot of the dimensionless ratioV/c s versus the dimensionless quantity y (Fm/F0)

1/5, with the

same symbols as before. The data all lie on a single linearcurve,

whose slope is found to be const0.84.



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the case. This result indicates that the Hertz interaction 1,in the quasistatic approximation 8, fully accounts for thepropagation of the pulse in the nonlinear regime toosee thediscussion above Eq. 26. Like in the previous case, nosystematic evolution of the velocity with the travel time ofthe pulse is observed. In Fig. 17 we show that formula 24gives rather accurately the velocity of the pulse. This behav-ior is linked to the fact that the solitary wave velocity tends

continuously toward Eq. 24 in the F0

0 limit, as weshowed in Sec. III B.

Even in the case of zero static force applied to the chain,we observe pulses that propagate over a great distance, witha constant velocity and shape, so that they may be identifiedwithsolitary waves. Their velocity and shape are both in fairagreement with the theoretical predictions of Nesterenko14,15 if we forget that in this singular limit nonlinear pe-riodic waves are predicted rather than solitary waves. Ofcourse the impact generator is not well suited to generate

periodic waves and we cannot expect this type of wave toappear spontaneously. Within the precision of our measure-

FIG. 13. Shape of the pulse at the end of the chain, when no static force is applied to the chain. Each graph displays the evolution of the

forcein Nwith time in s. Dots are experimental values recorded by the digital scope and solid lines the theoretical prediction 28fora wave of the same maximal amplitude. The velocities and amplitudes of the pulses are, respectively, a V629 m/s, Fm36.7 N; b

V659 m/s, Fm48.5 N; c V685 m/s, Fm61.4 N; and d V725 m/s, Fm86.4 N. The pulses are of small amplitude and the

agreement with the theory is rather poor, except for the highest amplitude pulse displayed ind.

FIG. 14. Same as Fig. 13, but for waves of greater amplitudes. The velocities and amplitudes of the pulses are, respectively, a V

811 m/s, Fm168.5 N; b V845 m/s, Fm216.2 N; c V877 m/s, Fm269.7 N; and d V919 m/s, Fm356.4 N. The negative

part of the force signal at the back of the pulses in bd is the signature of the sensor resonance. The agreement with the theory is very

satisfactory and much better than for the pulses of Fig. 13.



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ments, the pulses are able to travel more than ten times theirlength while keeping their shape and velocity, which meansthat they are rather stable.

D. Discussion

In this section, we compare our results to the previousexperiments of Nesterenko and Lazaridi 1618. They haveobserved the propagation of wave trains, built of several soli-tary waves, in chains of steel beads with zero applied static

force. The initial impact was much more violent in theirexperiments the impacting mass was five times that of abead in the chain and its velocity 1 m/s, more than twicewhat we get in our experiments, which explains that several

solitary waves were generated. Moreover, they did not varythe intensity of the first impact. They have also conductedsimilar experiments 17,18 with applied static forces of 2and 20 N. An interesting result is that the pulses observedwith a static force of 2 N are almost identical to the onesobserved without static compression of the chain; moreover,this behavior is confirmed by numerical simulations.

They recorded the arrival of the wave train at the end ofthe chain and compared its shape with numerical simulationsthat included the impact on the first bead of the chain withthe colliding mass used to generate the wave train. The ve-locity of the first pulse was not measured, but the time inter-vals between successive pulses, together with their respectiveamplitudes, compared well with the theoretical expectationsfor a short chain of 20 beads; the agreement was only quali-tative for a longer chain of 40 beads. In either cases, the

FIG. 16. Graph of the wave velocity in m/s as a function of

Fm1/6 in N1/6, where Fm is the maximum amplitude of the wave,

when no static force is applied on the chain; the relation is clearly

linear, which simply states that the Hertz interaction between adja-

cent beads is responsible for wave propagation.

FIG. 15. Same as Figs. 13 and 14, but for waves of greater amplitudes. The velocities and amplitudes of the pulses are, respectively, a

V

963 m/s, Fm

473.1 N; b V

1000 m/s, Fm

594.0 N; c V

1014 m/s, Fm

646.1 N; and d V

1029 m/s, Fm

704.6 N. Thenegative part of the force signal at the back of the pulse is the signature of the sensor resonance. As in Fig. 14, the agreement with the theory

is very good.

FIG. 17. Evolution of the wave velocityin m/swith the maxi-

mum amplitude of the wave Fm in N for no applied static force.

The solid line is the theoretical prediction 24.



13/14

comparison between the shape of the pulses was only quali-tative.

Our experimental setup allows the exploration of a largerange in the amplitude of the pulses, roughly from 1 to 20,and permits a systematic study of the shape and velocity ofthe waves as functions of their maximum amplitude. In ourexperiments, the chain contains 51 beads, and we find excel-lent agreement with the theory for either the velocity of the

pulses or their completeshape. We do not confirm the strongattenuation of the wave reported by Nesterenko and Lazaridifor the long chain of 40 beads.

A possible interpretation of this discrepancy is as follows.In their experiments, the beads are 4.75 mm in diameter,contained in a quartz tube with an inside diameter of 5 mm.The beads are thus allowed to move rather easily away fromthe tube axis and presumably they lose a lot of energy inimpacting the tube; this effect is reinforced if the length ofthe chain is increased. In our experiments, the beads are con-tained in a channel drilled in a framework of PTFE, with avery small clearance of 210 0 mm. Impacts on the walls aresuppressed and, moreover, the acoustic coupling between

steel and PTFE is much smaller than between steel andquartz.

V. CONCLUSIONS

We have conducted experiments on a chain of identicalbeads in contact with either moderate static forces or zerostatic force applied to the chain. With the help of an impactgenerator, we were able to generate high-amplitude pulses inthe chain, that is, with a dynamical amplitude much higherthan the static force.

With our experimental setup we were able to vary thestatic force applied on the chain, together with the pulsesamplitude. We added to the previous work of Nesterenko

and Lazaridi 1618 a systematic and quantitative study ofthe velocity and shape of the solitary waves.

At the end of the chain, we recorded the time evolution of

the force exerted on a dynamic force sensor and compared

this experimental shape of the pulse with the theoretical pre-

dictions of Nesterenko 14,15. The agreement was very

good when no static force was applied on the chain, except

for excitations of very small amplitude; with a nonzero static

force, excellent agreement was found between theoretical

predictions and experimental observations. In all cases, no

adjustable parameter was involved in the data analysis.

The velocity of the pulse seems to be accurately predicted

by the theory, either with or without static force applied on

the chain. Velocity measurements for different nonzero ap-

plied static forces may be displayed on a single curve with

proper rescaling of the variables. We stress that, as for the

pulse shape, no adjustable parameter has been used.

In all cases, the typical size of the excitation is about four

or five beads, which is sufficient to ensure the validity of a

long-wavelength approximation. This size is also about one-

tenth of the total length of the chain, which means that the

pulses propagate over a large distance while keeping their

velocity and shape. This is strong experimental evidence ofnonlinear solitary wave propagation in a chain of beads in

Hertzian contact, even in the limit of zero applied staticforce.

ACKNOWLEDGMENTS

We thank S. Gavrilyuk for many interesting discussionsand for an introduction to the Russian literature on the sub-

ject. We thank V. Nesterenko for kindly sending us his workon the subject, with commentaries about some papers writtenin Russian. We gratefully acknowledge D. Bouraya for tech-nical assistance in the building of the experimental setup. We

thank the MESR, through the Reseau de Laboratoires GEO,for financial support.

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Solitary Waves Under Hertz Contact