of 6
PHYSICAL REVIEW E
94
, 052907 (2016)
Wave propagation in one-dimensional microscopic granular chains
Wei-Hsun Lin
1
,
2
and Chiara Daraio
2
,
3
1
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
2
Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland
3
Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA
(Received 7 July 2016; revised manuscript received 4 October 2016; published 28 November 2016)
We employ noncontact optical techniques to generate and measure stress waves in uncompressed, one-
dimensional microscopic granular chains, and support our experiments with discrete numerical simulations. We
show that the wave propagation through dry particles (150
μ
m radius) is highly nonlinear and it is significantly
influenced by the presence of defects (e.g., surface roughness, interparticle gaps, and misalignment). We derive
an analytical relation between the group velocity and gap size, and define bounds for the formation of highly
nonlinear solitary waves as a function of gap size and axial misalignment.
DOI:
10.1103/PhysRevE.94.052907
I. INTRODUCTION
Ordered granular systems, often referred to as granular
crystals, are discrete arrays of solid particles arranged in
periodic lattice geometries. Macroscopic granular crystals,
composed of particles in the centimeter scale, have been
a subject of active research [
1
6
]. Granular crystals have
been shown to mediate nonlinear wave phenomena such as
solitary waves [
1
] and intrinsic localized modes [
7
], which
are of fundamental scientific interest. The dynamic response
of granular crystals is governed by highly nonlinear inter-
actions between neighboring particles, which is determined
by their contact geometry (e.g., Hertzian). More generally,
by modifying the contact geometry and particle arrangements
(i.e., changing the stress-strain relation and the direction of
contact forces) [
8
], granular crystals can be designed to behave
as highly flexible waveguides [
9
], focusing lenses [
10
], and
filtering materials [
11
]. These features of granular crystals
have inspired their use in engineering applications, such as
shock mitigation [
12
15
], acoustic rectification [
16
], and logic
elements [
17
]. However, the macroscopic size of the particles
tested experimentally to date imposes important limitations
to their applicability. The particle diameters determine the
length scale of the stress wave that can be transmitted through
the granular assemblies. For example, applications in acoustic
medical imaging or nondestructive evaluation, which employ
acoustic pulses in the ultrasonic range, require particle sizes in
the order of micrometers. Recent numerical work investigated
the stress propagation in nanometer-scale particle chains
composed of buckyballs, detailing important dynamic effects
rising at ultrasmall scales [
18
]. However, not much is known
to date about the response of granular chains in the micrometer
range.
In this work we study the stress wave propagation
in dry, one-dimensional microscopic granular chains us-
ing experiments and numerical simulations. Despite the
fundamental importance of understanding the dynamics of
microparticle assemblies, very little experimental work to
date has been conducted at these scales. The lack of
experimental investigation results from two major diffi-
culties: (i) the absence of reliable methods to assem-
ble and characterize dry microparticles in controlled con-
figurations, and (ii) the need for a systematic way to
excite microscopic granular particles without influencing
the response of the system. Experimental techniques em-
ployed for macroscopic granular systems include manually
assembled chains of particles, piezotransducers for applying
excitation [
19
], and sensors embedded in particles for force
measurements [
20
]. At the microscale, these techniques are not
applicable. The nonlinear interactions in granular systems [
21
]
are sensitive to the particle packing geometry and initial con-
ditions. To ensure uniform contacts between microparticles, a
high accuracy in particle positioning is necessary. In addition,
the application of controlled excitations and the measurement
of the propagating stresses cannot be achieved using contact
methods, which are not sufficiently accurate and intrusive.
To overcome these experimental challenges, we constructed
an experimental apparatus [Fig.
1(a)
], which employs a
computer-controlled micromanipulation system for the chain
assembly, and noncontact optical techniques for the character-
ization [
21
], as described in the next section.
II. EXPERIMENTAL SETUP AND
THEORETICAL MODELING
In this work, we study microscopic granular chains com-
posed of stainless steel particles (type 316 and 440c) with radii
of 150
μ
m. The steel 316 particles have a nominal surface
roughness and radius variation of
3
μ
m, while the steel 440c
particles have a roughness of 0.1
μ
m and a radius variation
of
1
μ
m, as provided by the manufacturer specifications
(New England Miniature Ball). The chains of particles are
assembled on supporting microgrooves, and aligned with
a micromanipulator, to ensure packing repeatability. The
microgrooves have a v-shaped cross section, which confines
the motion of the microparticles in one dimension. The grooves
are fabricated on silicon wafers by chemical etching with
potassium hydroxide and have opening widths of 240
μ
m
[Fig.
1(c)
]. To assemble the microparticles in one-dimensional
chains, the particles are first randomly deposited into the
grooves, and then compressed from both ends and positioned
adjacent to each other with the micromanipulator (Zaber
LSM025). The micromanipulator tip, controlled by a stepper
motor with a precision of 0.05
μ
m, is retracted after assembling
the chains, leaving the chains with free boundaries. The
2470-0045/2016/94(5)/052907(6)
052907-1
©2016 American Physical Society
WEI-HSUN LIN AND CHIARA DARAIO
PHYSICAL REVIEW E
94
, 052907 (2016)
FIG. 1. Schematic diagram of the experimental setup. (a) The
system consists of a laser excitation system, a high-speed microscopy
imaging system, two vibrometers, and a micromanipulator. (b) The
incoming focused laser beam (15-
μ
m beam waist) is used to excite
stress waves in the chains. The imaging system is used to record
the particle motion in high speed, and the vibrometers are used to
measure the velocity profiles of selected microparticles. (c) Scanning
electron microscopy image of the v-shaped groove supporting the
chain (the angle between two inclined planes is 70.6
°
). (d) Optical
microscope image of a microscopic granular chain assembled on the
supporting structure.
resulting granular chains are not precompressed. In macro-
scopic granular systems, the application of a static, external
precompressive force is commonly used to compact the chains
and tune their dynamic response [
22
]. In microscopic chains,
the application of a controlled, static compressive force is
more difficult and the use of fixed boundary conditions (e.g.,
a wall at the end of the chain) can interfere with the initial
excitation. In this work, we focus on the study of uncompressed
microgranular chains with free boundaries.
During assembly, the chains [Fig.
1(d)
] are inspected with
a high-speed microscopy imaging system (Vision Research
Phantom v12.1, Leica S6D) which can track their position
with a 2-
μ
m precision at 25 kHz. To reduce the effects of the
particle polydispersity, we assemble each chain multiple times
and measure the overall chain lengths. We select for testing
only the chains with shorter and equal lengths. The maximum
number of microparticles that can be included in a chain is
limited experimentally by the need for optical inspection of the
assemblies and the micromanipulator tip. The field of view of
our imaging system is
6 mm and allows for simultaneously
visualizing the assembly of chains with a maximum of 20
particles. We select for testing chains composed of 15 particles.
Macroscopic chains of similar lengths have been shown to
support the formation and propagation of nonlinear solitary
waves [
1
,
2
,
4
]. Also, numerical simulations (described below)
of propagating stress waves in an ideal 15-particle chain
showed that the pulse amplitude at the end of the chain differs
by
<
1% from pulses in an infinite chain [
2
].
The selected microparticle chains are excited by a striker
particle (particle 1 in our assembly), identical to the others,
set in motion by a Q-switched pulsed laser (Quantel Brilliant,
532 nm, 4 ns). The intense laser pulse induces vaporization
and ejection of material from the surface of the illuminated
particle [Fig.
2(a)
], transferring an initial momentum [
23
]. The
pulse laser energy intensity, the pulse duration, and the thermal
properties of the particle’s material all affect the momentum
transfer [
23
]. We characterize the relation between pulse
FIG. 2. Excitation of microparticles via pulsed laser ablation. (a)
The high-speed images of a particle before and after being illuminated
by a pulsed laser. The light pattern on the particle changes after
excitation due to laser-induced surface damages. (b) The observed
momentum gains of stainless steel type 316 (red square) and 440c
(blue circle) particle at varying pulse energy. The error bars represent
1 standard deviation for each data point derived from 20 experimental
runs.
energy and the momentum gain for our specific experimental
configuration, targeting individual particles initially resting
on the v-shaped groove and monitoring the resulting particle
trajectories. We vary the pulse energy of the laser beams
from 0 to 0.8 mJ and test the momentum gained by particles
made of both stainless steel type 316 and 440c [Fig.
2(b)
].
While these two materials have similar elastic properties,
such as Young’s modulus and Poisson’s ratio, they have
distinct thermal properties (for example, specific heat and
thermal conductivity for stainless steel type 316 at 100
C
are 0.50 kJ kg K and 16.2 W m K, and 0.46 kJ kg K and
24.2 W m K for type 440c) [
24
], which lead to a significantly
different response to laser ablation [Fig.
2(b)
]. Both types of
particles show a linear momentum gain within the range of the
tested pulse energy but with different gain-to-energy ratios.
We use this characterization to estimate the initial velocity of
the striker particles later used in the experiments with granular
chains.
We test microscopic granular chains at varying initial striker
velocity (
v
s
), from 0.005 to 0.1 m s, and measure the stress
wave propagation. To measure the wave propagation through
the chains, we employ two laser vibrometers (Polytec OFV-
534), which monitor the velocity profiles of the 2nd and 13th
particles. Because of the angles between the axis of the chains
and the beams of the vibrometers, the output of the vibro-
meters includes off-axis components (with respect to the
chains axis). To ensure the correct detection of the on-axis
velocities, we perform an independent calibration of both
vibrometers. For the calibration, we focus the vibrometers
on the targeted particles in the assembled chain and shift
the computer-controlled sample stage with known on-axis
velocities. The vibrometer output is monitored and compared
with the stage velocity to obtain a calibration factor.
We use numerical simulations to inform our experiments.
We model the granular chains with a discrete element model
using Hertzian contact interactions and simulate the evolution
of the system with a fourth-order Runge-Kutta solver. The
contact force
f
c
between two neighboring particles is
f
c
=
2
3
E
1
ν
2
R
2
(2
R
|
x
m
x
n
|
)
3
/
2
+
,
(1)
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FIG. 3. Measured particle velocity profile in a microscopic
granular chain of 15 stainless steel 440c particles with laser excitation.
(a) Experimental measurement (solid lines) and numerical prediction
(dashed lines) of velocities of the 2nd (blue) and 13th (red) particles
in the chain (normalized to the striker velocities). (b) Measured
maximum velocities at varying striker velocities compared with
the corresponding numerical prediction (dashed lines). Error bars
represent 1 standard deviation. (c) Ratio between maximum velocities
of the two monitored particles at varying striker velocities. The
corresponding numerical prediction is plotted with a dashed line.
Error bars represent 1 standard deviation.
where
x
m
and
x
n
are the coordinates of the two particles,
R
is the radius,
E
the elastic modulus, and
ν
the Poisson ratio
of the particles. The subscript to the bracket notation, defined
by (
x
)
+
max(
x,
0), indicates the tensionless behavior of the
system. All parameters used in the model are derived from
experimental data and we neglect dissipation.
III. MAIN RESULTS
A typical velocity profile measurement is shown in Fig.
3(a)
for a wave excited by a striker with velocity 0.1 m s.
The blue and red solid lines correspond to the velocity
profile measured for particles 2 and 13, respectively. The
corresponding numerical results are plotted for comparison
in the lower panel with dashed lines. The difference in
the pulse shape observed between the experiments and the
numerical simulations can be partially attributed to the limited
bandwidth of the laser vibrometers available for testing (2.5
MHz). The vibrometer’s limited bandwidth acts as a filter
to the recorded signal. Reconstruction of the original wave
form profile requires a detailed calibration of the frequency
and phase response of the vibrometer outside of its typical
working frequency in the particular experimental configuration
selected. Due to difficulties in obtaining such reconstruction,
we focus our discussions on the particle and group velocities
of the propagating waves (which are not affected by the
vibrometers’ bandwidth). We define
v
max
,
2
and
v
max
,
13
as the
maximum particle velocities of the 2nd and 13th particles,
respectively, and compare them with the numerical results
v
max
,
2
(sim)
and
v
max
,
13
(sim)
.InFig.
3(b)
, we plot the measured
v
max
,
2
and
v
max
,
13
at different striker velocities
v
s
, together
with the corresponding prediction from simulation (dashed
lines). We obtain the average values of
v
max
,
2
=
(0
.
57
±
0
.
09)
v
s
and
v
max
,
13
=
(0
.
46
±
0
.
07)
v
s
(with a 95% confidence
FIG. 4. (a) Group velocities of a signal traveling in uncompressed
microscopic granular chains. Experimental data for the two types of
stainless steel particles are shown as solid dots and squares with
error bars (1 standard deviation). The shaded bands represent the
group velocities of random samples of uncompressed chains with
average interparticle gap sizes of 0 (close-packed), 10, 20, 47, and
190 nm. The dashed lines are the corresponding prediction obtained
with Eq. (
2
). (b) Comparison of particle velocities (normalized to the
striker velocity) in a close-packed microscopic granular chain and in
a randomly generated chain with an average gap of 10 nm. The chains
consist of 15 stainless steel particles (440c) with a radius of 150
μ
m,
excited by an initial striker velocity of 0.1 m s. (c) Distribution of
group velocity at different average interparticle gaps based on 45
measurements and their corresponding experimental uncertainty. The
orange and white area represents the “allowed” and “forbidden” zone
for the measured group velocity defined by the analytical upper bound
from Eq. (
2
). The boundary between the allowed and forbidden zones
is illustrated with gradient color between orange and white due to the
uncertainty in estimation of the striker velocity in Eq. (
2
).
interval in a linear fitting model), which is 40% smaller
compared to the numerical prediction,
v
max
,
2
(sim)
=
0
.
89
v
s
and
v
max
,
13
(sim)
=
0
.
64
v
s
. However, their ratio
v
max
,
13
/v
max
,
2
=
0
.
80
±
0
.
08 agrees well with the corresponding numerical
prediction,
v
max
,
13
(sim)
/v
max
,
2
(sim)
=
0
.
72, shown in Fig.
3(c)
.
This suggests that the deviation of the measured
v
max
,
2
and
v
max
,
13
from numerical data is a result of the wave form
deformation due to the limited measurement bandwidth, in
addition to the presence of dissipation in experiments.
Although the measured relative particle velocity agrees
well with numerical simulations, the wave’s group velocity
is significantly lower in experiments than in numerical sim-
ulations. We plot the measured group velocity
v
g
(meas)
as a
function of the striker velocity [Fig.
4(a)
]. Here,
v
g
(meas)
=
(13
2)2
R/t
, where
t
is the time delay between the maxi-
mum amplitudes of
v
2
and
v
13
. For both types of stainless steel
particles tested, the measured group velocity (solid symbols
with error bar) varies nonlinearly with the striker velocity,
which is a clear indication that the wave propagation within
the granular assemblies is governed by nonlinear interactions.
However, the measured group velocities are significantly lower
than the numerical prediction for close-packed microscopic
granular chains [solid line in Fig.
4(a)
]. In addition, the
stainless steel 440c particles have higher group velocities than
stainless steel 316, despite that the two types of stainless steel
have similar elastic properties.
052907-3
WEI-HSUN LIN AND CHIARA DARAIO
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In order to explain this deviation between measurement
and numerical simulations, we study the role of defects and
packing imperfection in the wave propagation. We focus
on the presence of interparticle gaps, which are expected
to be in the order of magnitude of the particles’ surface
roughness (

R
), and as such below optical resolution of our
experimental characterization system. We model the dynamics
of one-dimensional chains with interparticle gaps varying
between 0 and 200 nm and investigate how the gaps affect the
group velocity of the propagating wave. We generate random
initial configurations of 15 particles with average gap sizes (
̄

)
per particle of 10, 20, 47, and 190 nm, and calculate the group
velocities for different striker velocities. For each gap size and
striker velocity, we simulate 100 configurations and plot the
distribution of the resulting group velocities [shaded bands
in Fig.
4(a)
]. As the average gap sizes increase, the calculated
group velocity deviates more from the group velocity expected
in a perfectly packed system. The differences in the measured
group velocities between the stainless steel 316 and 440c
microparticle chains could therefore be explained by assuming
bigger, more effective interparticle gaps between the 316 parti-
cles. This can be related to the difference in surface roughness
and size variation of the two materials, reportedly higher in
the 316 particles. While our sample preparation procedures
include micromanipulator compression and visual inspection
of the chain, an ideal close-packing condition cannot be
guaranteed. We thus expect larger interparticle gaps in chains
of free, noncompressed particles with higher polydispersity.
However, the average gap sizes suggested by the numerical
fitting [in Fig.
4(a)
] are smaller than the polydispersity for
the 316 and 440c particles. This can be partially explained
by considering three-dimensional effects in the particles’
positioning within the support grooves. Variations in the radii
(arising from the particles’ polydispersity) result in a vertical
misalignment of the particle centers from the main chain’s
axis. This misalignment reduces the effective center-to-center
distance between the particles and results in smaller gaps in
the one-dimensional model used in our analysis.
Accounting for the presence of average gaps in the chains,
the deviation of the group velocity from the close-packed
configuration can be explained by the following relation:
2
R
+
̄

v
g
(meas)
2
R
v
g
(0)
+
̄

v
s
,
(2)
where
v
g
(0)
is the group velocity of the solitary wave
propagating in an ideal, uncompressed close-packed granular
chain. Equation (
2
) assumes that the measured time of flight
for the pulse is the sum of the traveling time of a solitary wave
propagating in a close-packed granular chain and the traveling
time of the individual particles to close the gap to reach the
next particle. We plot the values predicted by Eq. (
2
) with
dashed lines in Fig.
4(a)
, which lies above the distribution
of group velocities obtained in simulation of chains with
corresponding average gaps. The particle velocity can lie
between two bounds: (i) For an ideal chain excited by a striker
particle which has the same mass as other particles in the
chain, the maximum particle velocity is
v
max
=
25
16
(
v
g
(0)
c
)
4
v
g
(0)
,
where
c
=
2
π
E
ρ
(1
ν
2
)
is the sound speed in the material of
the particle [
1
] and
ρ
is the density. (ii) For a chain with all
particles separated from each other, in the absence of losses,
momentum conservation imposes that each particle velocity is
equal to the striker velocity. Equation (
2
) defines the relation
between the group velocity and the presence of interparticle
gaps when chains in an experimental system lie between these
two bounds. By fitting Eq. (
2
) with different average gap sizes,
we can explain the reduction of group velocity in the different
steel chains tested experimentally. For example, the measured
group velocity for a chain composed of stainless steel 316
particles, with a rougher surface finish, agrees well with the
numerical model if we assume an average, effective, average
interparticle gap of 190.4 nm. Similarly, the effective, average
gaps for the chain of 440c steel particles (with a smoother
surface) is estimated to be 47 nm. Note that relatively small
gaps (in our case,
0
.
03%
0
.
1% of the radii of the particles)
can have a significant impact on the measured group velocities.
This results from the fact that the striker velocity is about 4
orders of magnitude smaller than the group velocity in an ideal
chain.
To gain more insight on the influence of interparticle gaps in
the dynamics of these systems, we compare the wave propaga-
tion properties in an ideal chain composed of 15 particles with
the response of the chain with only 10-nm average interparticle
gaps. We perform numerical simulations and plot the velocity
profiles of the 15 particles in Fig.
4(b)
. The particle velocities in
an ideal, close-packed chain follow the profile expected from
the formation of a highly nonlinear solitary wave [
1
]. However,
the particle velocity profile calculated for the chain with a
randomly distributed interparticle gaps shows irregular oscil-
lation as the wave propagates along the chain (and their value is
larger than the ideal case). We study experimentally the relation
between group velocity and the average gap sizes. While our
experimental apparatus is not able to resolve the nanometer-
size gaps between individual particles, it is possible to estimate
the effective, average gap size
̄

measuring variations in the
length of the chains assembled in our setup. We construct
loosely packed microscopic granular chains by assembling
chains of particles without the use of precise micromanipulator
compression, and measure the total distance
L
between the
2nd and 13th particles. We calculate the average gap sizes
in the different chains assembled as
̄

=
L/
(13
2)
2
R
and
estimate it to be between 250 and 500 nm, with a 30% accuracy
limited by the optical resolution of the system. We strike each
chain with a striker velocity of 0.01 m s and plot the distribution
of measured group velocity for 45 different chains with the
corresponding uncertainty. For each average gap, Eq. (
2
)
provides an upper bound for the corresponding group velocity.
We define the “allowed” (below the bound) and “forbidden”
(above the bound) zones for the measured group velocities at
varying average gaps and illustrate them in Fig.
4(c)
as the
orange and white areas, respectively. Due to the uncertainty in
the experimental estimation of the striker velocity in Eq. (
2
),
the boundary between the allowed and forbidden zones is not a
simple curve but shown as an area with gradient color between
orange and white, and width resulting from the uncertainty of
v
s
(1 standard deviation). The experimental data is plotted in
the graph with error bars, calculated with the uncertainty due
to the optical inspection of the total length of the chain. It is
evident that the measurements agree with the predicted zone.
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The results provide experimental evidence of the suppression
of group velocity in the presence of interparticle gaps.
In the presence of large interparticle gaps, the highly
nonlinear solitary waves that characterize close-packed gran-
ular chains (described by the Nesterenko [
2
]) cease to exist.
However, if the gaps are sufficiently small, the behavior of the
system approximates the ideal, close-packed case. This occurs
when
2
R
+
̄

v
g
(meas)
2
R
v
g
(0)
+
̄

v
s
2
R
v
g
(0)
, or when
̄

v
s

2
R
v
g
(0)
, and
̄


v
s
v
g
(0)
2
R
=
2
.
4
(
v
s
c
)
4
/
5
R.
(3)
Here, we use the relation between striker velocity and
maximum particle velocity,
v
max
0
.
64
v
s
, which is obtained
numerically for an uncompressed, close-packed, nondissipa-
tive granular chain with Hertzian interaction. This inequality
shows that for stainless steel particles (
c
=
4100 m s) excited
by a striker with velocity around 0.1 ms, the average inter-
particle gaps should be

0
.
05% of the particle’s radius in
order to approximate the ideal chain response. As the size
of the granular assemblies decreases, the required assembly
precision increases. Additionally, this requirement increases
even further if other types of imperfections are considered.
For example, misalignment between microparticles leads to
an offset (eccentricity) of the particle centers from the axis of
the chain, which also affects the wave propagation. For two
neighboring particles with slightly different radii (
R
m
+
1
and
R
m
), the off-axis component of the contact force is
f
c
e/
2
R
,
where
e
=|
R
m
+
1
R
m
|
. If this contribution exceeds the
particle’s confinements
f
confine
, the chain fragments laterally.
This suggests that in general, for the stabilities of granular
assemblies, the condition
f
c
e/
2
R

f
confine
needs to be
satisfied during the excitation of the systems. In our analysis,
we assume that the particle confinement is provided by the
particle’s own weight (
f
confine
=
mg
, where
m
is the mass of
the particle and
g
is the gravity of Earth) and neglect adhesive
and frictional forces, for simplicity. To estimate the maximum
contact force (
f
c,
max
) between the particles of a granular chain,
we consider collisions between two free particles, assuming
one is stationary and the other approaches with an initial
velocity
v
s
. If the particle interaction is Hertzian, as shown in
Eq. (
1
), the maximum interparticle force can then be derived
as
f
c,
max
=
1
6
(
15
2
)
3
/
5
(
E
1
ν
2
)
2
/
5
5
Rm
3
/
5
v
s
6
/
5
,
(4)
where
m
is the mass,
R
the radius,
E
the elastic modulus, and
ν
the Poisson ratio of the particles.
In this case, to ensure no instabilities in a one-dimensional
chain excited by a striker, this condition corresponds to an
upper bound for eccentricity,
e
:
e

5
.
3
g
v
s
6
/
5
c
4
/
5
R
2
.
(5)
Combining Eqs. (
3
) and (
5
), we obtain an expression,
̄
e
2
/
3

7
.
3
g
2
/
3
R
7
/
3
c
4
/
3
, to define an approximate bound for the
imperfections without dependence on the striker velocity.
Knowing fabrication limits, in terms of achievable average gap
sizes
̄

and misalignment
e
, we can determine the minimal
size of particles in a granular assembly that can support
the formation and propagation of highly nonlinear solitary
waves. In our experiment, we assume
̄
<
200 nm (which
is estimated to be the positioning precision of the particles
in a chain) and
e<
1
μ
m (which corresponds to the nominal
value of the particle radii variation for stainless steel 440c).
These parameters lead to an estimate for the minimum particle
radius
R
=
670
μ
m required to support highly nonlinear
solitary waves. This value is four times larger than the size of
our particles, partially explaining the discrepancies observed
between the experimental measurements of group velocities
and the corresponding values calculated for an ideal granular
chain (see Fig.
4
).
IV. CONCLUSIONS
In this work, we investigated the stress wave propaga-
tion in one-dimensional chains of microparticles. For the
experiments, we constructed an apparatus employing laser-
based excitation, noncontact high-speed microphotography
and vibrometry, and a micromanipulator system. The setup
allowed us to excite and study stress wave propagation
in microgranular assemblies. We informed our experiments
with discrete finite element simulations and modeled the
system with Hertzian contact interaction. We obtained good
agreement between experimental and numerical results for
the decaying of the stress wave amplitude. However, a large
deviation of the group velocity from theory was observed.
We found that microscopic granular chains can support
the propagation of highly nonlinear solitary waves under
ideal conditions but are very sensitive to the presence of
imperfections. We derived an analytical expression to relate the
group velocity to the presence of interparticle gaps and support
it with direct experimental evidence. In further analysis, we
characterized the role of defects (gaps and misalignments) in
the dynamic propagation of stress waves in a one-dimensional
granular chain and defined analytical bounds for the scalability
of a granular system with given assembly and fabricating
precision. These findings can be used as guiding principles
for implementation of engineering applications and scientific
studies of microscopic granular systems.
ACKNOWLEDGMENTS
The authors would like to acknowledge Prof. V. F.
Nesterenko for fruitful discussions and Gladia Hotan for help
in the experiments. This work has been supported by the MURI
program, Project No. US ARO W911NF-09-1-0436, and made
use of facilities supported by the Kavli Nanoscience Institute
at Caltech.
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