surajit sen et al- solitary waves in the granular chain

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Solitary waves in the granular chain

Surajit Sen a, Jongbae Hong b,∗, Jonghun Bang b, Edgar Avalos a,c, Robert Doney d

a Department of Physics, State University of New York at Buffalo, Buffalo, NY 14260-1500, USAb Department of Physics, Seoul National University, Seoul 151-747, Republic of Korea

c Graduate Institute of Biophysics and Centre for Complex Systems, National Central University, Chung-Li, 320 Taiwan, ROC d US Army Aberdeen Proving Grounds, Aberdeen, MD 21005, USA

Accepted 10 October 2007

editor: M.L. Klein

Abstract

Solitary waves are lumps of energy. We consider the study of dynamical solitary waves, meaning cases where the energy lumpsare moving, as opposed to topological solitary waves where the lumps may be static. Solitary waves have been studied in someform or the other for nearly 450 years. Subsequently, there have been many authoritative works on solitary waves. Nevertheless,some of the most recent studies reveal that these peculiar objects are far more complex than what we might have given them creditfor. In this review, we introduce the physics of solitary waves in alignments of elastic beads, such as glass beads or stainless steel

beads. We show that any impulse propagates as a new kind of highly interactive solitary wave through such an alignment and thatthe existence of these waves seems to present a need to re-examine the very denition of the concept of equilibrium. We furtherdiscuss the possibility of exploiting nonlinear properties of granular alignments to develop exciting technological applications.c 2008 Elsevier B.V. All rights reserved.

PACS: 45.70.-n; 05.45.Yv; 47.35.Fg; 07.05.Tp

Contents

1. Introduction: About solitary waves ..................................................................................................................... .. 21.1. Linear versus nonlinear forces ................................................................................................................ ... 21.2. History of solitary waves ...................................................................................................... ..................... 3

1.2.1. The Korteweg–de Vries equation ................................................................................................... 31.2.2. The solitary waves in the sine-Gordon equation ............................................................................... 31.2.3. Lack of thermalization and the Fermi–Pasta–Ulam problem ............................................................. 41.2.4. Solitary waves in Toda’s chain ................................................................................................... ... 41.2.5. The solitary waves of the nonlinear Schr odinger equation ................................................................. 41.2.6. The tsunamis ......................................................................................................... ...................... 4

1.3. Solitary waves in Hertz-type systems ....................................................................................................... ... 5

∗Corresponding author. E-mail address: [email protected] (J. Hong).

0370-1573/$ - see front matter c 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physrep.2007.10.007

Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007

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2. Nonlinear waves in a discrete system .................................................................................................................. .. 52.1. The Hertz potential ............................................................................................................... .................... 52.2. Nonlinear waves in a discrete Hertz-like system ........................................................................................... 72.3. Early studies on the solitary wave — Nesterenko’s solution and Chatterjee’s renements ................................. 72.4. Solitary waves in a discrete chain of power-law contact force ........................................................................ 9

3. Solitary waves in a granular chain .................................................................................................. ...................... 103.1. A new series solution to the equation of motion ........................................................................................... 103.2. Effect of gravitational loading on the granular chain ..................................................................................... 14

3.2.1. Role of gravity and nonlinearity .................................................................................................. .. 163.2.2. Analytical studies for the weakly nonlinear regime .......................................................................... 183.2.3. Summary ............................................................................................................... ..................... 20

4. Role of randomness and dissipation in amplitude attenuation ................................................................................... 215. Collision of solitary waves — secondary solitary waves .......................................................................................... 226. The quasi-equilibrium phase ............................................................................................................ .................... 257. Conning solitary waves and universal power-law decay ........................................................................................ 28

7.1. Solitary waves at granular interfaces ......................................................................................................... .. 287.2. Understanding total transmission .............................................................................................................. . 317.3. Connement of solitary waves and granular containers ................................................................................. 327.4. Universal power-law decay .................................................................................................. ...................... 35

8. Impulse absorption by tapered granular chains ...................................................................................................... . 378.1. Simple tapered chain ............................................................................................................ .................... 388.2. Decorated tapered chain ....................................................................................................... ..................... 41

9. Summary and discussion ................................................................................................................ ..................... 44Acknowledgements .................................................................................................... ........................................ 44References ................................................................................................................ ........................................ 44

1. Introduction: About solitary waves

1.1. Linear versus nonlinear forces

The concept of waves is routinely introduced to us in freshman physics [ 1]. The introduction typically involvesexamples of traveling waves in water, of sound waves and light waves. These waves are propagating entities that carryenergy. The waves carry both potential energy, which denes their shape and energy content, as well as kinetic energy,both of which along with the system characteristics describe their dynamical characteristics. It is conceivable then thatthere can be various kinds of waves — like waves on a tranquil pond where a little pebble has been dropped, to wavesas in some earthquake, to shock waves due to a large explosion or as in a tsunami wave like the devastating one of December 26, 2004 in the Indian Ocean.

Typical waves gradually disperse their energy and suffer attenuation of their amplitudes as they travel. Suchdispersion and attenuation happens because energy is slowly distributed among the available degrees of freedomof the system. Simple traveling waves can be described by a sinusoidal function describing some displacement froman equilibrium position, e.g.,

u( x, t ) = A sin(kx −ωt +φ0), (1.1)

where A is an amplitude, φ0 is a phase factor, the argument (kx −ωt +φ0) involves a linear relation between space x and time t via the angular frequency ω and the wave number k of the wave and ω =vk , where v is the wave speed.We next nd that such a sinusoidal function is the solution to a linear equation involving force and displacement of the form

¨u( x, t ) = −ω2u( x, t ), (1.2)

where ¨u ≡ d2u/ dt 2, and hence can be viewed as identical to the equation for a harmonic oscillator. Since linearequations have unique solutions, the concept of harmonic oscillation is unequivocally tied to that of sinusoidaltraveling waves. An important property of these waves is that the wave speed is unrelated to the wave amplitudePlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007



ARTICLE IN PRESSS. Sen et al. / Physics Reports ( ) – 3

and depends only upon the properties of the medium through which they propagate. In a typical curriculum, dampedand forced oscillations are next introduced [ 2].

An important feature of an interacting system of harmonic oscillators is that such a system can be “deconstructed”and “recast” as a system of independent harmonic oscillators, a procedure that we formally refer to as“diagonalization” [ 3]. In more advanced analyses, the concept of harmonic oscillators is used to analyze diverse

dynamical processes that seem amenable to theoretical analyses in condensed matter and statistical physics, quantumeld theory, etc [ 4].In the case of a nonlinear force, Eq. (1.2) must be replaced by an equation, say, of the form

¨u = f (u α ), α > 1, (1.3)

where f denotes some function of u α . In the remainder of this work, we shall be concerned with dynamical equationsinvolving nonlinear force laws [ 5–9]. Although there are systems where both linear and nonlinear forces are presentthat admit solitary waves, i.e., energy bundles that are nondispersive [ 5–9], our focus will be on systems where thelinear term would be absent and hence these systems can be thought of as “intrinsically nonlinear” or “stronglynonlinear” [ 10]. These discussions begin in Section 2. We briey review the body of existing work on solitary wavesbelow.

1.2. History of solitary waves

It is not uncommon to encounter traveling wave fronts through long and shallow water channels. These movingwave fronts may not decay or dissipate and can be extremely stable. In sharp contrast to the harmonic waves alludedto above, their velocities depend upon their amplitudes. Larger amplitude waves move faster than those with smalleramplitudes. We would refer to these waves as nonlinear waves, implying that the relationship between accelerationand displacement is nonlinear. In the literature, one refers to these waves as solitary waves. Two opposite propagatingsolitary waves can pass through each other without suffering any interaction except for a slight shift in their positionscompared to where they would have been had they not passed through one another. These waves were rst observed byJohn Scott Russell, a Scottish engineer, in the Union Canal linking Edinburgh and Glasgow in August 1834 [ 11,12].Sander and Hutter [ 13] have reviewed the evolution of the mathematical understanding of wave motion by describing

the work of Huygens in 1673 [14], Newton in 1687 [15], Euler [16,17] and Lagrange in 1759 [18,19], Gerstner in1809 [20], Russell in 1838 [ 11] and 1845 [12], Airy in 1845 [ 21], Stokes in 1847 [ 22], Earnshaw in 1849 [ 23],Boussinesq in 1872 [ 24,25], Lord Rayleigh (J.W. Strutt) in 1876 [ 26], de Saint Venant in 1885 [ 27], McCowan in1891 [28,29], Korteweg and de Vries in 1895 [ 30], Friedrichs in 1948 [ 31,32] and others (see [ 13] for details).

1.2.1. The Korteweg–de Vries equationThe propagation of solitary waves in water is described by what is known as the Korteweg–de Vries (KdV)

equation. The KdV equation is not easy to derive. There are several different ways to state this equation indimensionless form and here is one of the simplest ways

∂v∂t =v

∂v∂ x +

∂3v∂ x3 , (1.4)

where v( x, t ) is the dimensionless horizontal velocity of a pulse traveling through a long narrow channel of sufcientdepth and the wave amplitude is sufciently small.

1.2.2. The solitary waves in the sine-Gordon equationThe other known equations that admit solitary wave solutions are the sine-Gordon and sinh-Gordon equations and

the nonlinear Schr odinger equation. The sine-Gordon equation dates back to the works of Dehlinger in 1929 [ 33] andof Frenkel and Kontorova in 1939 [34] and can be stated in terms of displacement u( x, t ) as follows,

∂ 2u( x, t )∂t 2 −

∂2u( x, t )∂ x2 +sin u( x, t ) =0, (1.5)

and plays an important role in many branches of physics. The sinh-Gordon equation is the same as Eq. (1.5) aboveexcept that the sine term is replaced by a sinh term. It has been used to explore models of unied eld theory [ 35], isPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




encountered in the study of dislocations in crystal lattices [36], in studying properties of Josephson junctions [ 37] andso on.

1.2.3. Lack of thermalization and the Fermi–Pasta–Ulam problemNonlinear waves are not only encountered in continuous media, such as in uids and monodisperse alignments

of particles that interact via the Toda potential (mentioned in Section 1.3), but have also been seen in very earlystudies by Fermi, Pasta and Ulam when they were studying the role of anharmonic potentials in distributing theenergy of a perturbation within small chains, i.e., whether anharmonicity would in general lead to thermalization of the system [38]. Zabusky and Kruskal [ 39] made an effort to connect their studies on the solitary waves of the KdVequation with the ndings of Fermi, Pasta and Ulam. A signicant amount of work has been done on the dynamics of solitary waves and the exchange of energy between solitary wave-like entities and acoustic oscillations in systems of masses connected by springs that possess both harmonic and anharmonic features [ 7,8].

1.2.4. Solitary waves in Toda’s chainHere we briey mention a famous problem involving the propagation of mechanical energy through a chain of

equal masses connected by springs. The interaction force between the nearest neighbor masses possesses both linear

and certain nonlinear terms. A mechanical impulse initiated in this chain travels as a solitary wave. This problem isknown in the literature as the Toda lattice problem and surfaced in 1967 [ 40–42 ]. The Toda potential between nearestneighbors masses at a mutual distance r is given by

V (r ) =ab

exp(−br ) +ar +const ., a , b > 0, (1.6)

and the corresponding nonlinear equation of motion is given by

m ¨u i = −∂

∂r i [V (u i −u i−1) +V (u i+1 −u i )], (1.7)

where r i = u i −u i−1 . The faster the Toda soliton moves, the narrower it gets. The stability of the so-called Toda

soliton has been recently explored when the masses are not equal but possess random magnitudes [ 43]. This problemforms an example of a solitary wave bearing system where both linear and nonlinear force terms must be present.

1.2.5. The solitary waves of the nonlinear Schr ¨ odinger equationThe time evolution of the envelope of a weakly nonlinear deep-water wave train is also described by the nonlinear

Schrodinger equation. This equation was exactly solved by Zakharov and Shabat [ 44] pursuing a newly discoveredmethod called the inverse scattering technique. The Zakharov and Shabat solution was experimentally seen by Yuenand Lake in 1975 [ 45]. The nonlinear Schr odinger equation has the following form:

i∂u( x, t )

∂ t +∂ 2u( x, t )

∂ x2 +2|u( x, t )|2u( x, t ) = ε P (u ( x.t )), (1.8)

where u( x, t ) describes a wave envelope in some weakly dispersive nonlinear medium. Here P (u ( x, t )) is aperturbation and ε is some small parameter. The solitary wave solution to this equation has been extensively exploredin the literature. A recent study seems to suggest that the well-known dynamics of the Dufng oscillator [ 46] can berelated to the solution to the nonlinear Schr¨ odinger equation [ 47].

1.2.6. The tsunamisLarge and destructive nonlinear monster waves can originate as a byproduct of violent ocean bed seismic activities

around the world [ 48]. The December 2004 tsunami that was initiated by a catastrophic earthquake that caused a riftin the ocean oor near Banda Aceh in Indonesia initiated such a monster tsunami in the open Indian Ocean. Thistsunami was generated in and propagated in the open ocean as opposed to what one sees when a wave front movesthrough a channel. Studies on the generation of ocean waves and on the dissipation of such waves along coasts dene

an active and well-developed area of research in ocean science and engineering [ 49,50].Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 1.1. The dashed blue line shows V (δ)∝

δ2 (harmonic dependence) whereas the solid blue line shows V (δ)∝

δ5/ 2 (Hertz dependence). Theplot shows that the Hertz potential is softer than harmonic at small enough and is steeper than harmonic at large enough δ. The lines for n =5 andn

=10 show how the potential steepens with increasing n . (For interpretation of the references to colour in this gure legend, the reader is referred

to the web version of this article.)

1.3. Solitary waves in Hertz-type systems

We now turn our attention to the study of the most recently found incarnation of the solitary wave, namely, thesolitary wave that was rst probed by Nesterenko [ 51,52] and subsequently by Lazaridi and Nesterenko in 1983 [ 53].The properties of these waves were next veried in a publicly available US army report by Miller in 1986 [ 54]. Thiswave was also encountered in a 1995 particle dynamics based study on vertically downward sound propagation in1D and 2D gravitationally loaded granular systems by Sinkovits and Sen in 1995 [ 55,56]. Coste, Falcon and Fauvereported a comprehensive experimental study of Nesterenko’s wave in 1997 [57].

These are discrete systems of grains. In these granular systems, the force–displacement characteristics of eachgranular contact can be completely nonlinear, meaning that there is no linear term in the force at all [ 58] (see Fig. 1.1 ).Complete nonlinearity, meaning a potential with no harmonic term at all, arises when the grains in a chain are notpreloaded, i.e., when the grains barely touch each other. Such a system does not support the oscillations of each grainor particle about the mean position. Rather, the grain moves more in one direction than the other. Nesterenko hascharacterized these systems aptly as “acoustic vacua” — meaning no acoustic or sound propagation is possible inthese systems.

These solitary waves have nite and constant widths. The widths are dependent on the precise nature of thegrain–grain interface. The velocity of propagation of this solitary wave is depends upon its amplitude, a typicalcharacteristic of every nonlinear wave. These solitary waves that are found in granular alignments of discrete massesare not necessarily the same as those encountered in continuous media. The role of boundaries and initial conditionsin the generation and the dynamics of these waves also turn out to be important aspects that control their properties.

2. Nonlinear waves in a discrete system

2.1. The Hertz potential

When two elastic objects, are compressed, they repel each other as a function of increasing compression. Forsimplicity, let us consider two spheres, labeled i of radius Ri and i+1 of radius Ri+1, that are being pressed against oneanother by some force F . Let the corresponding potential energy be denoted by V . The distance between the centersof the spheres when they are barely touching is Ri + Ri+1. If they are pressed against each other, then the overlapparameter between the spheres is dened as δi, i+1 ≡ Ri + Ri+1 −(u i −u i+1) , where u i denotes the absolute positionof grain i . We assume that the spheres are made of different elastic materials, which are characterized by Young’smoduli Y i and Y i+1 and Poisson’s ratios σ i and σ i+1 , respectively. The potential energy due to the compression of thePlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




spheres is given by the Hertz potential [ 59–61 ] as follows —

V (δ i , i+1) =2

5 Di, i+1

Ri Ri+1

Ri + Ri+1δ

52i, i+1 ≡a i, i+1δ

52i, i+1, a i, i+1 ≡

25 Di, i+1

Ri Ri+1

Ri + Ri+1, (2.1)

and

Di, i+1 =34

1 −σ 2iY i +

1 −σ 2i+1

Y i+1. (2.2)

The Hertz law is derived in a few textbooks [60,61]. Spence has developed a generalization of Hertz law for arbitrarycontact surfaces between two grains [62]. The derivation centers on the calculation of the pressure felt at the graincontacts and the region of the overlap between the grains to estimate the force between the grains. Once the force isknown, the work done in accomplishing the compression can be calculated. Instead of a detailed derivation, whichis mathematically cumbersome, we discuss a “back of the envelope” derivation below. This derivation is sketched inFalcon’s thesis [ 63] and was introduced to one of us by Francisco Melo.

We start by expressing the pressure between two identical spherical grains of radius R, Young’s modulus Y and

sharing a contact area ∼˜r 2, where ˜r is the radius of the contact region. It is reasonable then to dene

r ∼

δ

˜r , (2.3)

as a measure of the deformation suffered by the grains in terms of the overlap parameter δ and the length scaleassociated with the contact area ˜r . The pressure P felt by each grain can then be described as

P =Y r =F

˜r 2, (2.4)

where F is the Hertz force between the grains and the object of our interest.We also note from purely geometrical considerations that

˜r 2∼δ R. (2.5)

Hence,

P =Y r =F

˜r 2 ⇒Y

δ

˜r ∼F

δ R ⇒Y

δ R

1/ 2

∼F

δ R, (2.6)

and hence,

F ∼

Y R1/ 2δ3/ 2 . (2.7)

This result implies that

V (δ)∼

Y R1/ 2

δ5/ 2

, (2.8)which is the result of Hertz law. In general, the index of the Hertz law, i.e., the number 5 / 2 is dependent on the contactgeometry between the grains and the grain shapes. For grains with irregular contacts, this number may be higher. Forgrains that squeeze more easily than two spheres would, this index may be lower.

It is important to observe that the Hertz law describes a compression force that is softer or weaker than that dueto harmonic springs for small enough overlaps and that builds up far more steeply than the harmonic force when theoverlaps become sufciently large. Fig. 1.1 shows a comparison between the potential energies as function of theoverlap for the (one-sided) harmonic and Hertz potentials.

We present here some simple arguments to address why an impulse in a granular chain travels as a solitary wave. Atan intuitive level, the grains are squishier at small enough compressions and they develop hard repulsions (comparedto the harmonic case) as compression increases. Thus, energy transport from one grain to the next starts off slowlyin time but then as the repulsion increases upon increasing overlap, the energy cost to continue to press against eachPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




other escalates strongly and the grains must hence get away from one another. The energy transfer from one grain tothe next hence becomes a progressively faster process as overlap increases. Such energy transfer tends to happen as a“one-shot deal”.

2.2. Nonlinear waves in a discrete Hertz-like system

We consider a monodisperse chain of elastic spheres of radius R in which the spheres barely touch one another,i.e., there is “zero loading” between the spheres. The spheres repel upon overlap by an amount δi, i+1 ≡ 2 R −(u i −u i+1) where zi describes the initial equilibrium position of some grain i in the chain, and u i is the displacement of the same grain from the equilibrium position. Then, according to Hertz law [ 59] the repulsive potential between twoadjacent spheres is given by,

V (δ i, i+1) =a δ5/ 2i, i+1, (2.9)

where a = (2/ 5 D(Y , σ ) ) ( R/ 2)1/ 2 and D (Y , σ ) = (3/ 2) 1−σ 2Y . In this work we study the dynamics for the potential

in Eq. (2.9) and for the general case where the index, which we will call n in Eq. (2.9) , and n can be any number > 2(note that Eq. (2.9) is a special case of Eq. (2.1) , when a i, i

+1

≡a ). The value of n depends upon the geometry of

the contact region between two adjacent elastic bodies. When one considers spheres, n = 5/ 2 [60–62 ]. Given themagnitude of n , it may be noted that the repulsive potential V (δ i, i+1) = a δn

i, i+1 , as stated in Eq. (2.9), is intrinsicallynonlinear in the sense that one cannot write the force between the compressed spheres as having any linear component.The absence of a linear component in the force law (i.e., absence of Hooke’s law) implies that sound propagation isnot possible in a chain of elastic beads at zero precompression, a phenomenon we referred to as “sonic vacuum” inSec. 1.3 [51,58].

To initiate sound propagation, one must introduce some precompression in the system. The simplest case is uniformprecompression, say by an amount ∆ , effected on every grain. The equation of motion of a bead in the chain (not atthe boundaries) then becomes,

md2u i (t )

dt 2

=na (∆

+u i−1(t )

−u i (t ))n−1

−(∆

+u i (t )

−u i+1(t ))n−1 , (2.10)

where the right-hand side (RHS) of Eq. (2.10) can be expanded as a perturbation against ∆ when ∆ > ( u i−1(t ) −u i (t )) . Nesterenko [51] showed that if u i varies slowly in space, i.e., if the long-wavelength limit is invoked, and if there is no external loading and 2 < n < ∞, then Eq. (2.10) can be approximated via a nonlinear equation, which canbe related to the KdV equation and which admits a solitary wave solution for a propagating perturbation in the chain.The derivation of the long-wavelength equation is presented in detail in Nesterenko’s book [ 58]. Nesterenko’s analysishas been subsequently revisited by Chatterjee [ 52], approached very differently by others, tested numerically and hasbeen experimentally veried [53,57]. We present below Chatterjee’s version of Nesterenko’s solution. Solitary wavescan be thought of as energy bundles. In other words, any perturbation travels as an energy bundle through a chain of elastic spheres that has not been precompressed. Below we sketch the main ideas associated with Nesterenko’s work.

2.3. Early studies on the solitary wave — Nesterenko’s solution and Chatterjee’s renements

Let us set the preloading parameter ∆ =0 in Eq. (2.10) . Eq. (2.10) can hence be rewritten as

u i = ( u i−1 − u i )n−1 −( u i − u i+1)n−1 , where i =2, 3, . . . , N −1, (2.11)

where u i (t ) ≡ (m/ na )1/( n−2) u i (t ) . We next assume that an innite granular alignment when perturbed by someimpulse admits a solitary wave solution. The displacements of every grain can then be described by the samemathematical function except that the function will be evaluated at different times. Then,

u i (t ) = u i+1(t +b), where b > 0 is a constant. (2.12)

We rewrite Eq. (2.11) then as

¨u (t ) = {u (t +b) − u (t )}n−1 −{u (t ) − u (t −b)}n−1 , (2.13)Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




where u (t ) can be any point on the chain. If we now insert the assumption that

u (t +b) >> u (t ) >> u (t −b), and u ∞− u (t +b) << u ∞− u (t ) << u ∞− u (t −b), (2.14)

where u ∞ ≡ u (t )|t →∞, Taylor expansion of Eq. (2.13) with n =5/ 2 yields,

d2 u (t )dt 2 =

32 b5/ 2

d u (t )dt

d2 u (t )dt 2 +

18 b9/ 2

d u (t )dt

d4 u (t )dt 4 +

18 b9/ 2 [

d2 u (t )/ dt 2

][d3 u (t )/ dt 3

]√ d u (t )/ dt

−1

64b9/ 2 (d2 u (t )/ dt 2)3

(d u (t )/ dt )3/ 2 +O (b13/ 2). (2.15)

If we drop terms of O (b13/ 2) , and then set b =1, we get

d2 u (t )dt 2 =

32

d u (t )dt

d2 u (t )dt 2 +

18

d u (t )dt

d4 u (t )dt 4 +

18[d2 u (t )/ dt 2][d3 u (t )/ dt 3]√ d u (t )/ dt

−1

64(d2 u (t )/ dt 2)3

(d u (t )/ dt )3/ 2 , (2.16)

where it is assumed that the series on the right is convergent. Nesterenko showed that

v ( t ) =d u (t )

dt =2516

cos4(2t / √ 10)

0for

t ∈

(−√ 10π/ 4, √ 10π/ 4)t < −√ 10π/ 4;t > √ 10π/ 4,

(2.17)

and the validity of the periodic solution in Eq. (2.17) above can be veried by direct substitution. The above solutioncaptures the solitary wave well for the case of elastic spheres, i.e., for n = 5/ 2. This solution is also valid when t issufciently small. Given that the solitary wave is nondispersive, and is a traveling energy pulse, the speed of the wavemust be constant. Thus, an equation in time is equivalent to an equation in space and the latter, originally derived byNesterenko is

d2udt 2 = c2 3

2 −dud x

d2ud x2 + b2

8 −dud x

d4ud x4 − b2

8

d2ud x2

d3ud x3

−dud x

− b264

d2ud x2

3

−dud x

3/ 2 , (2.18)

where −dud x > 0, and

c2 =2Y

πρ( 1 −σ 2) ≡ Ab5/ 2 (2.19)

with ρ being the density of the granular material. We refer the reader to Chapter 1 of Nesterenko’s book [ 58] for therelevant steps, transformations and arguments that allow one to establish that from Nesterenko’s solution the size of thesolitary wave is ≈5(2 R) . The solution reveals that the speed of the traveling solitary wave scales with the amplitudeof the solitary wave, a typical characteristic of nonlinear waves. We shall discuss the properties of the solitary wavein some detail below.

The speed–amplitude relation of Nesterenko’s solitary wave is given by simple scaling analysis for the equation of motion

md2u i (t )

dt 2 =na (∆ +u i−1(t ) −u i (t ))n−1 −(∆ +u i (t ) −u i+1(t ))n−1 . (2.10)

For xed mass and elastic constants, the scaling parameters of the equation of motion have the relation L/ T 2∝

Ln−1

or

L∝

T 2/( 2−n) , (2.20)

where L and T denote the scaling parameters of grain displacement and time, respectively. The impulse velocity vi

representing the amplitude of the solitary wave scales as vi∝

L/ T ∝

T n /( 2−n) using Eq. (2.20) , while the speed of Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




solitary wave scales as vs ∝1/ T . Thus, one nds the relation

vs ∝v (n−2)/ n

i , (2.21)

where vs is the speed of the solitary wave. For the Hertz chain, one obtains vs ∝v1/ 5

i . The speed of impulse can bemeasured by the maximum force F m exerted at the end of chain [57]. One can re-express the relation in Eq. (2.21)in terms of F m using the formula for contact force, F

∝δn−1 , which gives a scaling relation F

∝Ln−1. This and

Eq. (2.20) give rise to F ∝

T 2(n−1)/( 2−n) . Thus one gets vs ∝1/ T ∝

F (n−2)/ 2(n−1)m . For the Hertzian chain, one

has vs ∝F 1/ 6

m . This relation has been veried by experimentalists [57]. Since the detailed treatment of Nesterenko’ssolitary wave can be found elsewhere [58], we skip them in this review.

2.4. Solitary waves in a discrete chain of power-law contact force

In addition to Nesterenko’s analytical study of solitary waves in the continuum limit of a discrete chain withHertzian contact, an important advance in the study of the propagating waves in a discrete chain was made by Frieseckeand Wattis [ 64]. These authors provided a powerful theorem for the existence of the solitary wave solution of the form

zi (t ) = f (i −ct ) for the equation of motion ¨ zi = V (φ i ) −V (φ i−1) , where φi = zi+1 − zi , describes a discretechain with a general type of contact force. The theorem claims that the solitary wave solution exists in any discretechain system when the total energy of the system is positive and the contact potential energy V satises the followingfour conditions:

(i) V is twice differentiable in the range [0, ∞) ,(ii) V (0) =0 and V (0) =0,

(iii) V (φ)/φ 2 is strictly increasing for φ > 0, and(iv) V (0) =0 or V (φ) = (1/ 2)V (0) φ 2 +αφ β +O (φ β ) as φ decreases to 0 for some α > 0 and 2 < β < 6, where

φ is the distance associated with the squeeze between neighboring grains.

After Mackay [ 65] applied this theorem to prove the existence of solitary wave in a chain of grains interacting via theHertzian contact force, Ji and Hong [66] extended the proof to systems with arbitrary power-law type contact force.Since the details of the theorem of Friesecke and Wattis [64] are rather mathematical, we direct the interested readerto the proof in the original paper [64]. We, therefore, just use the theorem for nding the existence criterion of solitarywaves in a discrete chain with arbitrary power-law type contact forces between the particles.

Since the system we are interested in here is the chain of grains under a power-law type contact force, the equationof motion can be simply written as

¨ zi = (δ i )n−1 −(δ i−1)n−1 , (2.22)

where δi = ∆ − ( zi+1 − zi ) , and ∆ denotes the distance between adjacent grains in equilibrium and under externalloading. We set δi =0 when it has a negative value. We set all physical parameters, such as mass and elastic constant,unity for our convenience. This does not change our conclusions. Since this system does not have negative potentialenergy at any contact, the positivity condition of the total potential is satised. We can see that the function V (φ)introduced below satises condition (i). We now check the other conditions. If we choose

V (φ) =1n

δn −δ F +n −1

nF n / n−1, (2.23)

where δ = ∆ −φ and F = ∆ n−1 indicating the amount of external loading, we recover Eq. (2.22) . From Eq. (2.23) ,we have

V (φ) =δn−1 −F = (∆ +φ) n−1 −∆ n−1 , (2.24)

and

V (φ) = (n −1)δ n−2 . (2.25)Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




One can easily check V (0) = 0 and V (0) = 0 for any n from Eqs. (2.23) and (2.24) , respectively. Therefore,condition (ii) is satised. To test condition (iii), one can easily construct

V (φ)φ 2 =

1(δ −∆ )2

δ

∆(ξ n−1 −∆ n−1)dξ =

1(δ −∆ )2

δ

∆(δ −ξ )( n −1)ξ n−2dξ, (2.26)

using Eqs. (2.23)–(2.25) . Analysis of Eq. (2.26) reveals that the function V (φ)/φ 2 is constant for n = 2 and strictlyincreasing only for n > 2. Therefore, condition (iii) is satised only for n > 2. As for condition (iv), we obtainV (0) = (n −1) F (n−2/ n−1) . This does not give V (0) = 0 for F = 0 if n ≤ 2, while V (0) is unity for n = 2and diverges as F approaches zero for n < 2. The expansion coefcient α appearing in condition (iv) is given byα = (1/ 6)( n −1)( n −2) F (n−3/ n−1) , which shows that α is not positive if n ≤ 2. Therefore, condition (iv) can besatised only for n > 2. This proof shows that solitary waves exist in a discrete chain of power-law type contact forceas long as the power-law exponent in the force law is larger than unity.

3. Solitary waves in a granular chain

3.1. A new series solution to the equation of motion

The equation of motion of some bead i (excluding the edge grains) in a nite, monodisperse (i.e., all the massesare the same) chain of beads is given by

md2u i / dt 2 =na ∆ −(u i −u i−1) n−1

− ∆ −(u i+1 −u i ) n−1, (2.10)

where n > 2. Initially, every grain is placed barely in touch with one another such that ∆ = 2 R. Let us call thisthe no-loading case. An impulse dened by an initial velocity v0 at time t = 0 is initiated at the rst bead [ 67]. Asthe impulse propagates, one nds via numerical, experimental and analytical studies that a solitary wave develops inspace and time [67–69 ].

We now outline the procedure for constructing an approximate solution for ui(t ) in Eq. (2.10) . With reference to

earlier studies and the results from numerical simulations, we start by assuming that our system admits a solitary wavesolution. We further assume that the displacement of individual grains from equilibrium positions u i (t ) are continuousfunctions of time but are dened only at discrete positions zi . Since the solitary wave is nondispersive, we assume thatthis displacement can be obtained from a wave-type continuous function of both space and time, from the relation:

u i (t ) =u( zi −ct ) =u(α), with α = z −ct , (3.1)

where c is the constant velocity of the solitary wave.Exhaustive numerical studies on Eq. (2.10) and also other work indicate that, for a given n , the shape of the

solitary wave in space does not depend on the solitary wave amplitude. This implies that the function u is describedby u(α) = Aψ n (α) , where A represents the amplitude of the solitary wave and A = u(−∞) −u(+∞) = 1. Thequantity ψ

n(α) is an unknown generic function that describes the shape of the solitary wave and is expected to depend

upon the index n , which controls the stiffness of the potential. Because the solitary wave is localized in space, ψ n (α)should be necessarily zero for α → ∞( z → ∞for nite t , which represent a region that the solitary wave is yet toreach) and 1 for α → −∞(where grains have attained a new equilibrium position after the passage of the compressionproduced by the tsunami-like wave or the solitary wave). A function which respects this boundary condition and canonly take intermediate values between 0 and 1, can be always written as:

ψ n (α) =1

1 +exp( f n (α)), with f n (α) = ln

1ψ n (α) −1 . (3.2)

With this notation, the solitary wave function becomes:

u(α) = A2 (ϕn (α) +1), with ϕn (α) = −tanh

f n (α)2 . (3.3)





One can see from Eq. (3.3) that du / d z|t = −(1/ c)du / dt | z. Substituting Eq. (3.3) into Eq. (2.10) , we get, for t =0:

mc 2

na ( A/ 2)n−2 = {ϕn ( z −2 R) −ϕn ( z)}n−1 − {ϕn ( z) −ϕn ( z +2 R)}n−1

d2ϕn / d z2 ≡C 0(n), (3.4)

where the left-hand side of Eq. (3.4) is independent of z and the right-hand side is independent of m, a and A. Thus,

C 0 should be independent of z, m, a and A, which means that C 0 is a constant that depends only on n .Eq. (3.4) implies that ϕn ( z) is antisymmetric with respect to z = 0 or an arbitrary constant, which turns out to be

the center of solitary wave (recall that t was set to zero). This fact, combined with the asymptotic limits for ψ n ( z) andEq. (3.2) indicate that:

f n ( z) =∞

q=0C 2q+1(n) z2q+1. (3.5)

Since the function ϕn ( z) is independent of all quantities except n , knowledge of the coefcients C 0, C 1 , C 3 , C 5, . . . ,will completely solve the problem of pulse propagation for any system supporting this type of solitary wave. Inthe absence of a simple analytical approach for inferring C 0 and C 2q+1, one must resort to numerical methods forcomputing these coefcients. One approach is to numerically solve Eq. (2.10) and evaluate c and d pu/ du p

|α

=0,

where p is odd, as a function of the amplitude A and compare the results with the analytically obtained expressionsfor the same in terms of C 0 and C 2q+1. Direct comparison of the two results immediately reveals C 0 and C 2q+1. Thesecalculations can be pursued to any desired order. Recall that Eq. (3.4) implies that:

c =naC 0

m

1/ 2 A2

n−22

≡d 0 An−2

2 , (3.6)

which implies that the propagation velocity of the solitary wave scales with its amplitude except when n →2, theharmonic limit, where c becomes independent of amplitude, as expected. The higher-order derivatives behave as

vmax

=

du( z −ct )

dt t =0 = −c

du( z)

d z z=0 −

naC 0

m

C 1

2

A

2

n2

≡d 1 A

n2 , (3.7)

d3udt 3 α=0 =

nam

C 03/ 2

3C 3 −C 314

A2

3n−42

≡d 3 A(3n−4)/ 2 , (3.8)

and

d5udt 5 α=0 =

nam

C 05/ 2

60C 5 −15C 21 C 3 +C 512

A2

5n−82

≡d 5 A(5n−8)/ 2, etc. (3.9)

And by comparing the scaling properties of Eqs. (3.6)–(3.9) with the results obtained from numerically calculatedhigher-order derivatives, one can write C 0 and C 2q+1 in terms of d 0 , d 1 , d 3 , d 5 , etc. Here are the rst few relations,

C 0 =2n−2d 20 mna

, (3.10)

C 1 =4d 1d 0

, (3.11)

C 3 =23

d 3 +8d 31d 30

, (3.12)

C 5 =d 5 +80d 21 d 3 +384d 51

30d 50. (3.13)

Detailed calculations of up to order fth-order derivatives of displacement are shown in a 2001 work of Sen andManciu [ 68]. These calculations suggest that the series in Eq. (3.5) is convergent, albeit rather slowly. The agreementPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 3.1. φn ( z) and higher derivatives with respect to z are shown for the n = 5/ 2 case. The circles represent data obtained by integration of theequations of motion whereas the continuous line is obtained using the analytic solution discussed here.

Table 3.1The values of C 0 , C 1 , C 3 , C 5 are given as a function of n

n C 0 C 1 C 3 C 5

2.2 0.8709(6) 1.643(7) 0.08223(9) 0.0003257(8)2.35 0.6908(5) 2.3171(6) 0.2364(4) 0.0003407(4)2.5 0.85852(9) 2.3953(6) 0.26852(9) 0.006134(7)3.0 0.9445(1) 3.0168(2) 0.5971(1) 0.0376(4)4.0 1.3323(7) 3.5646(1) 1.331(4) 0.0676(3)5.0 2.0517(4) 3.79001(3) 2.177(5) 0.0665(1)

between the numerically obtained results and those obtained using our series solution with the coefcients in Table 3.1for displacement and its rst ve derivatives is highly impressive. These results are shown in Fig. 3.1 . The solutiongiven in Eq. (3.5) along with the contents of Table 3.1 allow the development of a solution to Eq. (2.10) that is moreaccurate than the one developed by Nesterenko (see Eq. (2.17) ). However, as we shall see, both of these solutionsare based on the assumption that the dynamical variables are continuous in z and in t . In a real granular system, thedynamical variables are continuous only in t . The consequences of the continuum approximation for z turn out tobe rather important, as we shall see when we discuss the problem of crossing of solitary waves in Section 5 and of quasi-equilibrium in Section 6.

It is important to address the issue of the spatial extent of a solitary wave. Although it has been claimed that thesolitary wave in a chain of elastic spheres is 5 grain diameters wide [ 57,58,68], the latest studies suggest that the extentis closest to 7 grain diameters although the movement of the grains at the edges of the wave may be hard to discernexperimentally [ 68]. It turns out that the difference between the velocity of the fastest moving grain in the solitarywave (which is at the center of the solitary wave) is 6 orders of magnitude more than the velocities of the grains at thePlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 3.2. The upper panel shows results in which the rst 7 grains have been perturbed at t = 0. The displacements and velocities of the 7 grainsare described in detail in [67]. In the lower panel, the formation of a solitary wave in time is shown when a δ function perturbation is initiated att =0.

leading edge and at the trailing edge of the wave [67]. The difference is 8 orders of magnitude when it comes to thegrain displacements from equilibrium, meaning that the grain in the center of the solitary wave is far more displaced

from equilibrium than those at the leading and trailing edges. The extent of this width brings to focus the many bodynature of the solitary wave problem and the importance of the collective effects in this problem. The analytical studiespresented above become unreliable when one is considering differences of the order of 10 −7 or so. One may ask herehow it is that we know that the solitary wave is indeed 7 grain diameters in spatial extent in a system with granularspheres? A logical worry would be that perhaps improvement in calculational precision will reveal that the solitarywave is wider. To address this concern, Sokolow, Bittle and Sen [ 67] decided to use the calculated grain displacementsand velocities of all 7 grains as initial conditions when starting out a mechanical energy perturbation at the edge of along granular chain. If there are more than 7 grains needed to dene the solitary wave pulse, it will stabilize into itsform over some space and time. Indeed when a delta function perturbation is used to perturb the system, studies haveconsistently shown [ 67,69] that the solitary wave takes some 10 or so grain diameters to form. The studies carried outwith the initial displacements and velocities of the rst 7 grains revealed that the solitary wave formed instantaneously

and hence conrmed that the solitary wave properties used must be quite reliable. The results are shown in Fig. 3.2 .The precision used in the actual calculations insured that the numbers used could be accurately tracked to 19 decimalplaces [67]. It should also be noted that as a solitary wave moves through a granular chain, its width uctuates slightlyas a function of time and space. An analytical understanding of such uctuations can only be obtained by carefullyexamining the spatial discreteness that is inherent in this problem and such analysis remains to be performed.

The width of the solitary wave, L(n) turns out to be sensitive to n and to the grain diameters. When one considersmonodisperse chains, n is the only parameter that controls the width of the solitary waves. When n → ∞, L(n) →1.When n →1, L(n) → ∞, i.e., the system no longer accommodates a solitary wave [ 70]. For the most common casewith n between 2.5 and 3, the solitary waves are about 7 to 9 grain diameters wide.

As we leave this section we note a particularly important feature of these solitary waves. Virial Theorem [ 3] allowsus to write an expression for the average kinetic energy of a system in terms of its average potential energy. By average

we mean averages in time. For potentials of the form in Eq. (2.1) , one nds that

12

n PE =KE , (3.14)

where KE and PE refer to the time-averaged kinetic and potential energies of the system, respectively. Eq. (3.14)implies that

KE =n

n +2, (3.15)

which means that for a system with n

=5/ 2, KE

=5/ 9

≈55.6%, a number that is in excellent agreement with

numerically obtained results [68].Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




3.2. Effect of gravitational loading on the granular chain

We now consider a granular medium that is placed under a constant force eld, such as gravity. One realization of such a system would be a vertically stacked granular chain. The constant eld makes the medium denser along thedirection of the applied force. This change of density of the medium due to the applied eld brings about important

changes in the propagation of the signal, in its speed and width, as well as to the grain velocity. This phenomenon isattributed to the effect of the constant force eld. We discuss the effect of gravity on the vertical granular chain withpower-law type contact force (as in the Hertz force) below.

An interesting feature in the propagation of a solitary wave in the vertical granular chain with power-law typecontact force is that the power-law behavior in depth or time appears in most propagation characteristics [71,72].This power-law behavior is a combined effect of gravity and the nonlinear power-law contact force. As we mentionedabove, the effect of gravity is rather simple, gravitational loading just changes the density of medium. But the role of nonlinearity turns out to be complicated.

One can change the effective strength of the nonlinearity by changing the strength of the initial impulse exertedat the top of the vertical granular chain. A detailed discussion of this issue can be found in Refs. [ 55,56]. Therefore,one can roughly discriminate between the strongly and weakly nonlinear regimes depending on the strength of theinitial impulse. As expected, under strong initial impulse the effect of gravity will be negligible and the solitary wavewill be similar to that seen in the horizontal chain. Thus, the effect of gravity will appear as a second-order effect ina strongly nonlinear chain. On the other hand, in the weakly nonlinear chain, the effect of gravity will appear in theleading order. We will show below that the vertical granular chain can be mapped onto a linear horizontal chain withdifferent elasticity between neighboring grains with the elasticity depending on the depth. This is the effect of gravityin the weakly nonlinear regime.

Our goal is to develop an analytical theory to capture the effect of gravity and the role of nonlinearity in a granularchain under gravitational loading. However, let us rst study the problem via direct solution of Newton’s equationswhen the gravitationally loaded chain is subjected to weak and to strong impulses at its “surface” at t = 0. A weak impulse yields a weakly nonlinear oscillatory signal, while a strong impulse yields a solitary wave-like signal. Thevarious power-law behaviors are shown by simulational analyses below. Power-law behavior in depth appears in bothoscillatory and solitary signals, but the characteristics of power-law exponents are quite different depending on theregime to which the signal belongs. In what follows (Section 3.2.1), we discuss the role of nonlinearity and the reasonwhy the power-law exponents are classied into two groups. Then we derive power-law exponents analytically for thesignal in the weakly nonlinear regime in Section 3.2.2 . The power-law behavior seen in this regime can be explainedby a linear dispersive wave equation, which maps the vertical granular chain onto the horizontal granular chain withvarying elasticity between the neighboring grains. In closing, we present a short discussion on the signal in the stronglynonlinear regime.

To get the simulation results, we focus on the motion of the grains. The equation of motion of the grain at the i thvertical position zi from the top is

md2 zi (t )

dt 2 =η (∆ + zi−1(t ) − zi (t ))n−1 −(∆ + zi (t ) − zi+1(t ))n−1 +mg , (3.16)

where η = na . We do not consider plastic deformation and viscoelastic dissipation in treating Eq. (3.16) . Forthe Hertzian chain n = 5/ 2. To perform numerical simulations for Eq. (3.16) , we choose a vertical chain of N grains, where N ranges between 1000 and 5000 according to our need. We choose 10 −5 m, 2.36 ×10−5 kg, and1.0102 ×10−3 s as the units of distance, mass, and time, respectively. These units give the gravitational accelerationas g =1 [55]. We set the grain diameter 100, mass 1, and the constant η of Eq. (3.16) as 5657 in the particle dynamicssimulations [ 55]. The equilibrium condition

mgi =ηδ n−1i , (3.17)

has been used for the i th contact of the vertical chain. Using the third-order Gear predictor–corrector algorithm [ 73] asa calculation tool, we perform numerical simulations for arbitrary n including n =5/ 2. We choose various strengthsof impulse for our study. Fig. 3.3 shows the snap shots of three typical types of grain velocity behavior propagatingdown the vertical chain of Hertzian contacts. Fig. 3.3 (a), which is obtained for the initial impulse velocity v I =0.001,Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 3.3. Snap shots of typical modes of propagating signals under (a) weak impulse, v I =0.001, (b) intermediate impulse, v I =1, and (c) strongimpulse, v I =1000, in the vertical granular chain with Hertzian contact. We set n = p +1.

Fig. 3.4. Upper two lines show (a) the leading and (b) the second to leading peaks of grain velocity versus depth drawn in log–log scale. Lowertwo lines show (c) the leading and (d) the second to leading peaks of displacement versus depth drawn in log–log scale. We set n = 5/ 2 = p +1

and v I =0.001.

shows typical grain velocity signals appearing in the weakly nonlinear regime, Fig. 3.3(b) obtained for v I = 1 is thesignal of intermediate regime, and Fig. 3.3(c) for v I = 1000 shows those of the strongly nonlinear regime in whichthe tail disappears.

The common features of propagation characteristics shown in Fig. 3.3 are increasing signal speed, decreasinggrain velocity, and increasing signal width as the pulse goes down the chain. The straight lines drawn in log–log scalein Fig. 3.4 certify that the depth-dependent behavior of the leading and second leading peaks of grain velocity anddisplacement are all power-laws. According to our analyses shown in Fig. 3.4 , the explicit expressions for the depth-dependent behavior of leading amplitudes of grain displacement and velocity are given by Amax (h)

∝h−0.0835±0.0003

and vmax (h)∝

h−0.2500±0.0001 for v I = 0.001 and n = 5/ 2. We obtain other depth-dependent power-laws showingthe dispersiveness of the signal. One of them is the number of particles participating at the leading part of velocitysignal, N (h) , which describes the length scale of the signal. The other is the elapsed time to reach the maximumamplitude, T max (h ), which describes the time scale of the signal. The power-law behaviors of these quantities witherror bounds are, for wavelength λ which increases with depth, λ

∝N ∝

h0.338±0.004 and T max (h )∝

h 0.170±0.002 .These numerical data well-satisfy the relations for oscillating signals, such as T = A/v and λ = T v p , where v p isthe phase velocity of the signal, even though the oscillation is not perfectly periodic. The phase velocity v p∝

h1/ 6 isthe well-known phase velocity of the signal propagating down the medium of Hertzian contacts [ 55,74].

One noticeable fact in our simulation is that nonlinearity leads to different classes of power-law behaviors. Weshow this by the plot of power-law exponents of grain velocity and signal speed v p for a wide range of impulsestrength from v I = 10−3 to v I = 103 for n = 2.2, 2.5, and 3.0 in Fig. 3.5 . The logarithmic scale has been used onlyfor the abscissa of the graph in Fig. 3.5 . One can see a clear trend that as v I decreases to less than 0.1, the exponentsapproach saturation values for given n , and the exponents vanish when v I is larger than 10 3 . The transition regionexists in between v I = 0.1 and v I = 103. These features are common to any value of n > 2. This indicates thatPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 3.5. Plots of power-law exponents of grain velocity versus v I for several n’s are shown in the left plot. Plots of power-law exponents of signalspeed versus v I for several n’s are shown on the right plot. Log scale has been applied only to the abscissa. We set n = p +1.

there is a region in which the characteristics of signal propagation are independent of the strength of impulse v I forany n > 2. We call this region the weakly nonlinear regime in which the signal is oscillatory. A typical form of thesignal in this regime was shown in Fig. 3.3 (a). The power-law behaviors of the signal in the at region of Fig. 3.5will be explained analytically in the next section using the linear dispersive wave equation as a limiting case of weak nonlinearity. Another region in which the exponent changes rapidly may be named as the strongly nonlinear regimewhere the signal is like that of a solitary wave as shown in Fig. 3.3(c). There is an intermediate regime where a weak oscillatory part remains in the tail of the signal as shown in Fig. 3.3 (b). The signal in the intermediate regime is acombination of nonlinear solitary and linear oscillatory waves.

3.2.1. Role of gravity and nonlinearityWe saw interesting phenomena driven by nonlinearity in the previous section. Investigating the role of nonlinearity

in any physical system is an attractive task. As mentioned in the beginning, we learn about the linear wave in a stringor the linear sound wave in air early in physics. The properties of these linear waves do not depend on the strength of external perturbation but depend only on the medium through which they propagate. The propagation of a nonlinearwave, however, depends on the strength of the external perturbation. These observations tell us that nonlinearity isresponsible for the dependence of propagation characteristics of the nonlinear wave on the strength of the externalperturbation. Fig. 3.5 clearly show that there are two regimes discriminated by the strength of the nonlinearity. In oneregime, i.e., weakly nonlinear regime, there is no dependence on the strength of external perturbation, but in the otherregime, i.e., strongly nonlinear regime, a strong dependence occurs.

In this section, we will discuss the role of nonlinearity in the propagation characteristics of the signal in a verticalgranular chain. The discrimination by nonlinearity may be readily understood by analyzing the equation of motionin both regimes. For this purpose, we introduce a new variable ψ i , denoting the displacement of i th grain from

equilibrium, dened by ψ i = zi − i∆

+il=1(mgl /η)

1/( n

−1)

. Then, Eq. (3.16) is transformed into the followingform,

m ψ i =ηmgi

η

1/( n−1)

+(ψ i−1 −ψ i )

n−1

−ηmg (i +1)

η

1/( n−1)

+(ψ i −ψ i+1)

n−1

+mg . (3.18)

One may expand Eq. (3.18) for the strongly nonlinear regime characterized by |ψ i−1 −ψ i | >> ( mg i/η) 1/( n−1) . Thisregime can be achieved by imparting a strong impulse such that the condition |ψ i−1 −ψ i | >> ( mg i/η) 1/( n−1) issatised. The weakly nonlinear regime is characterized by |ψ i−1 −ψ i | << ( mg i/η) 1/( n−1) , which can be achievedby a weak impulse.

In the strongly nonlinear regime, the continuum expression of Eq. (3.18) is written as

m ¨ψ i =η[(ψ i−1 −ψ i )n−1 −(ψ i −ψ i+1)n−1] +η( n −1)[ Di (ψ i−1 −ψ i )n−2 − Di+1(ψ i −ψ i+1)n−2], (3.19)Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




where D i = (mgi /η) 1/( n−1) denotes the grain overlap at the i th contact. The gravity term can be neglected in thehighly nonlinear regime, since the gravity effect appears in the coefcient D i . The scaling analysis introduced inSection 2 tells us that different order of ψ i in the left and right side of Eq. (3.19) implies v I -dependence of signalcharacteristics.

If we set Ai+1 = (ψ i −ψ i+1)n−1 and Ai = (ψ i−1 −ψ i )n−1 and use the relations

∂ Ai+1

∂ψ i+1 = −(n −1)(ψ i −ψ i+1)n−2 and∂ Ai

∂ψ i = −(n −1)(ψ i−1 −ψ i )n−2, (3.20)

Eq. (3.19) is written as

m ψ i = −η( Ai+1 − Ai ) +η Di+1∂ Ai+1

∂ψ i+1 − Di∂ Ai

∂ψ i. (3.21)

This equation is rewritten in the continuum form as,

ρ ψ( h) = −η∂ A∂h +η

∂∂h

D(h )∂ A∂ψ = −η 1 −

∂∂h

D(h)∂h∂ψ

∂ A∂h +η D(h)

∂h∂ψ

∂ 2 A∂h 2 , (3.22)

where ρ =m/ ∆ denoting the density, A = (−∂ψ/∂ h)n−1, and D (h) = (ρ gh /η) 1/( n−1) .The leading term in Eq. (3.21) , is strongly nonlinear giving rise to the solitary wave in the horizontal granular chain.

The dispersion effect due to gravity comes into the equation through the second term, where the parameter describingdispersion is D (h) . The strongly nonlinear regime is rather difcult to analyze. But through a rather simple analysis,one may understand the effect of gravity, which causes power-law increase of signal speed and width and decrease of signal height.

The rst term on the right-hand side of Eq. (3.21) is just the one describing a perfect solitary wave in the horizontalchain [51–53 ,58] and the second term is responsible for changing speed, height, and width of signal due to gravityas signal goes down. We re-express Eq. (3.21) into the form of Eq. (3.22) to separate the effects of damping anddispersion of a solitary wave. Both are governed by D (h) and the nonlinearity factor ∂h /∂ψ . The latter can berewritten as (∂ h /∂ t )(∂ t /∂ψ) which is equivalent to vs /v I or vs /v grain by dimensional analysis. Therefore, the signal

characteristics approach those of the solitary wave created in the horizontal chain as the strength of nonlinearity, i.e.,v I , increases in the strongly nonlinear regime. One can see this behavior in Fig. 3.5 .

The role of each term of Eq. (3.22) cannot be analyzed independently, since the equation is nonlinear. Since fullyanalytical treatment of Eq. (3.22) is not simple, we just show that the force term, in the strongly nonlinear regime, canbe divided into a gravity-independent part and a gravity-dependent part as shown in Eq. (3.21) and also divided intodamping and dispersion parts as shown in Eq. (3.22) . Therefore, one may understand that a solitary wave created bythe gravity-independent force −η (∂ A/∂ h) of Eq. (3.21) changes its speed, height, and width due to both gravity andnonlinearity. Fig. 3.3 explicitly shows the change of signal as v I increases.

The continuum form of Eq. (3.18) for the weakly nonlinear regime is written as

ρ ψ( h , t )

=

∂

∂h

τ ( h)∂ψ

∂h −

∂

∂h

N (h )τ ( h)2 ∂ψ

∂h

2

, (3.23)

where τ ( h ) = τ 1(h / ∆ )1−[1/( n−1)] is the tension at depth h , τ 1 = µ 1∆ is the tension at the rst contact, and N (h) = (1/ 2ρ gh )[1 − {1/( n − 1)}]. The leading term in Eq. (3.23) is linear giving rise to dispersion. Thenonlinear effect comes into the equation through the second term. Therefore, the parameter describing the strengthof nonlinearity is N (h ). Since the primary behavior in the weakly nonlinear regime is linear, one may understand theindependence of the power-law exponents of the propagation speed from the strength of the initial perturbation asshown in Fig. 3.5 . This independence is one of the characteristics of a linear wave.

The granular chain under gravitational loading admits various features of linear and nonlinear wave propagation.Such a chain also is strongly affected by the application of a constant force eld, such as a gravitational eld. Theconstant force eld only changes the density of the medium. This effect appears as a dispersion in the width of the signal. This system also shows both features of linear waves, i.e., independence from the perturbation strength,and nonlinear waves, i.e., dependence on the perturbation strength, thus signifying the simultaneous existence of Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




two different classes of power-law behaviors. The most interesting feature occurring in this system is the power-lawbehavior in depth or time. We will discuss this behavior analytically in the limiting case of the weakly nonlinearregime in the next section.

3.2.2. Analytical studies for the weakly nonlinear regime

The reader may nd it interesting to note that the absolute value of the power-law exponent of signal speed increasesand approaches a saturation value for a given exponent of the power-law contact force as the strength of the initialimpulse decreases as shown in Fig. 3.5 . The propagation mode in this saturated region, which we call the weaklynonlinear regime, is oscillatory. Interesting properties of the signal in this regime are that all characteristics, such asoscillation amplitude, frequency, and wavelength, of the signal follow certain power-laws as function of depth. Thesebehaviors can be derived at the limit of weak nonlinearity in which the equation of motion becomes a linear dispersivewave equation. For this purpose, we use the variable ψ i introduced above and set z0 = ψ 0 = 0. For the weaklynonlinear regime, Eq. (3.18) is transformed into

m ψ i = −µ i (ψ i −ψ i−1) +µ i+1(ψ i+1 −ψ i ), (3.24)

where µ i

=mpg (η/ mg )1/ pi 1−(1/ p) , where p

=n

−1, is the force constant of i th contact of the linear horizontal

chain. We drop weakly nonlinear terms in the expansion, since their effects are secondary. The linear equation, Eq.(3.24) , describes a horizontal granular chain with varying force constants in which the gravity effect is contained.Both left and right sides of Eq. (3.24) are linear in ψ i . Therefore, the scaling analysis introduced earlier tells us thatthere is no dependence on the strength of initial impulse v I .

To analyze this linear difference equation analytically, we rewrite Eq. (3.24) in the continuum limit, i.e.,

ρ ψ( h , t ) =∂

∂hτ ( h)

∂ψ( h , t )∂h

, (3.25)

when the intergrain distance →0, where τ ( h) = τ 1(h / ∆ )1−[1/( n−1)] denotes the depth-dependent tension, andρ =m/ ∆ and τ 1 =µ 1∆ are the density and the tension of chain at the rst contact, respectively. We set c1 =√ τ 1/ρ ,which is the well-known speed of wave in the string of tension τ 1 and density ρ .

Since Eq. (3.25) is a linear differential equation, we can apply Fourier analysis to this equation. Then we have thefollowing dispersion relation for the complex wave number k (ω) =k R +ik I :

ω( k ) =k Rτ ( h )

ρ1 +

τ 2

4τ 2k 2 R

1/ 2

. (3.26)

and k I = −τ / 2τ . From Eq. (3.26) , we obtain the phase and group velocities, respectively, as follows:

v p =ωk ≈

τ ( h )ρ

1 +τ 2

8τ 2k 2 R

1/ 2

, (3.27)

and

vg =dωdk ≈

τ ( h )ρ

1 −τ 2

8τ 2k 2 R

1/ 2

. (3.28)

Here and in what follows k means k R. The difference between v p and vg indicates that the wave is dispersive, and thefact k I =0 normally means the wave is diffusive [75].

Treating dispersive and diffusive waves are not simple. However, since τ /τ ∝

h−1 and therefore e −k i h is constantin h , the envelope function of the wave is not exponentially damping, so it is not diffusive for this special case.Therefore, the general solution of the linear equation (3.25) is written as

ψ( h , t ) = ω A(ω) ei(kh

−ω t )

, (3.29)Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




and the h -dependence of the envelope of the function ψ( h , t ) is solely given by the coefcient A(ω) . Now we solveEq. (3.25) again using ψ ς (h , t ) = u ζ (h) eiζ t as a normal mode solution, where u ς (h ) = A(ω) eikh and ω

∝ς . Then

u ς (h ) satises

d2

dh2 u ς (h )

+

1 − [1/( n −1)]h

d

dh

u ς (h)

+

ς 2

h1

−[1/( n

−1)

]u ς (h)

=0, (3.30)

which is a type of Bessel’s differential equations [ 76]. A solution of this equation that propagates in the positiveh -direction is given by the Hankel function [ 76],

u ς (h ) =h ξ H (1)ν (θ h γ ), (3.31)

where ξ =1/ [2(n −1)], γ = (1/ 2) +ξ = 12 1 + 1

n−1 , θ = ς/γ , ν =ξ/γ = 1n .

The asymptotic form of Eq. (3.31) at large h for a xed h is

u ς (h ) ≈2

πθh ξ−(γ / 2) ei[θ hγ −π

2 ν−π4 ], (3.32)

and the displacement is written as

ψ ς (h , t ) ≈ h ξ−(γ / 2) ei[ς γ hγ −ς t ]. (3.33)

Therefore, the depth-dependence of the coefcient A(ω) of the displacement signal is

A[ω ( h)] =h ξ−γ 2 =h−1

4 1− 1n−1 , (3.34)

for all ω .To obtain other depth-dependent properties of the signal, we need more information on the signal. This is given

by the asymptotic analysis for the wave of linear equation. The asymptotic form of the solution of the general linearequation is given by the saddle-point method or the steepest descent method. The result is presented as in [ 75],

ψ( h , t )∼=

√ 2π A(ω s ) exp[ik s h −iω( k s )t −α]√ t |ω (k s )|

, (3.35)

when ω (k ) = 0, where k s means the wave number at the saddle-point and α = (iπ/ 4)sgn ω (k s ) . When there aremany saddle-points, the asymptotic solution must be the sum over all the saddle-points. Here we encounter a singlesaddle-point. Therefore, the amplitude of the general solution of the linear wave equation in the asymptotic regime isgiven by A(ω s ){t |ω (k s )|}−1/ 2 where t = h /v g . Since we showed that A(ω s ) exhausts the depth-dependence of theamplitude of displacement ψ( h, t ) , {t |ω (k s )|}−1/ 2 must be depth-independent. This condition gives information onwavenumber k . Differentiating Eq. (3.26) once more, we get information on k ,

t |ω (k s )|∝hτ 2

τ 2k −3∝

h 0. (3.36)

This relation provides one of the key conclusions,

k ∝

h−1/ 3, (3.37)

in obtaining power-law behavior of the signal.We have already obtained the power-law behavior in depth for the amplitude of displacement in Eq. (3.34) and the

wave number in Eq. (3.37) . Using Eqs. (3.34) and (3.37) and signal velocity v p ∝h

12 1− 1

n−1 from Eq. (3.33) , weobtain the depth-dependent behavior of grain velocity and its oscillation frequency using the relation of linear wave,ω ( h) = k (h)v p(h ) , and the relation for oscillating signal, 1 / ω ( h)

∝A(h)/v( h ) . They are given by

v ( h)∝

h−14

13 + 1

n−1 , (3.38)

ω ( h)∝

h16 −

12(n−1) . (3.39)





Fig. 3.6. Dependence of power-law exponents of grain velocity (square) and displacement (circle) on the exponent of the contact force ( p =n −1).Solid and dashed lines are theoretical results. Data are obtained from the leading peaks of each signal.

The characteristic time of oscillation which is expressed by the period is given by the inverse of frequency or the ratioof displacement to grain velocity, i.e.,

T (h) = A(h)/v( h )∝

1/ω( h)∝

h−16 + 1

2(n−1) . (3.40)

The characteristic length of oscillation, on the other hand, which is expressed by wavelength is given by multiplyingT (h ) by phase velocity, i.e.,

λ( h) =T (h)v p(h )∝

h13 . (3.41)

The depth-dependent power-law behaviors obtained numerically for the weakly nonlinear regime in the last sectionagree well with Eqs. (3.38)–(3.41) obtained analytically. This analysis explains the damping and dispersive behavior

due to gravity for the weakly nonlinear regime in the vertical granular chain. To check the theoretical prediction givenabove, we obtain peak values of displacement and grain velocity signal for other values of n and compare with thetheory based results in Fig. 3.6 . One can see very nice ts to the theoretical curves. For large values of n , the deviationfrom theory occurs especially in grain velocity.

This is understandable because nonlinearity becomes stronger as n increases and grain velocity contains morederivative than displacement. A rather interesting fact is that the exponent of the characteristic length of the signal,Eq. (3.41) , does not depend on n .

3.2.3. SummaryWe saw above that the power-law behavior of the propagating signal in the gravitationally loaded granular chain is

generic for the whole range of the strength of impulse. For the strongly nonlinear regime in which the impulse is rather

strong, the simulation results show that power-law exponents depend on the strength of impulse, in other words, thestrength of nonlinearity of the system. The signal becomes more solitary wave-like as the impulse increases. But thisquasi-solitary signal changes its speed, height, and width as it goes into the gravitationally loaded chain. The grainvelocity also changes. Interestingly, all these changing features follow certain power-laws in depth. Absolute valuesof the power-law exponents approach zero as impulse increases. This fact implies that the role of gravity becomesnegligible under a strong impulse. One can also understand this phenomenon from Eq. (3.22) where one can see thatthe gravity factor D (h) is always coupled with the nonlinearity factor ∂h/∂ψ , which is inversely proportional tothe impulse velocity v I . This coupling tells us that increasing nonlinearity diminishes the gravity effect. It also tellsus that the signal becomes more linear under higher pressure. We separated the role of force into three regimes inEq. (3.22) , i.e., making the signal solitary, damping, and dispersing. One may guess the role of each term from itsform.

For a rather weak impulse regime, which we call the weakly nonlinear regime, the power-law exponent of thegrain velocity approaches saturation, i.e., it does not depend on the strength of nonlinearity for a given contactPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




force. The signal in this region is oscillatory. The equation of motion of the displacement of a grain in the limitof weak nonlinearity can be transformed into a linear differential equation to which Fourier analysis is applicable. Theasymptotic behavior of the linear differential equation is given by the saddle-point method. This provides independentinformation for the wave number of the signal. The normal mode analysis combined with this information gives riseto all the other needed information for the power-law behavior associated with signal propagation in the asymptotic

regime. The characteristics of the signal described by the linear dispersive equation can be applied to understand signalbehavior in the weakly nonlinear regime. The power-law exponents given by the analytical approach agree well withthe simulation data. The equation of motion for the weakly nonlinear regime is equivalent to those of a nonuniformtransmission line and a nonuniform string. Therefore, one may apply the analysis of this work to other areas. Thebehavior of the solitary wave under gravity or some other constant force eld may be useful in applied sciences.

4. Role of randomness and dissipation in amplitude attenuation

Thus far we have assumed that all the grains in the alignment are spherical and identical. We have also assumedthat there is no energy dissipated into the internal degrees of freedom of the grains, which of course is an unrealisticassumption. It is natural to ask how the propagation of solitary waves would be affected if the radii of the spherical

grains were to vary randomly while their centers still sit along an axis or if there were some variations in the nature of the granular contacts as the solitary wave propagates through the alignment.We summarize the propagation characteristics of solitary waves in granular alignments in the presence of random

variation of masses in the alignment and address the effects of restitutive losses in this section. Restitutive losses occurbecause the grains are macroscopic objects that have a molecular make up and because part of the energy transmittedthrough each grain is inevitably absorbed by the molecules that make up the grains. This loss per grain is obviouslymaterial dependent. For example lead is a more restitutive medium than stainless steel and hence will possess a highervalue of restitution than stainless steel. There is no simple way to estimate the effects of restitutive losses in a granularchain. These losses are typically estimated on the basis of experimental observations. In what follows we developthe simplest possible ways to incorporate restitutive losses and also to present calculations that capture the effect of random variations of mass distributions in a chain.

Let us consider a chain in which part of the chain has monodisperse grains and another part has grains that varyrandomly in their masses. The two parts are in contact. An impulse is initiated in the segment of the chain comprisingof monodisperse grains. This impulse propagates as a solitary wave, as expected. We now introduce a second segmentthat is attached to the monodisperse chain. In this segment, we allow the grains to have nite restitution coefcientsw dened following the famous work of Walton and Braun [ 77] as

F unloading

F loading =1 −w, (4.1)

where w in Eq. (4.1) describes the behavior that grain contacts decompress with less force than the force associatedwith their compression. When w = 0, the system is perfectly elastic. Our numerical studies [ 78] establish that theinitial kinetic energy E 0 of a propagating impulse attenuates as a function of its position with respect to the startingpoint in the presence of restitution as E ( z)

=E 0 exp(

−α E (w) z), where z denotes the distance traveled by the solitary

wave from its initiation point. Observe that the width of the solitary wave is assumed to remain unaffected due torestitution. This is expected in view of the fact that the width of the solitary wave depends only upon the Hertz lawindex n . If one assumes that ξ represents the fractional loss in kinetic energy as it is transmitted from one grain to thenext in a chain with restitutive loss at granular contacts, one nds that the energy at some position z must be

E ( z) = E 0(1 −ξ ) z = E 0 exp[ln(1 −ξ ) z] ≈E 0 exp(−ξ z), (4.2)

where the last approximation is valid when ξ << 1. To relate w to ξ , it is necessary to calculate the work done bythe Hertz force in moving the next grain when there is no restitution and when there is restitution and obtaining thedifference between the two energies. This calculation is reported elsewhere [ 79] and reveals that for small values of restitution and for n =5/ 2, ξ ≈0.38w . This approximation assumes that all the energy is transmitted from one grainto the next. In reality, the propagating energy bundle possesses a width that is a few grain diameters wide. The above(hard sphere) approximation is hence most meaningful as n → ∞and least valid when n →2.Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 4.1. Ratio between maximum energy and incident energy for a chain with uniformly randomly distributed masses for different degrees of randomness ε .

It turns out that the presence of polydispersity, i.e., variation in masses of grains also leads to an exponentialdecay of the kinetic energy of the propagating energy bundle [ 79]. Let us consider a system in which the masses aredistributed as a function of position z as

m( z) =m0(1 +r ( z)ε), (4.3)

where m0 is the mean mass, r is a random number that is uniformly distributed between −1 and 1 and ε is a xedparameter between 0 and 1.The data for different degrees of randomness are shown in Fig. 4.1 .

We nd that the maximum kinetic energy of each grain in a chain with a random distribution of masses dependsalso on the mass of that grain, which of course has random values. This is the reason why the points representingmaximum kinetic energy are distributed around the average exponential decay in Fig. 4.1 . One can also measure

the deviation σ E of these points from the average exponential decay of the energy that can be characterized by E ( z) = E 0 exp(−α E z) by using the data in Fig. 4.1 . Our extensive simulations indicate that the decay coefcient α E and the statistical dispersion σ E depend on the randomness ε , but are independent of the region of measurement. Thisimplies in turn that pulse propagation remains qualitatively unchanged over the whole interval studied, and again canbe reliably described by simple measurable parameters. In [ 79], we have described the values of α E and σ E obtainedfrom simulated data for different randomness and for n ranging from 2.2 to 5.0. While the statistical dispersion isapproximately a linear function of ε , the quantity α E ∝

ε β , where for small enough ε , β ≈ 2. A simple proof of this result can be constructed in the ballistic approximation by using mechanical energy and momentum conservationequations with the condition p

∼An / 2 (see Eq. (3.7)). The derivation is sketched in Appendix A of [ 79].

In summary, polydispersity and randomness lead to the decay of the amplitude of a propagating solitary wavewithout affecting its width. The energy chipped away by polydispersity and/or randomness in masses remains in thesystem as solitary waves of very small amplitude but of identical width. Small variations in the magnitude of n alsocontribute to the dispersion of the propagating solitary wave. More sophisticated dissipation schemes that have provenuseful for comparing simulations with experiment have been reported by Brilliantov and coworkers [ 80].

5. Collision of solitary waves — secondary solitary waves

In our discussions on solitary waves, we have touched upon how these waves form [ 67], what their physicalproperties are like [ 51–54 ,68], how they are affected by the presence of a uniform external eld (such as gravity) [71,72] and the inuence of polydispersity of the grains and of restitution on their propagation [77]. What we have learntfrom the earlier sections is that these waves are robust – it is not easy to destroy them – and perhaps this is also thereason why there are so many researchers who have studied these solitary waves theoretically, computationally andexperimentally. Let us now try to break these solitary waves to gain an understanding of how these waves interact witheach other, if at all, and with the boundaries.Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 5.1. We show a schematic setup of the numerical experiment to probe the collision of two identical solitary waves that start off at the ends of a monodisperse chain made of N grains and propagate through the system at equal speeds to collide in the center of the chain.

Let us focus on the problem of collision between solitary waves in a monodisperse granular alignment. Here, thedynamics of each grain satises the equation of motion (see Eq. (2.10) ). However, because the medium is discrete,there is a distinct time lag between the transfer of energy from one grain to the next (in the continuum limit, this timelag goes to zero). Thus, the physics of collision of two identical and opposite propagating solitary waves could differdepending on whether one considers a continuum system or a discrete system. It is well known that in continuummedia, solitary waves completely pass through each other without any interactions [5,6]. It is natural to ask whetherthis still is the case in discrete granular alignments?

If one inserts a hard boundary along the chain and a propagating solitary wave is incident on the hard boundary, itis easy to see that the time lag in energy transfer from one grain to the next will inevitably affect the collision of thesolitary wave with the boundary. The solitary wave will be expected to reverse its direction, but how does this reversalprocess actually happen? Is the original solitary wave recovered post collision? As we shall see below, the answer is“no”. The original solitary wave is lost in the collision with a boundary or with any other solitary wave for that matter.The collision leads to the formation of new solitary waves and while the new waves are related to the wave that existedbefore the collision, it is not easy to predict the outcome of the collision. Indeed, there is no theoretical formulation of the solitary wave collision problem to date.

Part of the difculty lies in the fact that in a discrete medium, a solitary wave collision problem is truly a many bodydynamical problem and in the absence of a closed form solution of the original Newton’s equations of motion for allthe grains in the system, it is difcult to construct a fully analytic picture. Our studies have been conned to collisionsof identical solitary waves in the center of the chain. Experimental studies carried out by Job and coworkers [ 81] havefocused on the process of collision of a single solitary wave with walls of various hardness and have reported on theprocess of break down of solitary waves at the walls.

This section is organized as follows: rst we briey mention the numerical procedure used in our simulations tosolve the equations of motion for all the grains in the chain. Next, we discuss the interaction of solitary waves and theformation of the so-called “secondary” or “baby” solitary waves.

We start with two identical impulses directed into the chain at the initial time t = 0 by assigning v1 = v0 andv N = −v0 , with v0 > 0, at the two ends of a chain made of N identical grains as shown in Figs. 5.1 and 5.2. Theother grains are kept at rest at t = 0. The kinetic energy in these impulses at t = 0 leads eventually to the formationof identical and opposite propagating solitary waves. Recall we have mentioned in Section 3 that for spherical beads,where n =5/ 2, the system takes about 10 grain diameters to form a solitary wave (see Fig. 3.2 ). The waves are about7 grain diameters wide. Thus, a collision of two waves involve a signicant number of grains as alluded to aboveand presents a challenging nonlinear dynamical problem involving many interacting grains. This problem is not yetamenable to a fully analytical treatment.

The system dynamics is obtained by time integration of the coupled Newtonian equations of motion via thethird-order Gear algorithm as before [ 73]. We choose 10 −5 m, 2.36 × 10−5 kg and 1 .0102 × 10−3 s as the unitsof distance, mass and time, respectively. The integration time step used was chosen to be small enough to resolvethe nest details (for spheres, we used d t = 5 ×10−7 , for steeper potentials, i.e., with n > 5/ 2, ner integrationtime steps were typically needed). The grain diameter is set to 100, i.e., 1 mm, 2 R = 1 and a = 5657, a value of 4.14 ×10−7 N/ m3/ 2 , which is in the range of elasticity of silicate materials. In the cases here, we use an even (odd)chain made of N =500(499) and v0 =5 ×10−4.

By using the method described above, one can numerically investigate the collision of two identical solitary waves.Fig. 5.2 shows the velocity proles of the two solitary waves in incoming trajectories prior to the collision. The widthsof the solitary waves are visible. In our study, the solitary waves approach each other and cross at the edge of twocentral grains in a chain with even N and at the center of the central grain in a chain with odd N . After the crossingoccurs, the solitary waves continue until they reach the opposite ends of the chain.Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 5.2. Two solitary waves are shown in a long granular chain at an early time, before the collision has taken place. The velocity of the grains asa function of position inside each wave is shown in the inside window. The vertical axis in the inside window is the same as in the vertical axis of the main plot.

Fig. 5.3. Snapshots of the two solitary waves shortly after collision. Observe the formation of multiple secondary solitary waves after the maincollision event. The need to form these waves is predicated at least in part by the discreteness of space in the chain (because the individual grainshave widths) and by the structure of the equations of motion.

The detailed dynamics of the crossing event progresses as follows. During the collision all the grains in the chainmove out of their original positions. Only the central grain in an odd chain remains static at the time of collision. In theeven chain all the participating grains, including the one in the region where the crossing occurs, move out from theirequilibrium positions during the complete interval of time. Therefore, immediately after the collision occurs, there ismore kinetic energy available in the even case as opposed to the odd case. This means that solitary wave collisionsresult in different outcomes depending upon whether one considers an even chain or an odd chain.

During the collision, solitary waves break down and reform; which means that during a short interval of timethe proles of the original solitary waves are lost and new solitary waves form that are identical to the precollisionwaves except that they carry less energy. In addition, several secondary solitary waves, each of which carry only aminiscule fraction of the energy of the original solitary wave are born [ 82,83]. In Fig. 5.3 one can see the times whensolitary waves form immediately after the actual collision event. In earlier studies it has been reported that solitarywaves can form at signicant intervals of time after the collision event as well [ 83]. To make sense of what we nd,we observe that the formation of a solitary wave must require seven grains to collaborate and such events must notbe very common. Thus, a signicant time after the collision, it is possible that post-collision solitary waves may beborn [83,84].

In Fig. 5.4 we see the solitary waves created as a by-product of the collision. Using higher resolution to observethe kinetic energies of the individual grains, one can nd that multiple solitary waves of progressively decreasingamplitudes form in the vicinity of the collision point as a by-product of the collision. These less energetic solitarywaves formed as a by-product of solitary wave collisions also move slower (recall Eq. (3.7)) as expected of solitarywaves carrying small amounts of energy [ 82–84 ]. The energies carried by the secondary solitary waves can be severalPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 5.4. The snapshot taken after the solitary waves have collided shows in excess of 4 secondary solitary waves. Many of these waves are sosmall that they are difcult to discern and even plot using a linear scale on the vertical axis.

orders of magnitude smaller than that of the parent solitary waves. However, energies carried by secondary wavesdepend upon the nature of the collision that created them — such as whether the collisions occurred at the centerof a grain or at the edge of two grains or anywhere in between or perhaps whether the collision was against a wall.Collisions of solitary waves with walls of variable softness have been experimentally probed by Job and collaboratorsand secondary solitary waves have been observed [ 81].

To see why these secondary solitary waves must form, we observe that our system cannot transport energy viasound waves. Therefore all energy must eventually be bundled into solitary waves. In Fig. 5.5 , we show snapshotsfrom extensive dynamical simulation runs that analyze how the grain velocities are distributed in space and time whentwo identical and opposite propagating solitary waves collide in an alignment with an even number of grains.

6. The quasi-equilibrium phase

We now look into the consequences of formation of secondary solitary waves in granular chains that are held withinboundaries. We will argue that the existence of secondary solitary waves implies the likely existence of some form of equilibrium-like state. But before we develop the arguments and present the calculations it is important to recall what

we mean by the concept of equilibrium.Solids, liquids and gases can often be found in the equilibrium state. Consider water in a bottle, for instance. If we

slightly perturb the bottle, the water quickly settles at its original level, no clue of the perturbation can be seen afterthe water level has settled and the temperature of the system can be easily measured to great accuracy by an ordinarythermometer. An equilibrium state is typically dened as one that does not depend upon the initial conditions as longas the initial perturbation is somehow “weak” enough to leave the system Hamiltonian unaffected, one in which theparticles show a Gaussian distribution of velocities, a consequence of the Central Limit Theorem [85], and in whichthe energy of the system is equally partitioned among the constituent molecules [ 85].

What leads to the equipartitioning of energy? Typically, a perturbation imparted to most systems would spreadthroughout the system in due course. We assume that the system consists of particles of the same mass and couplingsuch that there would not be any natural reasons for any of the imparted energy to get trapped in space.

The spreading of energy typically happens in a way such that energy exchange is readily possible between any twoparticles in the system. Thus, eventually, all the particles end up sharing the system’s energy more or less equally overtime [86]. One hence claims that energy is equipartitioned in the system or that the equipartition theorem is satised.

What if the interactions in a system are such that energy transport between any two particles in the systemis subjected to constraints — e.g., transport can only occur when certain specic many body collision/interactionconditions are satised. One such scenario may be as follows. All energy exchanges must happen collectively betweensmall groups of particles. Every particle then may not end up having more or less the same kinetic energy over time.Rather, the average energy per particle would lie within some bounds. The bounds may depend upon system size,the nature of the interactions between the particles, etc. It is possible that the temperature T ≡ KE / N , where KE refers to the time-averaged kinetic energy of the system, may not be a well-dened quantity in such systems. Yetconceivably, the particles may still have a Gaussian distribution of velocities (as shown in Fig. 6.1) and reach a nalsteady state that is independent of the initial conditions. Such a system would then exhibit an “equilibrium-like” statewith large temperature uctuations that are sustained in time and yet would be in a state that is different from thePlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 5.5. The gure shows a time sequenced schematic of a collision process between two identical and opposite propagating solitary waves. Thered and green arrows show secondary solitary waves that are born later in time. The outlined arrows show the process of mechanical energy transferfrom one grain to the next. (For interpretation of the references to colour in this gure legend, the reader is referred to the web version of thisarticle.)





Fig. 6.1. Velocities recorded across time for particle 19 in a 20 grain chain held between walls. All the grains in the system exhibit such a Gaussiandistribution of velocities.

usual equilibrium state encountered in conventional solids, liquids and gases. This new kind of equilibrium state hasbeen named the “quasi-equilibrium” state in the literature. In these systems, the equipartition theorem is not satised.That having said, when the system size goes to innity, the quasi-equilibrium state becomes indistinguishable fromthe equilibrium state because the temperature uctuations become vanishingly small in that limit.

A granular alignment presents a system in which such “lumps of energy” or solitary waves are transported between

groups of particles as a solitary wave propagates through the chain. There may be other examples of such behavior,such as chains with algebraically nonlinear inter-particle interactions as in the Fermi–Pasta–Ulam problem and inrelated studies [ 5–8,38,39] and in chains with purely quartic interactions [87]. In what follows, we revisit energytransport in bounded granular alignments as shown in Fig. 5.1 below and argue that the quasi-equilibrium state isrealized in a granular system.

It turns out that it is more difcult to quantify the violation of the equipartition theorem than one might imagine. Itis well known that in an innite system of harmonic oscillators, the average kinetic energy per particle is constant [ 88]and hence the equipartition theorem is respected. The dynamics of a chain of 2D granular disks, where n = 2, ina system without boundaries has recently been exactly solved [ 89]. We have studied the dynamics in a chain of 2Ddisks with boundaries. An initial impulse disperses in this system. Persistent collisions of these dispersive waveswith the two boundaries as well as with each other are eventually expected to distribute the total kinetic energy

equally among the constituent particles. However, for nite systems, such equipartitioning is not realized [ 88]. Thus,determining violation of equipartitioning, for practical purposes, becomes a challenge. The best one can presumablydo is to study whether or not equipartitioning of energy is attained in a system at late enough times since the initiationof a perturbation as the number of particles N → ∞. The expectation would be that for sufciently large N ,equipartitoning of energy would be attained.

How does one dene the temperature uctuations though? This turns out to be a tricky matter to which there maynot be a simple resolution. We discuss below our approach to the problem which captures the essential physics of uctuations as n is changed.

We consider the positive denite measure of total kinetic energy uctuations against the idealized kinetic energyof the system as stated below:

F (t ) ≡ N t

i=∆ t

√ (KE i (t )

−KE )

2

N t , (6.1)

where F i (t ) =√ (KE i (t )− KE )2

N t is the instantaneous value of the total kinetic energy, t is the time-step of the

integration (typically this is taken as 0 .2 µ s) and N t denotes the total number of time steps (typically around50,000 or so). One would expect these uctuations to approach zero as N → ∞since there are more particles todistribute the energy over with increasingly smaller changes about the mean. The current data however, indicates thatthis may not be the case for overlap potentials. It is found that F (t ) plateaus at a value that is a function of n —see for example Fig. 6.2 . Specically, for n = 2 and n = 2.5 differences between data is small when one looks atF (t ) and it was found that in both cases, F (t ) is about the same and decreasing in N . Thus, the distinction in

whether or not F →0 is hard to make when n = 2 and n = 2.5. It is pretty clear however that when n > 2.5,F appears to stabilize. Thus, these systems seem to have sustained uctuations that deviate signicantly from

equilibrium. The reason why this happens is because the solitary waves that carry energy span several particles andPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 6.2. Dependence of the kinetic energy uctuation F (i.e., deviation from the average kinetic energy) is shown as a function of system size N . The uctuations decay with growing N as expected. Results are shown for different power-law potentials. For n > 2.5, the data suggests that

F does not become vanishingly small with increasing N . It is difcult here to distinguish the differences between the n = 2 and n = 2.5 cases.

For better tests see discussion. The suggested expression for F is designed to t the data obtained.

their width is controlled by n . Thus, any two particles do not quite carry the same energy on average at any given time[90,91].

Thus, if Lt N →∞F →0, then equipartitioning of energy is attained. However, if Lt N →∞F = 0 and if Lt N →∞d F / d N = 0, then equipartitioning of energy will never be attained [90,91]. Such measurements thoughare difcult to do for strongly nonlinear systems.

It is worth recalling here (see Eqs. (3.4) and (3.5) ) that if the force can be derived from a potential of degree n , thenthe Virial theorem states that [ 3]

12

n PE =KE and KE =n

n +2, (6.2)

where PE is the average potential energy of the system and E ≡ 1. We nd in Fig. 6.2 that the above mentionedmeasure is “weak” when it comes to distinguishing between whether equilibrium in cases such as when n = 2and when n = 5/ 2. In fact, both these cases give about the same F . Perhaps this is to be expected becausein the n = 2 case, one must truly have N → ∞to realize equipartitioning [88]. Given the presence of solitarywaves and the absence of acoustic propagation, equipartitioning of energy is not expected when n = 5/ 2. Themagnitude of F , however, increases as n increases. Thus, for n ≥ 3, it is plausible that Lt N →∞F = 0 andalso that Lt N →∞d F / d N →0 and hence quasi-equilibrium is attained. The results of course are merely suggestiveof quasi-equilibrium for “large enough” n rather than denitive. Better measures may be necessary. One possibilitybeing currently explored is to study the time evolution of the system via the “zig–zag” or space–time plots shown inFig. 6.3 where the kinetic energy passing through each grain is recorded as a function of time for n = 5/ 2 and forn = 2. These space–time diagrams give a visual of how the solitary waves break down as shown in Fig. 6.3 (upper

panel) and how the waves in the n = 2 system behave as shown in Fig. 6.3 (lower panel). The differences betweenthe two cases are readily apparent by looking at the space–time plot at large times. If equipartitioning is achieved,the color of the space–time plot will become uniformly grey. If not, it would remain “patchy”. Studies are under wayto explore whether such space–time plots can prove useful in constructing stronger measures of attainment of quasi-equilibrium.

7. Conning solitary waves and universal power-law decay

7.1. Solitary waves at granular interfaces

Solitary waves are like shock pulses. It is hence natural to ask whether such pulses can be conned or dispersed.In Sections 2 and 8 we consider the problem of connement and of dispersion of shocks in much detail. We considerthe problem of shock containment, reection and transmission at granular interfaces below.Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 6.3. The upper panel depicts the space–time-dependent behavior of an impulse in a Hertz chain (i.e., with n = 2.5), whereas the lower panelshows the identical quantity when n =2. The discussion focuses on the differences between the two plots. This kind of “zig–zag” plots turn out tobe useful in the long-time analyses of the quasi-equilibrium phase.

One often wants to know what exists inside a medium. Various kinds of detection, from those in medicine relatingto the living body to geological detection for probing into the earth, are relevant to our lives. As a probing scheme,scientists normally use the properties of a wave that reects at an interface between the medium and the body to beprobed. Both linear and nonlinear waves are used as a probing tool. Even though we know most properties of the linearwave at an interface, such is not the case for the nonlinear wave. In many situations of detection, the elastic impulse,a typical nonlinear wave, is chosen as a good probing tool. This is especially the case when studying the inside of granular media. Another important aspect of the wave in our everyday life is in connection with protection from anexternal impulse. There could be various kinds of disastrous external impacts, such as impulses by earthquakes, bombexplosions, automobile collisions, land fall of tsunamis, and so on. These are phenomena associated with nonlinearwaves whose properties are not yet well understood. Even though people hope to protect something important from

these mechanical impacts, developing successful protection mechanisms remains an important issue.It may seem at rst glance that the detection in terms of a solitary wave and impulse protection in a specic wayhave no direct connection. These two subjects look quite different. However, they have the same origin when lookingfrom the point of view of physics. That is, both concern wave propagation at the interface. In this section, we willshow that as far as nonlinear solitary wave in a granular medium is concerned, interesting connection between impulseprotection and the property of the wave at the interface exists.

A known, but not widely known, anomalous feature of wave propagation in a granular chain is the total transmissionof a solitary wave along with impulse disintegration when it passes the interface from the granular region of large ratioof grain mass to elasticity to that of small ratio. This corresponds to the region of heavy mass to that of light masswhen the elastic properties of the grains are the same. We will discuss interface phenomena under the assumption of the same elasticity, i.e., we will discriminate the medium in terms of mass in what follows.

The number of disintegrated solitary waves depends on the strength of precompression. However, whenprecompression is weak enough the number of disintegrated solitary waves depends on the mass ratio of heavy tolight grains. An example of disintegration of a big solitary wave into smaller solitary waves is shown in Fig. 7.1 . Thearrangement of the grains and the direction of propagation are shown in Fig. 7.2(a). The leading solitary wave aftertransmission has the largest amplitude and moves the fastest, and the followers are smaller in amplitude and in energycarried and move slower as shown in Fig. 7.1 . However, no impulse disintegration occurs when the arrangement of grains and the direction of propagation are those shown in Fig. 7.2 (b). We show the snapshot of this case in Fig. 7.3 ,where only a single signal appears in both reection and transmission regions, which is just a normal feature occurringin the propagation of a linear wave. In other words, when a solitary wave propagates from the region of light grains tothat of heavy grains, impulse disintegration does not occur and the interface phenomena are just like those of a linearwave [52].

Other interesting phenomena of a solitary wave at an interface appear at the interface of grains and the linearmedium. Interestingly enough, the linear medium behaves as a granular chain of about ve times heavier mass inPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 7.1. Snapshots of solitary waves before (dashed line) and after (solid line) passing an interface between two granular chains of Hertziancontact. The mass of a granule on the left side of the 300th grain is 10 times larger than that on the right side. No precompression is applied.

Fig. 7.2. Two cases of the arrangement of granules along with the direction of incident impulse. The mass of black granule is larger than that of white granule.

Fig. 7.3. Snapshots of solitary wave before (dashed line) and after (solid line) passing an interface between two granular chains of Hertzian contact.The interface is at the 300th grain. The mass of a granule on the left side (incident region) is half of that on the right side (transmitting region). Noprecompression is applied.

the phenomenon of transmission and reection at the interface. Of course, the solitary wave disperses in the linearmedium as it propagates. We present snapshots of the “linear” granular medium (i.e., with n =2 or p ≡ n −1 = 1)of different mass ratios in Fig. 7.4 . One can see that reection occurs even for the same mass and total transmissionstarts to occur at a mass ratio mgran / m lin

≈5. We explicitly show the behavior of total transmission in terms of the

graph for the total energy ratio of incident to reected signals versus mass ratio in Fig. 7.5 .Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 7.4. Snapshots of signals before and after passing an interface from a Hertz chain to a linear granular medium (i.e., n = 2). The left side of the 300th grain (incident region) is the Hertz chain, while the right side (transmitting region) is a chain of linear contact force. The mass ratios are1 in (a) and 5 in (b). No precompression is applied.

Fig. 7.5. Plots of the ratio of total energy of transmitting to incident solitary wave versus mass ratio. Solid squares are the case of two Hertzianchains with initial impulses vi = 10. Solid triangles and dots are Hertzian linear chains with initial impulses vi = 10 and vi = 100, respectively.The mass mRight indicates that in transmitting region. No precompression is applied.

7.2. Understanding total transmission

The total transmission of a solitary wave at an interface of a granular chain is a unique feature. This occurs onlywhen the solitary wave propagates from the region of heavy grains to that of light grains if the elastic properties of the grains are the same or from the region of soft grains to that of hard grains if the mass and size of the grains arethe same. The meaningful parameter in the equation of motion (Eq. (2.10) ) is the ratio of grain mass m to elasticityη dened by η ≡ na as we have used in Eq. (3.12) . The physical quantity governing η is the Young’s modulus thatdetermines the amount of softness or hardness of the grain. For example, stainless steel of 316 type has a density of 8000 kg / m3, Young’s modulus of 193 GPa, and a Poisson ratio of 0.3 [ 92], while 21 polytetrauoroethylene (PTFE)has a density of 2200 kg / m3 , Young’s modulus of 1.46 GPa, and the Poisson ratio of 0.46 [ 92]. Because of the bigdifference in Young’s modulus, the steel ball is classied as a hard material, while the PTFE is classied as a softmaterial. In the propagation of solitary wave, however, the important factor is the ratio m/η . The ratio of m/η of PTFE and the steel ball of the same size is given by (m/η) PTFE /( m/η) STEEL =36.35.

This means that the PTFE behaves as a grain that is 36 times heavier than the steel ball, even though the steel ballis much heavier than PTFE in reality. Since the elasticity η is dominated by the Young’s modulus, the soft grain hasa smaller η than that of the hard grain. One can describe the contact of two elastic bodies in terms of a virtual springPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 7.6. Two arrangements of grains along with the direction of incident impulse. The arrangements of (a) and (b) are light–heavy–light andheavy–light–heavy, respectively.

connecting the two bodies. Therefore, the virtual spring connecting two soft grains is softer than that connecting twohard grains, which means that the grain connected by a soft spring moves slower than that connected by a hard spring.In other words, if the real masses are the same, the soft grain behaves like a heavy grain in solitary wave propagation.

The grains exert force by contact. In any college laboratory, one may try a head-on collision experiment involvinga steel ball at rest being hit by another steel ball. If the mass of the colliding ball is heavier than the mass of the ball atrest, one will see that both are moving forward. However, if the mass of the colliding ball is lighter than the mass of the ball at rest, one will see the reection of the colliding ball while the heavy one moves forward. Now imagine thecollision experiment with two granular chains of different masses in contact and keep two particle head-on collisions inmind. It is well known that one solitary wave in a granular chain spans about ve grains with observable displacementsfrom their equilibrium positions [ 52,53,72]. Therefore, one solitary wave created in a chain of grains of mass M maybe considered as a corpuscle of mass 5 M . If this corpuscle meets a chain of grains of smaller mass m = M / 3, forexample, fteen balls of mass m will respond to the impact of the corpuscle of mass 5 M . These fteen balls will makethree solitary waves in the granular chain of mass m . This may be an explanation for the total transmission shown inFig. 7.1 by simulation. The reverse propagation of a solitary wave in the example above corresponds to the head-oncollision of a light ball to a heavy ball, in which the light ball would recede after collision. In the case of the granularchain, when a solitary wave corresponding to a corpuscle of mass 5 m hits the end of granular chain of mass M =3m ,a receding solitary wave will be created as shown in Fig. 7.3 . It is also interesting that only one solitary wave is

created in the region of transmission composed of heavy grains as shown in Fig. 7.3 . We understand this phenomenonas follows. Since the solitary wave is the only mode of signal propagation in the granular chain, the solitary wave mustbe created in the chain of mass M = 3m even under the condition of insufcient momentum transfer from the lightgranular chain. In this situation, creating one solitary wave, which satises the energy and momentum conservationlaws is most probable. This is our understanding for the transmission and reection phenomena of solitary wavesshown in Figs. 7.1 and 7.3. Similar features happen for the contact of two granular chains of the same masses but withdifferent elasticities.

7.3. Connement of solitary waves and granular containers

The anomalous behavior of total transmission occurring at an interface between two granular chains of differentmasses or elasticities suggests the idea of conning the incident impulse inside a specic region of the granularmedium. Such an arrangement can be seen in Fig. 7.6(a). One can imagine that conning the impulse is impossiblefor the arrangement of Fig. 7.6(b). Recall the earlier discussion here in Section 7.1 regarding Fig. 7.2(b). In thissection, we treat these one by one.

We rst show the phenomenon of solitary wave connement in the vertical chain under gravitational compactionas an example [ 93]. Figs. 7.7 and 7.8 show snapshots of the grain velocity signal showing the backscattering processfor two typical cases, i.e., Fig. 7.7 depicts strong impulse for heavy impurities, and Fig. 7.8 depicts strong impulse forlight impurities. From the last snapshots of Figs. 7.7 and 7.8, one can see the connement of solitary waves explicitly.That is, there is no solitary wave connement for the arrangement of Fig. 7.6(b), while clear solitary wave connementappears for the arrangement of Fig. 7.6(a). The phenomenon of solitary wave connement in the region of small m/ηin the medium of relatively large m/η provides us an idea of a granular container in which the impulse is containedfor a short period. Furthermore, the property of disintegration of a big solitary wave into many smaller ones inside theconning region as shown in Fig. 7.6(b) provides an idea of effective impulse protection by breaking a strong impulsePlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 7.7. Snap shots of typical backscattering by light impurities. Initial impulse is vi = 100 (see text for details of program units). Connedsolitary waves are seen in (d).

into many weak ones of different propagation speed. The disintegrated solitary waves with different speeds leavethe granular container one by one with time lag. The mechanism of effective protection in terms of inhomogeneousgranular chains will be discussed in what follows. Conning a big impulse into a specic region, disintegrating it intomany weak impulses, and releasing it with time lag could be an effective way of protection against a strong externalimpact. This kind of protection mechanism could be realized in specially prepared granular containers.

We propose a standard type of effective granular container that is composed of a series of granular sections of different contact forces and masses as shown in Fig. 7.9(a). Three sections of the linear medium are added for thewalls of container. The scheme of effective protection is as follows. An impulse reaching one end of the protectorproceeds up to the central section without reection. But the impulse disintegrates into small solitary waves when itpasses through each interface, because the mass of grains in each section decreases. This impulse disintegration lastsuntil the leading solitary wave reaches the edge of the central section. Then, both transmission and reection occursimultaneously at each interface when the solitary wave proceeds from the central section to the side walls of thegranular container. The process of propagation inside the granular container of Fig. 7.9(a) allows time lag when thedisintegrated impulses leave the granular container. Several variations (see Fig. 7.9(b)–(e)), from the standard typewill be studied to compare the effectiveness of protection. Interesting experiments have been recently carried out byNesterenko’s group [ 94,95].

To study the time-dependence of energy leakage from the container, we focus on the motion of grains in a horizontalgranular chain with interfaces. Simulating the motion of grains and studying propagation characteristics of solitarywaves have been done elsewhere (see for example in [ 52,72,96,98,99]). Since we choose 10 −5 m, 2.36 ×10−5 kg,and 1 .0102 ×10−3 s as the units of distance, mass, and time, respectively, the integration time step 1 .25 ×10−5

corresponds to 1 .26 ×10−8 s and the snapshots taken every 104 integration time steps correspond to time intervalsof 0.126 ms in real time scale. The grain diameter used in this work is 100, i.e., 1 mm, and η = 5657. For the caseof stainless steel balls [57], η

=2618

×106 (N/ m3/ 2) for spheres with a diameter of 1 mm. To make our simulation

data realistic for stainless steel balls of diameter 1 mm, time rescaling by (5657 / 2618 )1/ 2 ×10−3t is required.Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 7.8. No conned solitary wave exists in (d). Snap shots of typical backscattering by heavy impurities. Initial impulse is vi =100 (see text fordetails on program units).

Fig. 7.9. Schematic diagrams of granular containers. Circle, grey octagon, and square represent the grain of p = 1.5 (i.e., n = 2.5), p = 2(i.e., n =3), and p =1 (i.e., n =2), respectively. Mass of grain is denoted by m . (a) is the standard type.

The phenomenon of solitary wave connement in the region of small m/η in the medium of relatively large m/ηprovides us an idea of the granular container in which the impulse is contained for a short period. Furthermore, thePlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 7.10. The 280th snapshot of energy leakage for the granular container of Fig. 7.9 (a). The container ranges from 121 to 300 in grain numberand grains of m =100 are placed outside the container.

property of disintegration of a big solitary wave into many smaller ones inside the conning region as shown inFig. 7.9 provides an idea of effective impulse protection by making a strong impulse into many weak ones of differentpropagation speed. The disintegrated solitary waves with different speeds leave the granular container sequentially.The mechanism of effective protection in terms of inhomogeneous granular chains will be discussed in what follows.Conning a big impulse into a specic region, disintegrating it into many weak impulses, and releasing it with timelag could be an effective way of protection against a strong external impact. This kind of protection mechanism couldbe realized in specially prepared granular containers.

We propose a standard type of effective granular container that is composed of a series of granular sections of different contact forces and masses as shown in Fig. 7.9(a). Three sections of the linear medium are added to serveas the walls of the container. The scheme of effective protection is as follows. An impulse reaching one end of the protector proceeds up to the central section without reection. But the impulse disintegrates into small solitarywaves when it passes through each interface, because the mass of the grains in each section decreases. This impulse

disintegration lasts until the leading solitary wave reaches the edge of the central section. Then, both transmission andreection occur simultaneously at each interface when the solitary wave proceeds from the central section to the sidewalls of the granular container. The process of propagation inside the granular container of Fig. 7.9(a) allows time lagwhen the disintegrated impulses leave the granular container. Several variations, Fig. 7.9(b)–(e), from the standard typewill be studied to compare the effectiveness of protection. To study the time-dependence of energy leakage from thecontainer, we focus on the motion of the grains in a horizontal granular chain with interfaces. Simulating the motionof the grains and studying the propagation characteristics of solitary waves have been done by many groups [ 52,57,94–99 ]. Fig. 7.10 shows energy leaking in the form of small solitary waves leaving out of the granular container of Fig. 7.9(a) with 20 grains in each section. But the mass of a grain in three p = 1 sections has been changed intom = 10 instead of m = 2. To see the leaking solitary waves from the container, we put heavy Hertzian grains of m =100 on either side of the container. Then we apply an initial impulse to the right end of the granular chain usinga grain of mass 100 with velocity 10 in our program units, which corresponds to 5 .4

×106 m/ s for the steel ball

mentioned above (though this number is too large of a velocity to be of much practical signicance). Fig. 7.10 is the280th snapshot of grain velocity which corresponds 0.112 ms after collision for the steel ball mentioned above.

7.4. Universal power-law decay

It is interesting to see the energies of small solitary waves leaving out of the granular container. The energy of asolitary wave is the sum of the mechanical energies of grains participating in solitary waves. Recall that the numberof grains composing a solitary wave is approximately 5 for any height of solitary wave [ 52,53]. The energies of leading solitary waves leaving the granular container appearing at the right and left ends of the snapshot of Fig. 7.10are, respectively, are 3.3% and 7.7% of the energy of the incident solitary wave. One can see that a strong initialimpulse incident on the granular container is broken into small solitary waves whose individual energy is less than10% of the original energy when they leave the granular container. Therefore, designing a specic granular containerPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 7.11. Impulse energies remaining inside the granular container of the types in Fig. 7.9 . Time scale denotes the number of snapshot.

is a way of constructing an effective protector. It is natural for us to pay interest to the remaining energy inside the

granular container as time elapses. A fascinating universal behavior is discovered in the energy leaking process fromthe granular container.Fig. 7.11 shows the plots of the energy remaining inside the granular container versus elapsed time for those

variations of Fig. 7.9(b)–(e) alongside the standard case of Fig. 7.9(a). Fig. 7.9 (b) has the same structure as thestandard one, but the sequence of sections in the right half is reversed. Fig. 7.9(c) has the same 60 grains of mass0.1 and p = 1.5 on each side of the central section. Fig. 7.9 (d) has the same structure as the standard type, but thenumber of grains in each section is 50 instead of 20. Finally, we introduce the simplest granular container composedof the same 180 nonlinear grains of p =1.5 and m =1.0 without linear walls as shown in Fig. 7.9(e). As one mightexpect, increasing the number of grains in each section, such as Fig. 7.9 (d), will cause slower leaking, while using thesame mass of grains, such as Fig. 7.9(c), will cause faster leaking. The reason for slower or faster leaking of energysurely stems from the time taken for a solitary wave to pass from one edge of the container to the other edge. To seethis more clearly, we add the one shown in Fig. 7.9(e) for comparison, which takes the shortest elapsed time whenpassing from one end to the other and results in the fastest leakage.

A remarkable universality behavior of energy leaking from different types of granular containers is seen if were-plot the same data of Fig. 7.11 on a log–log scale in Fig. 7.12 through time rescaling factors, 2 t , 0.22t , and 9t for(c), (d), and (e) of Fig. 7.9 , respectively, which correspond to the elapsed time inside pertinent granular containers.Falling on a single line after proper scaling stresses that the nature of energy leaking is universal, i.e., the same power-law exponent in time. In other words, the essential time-dependence of energy leaking from the granular containeris independent of its construction. The inverse scaling factors qualitatively describes relative time scale for a solitarywave to stay inside granular container.

The universal power-law behavior in time for the energy leaking from the granular container is an important recentrealization. Therefore, understanding of the universal power-law behavior is an interesting subject to pursue. Onecan understand that energy leakage from the container occurs when a solitary wave undergoes both transmission andreection at the interface in the edge of the container. This happens when a solitary wave passes through the interfacefrom a light- to a heavy-granular medium. In this process of energy leakage, the solitary wave decreases its heightand speed. Therefore, the energy remaining inside the granular container depends on the number of reections at bothinterfaces of the container walls, and the number of reections is proportional to the speed of the solitary wave andinversely proportional to the length of container. One may construct an equation of motion for the energy remaininginside the granular container based on this analysis.

If there were no reduction in the speed of solitary wave after reection, one may easily guess that the remainingenergy would decay exponentially. That is, the equation of motion for the remaining energy is written as

E R(t + t ) − E R(t ) = −k t E R(t ), (7.1)

where k is the inverse of the time constant. The energy decay in this study, however, is not this simple, becausethe speed of the solitary wave is not constant but decreases after reection. Therefore, the constant k is not quite aPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 7.12. The log–log plots of the data of Fig. 7.11 expressed by time rescaling. Times are rescaled by multiplying 2, (2 / 9), and 9 for the datadesignated by stars, open squares, and solid squares, respectively.

constant any more but time dependent. According to the analysis above, k must be replaced by N r (t ) that is theaverage number of reections per unit time at time t . Thus, the change in the energy remaining inside the granularcontainer is written as

E R(t + t ) − E R(t )∝− N r (t ) t E R(t ). (7.2)

Since the speed of the solitary wave decreases after every reection, N r (t ) must decrease as well. One can infer thetime-dependent behavior of N r (t ) as N r (t )

∝1/ t from the data of Fig. 7.12 . Then, we nally set up the equation

of motion for the change in the remaining energy as follows

E R(t + t ) − E R(t ) = −γ ( t / t ) E R(t ), (7.3)

which gives the solution

E R(t ) = A t −γ , (7.4)

where γ is a universal dimensionless constant and the constant A depends on the structure of the granular container,such as the length of the container and the arrangement of the grains. Fig. 7.12 gives gives γ = 0.70 and A =65.31 ×α γ , where α is the time scaling factor. We choose α =1 for our standard type of Fig. 7.9(a).

In conclusion, we have discussed an interesting universal behavior on energy leakage from a granular container.It is noteworthy that various types of granular containers show the same power-law type energy-leakage behavior intime. We understand that this power-law behavior stems from the decrease in the speed of the solitary wave uponreection accompanying any transmission. Reduction of the speed of solitary wave in time causes a reduction in theaverage number of reections per unit time as time elapses. The universal power-law behavior in time of the energyremaining inside the granular container implies that the rate of energy leakage becomes progressively slower due to theslowing down of the solitary waves after the reection accompanying each transmission and it is proportional to (1/ t ) .A big solitary wave produced by a strong external impulse is broken into many small solitary waves when it passesan interface from a large m/η - to a small m/η -granular medium. Therefore, a granular container having multipleinterfaces with an appropriate alignment of m/η may play a role as an effective protector for an external impulse. Themechanism of protection concerns holding on to the energy carried by solitary waves inside the container and releasingthe energy in small installments in the form of separate solitary waves as shown in Fig. 7.10 . Impulse connement anddisintegration in a granular chain discussed here may also appear in other systems, such as electromagnetic devicesand bio-molecular chains, if a power-law type nonlinear interaction exists between the elements of the system.

8. Impulse absorption by tapered granular chains

Shock waves concentrate signicant amounts of mechanical energy into small regions of space across shorttimes [100–104 ]. There is no accepted way to shock-proof a surface. In the previous section we discussed thePlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




properties of the granular container that can “absorb” a solitary wave and then leak it slowly through a desired part of the system. Another proposal involves the use of functionally graded materials (FGMs) [ 105,106] where the systemdensity is varied to alter the elastic properties. A third approach is to arrange for an incident shock to travel along analignment (e.g., horizontal) of elastic spheres, where the spheres progressively shrink in radius [ 107–114 ].

8.1. Simple tapered chain

Let us take two spheres of radii R1 and R2 = R1(1 − q ), 0 < q < 1, of some material of density ρ . Letthe sphere with R1 , velocity v1 collide with the sphere with R2 , which is at rest. Momentum conservation reveals(1−q )−3 =v2/(v 1 −v1) , where v2 is the velocity acquired post collision by R2 and v1 is the recoil velocity of R1. Asq →1, v 2/(v 1 −v1) diverges. Thus, when a massive sphere hits a smaller sphere, only part of the kinetic energy (KE)is transferred and v2 (v 1 −v1) . Any progressive reduction in radii of successive spheres results in break down of the propagating energy through collisions. Hence, a tapered chain (TC) with several hundred spheres should “absorb”a shock wave [107,108].

It would be desirable to explore much smaller alignments of elastic spheres to achieve the desired shock absorption.One may then use many embedded TCs on a wall to protect the same from shocks. Here, we discuss the physics of

small TCs with shock absorption properties that are comparable to TCs with several hundred spheres [ 107,110].Using hard spheres , KE and momentum conservation laws, we can nd in a TC of N spheres, the ratio of KE of the smallest grain ( KE out) to the same for the largest grain ( KE in). This calculation is as follows. Ignoring any energyloss during a collision, masses and radii are expressed as

Ri+1 = Ri − Ri q = (1 −q ) Ri = ε Ri and m i =ρ V i =43

π R3i ρ ≡η R3

i , (8.1)

m i+1 =η R3i+1 =η ε 3 R3

i , (8.2)

where ε = 1 −q . Evaluating the conservation of momentum with a single prime denoting post collision values andthe initial condition that the (i +1)th particle is stationary before a collision ( vi+1 =0), all η cancel and we obtain

m i vi +m i+1vi+1 =m i vi +m i+1vi+1

R3i vi = R3

i vi +ε 3 R3i vi+1

vi =vi +ε 3vi+1,

(8.3)

where Eqs. (8.1) and (8.2) have been used. Following the same procedure for the conservation of energy while ignoringthe factor of one-half yields,

v2i =vi

2+ε3v 2

i+1. (8.4)

Letting A = ε 3vi+1 we can rewrite Eq. (8.3) in terms of vi and substitute the resulting expression into Eq. (8.4),

v2

i =(v

i −A)2

+Av

i+1= v2

i −2 Avi + A2 + Avi+1

2vi = A +vi+1

vi+1

vi =2

1 +ε 3 . (8.5)

Note that for one collision

KE out

KE in =K E i+1

KE i =m i+1

m i

vi+1

vi

2

= ε 3 vi+1

vi

2

K E i+1

KE i =4ε 3

1 +ε 3 2. (8.6)





For N particles there will be N −1 collisions, each of which has the ratio in Eq. (8.6) . Therefore, the normalized KE,KE N , for the lossless TC in the hard-sphere approximation is given as

KE N =4 (1 −q )3

1

+(1

−q )3 2

N −1

. (8.7)

The same approximation can be performed with some amount of energy loss, ˜ E L , included such that the system isstill effectively conservative and may represent a better approximation. Consequently, the momentum equation (Eq.(8.3)) is unchanged, but Eq. (8.4) becomes

v2i =vi

2+ε3vi+1

2+ ˜ E L . (8.8)

One then obtains a more complicated expression replacing Eq. (8.5):

vi+1

vi =2 − ˜ E L

ε 3vi vi+1

1 +ε 3 . (8.9)

We can make the substitution, ˜ E L∝

vi vi+1 or ˜ E L = E Lvi vi+1 , where E L is the constant of proportionality. Thisadjustment yields

vi+1

vi =2ε3 − E L

ε 3 1 +ε 3 .

The corresponding result for the normalized KE for N particles is

KE N =2 (1 −q )3 − E L

2

(1 −q )3 1 +(1 −q )3 2

N −1

. (8.10)

In the limit E L

=0, Eq. (8.10) reduces to the lossless case, Eq. (8.7) , as one would expect. Note that results are

independent of initial velocity and the size of the grains.The rich and complex nonlinear dynamics of grains with soft-sphere potentials, held in horizontal alignment

between xed boundaries, has been extensively discussed in this paper. We focus now on the ability of a TC with N =20 to absorb a δ-function impulse incident at the large sphere and probe the force that will be experienced by thesmallest grain as it collides with a rigid end wall. Such calculations require accurately solving the Newton’s equationsfor a dissipative, strongly nonlinear system across many decades in time . The force calculations could be a usefulguide for experimental measurements of forces felt at a force sensor placed at the tapered end of the system.

Let the spheres be held within walls at zero external loading, that is the grains are not squeezed to begin with. Thespheres interact via the repulsive Hertz potential (see Eq. (2.1)). We assume that the spheres are made of Ti–6Al–4V (ahard material) with density ρ =4.42 mg / mm3, σ =0.34 and Y =110 Gpa. As we shall see, our results in Fig. 8.1(a)and (b), which report the system properties, are insensitive to the choice of materials. The radius of our largest sphere

is 5.0 mm. We introduce dissipative effects via the restitution coefcient which reduces the force between the grainsduring decompression ( F decompression ) and compression ( F compression ) as follows F decompression / F compression = 1 −w ,where w 1 [77,115].

The equations of motion for each grain are solved via the Velocity Verlet algorithm [ 116]. The initial conditions aredened by assigning a velocity v1 =10 m / s at t =0 to the largest sphere, vi =0 (i > 1) . Larger velocities, resultingin signicant compressions between adjacent grains (

∼5% of grain radii) have also been employed and approximately

the same results as in Figure 2 obtained. The time step of integration, t = 10−5µ s, with runs across

∼5 ×103

µ s.Energy conservation holds up to eight decimal places when w =0.

The force recorded at the tapered end is computed by calculating the momentum changes of the smallest grainduring its collisions with the wall. When w =0, the rst collision with the wall typically does not record the maximumforce felt by the smallest grain. Long time data reveals the existence of collective effects that lead to slightly largerforces being felt later by the smallest grain. However, when w

=0, the largest force is typically felt during the rst

end wall collision. Fig. 8.1(a) presents a phase diagram to show how the maximum force, f max = F max / F in variesPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 8.1. Phase diagram showing (a) f max and (b) KE out / KE in felt at the tapered end versus restitution w and tapering q . Observe thatKE out / KE in < 1 for q = 0, w = 0 because part of the energy is carried as potential energy. In (c) KE out / KE in is dramatically affected uponprecompression (which causes 0.3% reduction in the radius of the largest sphere).

with q and w , where F max is the maximum force recorded at the sensor at the tapered end and F in is the maximuminput force for the w =0 case. We nd that

∼80% of F in can be absorbed by TCs with q =0.09, provided restitutive

losses can be articially controlled via appropriate cushioning.The initial impulse is initiated into the system in the form of KE. As the impulse starts to propagate, approximately

55% of this energy is transmitted in kinetic energy form and the rest as potential energy. In Fig. 8.1(b), we presentcalculations of KE out / KE in as functions of q and w . Our extensive analyses reveal that 90% of the input energy(which is all in KE in) can be absorbed by TCs with q = 0.10. Further improvement in shock absorption of a TC canbe effected by subjecting the system to external loading. Simulations show ( Fig. 8.1 (c)) that an external loading forcecausing 0.3% compression of the largest sphere reduces KE out / KE in by a factor of 4.

When w →0 and q = 0, i.e., for monodispersed chains, energy loss in the chain is roughly constant everywhere(Fig. 8.2(a)). However, increase in w , enhances loss where the forces are largest, i.e., at the end where the impulse isincident ( Fig. 8.2(g)). Tapering leads to high velocities at the tapered end. The presence of a force sensor at the taperedend leads to a large number of collisions with the sensor and hence large energy loss at the tapered end ( Fig. 8.2(c)).When both q and w are large, the bulk of the energy loss is at the central part of the TC (Fig. 8.2(f, i)). The use of nitinol spheres, in a phase where nitinol shows strong hysteresis or energy absorption [ 117], in the middle part of suchTCs, can dramatically enhance shock absorption.

We have discussed the shock absorption properties of TCs and have depicted the effects of radius shrinkage ( q ) andrestitutive losses ( w ) in terms of “dynamical phase diagrams”. Our studies demonstrate for the rst time that smallTCs with 20 spheres can absorb

∼80% (∼

90%) of the incident force (energy) pulse ( Fig. 8.1(a)–(c)). Enhancementin energy absorption due to increasing N is well approximated by Eq. (8.7). It is conceivable that arrays of TCs,

embedded on a wall matrix, could help design impulse resistant walls [ 118].Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




Fig. 8.2. The panels (a)–(i) show spatially resolved collisional energy loss between granular contacts in a 20 grain tapered chain. For large w andq , the highest losses are at the central part of the tapered chain, which is not something entirely obvious unless one probes the detailed dynamics of the system.

8.2. Decorated tapered chain

Larger and more frequent inertial mismatches lead to better momentum traps and energy dispersion. We thereforebriey discuss an improved tapered chain design in which we place interstitial grains between contacts of a simple TC(STC). We refer to this as as the decorated tapered chain (DTC — examples shown in detail later in Fig. 8.3(d)–(i)).The DTC is essentially a modied STC where we have introduced a single-sized interstitial grain of radius, f R N ,between every member of the STC, where 0 < f ≤1.0 and R N is the radius of the smallest member of the STC. Therelation, f R N , was chosen for convenience in deriving a hard-sphere approximation. From this point on, q appearsin both STC and DTC but is dened differently. As such they are denoted as qs and qd , respectively. We constrainthe DTC system to an odd number of particles such that the grains that formed the ends of the STC are still theouter members in the DTC. Derivation of a hard-sphere approximation for the DTC is more cumbersome than that of the STC and only key points are presented here. The conservation equations for mass and energy are carried out forseveral terms until a pattern emerges. The velocity ratio is given by

v N

v1 = 2 A1/ 2 N −1 N −1

2

j=1

ε3( j−1)

A

+ε 3( j

−1)

1ε 3 j

+A

, (8.11)

where A ≡ f 3ε 3( N −1)/ 2 , ε ≡ (1 −qd ) and qd is now dened as Ri+2/ Ri = (1 −qd ).Turning to the mass ratios, we nd that,

m N

m1 = ε 3( N −1)/ 2. (8.12)

We can now identify the normalized kinetic energy, by squaring Eq. (8.11) and combining it with Eq. (8.12) to form

K N = 4 Aε32

( N −1) N −1

2

j=1

ε 3( j−1)

A +ε 3( j−1) ε 3 j +A

2

. (8.13)

Fig. 8.4 sketches the hard-sphere approximation for the DTC.Please cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




(a) HSA (top); Numerical (bottom). (b) Monodisperse chain: qs =0, N =10.

(c) Tapered chain: qs =0.1, N =10. (d) Difference in HSA and numericalsolution.

Fig. 8.3. Hard-sphere analysis and numerically-solved normalized kinetic energy surfaces, K N ≡ K OUT / K IN , for the simple tapered chain asfunctions of the number of spheres N and tapering qs . Their difference is plotted in (d) with a reduced z-axis. Sample tapered chains are identiedin panels (b), (c).

(a) f =1. (b) f =0.7. (c) f =0.3.

Fig. 8.4. Here we show the normalized kinetic energy surfaces, K N ≡ K out / K in , for the decorated chain under the hard-sphere approximation asfunctions of the number of spheres, N , fractional size of interstitial sphere, f , and tapering, qd .

It is difcult to draw any physical intuition from Eq. (8.13) . However, a very curious and astonishing result occursin the limit qd =0. Under that condition, Eq. (8.13) reduces to

K N |qd =

0

=4 f 3

f 3 +1 2

N −1

. (8.14)





(a) f =1. (b) f =0.7. (c) f =0.3.

(d) f =1; N =13;qd =0. (e) f =0.7; N =13;qd =0. (f) f =0.3; N =13;qd =0.

(g) f =1; N =13;qd =0.1. (h) f =0.7; N =13;qd =0.1. (i) f =0.3; N =13;qd =0.1.

Fig. 8.5. (a)–(c): Numerically produced normalized kinetic energy surfaces are shown for the decorated, tapered chain system as functions of thenumber of spheres, N , fractional size of the interstitial spheres, f , and tapering, qd . Several examples of decorated, tapered chains are shown in(d)–(i).

This limit is equivalent to Eq. (8.7) under the exchange f ⇔

(1 −qs ). As a result K N decays as a half-gaussian orsigmoid with increasing f , and exponentially with increasing N . It is clear that f =1 should imply qs =0 since theyboth generate monodisperse chains.

That this equivalency goes beyond that special case is quite unexpected. One can now begin to see the incredibleeffect f has on the energy mitigation capability when an innite potential is invoked: for f

=0.3 – a typical value we

might consider – the equivalent tapering in the STC would be qs = 0.7. This value is 7 times larger than any systemwe had previously considered and could be a signicant system integration challenge. Visually, for hard spheres, theenergy mitigation capability of the STC shown in Fig. 8.3(c) (qs = 0.1) is identical to that for a DTC similar to thatshown in Fig. 8.5(d) but with qd =0, N =10, f =0.9.

Fig. 8.5 highlights the results from numerically solving the equations of motion for the DTC. The lower-half of the gure demonstrates the wide variety of DTCs possible given f , qd , and N . It is immediately clear that the inertialmismatch changes as a function of position along the DTC — a dynamic not present for the STC. It is possible thento have DTC chains that appear monodisperse ( qd = 0) for only a subset of the chain. This is what we believe to bethe cause of a ripple in the surface of the K N plots that propagate toward the origin as f decreases. As one mightexpect, such behavior would be functions of N , q , d and f . The effect vanishes for f ≤0.6, approximately. At aboutthis threshold, the interstitial grain is not much smaller (less massive) than the grains toward the end of the chain. Theexplanation is that as an impulse propagates down the DTC, energy transmission becomes increasingly efcient dueto smaller inertial mismatches — a prerequisite for admitting solitary waves. Thus the system changes from a shock absorber to shock transmitter. This effect however must compete with compressive effects in some manner sinceno such behavior is present for the hard-sphere approximation even though it too has a position-dependent inertialmismatch.

Simulations suggest that for f = 0.3, N = 5, qd = 0.1, one can disperse energy within the chain such thatonly about 10% of that put into the system is transmitted to the end with the initial pulse. At later times, the pulse isconverted into noise.

To conclude, granular alignments are rich, highly scalable, nonlinear dynamical systems that can be constructedto act as shock absorbing systems. They can be tuned by modifying the material properties and contact geometries,producing fascinating and sometimes unexpected outcomes. Both system types can be realized — and have beencorroborated experimentally. In point, a preliminary experimental study by J. Agui at NASA–Glenn Research CenterPlease cite this article in press as: S. Sen, et al., Solitary waves in the granular chain, Physics Reports (2008), doi:10.1016/j.physrep.2007.10.007




on a DTC with qd =0.05, f ≈0.3, N =9 suggests that the force felt by the last grain may be close to some 50% of that compared to a STC of similar length.

A hard-sphere approximation for the STC correctly describes the functionality of N and qs for the normalizedenergy parameter space. The softness of the potential is a factor but not a dominant one. The DTC, much to oursurprise however, cannot be described by a hard-sphere approximation because shock transmission properties vary

with position along the chain and softness of the spheres – due to the Hertz potential – strongly inuence that behavior.This particular system, consequently, cannot be treated by an independent collision model. As a note of academicinterest, the limit of qd = 0 for the DTC hard-sphere approximation surprisingly reduces to the STC hard-sphereapproximation under the exchange f

⇔(1 −qs ) . This says that a hard-sphere chain consisting of an alternating

series of radii (where Rsmall = f Rlarge ) has the kinetic energy absorption equivalency of a STC of tapering qs .

9. Summary and discussion

In this review we have tried to dene the differences between the solitary waves in continuous media and those indiscrete media such as in granular media. The latter behave very differently when they cross each other. And the lateststudies strongly suggest that it is the formation of secondary solitary waves as byproducts of solitary wave collisionsin a discrete medium that could give rise to an equilibrium-like phase that does not satisfy the equipartition theorem.We point out that there are underlying similarities between the problem of solitary wave formation and breakdown innite chains with boundaries and the celebrated Fermi–Pasta–Ulam problem. The effects of gravitational loading andconstant loading on the propagation of solitary waves have been addressed in detail in this work.

In the last two sections we have discussed two interesting ways in which impulses or shock waves can be decimatedwithin a granular medium. The concept of a granular container involves trapping an impulse and then slowly releasingit whereas that of the tapered chain involves chipping away at the front of an impulse as it propagates through agranular chain by using mass and size mismatch of grains. It is conceivable that in individual and in combined forms,these two approaches can help develop effective shock mitigation systems. Indeed such studies are under way andsome of our recent work along these lines will appear in due course.

Acknowledgements

SS acknowledges support of the US Army Research Ofce during the course of this work. EA was supportedby a Fulbright Postdoctoral Fellowship at SUNY-Buffalo during the course of this study. We are deeply grateful forcountless discussions we have had with Juan Agui, Jr., Emily Bittle, Stephan´ e Job, V. M. (Nitant) Kenkre, BruceLaMattina, Daniel C. Mattis, Francisco Melo, Masami Nakagawa, Vitali Nesterenko, Jan M.M. Pfannes, Robert P.Simion, Adam Sokolow, Diankang Sun, Otis Walton, and David Wu.

References

[1] H. Young, R. Freedman, University Physics, 11th edition, Pearson, San Francisco, 2004.[2] L.E. Kinsler, A.R. Frey, A.B. Coppens, J.V. Saunders, Fundamentals of Acoustics, 4th edition, Wiley, New York, 1999.[3] H. Goldstein, C. Poole, J.L. Safko, Classical Mechanics, 3rd edition, Addison-Wesley, San Francisco, 2002.

[4] G.D. Mahan, Many Particle Physics, Academic, New York, 1981.[5] M. Toda, Theory of Nonlinear Lattices, Springer, Berlin, 1989.[6] M. Romoissenet, Waves Called Solitons, Springer, Berlin, 1996.[7] E.A. Jackson, Perspectives of Nonlinear Dynamics, vols. I and II, Cambridge, Cambridge, 1990.[8] J. Ford, Phys. Rep. 213 (1992) 271.[9] S. Sen, M. Manciu, R.S. Sinkovits, A.J. Hurd, Gran. Matt. 3 (2001) 33.

[10] V. Nesterenko, Dynamics of Heterogeneous Materials, Springer, New York, 2001.[11] J.S. Russell, Rept. Of the Comm. On Waves, Repts of the Seventh Meeting of the British Association for the Advancement of Science,

Liverpool, 1838, p. 417.[12] J.S. Russell, On Waves, Rept. Of the 14th Meeting of the British Association for the Advancement of Science, York, 1845, p. 311.[13] J. Sander, K. Hutter, Acta Mech. 86 (1991) 111.[14] C. Huygens, Horologium oscillatorium sire de motu pendulorium ad horologio aptato, demonstrations, geometricae, Paris, 1673.[15] I. Newton, Philosophiae naturalis principia mathematica, London, 1687.[16] L. Euler, D ecourverte d un nouveau principle de m´ecanique, M em. Acad. Sci. Berlin 6 (1750) 185. Publ. (1752).[17] L. Euler, De la propagation du son, M´ emoires de l’ Academie des Sciences de Berlin 15 (1759) 185. Publ. (1766).





[18] J.L. Lagrange, M´ecanique Analytique, Paris, 1788.[19] J.L. Lagrange, Recherches sur la nature, et al. propagation du son, Miscellania Taurinensia (Turiner Akademic berichte) 1 (1759) 1.[20] F. Gersten, Ann. Der Phys. 32 (1809) 412.[21] G.B. Airy, Tides and waves, Encyclopaedia Metropolitana, London 5 (1845) 241 (Sect 392).[22] G.G. Stokes, On the Theory of Oscillatory Waves, in: Mathematical and Physical Papers, vol. 1, Cambridge University Press, Cambridge,

1880, p. 197.

[23] S. Earnshaw, Trans. Cambridge Philos. Soc. 8 (1849) 326.[24] J. Boussinesq, C. R. Hebd. S´eance Acad. Sci. 72 (1871) 755.[25] J. Boussinesq, C. R. Hebd. S eance Acad. Sci. 73 (1871) 256.[26] Lord Rayleigh, London, Edinburgh and Dublin. Philos. Mag. J. Sci., Ser. 5 1 (1876) 257.[27] J.-C. de Saint-Venant, C. R. Hebd. S´ eance Acad. Sci. 101 (1885) 1101, 1215.[28] J. McCowan, London, Edinburgh and Dublin. Philos. Mag. J. Sci., Ser. 5 32 (1891) 45.[29] J. McCowan, London, Edinburgh and Dublin. Philos. Mag. J. Sci., Ser. 5 38 (1894) 351.[30] D.J. Korteweg, G. de Vries, London, Edinburgh and Dublin. Philos. Mag. J. Sci., Ser. 5 39 (1895) 422.[31] K.O. Friedrichs, Comm. Pure Appl. Math. Mech. 1 (1948) 81.[32] K.O. Friedrichs, D.H. Hyers, Comm. Pure Appl. Math. Mech. 7 (1954) 517.[33] U. Dehlinger, Ann. Phys. (Leipzig) 2 (1929) 749.[34] J. Frenkel, T. Kontorova, Zh. Eksp. Teor. Fiz. 8 (1938) 1340; J. Phys. (Moscow) 1 (1939) 137.[35] J. Frohlich, Phys. Rev. Lett. 34 (1975) 833.[36] T.C. Frank, J.H. van der Merwe, Proc. Roy. Soc. Lond. Ser. A 198 (1949) 205.

[37] V.N. Belykh, N.F. Pedersen, O.H. Sorensen, Phys. Rev. B 16 (1977) 4853; 16 (1977) 4860.[38] E. Fermi, J. Pasta, S. Ulam, Studies of nonlinear problems, Los Alamos National Laboratories Rept. LA-1940, 1955.[39] N.J. Zabusky, M.D. Kruskal, Phys. Rev. Lett. 15 (1965) 240.[40] M. Toda, J. Phys. Soc. Japan 22 (1967) 431.[41] M. Toda, J. Phys. Soc. Japan 23 (1967) 501.[42] H. Flaschka, Phys. Rev. B 9 (1974) 1924.[43] J. Garnier, F.K. Abdullaev, Phys. Rev. E 67 (2003) 026609.[44] V.E. Zakharov, A.B. Shabat, Zh. Eksper. Teor. Fiz. (JETP) 61 (1971) 118.[45] H.C. Yuen, B.M. Lake, Phys. Fluids 18 (1975) 956.[46] G. Dufng, Erzwungene schwingungen bei veranderlicher Eigenfrequenz, Vieweg, Braunschweig, 1918.[47] K. Hizanidis, D.J. Frantzeskakis, C. Polymilis, J. Phys. A 29 (1996) 7687.[48] E.L. Geist, C.E. Synolakis, Sci. Amer. 294 (2006) 56.[49] S.T. Grillo, J. Horrillo, J. Engng. Mech. 123 (1997) 1060.[50] P. Guyene, S.T. Grilli, J. Fluid Mech. 547 (2006) 361.[51] V.F. Nesterenko, J. Appl. Mech. Tech. Phys. 24 (1983) 733.[52] A. Chatterjee, Phys. Rev. E 59 (1999) 5912.[53] A.N. Lazaridi, V.F. Nesterenko, J. Appl. Mech. Tech. Phys. 26 (1985) 405.[54] L.D. Miller Jr., Proc. of Army Res. Conf. (declassied), West Point, New York, 1986.[55] R.S. Sinkovits, S. Sen, Phys. Rev. Lett. 74 (1995) 2686.[56] S. Sen, R.S. Sinkovits, Phys. Rev. E 54 (1996) 6857.[57] C. Coste, E. Falcon, S. Fauve, Phys. Rev. E 56 (1997) 6104.[58] V.F. Nesterenko, J. Phys. IV 4 (1994) C8.[59] H. Hertz, J. Reine. Angew. Math. 92 (1881) 156.[60] L.D. Landau, E.M. Lifshitz, Theory of Continuum Media, Pergamon, Oxford, 1970, p. 30.[61] A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover, New York, 2003.[62] D.A. Spence, Proc. Roy. Soc. Lond. Ser. A 305 (1968) 55.[63] E. Falcon, Ph.D. Thesis, L Universite Claude Bernard – Lyon I, 1997.

[64] G. Friesecke, J.A.D. Wattis, Commun. Math. Phys. 161 (1994) 391.[65] R.S. MacKay, Phys. Lett. A 251 (1999) 191.[66] J.Y. Ji, J. Hong, Phys. Lett. A 260 (1999) 60.[67] A. Sokolow, E.G. Bittle, S. Sen, Europhys. Lett. 77 (2007) 24002.[68] S. Sen, M. Manciu, Physica A 268 (1999) 644;

S. Sen, M. Manciu, Phys. Rev. E 64 (2001) 056605;J.M.M. Pfannes, M.S. Thesis, SUNY-Buffalo, 2003.

[69] D. Sun, S. Sen, unpublished;D. Sun, Ph.D. Thesis, SUNY-Buffalo (in progress);R. Baumer, C. Daraio, D. Sun, S. Sen, unpublished.

[70] M. Manciu, S. Sen, A.J. Hurd, Physica A 274 (1999) 588.[71] J. Hong, J.Y. Ji, H. Kim, Phys. Rev. Lett. 82 (1999) 3058.[72] J. Hong, A. Xu, Phys. Rev. E 63 (2001) 061310;

M. Manciu, V. Tehan, S. Sen, Chaos 10 (2000) 658;E. Avalos, J.M.M. Pfannes, T.R. Krishna Mohan, S. Sen, Physica D 225 (2007) 211.





[73] C.W. Gear, Numerical Initial Value Problem in Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, NJ, 1971.[74] J.D. Goddard, Proc. Roy. Soc. Lond. Ser. A 430 (1990) 105.[75] P.L. Bhatnagar, Nonlinear Waves in One-Dimensional Dispersive Systems, Clarendon, Oxford, 1979.[76] M. Abromowitz, I. Steghun, Handbook of Mathematical Functions, in: American Math. Soc., vol. 55, National Bureau of Standards, 1972.[77] O.R. Walton, R.L. Braun, J. Rheol. 30 (1986) 949.[78] M. Manciu, S. Sen, A.J. Hurd, Physica D 157 (2001) 226;

A. Rosas, K. Lindenberg, Phys. Rev. E 68 (2003) 04134.[79] M. Manciu, Ph.D. Thesis, SUNY-Buffalo, 2001.[80] N.V. Brilliantov, F. Spahn, J.M. Hertzsch, T. P˜ oschel, Phys. Rev. E 53 (1996) 5382.[81] S. Job, F. Melo, A. Sokolow, S. Sen, Phys. Rev. Lett. 94 (2005) 178002.[82] M. Manciu, S. Sen, A.J. Hurd, Phys. Rev. E 63 (2001) 016614.[83] F. Manciu, S. Sen, Phys. Rev. E 66 (2002) 016616;

Z.Y. Wen, S.-J. Wang, X.-M. Zhang, L. Li, Chinese Phys. Lett. 24 (2007) 2887.[84] E. Avalos, R.L. Doney, S. Sen, Chinese J. Phys. 45 (2007) 666.[85] C.J. Thompson, Classical Equilibrium Statistical Mechanics, Pergamon, Oxford, 1988.[86] J.-P. Boon, S. Yip, Molecular Hydrodynamics, Dover, New York, 1992.[87] T.R. Krishna Mohan, S. Sen, Pramana J. Phys. 64 (2005) 423.[88] J. Florencio Jr., M.H. Lee, Phys. Rev. A 31 (1985) 3231.[89] W. Ma, C. Liu, B. Chen, L. Huang, Phys. Rev. E 74 (2006) 046602.[90] R.L. Doney, Ph.D. Thesis, SUNY-Buffalo, 2007.

[91] R.L. Doney, S. Sen, in press.[92] ASM Metals Reference Book, p. 268; ASM Metals Handbook, Properties and Selection of Metals, 8th edition, vol. 1, p. 52.[93] J. Hong, A. Xu, Appl. Phys. Lett. 81 (2002) 4868.[94] V.F. Nesterenko, C. Daraio, E.B. Herbold, S. Jin, Phys. Rev. Lett. 95 (2005) 158702.[95] C. Daraio, V.F. Nesterenko, E.B. Herbold, S. Jin, Phys. Rev. Lett. 96 (2006) 058002.[96] J. Hong, Phys. Rev. Lett. 94 (2005) 108001.[97] S. Sen, M. Manciu, J.D. Wright, Phys. Rev. E 57 (1998) 2386.[98] E. Hasco et, H.J. Herrmann, Eur. Phys. J. B 14 (2000) 183.[99] J. Hong, H. Kim, J.-P. Hwang, Phys. Rev. E. 61 (2000) 964.

[100] A.R. Rao, G.A. Engh, M.B. Collier, S. Lounici, J. Bone Joint Surg. Am. 84 (2002) 1849.[101] S. Abrate, Impact on Composite Structures, Cambridge, Cambridge, 1998.[102] M.Y.H. Bangash, Impact and Explosion, CRC, Boca Raton, FL, 1993.[103] D.C. Lagoudas, K. Ravi-Chandar, K. Sarh, P. Popov, Mech. Mater. 35 (2003) 689.[104] Y.M. Gupta, S. Sharma, Science 277 (1997) 909.[105] T. Hirai, in: R.J. Brook (Ed.), Materials Science and Technology, in: Processing of Ceramics Part II, 17B, Wiley, New York, 1996, p. 293.[106] H.A. Bruck, Int. J. Solids Struct. 37 (2001) 6383.[107] S. Sen, M. Manciu, F.S. Manciu, Physica A 299 (2001) 551.[108] D. Wu, Physica A 315 (2002) 194.[109] M. Nakagawa, J. Agui, D. Wu, D. Extramiana, Granul. Matt. 4 (2003) 167.[110] A. Sokolow, J.M.M. Pfannes, R.L. Doney, M. Nakagawa, J.H. Agui, S. Sen, Appl. Phys. Lett. 87 (2005) 254104.[111] R.L. Doney, S. Sen, Phys. Rev. E 72 (2005) 041304.[112] F. Melo, S. Job, F. Santibanez, F. Tapia, Phys. Rev. E 73 (2006) 041305.[113] R.L. Doney, S. Sen, Phys. Rev. Lett 97 (2006) 155502.[114] R.L. Doney, Ph.D. Thesis, SUNY-Buffalo, 2007.[115] W.J. Stronge, Impact Mechanics, Cambridge, Cambridge, 2000.[116] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon, Oxford, 1987.[117] J.L. Nichols, J.S. Cory, J. Appl. Phys. 61 (1987) 972. Sec. 6.

[118] R.L. Doney, S. Sen, in: W.J. Flis, B.R. Scott (Eds.), Ballistics 2005, Vol. 2, DESTech, Lancaster, PA, 2005.

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