CHAOS
VOLUME 10, NUMBER 3
SEPTEMBER 2000
Dynamics of a gravitationally loaded chain of elastic beads
Marian Manciu, Victoria N. Tehan, and Surajit Sena)
Department of Physics, State University of New York at Buffalo, Buffalo, New York 14260-1500
共Received 10 August 1999; accepted for publication 3 January 2000兲
Elastic beads repel in a highly nonlinear fashion, as described by Hertz law, when they are
compressed against one another. Vertical stacking results in significant compressions of beads at
finite distances from the surface of the stack due to gravity. Analytic studies that have been reported
in the literature assume acoustic excitations upon weak perturbation 关J. Hong et al., Phys. Rev. Lett.
82, 3058 共1999兲兴 and soliton-like excitations upon strong perturbation 关V. Nesterenko, J. Appl.
Mech. Tech. Phys. 5, 733 共1983兲; S. Sen and M. Manciu, Physica A 268, 644 共1999兲兴. The present
study probes the position, velocity and acceleration and selected two-point temporal correlations and
their power spectra for individual beads for cases in which the system has been 共i兲 weakly, 共ii兲
strongly, and 共iii兲 moderately perturbed at the surface in the sense specified in the text. Our studies
reveal the existence of distinctly different dynamical behavior of the tagged beads, in contrast to
conventional acoustic response, as the strength of the perturbation is varied at fixed gravitational
loading. We also comment on the effects of polydispersity on system dynamics and probe the
relaxation of isolated light and heavy beads in the chain. © 2000 American Institute of Physics.
关S1054-1500共00兲00902-2兴
time evolution behavior of weakly perturbed systems and of
their relaxation to their original equilibrium states can be
formally described in terms of Kubo’s linear response
theory.1
When the initial perturbation imparts a significant
amount of energy to a subset of particles of the system, the
Hamiltonian describing the system may need to be modified
to account for the interactions that may be invoked due to the
additional energy acquired by the system. As far as we are
aware of, there is no general and well tested theoretical infrastructure to describe time evolution processes and the
long-time behavior of many particle systems that have been
subjected to strong perturbations.
In the present work, we focus upon a strongly anharmonic system that is subjected to ␦-function perturbations of
variable amplitudes at a boundary. The response of the system is then probed by numerically solving the relevant system of nonlinear equations of motion. We start with a finite
chain of elastic beads.5,6 The calculations are structured in
such a way that the boundary effects do not affect the dynamics. The chain can be thought of as being oriented horizontally, vertically and in any angle that is greater than 0
degrees and is less than or equal to 90 degrees.7 For all
orientations except the horizontal one, an external gravitational force acts on each bead. The role of the gravitational
force, is to progressively load the beads as one gets farther
from the ‘‘surface’’ bead. Dynamical behavior of chains of
elastic beads with progressive loading is markedly different
from that with constant loading and is discussed in Ref. 7. As
we shall see, the strength of the gravitational force and the
magnitude of the perturbation imparted to the system at time
t⫽0 strongly affect the dynamical behavior of the system.8,9
In our studies, without any loss of generality, we shall keep
the magnitude of the gravitational force constant and vary
the amplitude of the ␦-function perturbation. Also, the per-
In 1881, Hertz showed that elastic beads repel upon intimate contact in a strongly nonlinear fashion. We have
shown that a ␦-function perturbation of arbitrary magnitude imparted at a boundary of a chain of elastic beads
that barely touch one another, develops into a traveling
solitary wave of finite width. When the beads are initially
in equilibrium and subjected to progressive gravitational
loading, however, the ␦-function perturbation becomes
dispersive as it travels in space and time. In this study, we
probe the nature of spreading of the perturbation for
various amplitudes of the initial perturbation in a gravitationally loaded chain of beads. We show that regardless
of how weak the amplitude of the perturbation may be,
the beads do not return to their original equilibrium configurations. Our studies recover expected dispersion behavior in the limits of vanishingly weak and overwhelmingly dominant initial perturbations. We probe the
relaxation behavior of the beads for various impulse amplitudes and show that weak polydispersity introduces
very slow relaxation of the beads in the chain. We close
with a discussion of the sojourn of a perturbation as it
backscatters off a light or heavy mass impurity.
I. INTRODUCTION
When a many particle system is weakly perturbed, a
vanishingly small amount of energy is imparted to the system. This infinitesimal additional energy typically resides on
a subset of particles of the system at time t⫽0. As t
progresses, the additional energy disperses in space.1 Eventually, as t→⬁, if the system is ergodic, every particle returns to its original equilibrium state.2–4 The study of the
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
1054-1500/2000/10(3)/658/12/$17.00
658
© 2000 American Institute of Physics
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Chaos, Vol. 10, No. 3, 2000
Dynamics of elastic beads
turbation will always be applied at a boundary of the system.
We choose this boundary to be the surface bead, which does
not suffer any gravitational loading by virtue of its position.
It has been suggested in the literature that when the initial perturbation is infinitesimally weak, the time evolution of
any bead, which is far enough from the surface, can be described using linearized equations of motion.10 It is also
known that the dynamical behavior has interesting properties
and exhibits soliton-like features for sufficiently strong
perturbations.5,11 The solitons were first observed experimentally by Lazaridi and Nesterenko.6 Recent experiments carried out on a horizontal chain of steel spheres, in which the
spheres are barely in contact, have reconfirmed the presence
of soliton-like excitations in this system12 and have improved
upon earlier work.
We find that for weak perturbations, where the bead displacements are assumed to be significantly less than the ambient compression of beads, the system responds in a manner
that is distinctly nonlinear. Hence, we contend that linearization of the equations of motion is of limited value. For strong
perturbations, we find that our model system supports the
propagation of shock-like fronts that possess some of the
characteristics of soliton-like behavior reported by
Nesterenko5,6 and by Sen and Manciu.10
We also address the dynamics of light and heavy impurity beads located at finite distances from the surface grain in
the system. It turns out that impurity beads backscatter a part
of the incoming perturbation. Hence, the dynamics of impurity grains are expected to provide valuable insights into the
process of backscattering of the perturbation for various amplitudes of the initial impulse. Finally, we extend our studies
to explore the effects of randomness in the size distribution
of beads.
This study is arranged as follows. In Sec. II we discuss
the equations of motion. Section III presents a summary of
earlier analytical work. The results for the dynamics of the
displacement function of a tagged bead in a monodisperse
chain, of correlation functions in the same system and of the
dynamics of impurity beads are presented in Sec. IV–VI,
respectively. The work is summarized in Sec. VII.
II. EQUATIONS OF MOTION
A single elastic bead can be characterized by two intrinsic parameters; the Young’s Modulus Y and the Poisson’s
ratio ␴. In addition, parameters may be needed to describe
the geometrical properties of the bead itself. For spherical
beads, only one parameter, namely the radius R, is needed. It
turns out that when two identical beads are in mutual contact
and are pressed against one another, potential energy is
gained by the two-bead system. This potential leads to a
repulsive force between the two beads.13 In one-dimension,
let z i and z i⫹1 denote the locations of two adjacent beads of
radii R and R ⬘ , and Young’s moduli and Poisson ratios Y, Y ⬘
and ␴ and ␴ ⬘ , respectively. Then one can define an ‘‘overlap
function’’ ␦ ⬅R⫹R ⬘ ⫺(z i ⫺z i⫹1 )⬎0, if the beads are compressed and is 0 when they are barely in contact. One can
show that for spherical beads13–15
V 共 ␦ 兲 ⫽a ␦ 5/2,
共1兲
659
and in general V( ␦ )⫽a ␦ p⫹1 ,p⬎1, where
a⫽ 共 52 D 共 Y ,Y ⬘ , ␴ , ␴ ⬘ 兲兲关 RR ⬘ / 共 R⫹R ⬘ 兲兴 1/2,
D 共 Y ,Y ⬘ , ␴ , ␴ ⬘ 兲 ⫽ 共 3/4兲关共 1⫺ ␴ 2 兲 /Y ⫹ 共 1⫺ ␴ ⬘ 2 兲 /Y ⬘ 兴 .
共2a兲
共2b兲
In Eq. 共1兲, the repulsive potential and hence the repulsive force are nonlinear. Thus, when two elastic beads that
are initially barely in contact, are compressed against one
another, no matter how small the compression is, an intrinsically nonlinear repulsive force comes into play. As we
shall see below, if the beads were initially slightly compressed, i.e., loaded, then further compression by the same
overlap ␦ would result in a repulsive force that is less in
magnitude compared to the nonloaded case. Thus, it matters
greatly as to whether the beads in the one-dimensional chain
are initially nonloaded or loaded 共i.e., precompressed兲.
In the nonloaded case, the dynamics of the elastic beads
is highly nonlinear in nature. As we shall see, it is in this
limit that one realizes the propagation of soliton-like excitations or shock fronts in a chain of elastic beads in contact. In
the loaded case, the soliton-like excitations become quickly
dispersive as the precompressed grains are able to ‘‘rattle’’
back and forth about their original positions. It is important
to observe that in the analyses of impulse propagation carried
out in the continuum approximation, the precompressed state
is formally required to obtain soliton behavior. While the
two claims are in apparent disagreement, the point to remember is that our study is conducted for the original equation of
motion while that of Nesterenko and others have been carried out with a modified equation and in the long wavelength
approximation. As far as we know, our study does not possess a continuum limit and hence cannot be mapped to that
of Nesterenko via some manipulation of a control parameter.
The linear response regime with acoustic propagation of the
perturbation is realized in the loaded regime.
In addition to the Hertz potential,13 the grains also experience a gravitational potential. We define the zero of this
potential at the bottom grain of the chain.
The equation of motion of a spherical grain, in a chain of
monodisperse spherical grains in contact, labeled i at location z i and moving with acceleration d 2 z i /dt 2 can be written
as
md 2 z i /dt 2 ⫽A 关共 A 0 ⫺z i ⫹z i⫺1 兲 3/2⫺ 共 ⌬ 0 ⫺z i⫹1 ⫹z i 兲 3/2兴
⫹mg,
共3a兲
where A⫽5a/2, g is the gravitational acceleration and R is
the radius of each grain 共assumed to be homogeneous for the
sake of simplicity兲. The position of each bead can be readily
obtained by using the condition of static equilibrium on the
bead, namely
␯ mg⫽ 共 p⫹1 兲 a ␦ 3/2,
共3b兲
where ␯ denotes the number of grains above the grain for
which the overlap and hence the position coordinate upon
gravitational loading is being determined.
The quantity ⌬ 0 ⬅2R. It is convenient to carry out a
change of coordinates and define the following quantity:
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660
Chaos, Vol. 10, No. 3, 2000
i
␺ i ⫽z i ⫺i⌬ 0 ⫹⌺ l⫽1
共 mgl/A 兲 2/3.
Manciu, Tehan, and Sen
共4兲
The equation of motion can now be rewritten as
md 2 ␺ i /dt 2 ⫽A 关 兵 ␺ i⫺1 ⫺ ␺ i ⫹ 共 mgi/A 兲 2/3其 3/2⫺ 兵 ␺ i ⫺ ␺ i⫹1
⫹ 共 mg 共 i⫹1 兲 /A 兲 2/3其 3/2兴 ⫹mg.
共5兲
Equation 共5兲 cannot be solved analytically. Nesterenko has
analyzed the nonlinear behavior of Eq. 共5兲 in the long wavelength regime5,6 and have reported that perturbation imparted
at one end of a chain of elastic beads propagates as a soliton
at large distances from the interface. Recently, Sen and
Manciu11 have solved Eq. 共5兲 approximately analytically for
arbitrary p with g⫽0. In this limit, they have reported that an
impulse propagates as a soliton-like bundle of energy that is
nondispersive in space and time. The results obtained agree
well with the long wavelength analysis based results of Nesterenko and co-workers5,6 for small p 共i.e., for p not exceeding two兲. The long wavelength approximation is not appropriate for large p. The predictions between the two studies
therefore differ significantly for large p, i.e., when the repulsive force between the grains become very steep. The problem of crossing of soliton-like energy bundles has also been
probed by Manciu et al.16 It turns out that two opposing,
identical soliton-like objects traveling along a chain completely negate one another at the point of intersection but not
at the immediate vicinity of the point of intersection. This
behavior is unusual and due to the strongly nonlinear nature
of the Hertzian interaction and is a property that could be
related to the small size of the soliton-like objects in a chain
of discrete grains.16
III. SUMMARY OF EARLIER ANALYTIC WORK
We remind the reader that Hong et al.10 have studied Eq.
共5兲 by linearizing the right-hand side, i.e., by assuming p
⫽1. These authors report the propagation of acoustic-like
waves, which evolve in space and time due to the presence of
gravity. Analyses of equations that are structurally similar to
or that are identical to Eq. 共5兲, in the ‘‘acoustic limit’’ 共i.e.,
when p⫽1兲, are also reviewed in the works of Boutreux
et al.17 and of Hill and Knopoff.18
There is no solution presently available that interpolates
between the limiting results discussed in Refs. 10 and 11.
The present study discusses a detailed numerical study of the
dynamics in the intermediate regime.
The approximate solution to Eq. 共5兲 for g⫽0 and arbitrary nonlinearity 共and hence arbitrary p兲, has been constructed in Ref. 11 and reads as follows:
␺ i 共 t 兲 ⫽ 共 A/2兲关 tanh f p 共 t⫺i⌬ 0 /c 兲 ⫹1 兴 ,
␶ i ⬅ 共 t⫺i⌬ 0 /c 兲 .
m ⳵ 2 ␺ i / ⳵ t 2 ⫽⫺ ␮ i 共 ␺ i ⫺ ␺ i⫺1 兲 ⫹ ␮ i⫹1 共 ␺ i⫹1 ⫺ ␺ i 兲 ,
共8兲
关 ⫺1⫺1/p 兴
is the force constant of the ith conwhere ␮ i ⫽ ␮ 1 i
tact and ␮ 1 ⫽mpg(A/mg) 1/p is the force constant of the first
contact. The description of ␺ i (t) is hence significantly different from that given in Eq. 共6兲. The displacement function
has strongly oscillatory components in space and time and is
dispersive as the wave propagates.10 According to Ref. 10
␺ i 共 t 兲 ⫽ ␺ 共 h,t 兲 ⬀h 共 ⫺1/4兲 , 关 1⫺1/p 兴 exp关 ⫺k 共 h 兲 h⫺i ␻ 共 h 兲 t 兴 ,
共9兲
where h denotes the depth of the bead from the surface
where h⬅0. The quantities k(h) and ␻ (h) are important to
obtain. It can be shown that
k 共 h 兲 ⫽ v ph 共 h 兲 / ␻ 共 h 兲 ,
共10兲
where
v ph 共 h 兲 ⫽c 1 共 2R 兲 共 1/2兲关共 1/p 兲 ⫺1 兴 h 共 1/2兲关 1⫺1/p 兴 ,
c 1 ⫽ 共 2R ␮ i / ␳ 兲 1/2
共11兲
and
␻ 共 h 兲 ⬀h 关共 61 兲 ⫺ 共 21 p 兲兴 .
共12兲
The physical description of ␺ i (t) or ␻ (h,t) will be discussed
in Secs. IV and V below.
The primary limitation of the above treatment is that Eq.
共8兲 is not valid near that end of the chain where the impulse
is generated. This is because of the fact that gravitational
loading is vanishingly small at the surface and hence linearization of Eqs. 共5兲–共8兲 is invalid in the vicinity of the surface. Thus, the best that one can hope for is that Eq. 共9兲 will
provide insights into the properties of the propagating pulse
at some very large depths inside the chain or for perturbations that are so weak that the ‘‘acoustic limit’’ is realized.
As we shall see below, our numerically calculated behavior
of displacement, velocity, acceleration, etc. suggest that at
finite depths (⬃10␣ ,1⬍ ␣ ⬍2), no perturbation can be regarded as ‘‘weak’’ in a meaningful way. We shall return to
this issue in Sec. IV below. Thus, it is unclear as to whether
the acoustic limit is realized in gravitationally loaded chains
of N(N⬍⬁) beads.
共6兲
where c is the velocity of propagation of the soliton-like
impulse through the chain of beads. For tⰆi⌬ 0 /c, tanh f p
→⫺1 and for tⰇi⌬ 0 /c, tanh f p→1. The function
⬁
C 2q⫹1 ␶ 2q⫹1
,
f p ⫽⌺ q⫽0
i
d m ␺ i (t⫽i⌬ 0 /c)/dt m ⬀A ␰ , where ␰ ⫽(p⫺1)/2, (3/2)(p⫺1)
⫺2, 3(p⫺1)⫺4,... for m⫽1,3,5,... . Manipulations of the
above mentioned condition yields the quantities C 1 ,C 3 ,C 5 ,
etc. The soliton velocity c⫽c( 兵 C 2q⫹1 其 ) and can be subsequently computed.16
If one assumes that ( ␺ i ⫺ ␺ i⫺1 )Ⰶ(mgi/A) 2/3 共we define
this statement as the ‘‘acoustic limit’’兲, Eq. 共5兲 simplifies to
共7兲
The coefficients C 2q⫹1 can be obtained in several ways.11 A
preferred approach is to obtain the higher time derivatives of
␺ i (t) at the symmetry point ␶ i ⫽0, i.e., t⫽i⌬ 0 /c for various
values of the soliton amplitude A. These results then yield
IV. DISPLACEMENT OF A TAGGED BEAD
Our studies are based upon particle dynamical simulations carried out using the Velocity-Verlet algorithm.19 Analytic description of the dynamics of tagged beads in the case
of variable strengths of initial impact to one end of the chain
is presently unavailable. We have used numerical constants
that are appropriate for a chain of quartz beads in our studies.
The bead diameter 2R⫽10⫺3 m, masses are set to m
⫽1.41⫻10⫺6 kg, the Young’s modulus Y ⫽7.87⫻1010
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Chaos, Vol. 10, No. 3, 2000
FIG. 1. Normalized displacement vs time for a tagged bead 共bead number
50 from the surface at which the perturbation is imparted兲 for various magnitudes of v at fixed g.
N-m⫺2 and the Poisson’s ratio ␴ ⫽0.144, both of the material
dependent constants being appropriate for quartz. It is helpful to note that the velocity of the impulse is a few m/s and
is much slower than sound speed in a given material 共e.g.,
sound speed in quartz is ⬇6000 m/s兲. Therefore, for our
analyses, it is reasonable to suppose that the quartz beads can
be replaced by point particles and the internal degrees of
freedom of the beads are decoupled with the physics of impulse propagation in the chain.20,21
In Fig. 1 we show the behavior of ␺ i (t)⬅ ␺ (t) for a
tagged grain, chosen to be the 50th grain when counted from
the end of the chain at which the perturbation has been generated. The data present results for fixed g
⫽9.8 ␮ m/ms2⫽9.8 m/s2. The quantity v denotes the initial
velocity of the bead that has been perturbed. The negative
sign of v indicates that the direction of the velocity is such
that it compresses the beads. We have varied v across four
decades in our studies.
The key feature of the displacement function is that in
the region t⫽t * and t⫺t * ⫽ ⑀ →0, t * being the instant at
which the tagged bead starts to move, ␺ behaves like a hyperbolic tangent function as in Eq. 共6兲 above. To leading
order, the magnitude of ␺ scales linearly with v . At larger
values of ⑀, the function ␺ begins to decay in an oscillatory
fashion. The time window across which the oscillatory behavior survives is longer when v is weaker. This behavior is
qualitatively consistent with the form in Equation 共9兲. As v
is increased, one can see that the oscillation decays progressively rapidly and then begins to exhibit rather slow decay.
Further weakening of v does not significantly change the
behavior reported for v ⫽⫺1⫻10⫺3 ␮ m/ms case. Our studies suggest that the system remains in a compressed state for
many decades in time. It is presently unclear as to how a
tagged bead settles at a position long after the initial pulse
has passed through it. Simulations across many decades in
time and for extremely large system sizes are needed to ascertain the asymptotic behavior of ␺.
The spatial extent of the impulse is ⬇5 grains in the
limit when v →⬁ 11 and the soliton regime is recovered.
Dynamics of elastic beads
661
FIG. 2. Width of the traveling pulse ␭ vs t for various magnitudes of v .
Observe the approximately linear increase of ␭ in t, with the slope remains
approximately constant between v ⫽⫺1⫻10⫺3 ␮ m/ms and v ⫽1
⫻10⫺2 ␮ s/ms 共not shown兲 and then decreasing for larger magnitudes of v .
The plot at the bottom right panel records the increase in d␭/dt as v becomes weaker.
However, for v ⬍⬁ and g⫽0, the spatial extent of the perturbation widens as it travels through the chain. We have
probed the spatial extent ␭ of the perturbation as a function
of space between v ⫽⫺1⫻10⫺3 ␮ m/ms and ⫺1
⫻102 ␮ m/ms. The results are shown in Fig. 2.
We make the following observations on the basis of the
results reported in Fig. 1.
共i兲
共ii兲
共iii兲
It may be possible to represent ␺ (t) for arbitrary v
and g approximately using Eq. 共6兲 and the oscillatory
behavior obtained from studies using Eq. 共9兲. The
available studies are unable to describe the fast oscillatory decay of ␺ (t) to a slow and nonoscillatory regime which appears to last for macroscopically large
times.
The exponent associated with the decay of the maximum amplitude of ␺ (t⫽t * ) to its original or some
new equilibrium state and the behavior of the dispersion of the perturbation as function of space depend
upon the details of v for fixed g.
Even for very weak perturbations as in v ⫽⫺1
⫻10⫺3 ␮ m/ms, where the displacements are negligible 共⬃10⫺2 Angstroms兲, our studies suggest that
displacement remains at some 50% of the maximum
compression suffered up to relatively long times after
the perturbation reached the tagged bead. Thus, we
contend that the acoustic approximation,10 which implicitly assumes that the perturbation is weak enough
such that the system monotonically relaxes to its
original equilibrium state, is of limited value in analyzing the problem of propagation of impulses in
gravitationally loaded columns.
The data shown in Fig. 2 suggests that for weak perturbations, the spatial width of the propagating pulse, ␭, increases linearly in time. Due to the oscillatory nature of the
propagating impulse, it is difficult to estimate ␭ with absolute certainty. The slope d␭/dt decreases with increasing v ,
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662
Chaos, Vol. 10, No. 3, 2000
Manciu, Tehan, and Sen
FIG. 3. Velocity, normalized velocity correlation function, and velocity
power spectrum for the tagged bead for v ⫽⫺1⫻10⫺3 ␮ m/ms. Observe
that the magnitude of v dictates the magnitude of velocity and is distinctly
oscillatory.
which is an expected result. The data shows that d␭/dt→0
with ␭⬇5 beads when v becomes sufficiently large and
again becoming approximately constant for v →0. The precise decay law for ␭ 共measured in bead diameters兲 is difficult
to evaluate reliably using numerical analyses due to the oscillatory nature of the propagating impulse.
V. CORRELATION FUNCTIONS OF TAGGED BEADS
A. Monodisperse chain
A standard approach to probing the relaxation of a
tagged particle or bead in a perturbed many body system is
to characterize the behavior of correlation functions. We
have probed the velocity and acceleration of our tagged grain
for various values of v . We have also computed the normalized velocity and acceleration relaxation functions that are
defined as follows:
C 共 t 兲 ⫽⌺ 兵 ␶ i 其 关 ␽ 共 t⫹ ␶ i 兲 ␽ 共 ␶ i 兲兴 / 关 ⌺ 兵 ␶ i 其 ␽ 共 ␶ i 兲 ␽ 共 ␶ i 兲兴 ,
FIG. 4. Velocity, normalized velocity correlation function, and velocity
power spectrum for the tagged bead for v ⫽⫺1⫻10⫺1 ␮ m/ms. The behavior of v is identical to that in Fig. 3, which indicates that for the regime of
v being studied, the behavior probably remains functionally unchanged as v
is weakened.
where the tail of the velocity function in time is extended,
the time-dependent behavior of the envelope function is not
describable by some simple law.
Given that the perturbation is dispersive in space and
time, the data for various magnitudes of v , from large to
small, carry information about impulse propagation as a
function of increasing depth and hence of increasing time.
This is evident upon comparing the upper panels of Figs.
3–6. The spatio-temporal stretching of the tail is clearly related to the spatio-temporal reduction in the leading amplitude of the velocity function. Our calculations indicate that
the leading amplitude of the velocity function decays with
depth algebraically with slope ⫺0.250. We have also carried
out the corresponding check for the algebraic decay of the
leading amplitude of displacement function with depth and
find a value of ⫺0.0835. Both of these results are consistent
with the calculations of Hong et al.10 However, the theoret-
共13兲
where ␽ ⫽velocity or acceleration. If the velocity or acceleration of the tagged bead becomes uncorrelated with its initial velocity or acceleration, then C(t)→0. Observe that
C(t)→0 does not necessarily imply that the system reaches
equilibrium. We have also computed the Fourier transforms
of each of the velocity and acceleration correlation functions
that we have computed. The velocity and selected acceleration data with the appropriate C(t) and C( ␻ ) are reported in
Fig. 3 for v ⫽⫺1⫻10⫺3 ␮ m/ms, in Fig. 4 for v ⫽⫺1
⫻10⫺1 ␮ m/ms, in Fig. 5 for v ⫽⫺1 ␮ m/ms and in Fig. 6 for
v ⫽⫺10 ␮ m/ms, i.e., across four decades in the magnitude
of v .
The upper panel of Fig. 3 describes the velocity of the
tagged bead as a function of time. As can be seen, the velocity is oscillatory and decaying as a function of time.
Given the short decay period in time, we have not been able
to obtain the temporal decay law 共i.e., whether it is exponential, algebraic, etc.兲. Our data shows that even at large depth,
FIG. 5. Velocity, normalized velocity correlation function, and velocity
power spectrum for the tagged bead for v ⫽⫺1 ␮ m/ms. The time window
through which the bead moves is much shorter than those in Figs. 3 and 4.
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Chaos, Vol. 10, No. 3, 2000
FIG. 6. Velocity, normalized velocity correlation function, and velocity
power spectrum for the tagged bead for v ⫽⫺10 ␮ m/ms, a rather large
initial impact. Comparison with data in Ref. 11 reveals that the behavior of
velocity is nearly characteristic of the soliton-like regime. The Gaussiantype pulse is not quite symmetric in shape and hence the departure from
purely soliton-like behavior. There are some oscillations in v and C(t),
presumably because of gravitational loading and the production of secondary impulses at the surface bead.
ical calculations of the displacement functions at large times
for the various cases of v , as carried out by Hong et al.,10
may need some refinement. This is to be expected in view of
the fact that the displacement function is the actual solution
to the equation of motion and hence contains more information than the velocity function.
Our results on the velocity relaxation behavior for monodisperse chains reveal the following behavior.
共i兲
共ii兲
The velocity correlation function, C(t), for the tagged
bead relaxes rapidly in less than 0.1 ms, for all v
values except for v ⫽⫺10 ␮ m/ms, i.e., at large initial
impacts, when the propagating pulse is compact and
presumably followed by aftershocks initiated during
the generation of the original impact itself 共see Figs. 1
and 2 in Sen et al.7兲.
The power spectrum is broad, indicating the presence
of a continuum distribution of system frequencies 共associated with quartz–quartz interaction兲 centered between 1.0⫻105 and 1.5⫻105 Hz. In this connection it
may be noted that the harmonic frequency associated
with the bead–bead interaction is ⬇107 Hz. Thus, the
frequencies reported in the lower panel of Figs. 3–6
are related to the bead dynamics associated with the
propagating impulse itself and not perturbation of the
harmonic frequencies of the system. It may be noted
that this power spectrum is remarkably stable given
the significant variation in the magnitude of the perturbation used in the calculations.
Figures 7 and 8 describe the acceleration function, its
correlation and the acceleration power spectrum for v ⫽⫺1
⫻10⫺3 ␮ m/ms 共weak perturbation兲 and v ⫽⫺10 ␮ m/ms
共strong perturbation兲, respectively. The overdamped and oscillatory decay of the acceleration correlation function,
which is proportional to the memory function in the linear
Dynamics of elastic beads
663
FIG. 7. Acceleration, normalized acceleration correlation, and acceleration
power spectrum for the tagged bead with v ⫽⫺1⫻10⫺3 ␮ m/ms. Observe
the differences in the plots of acceleration and velocity.
response limit, is reminiscent of that of particles in a liquid.22
The acceleration correlation functions and its Fourier transforms differ significantly between Figs. 7 and 8. The solitonlike behavior of the acceleration that is evident in the upper
panel of Fig. 811 yields sharply decaying C(t), indicating
that the perturbation leaves the tagged bead within a much
shorter time compared to the case in Fig. 7.
It is, however, important to observe that for the case in
Fig. 8, the system does not return to its original equilibrium
state when C(t) appears to vanish but rather strives to settle
in a very different state. We expect that if one waits long
enough, the system will evolve further, from the compressed
state to a less compressed state. The probing of such extended scale time evolution is likely to take enormous
amount of computing time and accuracy and hence has not
been attempted in this study. Thus, since ␺ is not constant,
v ⫽0, but very small, up to macroscopically large times 共al-
FIG. 8. Acceleration, normalized acceleration correlation, and acceleration
power spectrum for the tagged bead with v ⫽⫺10 ␮ m/ms. The behavior of
acceleration is nearly soliton-like. The fast decay of C(t) indicates that the
pulse leaves the bead within a very short time 共about 0.025 ms兲.
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664
Chaos, Vol. 10, No. 3, 2000
Manciu, Tehan, and Sen
though our data apparently suggest that v ⫽0 for sufficiently
large t兲. The power spectrum in Fig. 8 reveals a broader
hump compared to the same in Fig. 7 and exhibits the presence of many high frequency modes. The high-frequency
modes are associated with the finiteness of the energy bundle
itself 共as is the case for soliton-like wave兲.
We summarize the results below.
共i兲
共ii兲
共iii兲
The leading amplitudes of the displacement and velocity functions of a tagged bead decay in depth algebraically with slopes ⫺0.0835 and ⫺0.25, respectively.
The rapidly decaying velocity correlation function,
which is bounded between 1 and 0, is not a revealing
measure of system relaxation in nonlinear systems
that do not generally relax to their original equilibrium states 共as evident upon comparing the central
panels of Figs. 3 and 6兲.
The acceleration correlation function shows significant differences between the weak and strong initial
impact cases, being short-lived and exhibiting significant high-frequency modes in the latter instance.
B. Polydisperse chain
In this section, we present a brief discussion on the dynamics of a polydisperse chain of tagged beads in which the
bead radii vary uniformly randomly within chosen limits.
Since mass is proportional to R 3 , the corresponding variation
in bead masses is more significant. The data we present is for
a randomness of R(1⫾0.05). Although this may not seem
like a significant randomness at first sight, it is important to
note that in one-dimensional systems the path of propagation
of any perturbation is highly constrained. Any excitation
must travel through all the grains. In higher dimensions, the
excitation can select the path that can avoid the most significant size variations of the beads. Thus, the effects of randomness on impulse propagation are more dramatic in onedimensional systems when compared with propagation in the
same system in two and three dimensions.
The equations of motion for systems with randomly
varying masses differ from Eq. 共3a兲 in the sense that m and A
are replaced by m i and A i , respectively. Furthermore, Eq.
共3b兲 needs to be changed to the following:
i⫺1
m i ⫽ 共 p⫹1 兲 a i ␦ 3/2
g⌺ i⫽1
i .
FIG. 9. Acceleration, normalized acceleration correlation, and acceleration
power spectrum for the tagged bead with v ⫽⫺1⫻10⫺1 ␮ m/ms when the
grain radii vary randomly as R(1⫾0.05). Observe the long-lived time dependence in C(t) and the sharp peaks in C( ␻ ) due to the lack of translation
invariance. The frequency window in which C( ␻ ) remains finite is about
the same as that in Fig. 7 but is different compared to the same in Fig. 8.
functions are significantly altered at large times due to the
presence of randomness in the grains. Upon comparing the
upper panels in Figs. 9 and 10 with the same in Figs. 7 and
8, we see that the magnitude of the fluctuations in acceleration, at times after the impulse has passed through, depends
upon v . Hence the response of the random system to differences in radii is also strongly nonlinear. Thus, increasing v
for fixed randomness increases the noise in the system.
One way to interpret the observation that increasing v
leads to increasing noise is to postulate that any propagating
perturbation prefers to travel as a weakly dispersive energy
bundle. The frequencies that constitute the bundle are not
strongly sensitive to v . However, the displacement amplitudes of the beads increase upon increasing v . Thus, increas-
共14兲
It is technically challenging to establish equilibrium in this
system. The slightest deviation from the condition in Eq.
共14兲 introduces significant errors in computing the forces on
the beads and hence on the system dynamics.
Figures 9 and 10 show the acceleration, acceleration correlation function and the corresponding power spectrum for
cases in which v ⫽⫺1⫻10⫺1 ␮ m/ms 共moderately weak perturbation regime兲 and v ⫽⫺10 ␮ m/ms 共strong perturbation
regime兲. Upon comparing the data in Fig. 7 with Fig. 9 and
the same between Figs. 8 and 10, we find that there are no
significant differences between the plots for the monodisperse and random cases 共note that the magnitudes of v in
Figs. 7 and 9 differ significantly兲 except that the correlation
FIG. 10. Acceleration, normalized acceleration correlation, and acceleration
power spectrum for the tagged bead with v ⫽⫺10 ␮ m/ms for the polydisperse system. The frequency window in which C( ␻ ) remains finite is about
the same as that in Fig. 8.
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Chaos, Vol. 10, No. 3, 2000
FIG. 11. Displacement vs time for a light impurity for various v ’s spanning
4 decades. Oscillatory decay in ␺ is observed at low v ’s. Oscillations are
suppressed but relaxation is slower at large v ’s.
ing v leaves the frequency response of the system largely
invariant but affects the degree of noise in the system.
The behavior of C(t) in Figs. 9 and 10 indicate that the
correlations are long lived, lasting up to a decade longer than
that in the monodisperse cases. The power spectra remain
surprisingly similar between the monodisperse and random
cases. The similarity of the power spectra is also indicative
of the fact that the presence of randomness spawns nonlinear
excitations that perhaps differ only in magnitude from the
behavior of the propagating primary perturbation.
It is difficult to carry out simulations for chains with
significantly larger randomness in one dimension. The key
reason for the difficulty being that the signal and the noise
become comparable and highly sophisticated spectral analyses tools are then needed to identify the signal itself. In that
regime, it is possible that one may find the analyses of correlation functions and spectral data more useful than the depiction of the dynamical variable itself.
VI. IMPURITY DYNAMICS
The dynamics of an impurity mass in a chain has been a
topic of fundamental interest in statistical physics.2,3,23,24 It is
well known, for instance, that a light impurity in a chain of
harmonic oscillators leads to the presence of the ‘‘plasma
mode’’ that has an infinite lifetime. On the other hand, a
heavy impurity exhibits algebraically decaying but overdamped oscillatory behavior.
In that spirit, we discuss the time evolution of an impurity bead of mass m ⬘ for m ⬘ Ⰶm, m ⬘ ⬇m and m ⬘ Ⰷm for this
gravitationally loaded chain. The impurity is placed at the
center of the chain and the tagged bead now serves as the
impurity. The calculations reveal that it is most useful to
probe the displacement functions of the tagged impurity
beads.
Figures 11 and 12 show the time evolution of the displacement function of the impurity bead for m ⬘ ⫽0.1 m and
m ⬘ ⫽10 m, respectively. In all instances, as before, the bead
is at rest until it is excited by the perturbation. The perturba-
Dynamics of elastic beads
665
FIG. 12. Displacement vs time for a heavy. The displacement function ␺
relaxes to a larger displacement, which is similar to sagging of the heavy
bead after it has been perturbed.
tion compresses the bead against the next bead 共along the
direction of propagation兲 and hence the displacement from
the original equilibrium position is negative. The data reveal
the following features.
共i兲
共ii兲
共iii兲
The data shows that in each of the panels, the nonlinear perturbation results in grain compression that is
proportional to the perturbation itself.
The restoration of the grain position to some longlived metastable phase depends upon the magnitude
of v and typically requires some 0.35 ms for our system 共with quartz host beads兲. Observing the panels in
Figs. 11 and 12 reveals that for large v , the magnitude
of displacement at the metastable state 共reached after
the perturbation has passed through the system兲 is
sensitive to the impurity mass, being larger in magnitude for larger m ⬘ .
The increase in the slope of ␺ as a function of time
depends upon v . For small v , the slope increases
more gradually, as one would intuitively expect.
We have probed the velocity, acceleration, the corresponding correlation functions and the corresponding power
spectra for the impurity bead for m ⬘ ⫽0.1 m, m ⬘ ⫽1.1 m, and
m ⬘ ⫽10 m. There is no significant difference between the
respective quantities for the impurity bead and those of the
tagged bead of the monodisperse chain reported earlier except for large v cases.
Figures 13 and 14 describe acceleration, acceleration
correlation and its power spectra for cases m ⬘ ⫽0.1 m and
m ⬘ ⫽10 m for v ⫽10 ␮ m/ms. The light mass case with m ⬘
⫽0.1 m supports the propagation of a soliton-like pulse, as is
evident from the upper panel of Fig. 13.11 Clearly, for the
chosen magnitudes of v and g, the dispersive nature of the
pulse is minimal. The perturbation, however, is significantly
damped for the m ⬘ ⫽10 m case. In both instances, examination of the acceleration data reveals that the compact solitonlike wave is split by the impurity and leads to the formation
of several smaller soliton-like waves. This observation is
consistent with the experiments of Nesterenko and
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666
Chaos, Vol. 10, No. 3, 2000
Manciu, Tehan, and Sen
共i兲
共ii兲
共iii兲
FIG. 13. Acceleration, normalized acceleration correlation, and acceleration
power spectrum for the light impurity for large v showing the characteristic
behavior of the soliton-like regime.
collaborators.25 The process of such splitting is complex and
dedicated analyses of the dynamics of transport of solitonlike waves through impurities need to be carried out. A key
difficulty in the analyses of splitting of the soliton-like waves
enters from the fact that most of the spawned solitons are
very weak and hence move extremely slowly. Thus, extremely high precision calculations are needed to study them.
Such calculations require exceptionally fast computing ability because the simulations must run for very long times.
Figures 15共a兲–15共c兲 present snapshots in time of velocities of individual beads for m ⬘ /m⫽0.1, 1.1, and 10, respectively, when subjected to initial perturbations of various
magnitudes. Each figure presents data in six panels with each
panel representing the identical snapshot in time for various
magnitudes of v spanning 4 decades in the magnitude of v .
We make the following observations.
FIG. 14. Acceleration, normalized acceleration correlation, and acceleration
power spectrum for the heavy impurity for large v showing the strong
damping in time of the acceleration function that is characteristic of the
soliton-like regime. Observe that secondary oscillations 共see text兲 are
present long after the perturbation has traveled through the heavy bead.
Regardless of the magnitude of v , presence of any
m ⬘ ⫽m leads to a backscattered pulse.
The amplitude of the velocity of the leading edge of
the backscattered pulse possesses the same sign 共i.e.,
both negative兲 as that of the in going pulse when m ⬘
⬍m and the opposite sign when m ⬘ ⬎m, consistent
with observations noted earlier by Sen et al.7
At shallow depths 共such as the depths studied兲, the
ratio of the amplitude of the leading edge of the backscattered and incoming pulses at any given instant of
time, referred to as A(b) and A(i), respectively, is
devoid of any measurable time dependence and is
proportional to the ratio m ⬘ /m. The sign of the ratio is
positive for m ⬘ ⬍m and negative otherwise.
Figure 16 presents a detailed study of the observation in
共iii兲 above and reveals that A(b)/A(i)⫽g 1 ( v )⫺g 2 ( v )
⫻(m ⬘ /m). Our analyses show that g 1 ⫽0.059, 0.068, 0.086,
0.096, and g 2 ⫽0.065, 0.073, 0.089, and 0.101 for v ⫽⫺1
⫻10⫺2 ␮ m/ms, ⫺1⫻10⫺1 ␮ m/ms, and ⫺10␮m/ms, respectively. The data fits the following behavior, g 1 ( v )
⫽0.084⫹0.013 log10(⫺v) and g 2 ( v )⫽0.088⫹0.0124 log10
(⫺ v ), which is roughly consistent with the requirement
A(b)/A(i)⫽0 for m ⬘ ⫽m.
We have also probed the behavior of the forward propagating and backscattered pulses for a polydisperse chain with
5% random variation bead radii with an impurity in the
above mentioned range of m ⬘ . Our results do not reveal any
significant differences with the case of the monodisperse
chain except for the presence of a noisy background in all the
data. It would be interesting to probe the backscattering
problem for chains with a high degree of polydispersity.
Such studies are in progress and will be reported in a separate publication.25
VII. SUMMARY AND CONCLUSIONS
We have presented a numerical study of the Newtonian
dynamics of tagged beads in a gravitationally loaded chain of
elastic beads. The studies have been carried out for various
magnitudes of initial ␦-function perturbations, characterized
by the velocity v of the surface bead at t⫽0. We have studied cases in which the chain is 共i兲 monodisperse, 共ii兲 polydisperse, and 共iii兲 the tagged bead is a mass impurity in the
chain.
Our calculations shed light on the dynamics of beads in
the gravitationally loaded elastic bead chain. One would expect that for v →⬁, the perturbation would propagate as a
soliton-like object as demonstrated in earlier studies of Nesterenko and co-workers,5,6 Sen and co-workers7–9,11 and in
the experimental studies of Nesterenko5,6 and of Coste
et al.12 Our calculations are consistent with this expectation
and we reach the special ‘‘soliton-like object limit’’ for the
largest magnitudes of v that we have considered. This is
important in view of the fact that loading is always present in
our study. In the opposite limit of v →0, a recent study by
Hong et al.10 suggests that an acoustic analysis of the dynamics of beads would be appropriate. Indeed, the descriptions of velocity and acceleration of beads obtained within
the acoustic approximation appear to be consistent with nu-
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Chaos, Vol. 10, No. 3, 2000
Dynamics of elastic beads
667
FIG. 15. Velocity of beads at fixed time showing forward propagating and backscattered perturbations for various v ’s between ⫺1⫻10⫺3 ␮ m/ms and ⫺10
␮m/ms for 共a兲 m ⬘ /m⫽0.1 共light impurity兲; 共b兲 m ⬘ /m⫽1.1, and 共c兲 m ⬘ /m⫽10 共heavy impurity兲. Clearly, heavier objects are most easily detectable.
merical analyses. However, the behavior of the displacement
function for any bead appears to be different from what one
would expect. Thus, we contend that more studies are needed
to understand bead dynamics at finite g and any v ( v ⬍⬁).
In the regime where loading is constant and v is arbitrary, we find the following key properties that characterize
the dynamics of beads in the system.
共i兲
Regardless of the magnitude of v , the normalized displacement function ␺, does not return to the original
equilibrium position up to times that are significantly
larger than the relaxation time for the system, where
the relaxation time is estimated based upon the time
needed for the normalized velocity and acceleration
correlations to decay to 0.
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668
Chaos, Vol. 10, No. 3, 2000
Manciu, Tehan, and Sen
FIG. 15. 共Continued.兲
共ii兲
共iii兲
For finite v , the width of the soliton, ␭, which can
only be crudely measured from the data, increases as
the energy bundle propagates, as a function of time.
The quantity d␭/dt remains approximately constant
for small enough v 共see Fig. 2兲 and then decreases
nonlinearly with increasing v , as expected. We found
that as v →⬁,d␭/dt→0.
The velocity and acceleration functions that we obtain
agree well with the results expected from the works of
Hong et al.10 and of Sen and Manciu.11 This suggests
共iv兲
共v兲
that the analyses of Hong et al.10 in the acoustic approximation correctly describe the higher derivatives
of the displacement function.
Our studies indicate that the key features of the displacement, velocity, and acceleration functions are
preserved when weakly polydisperse chains are considered. The functions, however, possess sizeable
fluctuations long after the primary pulse has passed
through the system and exhibit long-lived correlation
functions.
We have probed the dynamics of isolated light and
heavy impurities in gravitationally loaded elastic bead
chains. Any traveling perturbation that passes through
an impurity is backscattered by it. Our study shows
that the ratio of the leading amplitudes of the backscattered and incident perturbations is proportional to
m ⬘ /m, where m ⬘ is the impurity mass. We do not
consider the case in which m ⬘ →⬁.
Our work in progress26 is focused upon the construction
of a functional form for ␺ that incorporates the key features
of the displacement function for the gravitationally loaded
chain of elastic beads.
ACKNOWLEDGMENTS
FIG. 16. Plots show the ratio of the leading velocity amplitude of the backscattered vs propagating pulses for various v ’s. Observe that the ratio vanishes at m ⬘ /m⫽1, as expected.
This research has been partially supported by the U.S.
Department of Energy contract from Sandia National Laboratories. We are grateful to Professor Francis M. Gasparini
and to Professor Jongbae Hong for valuable comments.
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Chaos, Vol. 10, No. 3, 2000
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