Table Of ContentLocalizing energy in granular materials.
Michelle A. Przedborski,1,a) Thad A. Harroun,1,b) and Surajit Sen2,c)
1)Department of Physics, Brock University, St. Catharines, Ontario, Canada L2S 3A1
2)Department of Physics, State University of New York, Buffalo, New York 14260-1500,
USA
(Dated: 13 January 2016)
A device for absorbing and storing short duration impulses in an initially uncompressed one-dimensional
granular chain is presented. Simply stated, short regions of sufficiently soft grains are embedded in a hard
6
granular chain. These grains exhibit long-lived standing waves of predictable frequencies regardless of the
1
timing of the arrival of solitary waves from the larger matrix. We explore the origins, symmetry, and energy
0
content of the soft region and its intrinsic modes.
2
n
PACS numbers: 45.70.-n,05.45.Yv
a
J
1
Methods fortrappingenergyingranularmaterialshas Consideraone-dimensionalchainofN sphericalgrains
1
been a topic of interest to the scientific community in betweentwofixedwalls,underzeropre-compression. Ini-
recent years. Controlling energy localization processes tially the grainsarebarelytouching,but anyoverlapbe-
]
h insuchmaterialsaffords the opportunity to developnew tweengrainsresultsinaHertz12–14potential,V(δ )=
i,i+1
p energy-harvesting or filtering devices. One direction of a δ5/2 , where δ =R +R (x x ) is the
p- this research is into the formation of intrinsic localized gri,ai+in1oiv,ie+r1lap with xi,i+th1e absiolutei+p1o−sitioni+,1a−ndi
i
m modes(ordiscretebreathers)1,2,andtherehavebeensev-
eral experiments and simulated demonstrations of these
o 2 RiRi+1
.c wexocriktahtaiosnfsocinusoendeo-dnimeneenrsgiyontaralpgprainngulianrgcrhaaniunlsa3r–5c.onOttahiner- ai,i+1 = 5Di,i+1sRi+Ri+1, (1)
s
c ers6,7 and composite granular systems8,9. with R the radius. The softness10 is characterized by
i Recently,weshowedthatsoftergrainsinamonoatomic i
s pre-factor D , which is related to the Young’s modu-
y Hertzian chain results in slowing solitary wave (SW) i,i+1
lus Y and Poisson’s ratio σ of the grains as
h propagation,andthatsoftboundariescausesasignificant
p delay in the reflection of SWs due to the introduction of 3 1 σ2 1 σ2
[ an inertial mismatch at the boundaries of the system10. Di,i+1 = 4 −Y i + −Y i+1 . (2)
1 This leads to the straightforward idea of localizing both (cid:20) i i+1 (cid:21)
v kineticandpotentialenergyonadefectofsoftandsmall Fixedwalls use R , relaxingthe conditionthat the
j
6 grains, in a manner distinct from previous studies. wallmustbe flat. It→is∞thenpossibletoexpressthe equa-
5 Our goal is to engineer a simple system that can trap tions of motion for any grain in the chain as described
6
a significant fraction of the system energy. We propose previously10. We model our chains with soft grains cen-
2
that a small core of central grains, much softer than the trallyembeddedinalongerchainofstainlesssteelgrains
0
. mainlattice,candothis,suchasillustratedinFig.1a. In of twice the radius (6 mm). Choosing the central soft
1 suchasystem,the softcentralgrainsremaincompressed grains to be smaller than the hard grains creates an ef-
0
for long times because it is energetically more favorable fective potential well at the center of the chain, since a
6
forthesystemtokeepthesoftgrainscompressedthanto sharp decrease in radius ensures complete transmission
1
: have a nonzero overlap between the harder grains. This of momentum and energy, and partial reflection in the
v introduces a restoring term to the force among the soft reverse case15–19.
i
X grains,andwecanwell-modeltheembeddedsystemwith To avoidSW formation in the center of the chain, the
r a linearized force, possibly allowing us to tune the de- number of soft central grains should remain small (typi-
a vice’s frequency response11. cally 3 N 6). For brevity, we report on the results
0
≤ ≤
To excite this setup, we perturbthe systemasfollows. only for N =3,4. The total number of grains is chosen
0
Weapplyasingle,discrete,symmetricperturbationfrom to be N=39 or 40, to have an even number (N N )
0
−
both ends, and this causes the two SWs to meet at the of stainless steel grains and preserve the symmetry. We
centerofthechain,wheretheystaytrappedforextended then perturb the endmost grains at t = 0 with a small
periods of time. Standing waves are then excited in the initial velocity directed toward the center.
soft regions. We find that standing waves with distinct frequencies
first form for systems whose central region corresponds
to a D-value, see Eq. 2, which is at least two orders of
magnitude larger than the host chain, i.e D /D 102,
s h
∼
a)Electronicmail: [email protected] whereDs(h) correspondstothesoft-soft(host-host)grain
b)Electronicmail: [email protected] interaction. Further softening of the center slows energy
c)Electronicmail: sen@buffalo.edu propagation and leads to larger fractions of the system
2
energy residing in central grains. This leads to beau- showthesepatternstobesimilartostandingwavesbeat-
tiful harmonic behavior with clean, fast oscillations in ing with quite distinct frequencies. Each grain oscillates
the central region. For this reason, we present results with a similar high frequency while a lower frequency
for systems with central region D 38, consistent with wave unites the movement of the central and edge soft
s
≃
material properties of rubber (stainless steel host chain grains.
corresponds to D 0.007). While these systems may
h
≃
be inevitably lossy,we focus onthe shorttermdynamics
1
and thus neglect dissipation in our studies. kinetic
n0.8 potential
o total
(a) acti0.6
r
f
... gy 0.4
r
e
n0.2
e
e 0
uls ... 0 0.01 0.02 0.03 0.04 0.05
mp time (s)
i
(b) FIG.2. Fractionofsystemenergy(kinetic,potentialandtotal
energy) contained in the central soft grains in the N = 40
system with four soft central grains for the first 0.05s. From
n
gio the virial theorem, the total system energy is split into 5/9
e
oft r kineticenergy and 4/9 potential energy.
s
The density of energy in the soft region remains high
from the moment of initial compression by the first SW.
Fig.2showsthefractionofthetotalsystemenergyfound
in just the central region for the same configuration as
FIG.1. Kineticenergy densityplotsfor theN =40 granular
Fig. 1, over time. The amount of stored compression
chainwith fourcentralsoft grains. (a)showshowthekinetic
energyismaintainedinthecentralregionlongaftertheinitial (potential) energy in the soft grains fluctuates with the
perturbationtothesystem(b)showsazoomed-inviewof(a) absorption and emission of SWs from the hard grains,
foronlythesixcentralgrainsandoverasmallertimeinterval. however, on average more than 80% of the system’s
∼
Note that grains 18 and 23 act as effective walls for the four entire potential energy is found here. The soft grains
centralsoftgrains, andthemotionofthese“walls” isevident remain compressed for & 1 sec, the duration of the sim-
in thekineticenergy density plot. ulation, explaining the significant gaps among the hard
grains that lead to the erratic paths of the SWs.
Fig. 1 shows the evolution of the N =40 system with The discrete frequencies of the soft grain motion are
N =4softgrainsbymappingthekineticenergyateach clear in the oscillations of the kinetic energy fraction in
0
grain. PrincipalSWsapproachthecenterregionfromthe Fig.2. InFig.3weshowthecosinespectrumofthetotal
edges and are stopped at the edge of the soft grains for kinetic energy of soft grains. The spectra exhibit several
several ms before re-emerging. Energy transmission at low(<700Hz)characteristicfrequenciesthatcorrespond
hard/soft grain boundaries, as well as reflections at the to the periodic arrivals and departures of the SWs from
hard system boundaries, create many secondary solitary the hard grains. From the kinetic energy density plots,
waves (SSWs), howeverthe transition to the well known the frequency of the initial SWs approaching the wall is
quasi-equilibrium phase is greatly delayed10. In Fig. 1a, 170 Hz for the four soft grainsystem. Note that there
∼
we see that the emission of SWs from the soft region is are also SSWs that frequently approach the soft grains,
slow compared to reflections at the walls, as SWs pick andthesewillcontributeadditionalpeakstotheDCTas
up speed until entirely free of the center grains. SSWs well.
donothaveconstantspeeds,travelingwitherraticpaths Moreinterestingly,thespectraalsoshowabroadband
through the steel grains,indicating significantgaps have athigherfrequencies(>3000Hz)thatcorrespondstothe
opened. standing wave pattern of Fig. 1b. This band decreases
As the principal SW comes in the vicinity of the soft in frequency with increasing number of soft grains, and
region, it slows down and remains nearly localized for a second band with frequencies > 5000 Hz appear with
a period of time before entering the soft region. This four or more softgrains. We show that these frequencies
shows up as the dark horizontallines seen bordering the are harmonic modes induced in the soft grains as a re-
soft region for long time spans in Fig. 1a. Once the SW sult of their large degree of compression, and that they
enters the softregion,one cansee severalback and forth contribute to the stored energy as well.
reflectionsbeforeanyenergyistransmittedbackintothe To understand the origin of these frequencies, we also
hardgrains. AcloserlookatthecentralregioninFig.1b simulated a subset system comprised only of soft grains
3
where subscript S corresponds to symmetric modes in
which motion of the central grains is mirror reflected
about the center of the chain, and subscript A to anti-
symmetric modes in which there is no mirror reflection
symmetry. In this last expression, k is the grain-wall
w
spring constant, and k is the grain-grain spring con-
g
stant, noting that the grain-wallconnection corresponds
tothesoft-hardgraininterfacefortheembeddedsystem.
Theangularfrequencies,ω,arerelatedtothefrequencies
measuredinkineticenergyspectraviaω/π=f,sincethe
positionsandvelocitiesofthegrainsoscillateascos(πft)
while kinetic energyoscillatesas the velocitysquared,or
cos(2πft).
The averageforce between the softgrains and the soft
grainsandwallsisthe same. We findthe effectivespring
constantsforthegrain-grain andgrain-wall forcesatthe
pre-determined average force magnitude using the force
vs. δ plots(notshown). Inparticular,theslopesofthese
plots at the average force magnitude give k and k ,
g w
respectively. The fluctuations about the average force
are wide. If we take the value of the slope of the grain-
wall force at these limits of one standard deviation from
FIG.3. Discretecosinetransformsofsoftgrainkineticenergy.
IneachDCT, wepresentbothresultsof simulationsbetween the average, we obtain extrema of kw. Likewise we can
fixed walls and embedded in a steel grain matrix. (a) Three find extrema of kg.
soft grains in an N = 39 system and corresponding N = 3 Using Eq. 3, the average values of kw and kg predict
pre-compressed system for comparison. (b) Four central soft the center frequency and the two extrema predict the
grains in an N = 40 chain, and corresponding N = 4 pre- bandwidthofthe kinetic energyspectrum. These results
compressedsystem. Arrowsindicatethefrequenciespredicted areindicatedbythearrowsinFig.3b,wherethesolidar-
by the harmonic model (dashed arrows indicate asymmetric
rowsindicate the predicted frequencies of the symmetric
modeswhicharenotaccessiblewhenthesystemisperturbed
symmetrically). The full width at half maximum (given by modes ωS±, and the dashed arrows indicate the asym-
thedottedline)ispredictedfromthewidthofthedistribution metric modes ωA±. The frequencies of both the embed-
dedandsmalltestsystemsarepreciselypredictedbythe
in spring constants.
harmonic model. Furthermore, the bandwidth of ωS− is
indicated by the vertical dashed lines, and does encom-
between fixed walls. We pre-compress the subset sys- passamajorityoftheprincipalfrequencyband. Thusthe
tems with an initial loading approximately equal to the broadbandwidthoftheharmonicmotionofthesoftcen-
average force experienced by soft grains in the embed- tral region is due to changing compression forces caused
ded systems. In particular, we simulated N =N =3,4 by its variable boundaries.
0
grain systems with F0 = 3.5 10−4 kN. As reference, These values of kw and kg also agree with the N =
×
the average forces at soft-grain interfaces in embedded N0 = 4 soft system. The inter-grain force is Fij =
ssayhnsoTdtwehhmneF4ssinipwe=Fecrti3egr.a.1c73ao7lfca3utth×loaept1es0tmdh−eat4ollkspbNsyee.cshttFeram3iok=fint3he.te7ic8e9me5nb×eerdg1dy0e−ids4askylNsso-, ic25naantδhi3bje/e2m,TwoadhyeilclohrHieenxrttpzhaieannldimesdyitstttoehmbaetistFhliaejrig=neit,1i4i5a.eal.pδ01rδ/e02-≫c(oumiu−pi1−r−e1s−suiuoi)in.,
The effective spring constant is then given by k =
eff
tems. These show one or two very narrow bands whose 15aδ1/2. Using the values for a and δ for rubber
frequencies correspond very well to the bands of the 4 0 0
with steel walls, we find k = 0.02141 kN/mm and
largersystem. Pre-compression,andtheresultingrestor- w
k = 0.010707 kN/mm for the N = N = 4 soft system,
ing force, is required to obtain these spectra; without g 0
whichisinagreementwithlinearfitstotheforcevs. dis-
pre-compression,frequency bands may not appear11.
placement curve described abovefor the N =4,N =40
We can predict the frequency response of the embed- 0
system.
dedsoftgrainswithacoupledharmonicoscillatormodel.
Due to howthe systemis perturbed, the resultingmo-
For a four coupled mass-spring system between fixed
tion of the soft grains is a combination of only the sym-
walls, we obtain the following eigenfrequencies20:
metric modes with corresponding eigenvectors
1/2
1 1
ωS± = 2m kw+4kg± kw2 −4kwkg+8kg2 , ~ωS± = 2√m(1,∓1,±1,−1)T , (4)
(cid:20) (cid:16) q (cid:17)(cid:21)
1 1/2 wherewehaveusedthefactthatk 2k inthelastex-
ωA± = 2m kw+2kg± kw2 +4kg2 , (3) pressions. Theasymmetricmodeswca≈nbegpopulatedonly
(cid:20) (cid:16) q (cid:17)(cid:21)
4
with asymmetric perturbations, e.g. single edge pertur- ACKNOWLEDGMENTS
bations or unequal perturbations from both ends (data
not shown). It can be shown that the total (kinetic plus This work was supported by a Vanier Canada Gradu-
potential) energy of each harmonic mode is exactly the ate Scholarshipfromthe NaturalSciences andEngineer-
same and equal to 1mv2, where v is the initial velocity
2 0 0 1.5
of the first mass. average fraction of
PE in central region:
m 0.86 (e)
psteeFneodlregnrttahioennsetmchobenetidnadisnteaidnngtsaytnhsteeeomsuos,ftbvog0urainsidniasll.r-ydTechfioennedadritriibovunatslooisrftddheee-- E spectru0.15 00..7416 ((dc)) ng softnessal region
partureofeverySWtransmitsanadditionalacceleration K 0.18 (b) asintr
ee
porernteugrabtaitvieondetpoenthdeinsgooftngwrahientsh,erwthhicehbmouanydabreiepsoosfittihvee 0 0.14 (a) incrof c
0 1 2 3 4 5
softgrainsarecontractingorexpandingatthatmoment. frequency / (Hz) 4
Thus the amount of energy in the harmonic modes can (cid:0)✁ x 10
vary, however, this is just a small fraction of the total
FIG.4. ComparisonofkineticenergyspectraforN =4,N =
system energy. Since the soft grains are much softer and 0
40 chains with varying central grain softness; (a) stainless
havelessmassthanthesteelgrains,mostofthesystem’s
steel control experiment, (b) Pyrex, (c) PVC, (d) PTFE, (e)
energyiscontainedintheformofstoredpotentialenergy
rubber. The average fraction of the total system potential
due to the initial compression of the soft grains.
energy (PE) that is contained in the central region is also
shown. SeeRef. 10 for a table of material values.
For the system potential energy to be located almost
exclusively in the central region, the central grains must
ing Research Council. S.S. thanks US Army Research
be sufficiently softer than the matrix. We tested a range
OfficeforpartialsupportofthisresearchthroughaShort
of materials with D 0.007 38.0 (larger is softer).
s ≈ → Term Innovative Research Grant.
Fig. 4 illustrates that pseudo-harmonic frequencies ap-
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