Table Of ContentBreathers and surface modes in oscillator chains with Hertzian interactions
GuillaumeJames∗⋆,Jesu´s Cuevas† and PanayotisG. Kevrekidis‡
∗ LaboratoireJeanKuntzmann,Universite´deGrenobleandCNRS
BP53,38041GrenobleCedex9,France
⋆INRIA,BipopTeam-Project
ZIRSTMontbonnot,655Avenuedel’Europe,38334SaintIsmier,France
3 †GrupodeF´ısicaNoLineal,DepartamentodeF´ısicaAplicadaI,EscuelaPolite´cnicaSuperior,UniversidaddeSevilla
1
C/VirgendeA´frica,7,41011Sevilla,Spain
0
2 ‡DepartmentofMathematicsandStatistics,UniversityofMassachusetts
Amherst,Massachussets01003-4515,USA
n
Email:[email protected],[email protected],[email protected]
a
J
9 Abstract—Westudylocalizedwavesinchainsofoscil- waves exist in diatomic granular chains under precom-
latorscoupledbyHertzianinteractionsandtrappedinlocal pression [2, 13] (with their frequency lying between the
] potentials. This problem is originally motivated by New- acoustic and optic phonon bands) and can be generated
S
ton’scradle, a mechanicalsystemconsistingofa chainof e.g. throughmodulationalinstabilities. However,because
P
. touchingbeadssubjecttogravityandattachedto inelastic precompressionsuppressesthefullynonlinearcharacterof
n
strings. We consider an unusual setting with local oscil- Hertzian interactions, these excitations inherit the usual
i
l lations and collisions acting on similar time scales, a sit- properties of discrete breathers, i.e. their spatial decay is
n
[ uation corresponding e.g. to a modified Newton’s cradle exponentialandtheirwidthdivergesatvanishingamplitude
withbeadsmountedonstiffcantilevers.Suchsystemssup- (forfrequenciesclosetothebottomoftheopticband).
1
portstaticandtravelingbreatherswithunusualproperties, The situation is sensibly different for granular systems
v
including double exponential spatial decay, almost van- withoutprecompression.Inthatcase,localizedoscillations
9
6 ishing Peierls-Nabarro barrier and spontaneous direction- can be generatedonshorttransientsin the formof transi-
7 reversing motion. We prove analytically the existence of tory defect modes induced by a mass impurity (see [11]
1 surface modesand static breathersforanharmonicon-site and references therein) but never occur time-periodically
.
1 potentialsandweakHertzianinteractions. asprovedin[5]. Indeed,uncompressedgranularchainsare
0 described by the Fermi-Pasta-Ulam lattice with Hertzian
3 Granular media are known to display a rich dynami- interactions
1 cal behavior originating from their complex spatial struc-
v: ture and different sources of nonlinearity (Hertzian con- y¨n =V′(yn+1−yn)−V′(yn−yn−1) (1)
i tact interactions between grains, friction, plasticity). In
X (orspatiallyinhomogeneousvariantsthereof),where y (t)
the case of granularcrystals (i.e. for grains organizedon n
r denotesthenthbeaddisplacementfromitsreferenceposi-
a a lattice), nonlinear contact interactions lead to different
tion. ForallT-periodicsolutionsof(1),theaverageinter-
types of localized wave phenomena that could be poten-
tiallyusedforthedesignofsmartmaterialssuchasacous- actionforcesR0TV′(yn+1−yn)dtareindependentofn(this
is immediate by integrating(1)). Consequently, localized
tic diodes [1]. Among the most studied types of excita-
oscillations would yield a vanishing average interaction
tions,solitarywavescanbeeasilygeneratedbyanimpact
force between grains, which is impossible since Hertzian
atoneendofachainoftouchingbeads(see[7,8]andref-
interactions are repulsive under contact and vanish other-
erencestherein).Intheabsenceofanoriginalcompression
wise.
in the chain(the so-calledprecompression),these solitary
waves differ from the classical KdV-type solitary waves, In contrast to the above picture, we have numerically
sincetheyarehighly-localized(withsuper-exponentialde- established in [5] the existence of time-periodiclocalized
cay) and their width remains unchanged with amplitude oscillationsinHertzianchainswithsymmetriclocalpoten-
(see e.g. [12]). These properties originate from the fully tialsdescribedbythesystem
nonlinear character of the Hertzian interaction potential
V(r) = 52γ(−r)5+/2 (with γ > 0 and (a)+ = max(a,0)), y¨n+W′(yn)=V′(yn+1−yn)−V′(yn−yn−1) (2)
which yieldsa vanishingsoundvelocityinthe absenceof whereW(y)= 1y2+ s y4ands∈Rmeasuresthelocalan-
precompression. 2 4
harmonicity. System (2) describessmallamplitudewaves
Discrete breathers(i.e. intrinsic localized modes) form inaNewton’scradle[4](figure1,left)orothermechanical
anotherinterestingclass of excitationsconsistingof time- systemsconsistingofbeadsmountede.g.onanelasticma-
periodic and spatially localized waveforms [6, 3]. These trix[10]orcantilevers[5](figure1, right). Inthelasttwo
cases, local oscillations and collisions between beads can breather),foranharmonicon-sitepotentials,butmorecriti-
occuronsimilartimescalesforrealisticmaterialparameter callyevenintheharmoniccase.Thisphenomenonislinked
values,whichallowsforbreathingdynamicstotakeplace. with an extremely small difference between the energies
Static and moving breathers can be generated from stan- H of site- and bond-centered breathers having the same
dardinitialconditionssuchasalocalizedimpact[5,10]or frequency(theso-calledapproximatePeierls-Nabarrobar-
perturbationsofunstableperiodictravelingwaves[4]. rier).Typicalvaluesofbreatherenergiesaregiveninfigure
3,whereH = e ande = 1y˙2+W(y )+V(y −y ).
Pn n n 2 n n n+1 n
Anothermanifestationofthehighbreathermobilityisthe
systematic formation of a travelling breather after an im-
pactatoneendoftheoscillatorchain(seefigure4).
8·10−5 10−4
) )
Figure 1: Left : prototypical Newton’s cradle consisting 0 0
(0 (0
n n
of a chain of beads attached to pendula. Right : array of y y
clampedcantileversdecoratedbysphericalbeads.
−8·10−5 −10−4
5 10 15n20 25 30 5 10 15n 20 25 30
Let us summarize the results of [5]. Two fami-
lies of static breather solutions of (2) (parametrized by
their frequency ω > 1) have been computed using the Figure2: Bond-centered(leftplot)andsite-centered(right
b
Gauss-Newton method and path-following. The first one plot)breathersolutionsof(2)forωb = 1.01(potentialpa-
consists of bond-centered breathers satisfying y (t) = rameters are γ = 1 and s = 0). The numerical solutions
n
−y (t) := S y (t), and the second correspondsto site- (marks)are comparedto the quasi-continuumapproxima-
−n+1 1 n
centeredbreatherswith y (t) = −y (t+T /2) := S y (t), tionsy(1),y(2)(continuouslines).
n −n b 2 n n n
where T denotes the breatherperiod (each solution fam-
b
ily is invariantby a symmetry S of (2)). Profilesof both
i
breather types are given in figure 2 for a particular fre- 100 100
quency close to unity (i.e. the linear frequency of lo-
10−5
cal oscillators), a limit where the breatheramplitude van- 10−5
itshheebsr.eNatehaerrtphriosfilliemsict,aqnubaesid-ecroinvteidnu[u5]m,naapmpreolyximationsof Energies10−10 Energies10−10
10−15
y(1)(t)=2ǫ(−1)n[g(n)+g(n−1)] cos(ω t) (3)
n b 10−151 1.1 1.2 ω 1.3 1.4 1.5 10−201 1.1 1.2 ω 1.3 1.4 1.5
b b
forbond-centeredbreathersand
1 1
y(2)(t)=2ǫ(−1)n[g(n+ )+g(n− )] cos(ω t) (4) Figure3: Breatherenergiesversusfrequencyωb,whenthe
n 2 2 b on-site potential is harmonic (s = 0, left plot) or anhar-
monic (s = −1/6, right plot) and γ = 1. The red curves
for site-centered ones, where ω = 1+ ǫ1/2 (τ ≈ 1.545),
b 2τ0 0 give the energy Hbc of bond-centered breathers, the blue
3
g(x) = (cid:16)130(cid:17) cos6(cid:16)3x(cid:17)for|x| ≤ 32π and g = 0 elsewhere. curves the approximate Peierls-Nabarro barrier EPN (en-
Figure2comparesthenumericalsolutionswiththeabove ergydifferencebewteensite-andbond-centeredbreathers)
approximationsforω ≈ 1, showingexcellentagreement. andtheblackcurvestherelativeenergyratioE /H .For
b PN bc
Obviously discrepanciesappearat largeramplitude, since smallamplitudebreathers(i.e. ω ≈1),thedifferentvalues
b
approximations (3) and (4) retain only one Fourier mode ofsyieldcomparablevaluesofE .ClearlyE increases
PN PN
and are independentof the local anharmonicity(which is withbreatheramplitudebutremainsverysmallinthispa-
dominated by the Hertzian nonlinearity at small ampli- rameterrange(e.g. E iscloseto10−4 forω = 1.5and
PN b
tude).Theaboveapproximationspossesscompactsupports s=−1/6).Theharmoniccaseyieldsevensmallerbarriers,
(with a width independentof ω ), but the breathers com- by3−4ordersofmagnitudeforω =1.3.
b b
putednumericallydonotsharethispropertystrictlyspeak-
ingandinsteaddisplaysuper-exponentiallocalization(this Differentphenomenahavebeenidentifiedin[5]depend-
isinanalogywiththecaseofhomogeneouspolynomialin- ing on the softening or hardening character of the local
teractionpotentials,seesection4.1.3of[3]). potential W. Firstly, the stability of both site-centered
Dynamical simulations of [5] indicate that extremely andbond-centeredbreathersiscriticallydependentonthe
small perturbations of the static breathers can lead to strength(andsign)oftheanharmonicity(wereferto[5]for
theirtranslationalmotion(generatingaso-calledtraveling moredetails).Secondly,dependingwhethers<0ors>0,
theoccurenceofsurfacemodeexcitations(i.e. oscillations theenergyoftheinitialexcitationiswellbelowtheexcita-
localizednearaboundary)ordirection-reversingtraveling tionthreshold.Thissituationcontrastswiththecaseofsoft
breatherswas observedafteran impact. Both phenomena local potentials illustrated in figure 6 (right column). For
are illustrated by figures 4 and 5. The origin of direction s = −0.7, surface modes exist for ω ∈ [0.705,1). They
s
reversal is still unclear at the present stage, although we are spectrally stable for ω ≈ 1 but oscillatory instabili-
s
think it might originate from the interaction between the tiesappearatsmallerfrequencies(spectranotshownhere).
traveling breather and nonlinear waves confined between Whenω →1,theirenergyandamplitudevanish,opening
s
thebreatherandtheboundary(figure5,rightplot). thepossibilityofexcitingsuchmodesforarbitrarilysmall
initialvelocitiesofthefirstparticle.
70 0 50
−0.05 45 −0.05
60 −0.1 40 −0.1 2 0.35
n4500 −−−000...22155n233505 −−−000...22155 1.5 0.02.53
30 −0.3 20 −0.3 1 0.2
20 −−00..435 1105 −−00..435 yn 0.5 yn0.15
10 −0.45 5 −0.45 0.1
−0.5 0
0 50 100 1t50 200 250 300 0 50 100 1t50 200 250 300 0.05
−0.5 5 10 15 20 25 30 0 5 10 15 20 25 30
n n
35 0.5
Figure 4: Space-time diagrams showing the interaction 0.45
30 0.4
forces fn = −(yn −yn+1)3+/2 in system (2) with γ = 1 and 25 0.35
fgrreeeyelenvdelbso,uwnhdiatreyccoorrnedsiptioonndsi.ngFtoorcveasniasrheinrgepirnetseernatcetdionins Energy20 Energy0.002..523
(beadsnotincontact)andblacktoaminimalnegativevalue 15 0.15
ofthe contactforce. Theinitialconditionis yn(0) = 0for 10 0.00.51
n ≥ 1,y˙n(0) = 0forn ≥ 2,y˙1(0) = 0.94. Foranharmonic 5 2 2.2 2.4ωs 2.6 2.8 3 0 0.75 0.8 0ω.8s5 0.9 0.95 1
localpotential(s=0,leftplot),atravelingbreatherisgen-
erated. Forasoftlocalpotential(s= −0.7,rightplot),one
observesinadditiontheexcitationofasurfacemode. Figure 6: Top : profile of a surface mode at the instant
of maximal amplitude, for a hard local potential (s = 1,
ω = 2, left plot) and a soft one (s = −0.7, ω = 0.99,
s s
right plot). Bottom : energy of the surface mode versus
0 1.8 0.8
500 1.6 frequencyωs,fors=1(leftplot)ands=−0.7(rightplot).
1000 1.4
t12500000 011..82 ˙y(t)n0 Having numerically analyzed the properties of surface
2500 0.6 modes of (2) when γ (the stiffness constant of Hertzian
3000 0.4 interactions) equals unity, we now prove the existence of
3500 0.2
40000 20 40 n60 80 100 0 -10 20 40 n60 80 100 time-periodic and spatially localized solutions when γ is
small. The small coupling (anticontinuum) limit was in-
troducedinreference[6](andconsiderablygeneralizedin
Figure5: Left: space-timediagramoftheenergydensity [9])andappliesbothtobreathersandsurfacemodes.From
e forsystem (2)with γ = 1 anda hardanharmoniclocal a physicalpointof view, thisparameterregimecanbe re-
n
potential (s = 1). The initial condition is the same as in alized with an array of clamped cantilevers (see figure 1)
figure4excepty˙ (0) = 1.9. Right: correspondingparticle decoratedbysphericalbeadsmadefromasufficientlysoft
1
velocitiesclosetodirection-reversing(t≈241). material(e.g. rubber). Inaddition,theanticontinuumlimit
requiresananharmoniclocalpotentialW,i.e. s,0.
In orderto understandthe absenceof surfacemodeex- Weconsidersystem(2)forn∈Γ,withΓ=Zfordiscrete
citation observed in figure 5 for s = 1, we now analyze breathers(caseofa doublyinfinite chain)and Γ = N for
0
the properties of surface modes computed by the Gauss- surface modes (semi-infinite chain, with free-end bound-
Newtonmethod(allnumericalcomputationsareperformed ary conditions at n = 0). For γ = 0, system (2) admits
forγ=1). Theresultsareshowninfigure6(leftcolumn). localized periodic solutions satisfying y = 0 ∀n , 0,
n
For s = 1, surface modes exist for frequencies ω lying y¨ +W′(y ) = 0andy (t+T ) = y (t). Settingy˙ (0) = 0,
s 0 0 0 0 0 0
above a critical frequency ω ≈ 1.96, where a pair of a := y (0) > 0andT ≡ T (a), itisa classicalresultthat
min 0 0 0
Floquet eigenvalues converges towards unity (spectra not sT′(a) < 0. Consequently, the time periodic oscillations
0
shownhere). For ω ≈ ω thesesolutionsarespectrally y can be parametrized by their period T , with T < 2π
s min 0 0 0
stable and one can observe an energythreshold. This ex- for s > 0 and T > 2π for s < 0. We shall denote by
0
plainswhyasurfacemodeisnotobservedinfigure5where Y := (y ) the even T -periodic solution constructed
0,T0 n n 0
above for γ = 0. Now let us assume T < 2πN (this Acknowledgments
0
condition is automaticallysatisfied for s > 0). Thanksto
J.C.acknowledgessupportfromtheMICINNprojectFIS2008-
thisnonresonancecondition,onecanapplythepersistence
04848&P.G.K.fromtheNSF(grantCMMI-1000337),fromthe
theorem for normally non-degenerate discrete breathers A.S. Onassis Public Benefit Foundation (grant RZG 003/2010-
provedinreference[9](section3.5),whichreadilyapplies
2011)andfromAFOSR(grantFA9550-12-1-0332).
to system (2) since V ∈ C2(R) has sufficient smoothness.
More precisely, for γ small enoughin system (2), this re-
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