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Wave localization in strongly nonlinear Hertzian chains with mass defect Stéphane Job1, Francisco Santibanez2, Franco Tapia2, and Francisco Melo2 1 Supmeca, 3 rue Fernand Hainaut, 93407 Saint-Ouen Cedex, 2 France. Departamento de F´ısica and Center for Advanced Interdisciplinary Research in Materials (CIMAT), Universidad de Santiago de Chile, Avenida Ecuador 3493, Casilla 307, Correo 2, Santiago de Chile. (Dated: January 22, 2009) Weinvestigatethedynamicalresponseofamassdefectinaone-dimensionalnon-loadedhorizontal chain of identical spheres which interact via the nonlinear Hertz potential. Our experiments show 9 that the interaction of a solitary wave with a light intruder excites localized mode. In agreement 0 with dimensional analysis, we find that the frequency of localized oscillations exceeds the incident 0 wave frequency spectrum and nonlinearly depends on the size of the intruder and on the incident 2 wavestrength. Theabsenceoftensilestressbetweengrainsallowssomegapstoopen,whichinturn induceasignificantenhancementoftheoscillationsamplitude. Weperformednumericalsimulations n that precisely describe our observations without any adjusting parameters. a J PACSnumbers: 05.45.-a,83.80.Fg,62.50.-p 2 2 Wave localization has been studied for long in linear purity slowly translates, leading to a large transmitted ] i systems[1]orlattices[2]andbecamewidelystudiedmore solitary waves train in the forward direction [16], simi- c s recently in nonlinear systems [3]. For example, the pres- larly to what was observed in stepped chains [5, 8, 12]. - ence of an isotope in a perfect linear crystal is known Alightintruderoscillatesandscattersforwardandback- l r to enhance optical waves absorption at given frequen- ward weak delayed solitary waves trains [16]. t m cies [4]. One-dimensional chains of beads interacting via In this letter, we investigate experimentally the inter- . the Hertz potential are systems suitable to observe non- action of a solitary wave with a light intruder in a non- t a linear localization effects. A loaded chain of identical loaded monodisperse chain of beads. We observe that m beadsisdispersive,allowingsmallperturbationstoprop- the impurity oscillates while interacting with a solitary - agateaslinearorweaklynonlinearacousticwaves[5]. In wave. Oscillations remain localized and fade as soon as d contrast, when grains in a chain barely touch one an- the wave travels away. The presence of a small distance n o other, the energy of an impulse only propagates as fully gap between the intruder and its nearest neighbors en- c nonlinear solitary waves [5, 6, 7] resulting from the bal- hances the oscillation amplitude. A multiscale analysis [ ancebetweendispersionandnonlinearityofthemedium. predicts the oscillations frequency as a function of in- 1 Nesterenkoearlydescribedthisregimeasasonicvacuum trudersizeandwavestrength. Highresolutionnumerical v limit[8]. Dissipativeeffectssuchasviscoelasticityorfric- simulations capture all the experimental features. 2 tiononlyattenuateandspreadthesesolitarywaves[7,9]. Beads are 100C6 steel roll bearings [17] with density 3 In contrast, any heterogeneity of the medium capable of ρ = 7780 kg/m3, Young’s modulus Y = 203 4 GPa 5 unbalancingdispersionandnonlinearityresultsinbreak- ± 3 and Poisson ratio ν = 0.3. The chain is made of 20 ing the solitary wave symmetry. For example, a narrow . equal beads (radius R = 13 mm) and contains an in- 1 pulse propagating in a chain of beads with decreasing 0 truder (2.5 mm Ri 10 mm) in the middle, as shown sizes develops a long tail which spreads in time the mo- ≤ ≤ 9 in Fig. 1. The beads, barely touching one another, are mentum transfer [10, 11, 12]. Designing powerfull im- 0 aligned on a horizontal Plexiglas track. A special short : pact protection systems takes advantage of these fea- v track automatically aligns the intruder on the axis of tures [10, 11, 13]. Granular chains made of successions i the chain [11]. The chain is ended by a flat, fixed and X of heavy and light beads also proved valuable efficiency heavy piece of steel. A nonlinear compressive wave (see r in energy absorption [14]. More recently, fully nonlinear a below) is initiated from the impact of a small striker waves with finite-width were observedin chains contain- (R =4 mm). The pulse is monitored by measuring the s ing periodic mass defects or soft inclusions [15]. Such load with a piezoelectric transducer (PCB 200B02, sen- nonlineardimerchainsareexpectedtosupportadditio n- nal optical modes and forbidden band gap when su b- jected to a static load [15]. trigger force PC The elementary interaction of either lighter or heavier A/D intruderswithsolitarywavesinnon-loadedmonodisperse chains of beads has been investigated numerically [1 6]. switch When a solitary wave reaches a mass defect, energy is partiallyreflectedintoabackwardtravelingsolitarywa ve FIG.1: Upperpanel: Experimentalsetupshowing achainof andispartiallytransmittedtotheintruder. Aheavyim - beads with an intruder,sensors and acquisition facilities. 2 sitivity 11.24 mV/N and stiffness 1.9 kN/µm) inserted N]12 iatrnhicasetliidvrpeeirgoaibdpeibeatryedtaiodemfs.actuhtTctehhiesneesnettsmwhoerobeimpsdaadcrseotsdms.opsfeaTnrahasnoebrlotertoihttgoauinlsbamealalalbdose’wssasodmfnaatotnhnede-- ch [V]Force [ 1048 (a) Force [N] 48 (c) LHFF intrusive measurements of the force inside the chain [7]. Swit 0 (b) 0 Typicalforcesignalsmeasuredattheintrudercontact, 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1.5 2.0 2.5 3.0 plottedin Fig.2a1,showsthatlargeandwelldefinedos- Time [ms] Time [ms] cillations appearin the tail ofthe incident solitarywave. FIG.3: Forceversustimeasdetectedbytheembeddedsensor The tail correspondsto a slight reflectionof the incident incontactwitha3mminradiusintruder. Thefirstlargepeak in(a) istheincidentsolitary wave,andthesecondoneisthe pulse on the intruder mass defect. Measured oscillations wave after being reflected by therigid end. The gap opening in the tail of the force are displayed in more details in between intruder and nearest beads is indicated in (b) by Figs. 2a3-2a6 for increasing amplitudes of the incident jumps from upper to lower level in the contact switch. High solitary wave. Dissipative processes being negligible [7], andlow frequencycomponentsoftheforce, shown in (c),are conservative numerical simulations (see below) shown in obtained by using a tenth order Butterworth filter with zero Fig. 2b1 demonstrate satisfactory agreements with ex- phase distortion. periments, withoutany adjusting parameter. Calculated overlapsbetweentheintruderandneighborbeads,shown inFig.2b3(seealsotherelativedisplacementsinFig.4a), time scales, that are separated by filtering low or high demonstrate that force oscillations correspond to local- frequency contents, as shown in Fig. 3c. Oscillations are izedoscillationsoftheintruder. Figs.2a2and2b2depict onlyobservedwhentheintruderisloadedbythesolitary power spectral densities of experimental and calculated wave. In Fig. 3a, intruder oscillations are only visible forces, respectively, in which the high frequency compo- in the tail of the incident pulse (0.0 ms < t < 1.0 ms). nent corresponds to the observed oscillations. Strongerandrelativelyfasteroscillationscanbeobserved Fig. 3 presents a longer force acquisition, in which the in the reflected pulse (1.5 ms < t < 3.0 ms), both dur- incident pulse and the pulse reflected by the rigid end ing the main compression and in the tail. Gap openings areshown. Forcesignalsexhibit featuresattwodifferent between the intruder and neighbor beads might explain the differences betweenincident andreflected pulses features. We designedanelectricalswitch(smooth metallic brushes in contact with neighbor beads, connected to a N] force [N] 23450000 (a1) force [force [N]012012 1.t4ime (([m1aa.s835])) 1 ti.m4e [((m1asa.8]64)) force [N] 23450000 (b1)m]overlap [0120.5(b3) tim1e. 0[m iisnn]ttrruuddeerr ((LR)1).5 fotFsaaifioscgcntt.oTa3(ntnbtTad,Lcgota0e.ffp.l5eT(cfi0hmtr)ressiwtcsaahwolnepidcntecinrhltcosusirssiitg2o,oh.fns1tec(eoam1nfFt)ste)aiwrgca.httnh1eoden)csctibneoueccraodisddn.eetsdAneactasrttcseaohtinmhonywepcnlerooennissdns-- ≃ ≃ incident SW incident SW of tail (t 1.1 ms and t 2.7 ms). The first opening 10 intruder (L) 10 intruder (L) ≃ ≃ is due to the weak reflection of the pulse on the mass 0 0 defect and the second one is due to the momentum and 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 energy transfer from the oscillating intruder to neigh- 10-1 time [ms] 10-1 time [ms] bors. Higher amplitude oscillations appears provided a Hz]10-3 HF oscillations Hz]10-3 HF oscillations gap between the intruder and neighbor exists. 2 D [N/10-5 (a2) 2 /D [N1 0-5 (b2) Theseobservationsarecorroboratedbynumericalsim- PS -7 incident SW PS -7 incident SW ulation. Calculated displacements of the intruder and 10 intruder (L) 10 intruder (L) -9 -9 neighbors shown in Fig. 4a and the overall gap around 10 3 4 10 3 4 10 frequency [Hz] 10 10 frequency [Hz] 10 the intruder(the difference betweenleft andrightneigh- FIG. 2: Insets (a1-a6) are experiments and (b1-b3) are nu- bor beads displacements) shown in Fig. 4b demonstrate mericalsimulations, bothperformed inamonodispersechain that gaps open twice at the intruder, consistently with containing a3mm in radius intruder. Dashed linein (a1,b1) our experiments. Fig. 4c, showing the intruder displace- is the 6 beads far the intruder incident solitary wave force ment relative to the center of mass of the two neigh- versus time, and solid line is the force versustime at the left bors beads, reveals that intruder oscillations even ex- intruder’scontact. Powerspectraldensitiesoftheseforcesare ist during the main compression of the incident pulse plotted in (a2,b2), respectively, showing high frequency con- (0.5 ms < t < 1.0 ms, see magnified view in Fig. 4d). tentintheintruderforce. Closerviewsofoscillatingtailinthe intruder’s force are shown in (a3-a6) for increasing incident Momentum transfer is enhanced by larger strain gradi- force magnitude (14.1, 17.3, 22.7 and 26.3 N, respectively). ent in the presence of gaps. Solidanddashedlinesin(b3)representoverlapsbetweenthe Next, we analyzes how oscillations depend on wave intruderand left and right neighbor beads, respectively. strength and on intruder size. Frequency of oscillations 3 m] turbations in the acoustic limit propagate at zero speed, m]displacement [246 uuLI pl. [m]overall gap [-- 0042....3000 (d) 1st gap opening uI-[u R [+uRu-Lu]/L2] 2nd gap opening (b) araeniacduoordmdusiiteyscirtsoifitcncueanmmltcoostocidoafhennEesamqralesyer.pte1[ri1ctebah8syel]u,nseuttsfhsstoieinarmbgesaimodatldibifotoeenanudrs.rdy,teOhwwdnoeafervosdehroeculsrveomRelsupeuttnhnoiosegfonetrn-h[Ko8bisn]ue.lpitinIntenga-- uR e dis0.0 0.6 0.8 1.0 incorporated in simulations for closer comparisons [19]. ativ Numerical time step is few orders of magnitude smaller 0 (a) rel-0.3 (c) thantheshortestphysicaldurationandenergyconserva- 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 tion is fulfilled within a relative error better than 10−9. time [ms] time [ms] The characteristic frequency of localized oscillations FIG. 4: Numerical simulations of the incident pulse, under can be estimated through a multiscale analysis of Eq. 1. sameconditionsasinFig.2b1. (a)Positionsversustimeofthe intruder and left and right neighbor beads. (b) Overall gap Here, index n denotes the intruder with radius Ri and around theintruderdemonstrating gap openings occur when mass mi = (4/3)πρRi3, and κi = κ(R,Ri) = κ(Ri,R) is asolitarywavecrossestheintruder. (c)Intruderdisplacement the elasticconstantthatdepends onradiiandproperties relative to the center of mass of the two neighbor beads. (d) ofthe beads. Index n 1 denote the two neighborbeads Magnified relative displacement duringthecompression. ± withradiusR>R . Weconsidertwodistincttimescales: i a slow variation of the order of solitary wave duration, and a fast time scale associated with intruder oscilla- is obtained from the analysis of power spectral densities tions. Displacements of the intruder and neighbor beads of the force. Following measurements are repeated nine timesandaveragedtominimizeerrors. Wefirstrunaset arewrittenasun un+uneiωt andun±1 un±1,where ≃ ≃ u andu areslowlyvaryingfunctionsoftime. Harmonic of experiments at constant intruder size (R = 2.5 mm) n n e i oscillationsamplitude is assumednegligible comparedto whilevaryingtheamplitudeoftheincidentsolitarywave. e solitary wave amplitude, u u . Replacements into Theforceamplitudeisobtainedfromthelow-passfiltered n n ≪ Eq. 1 leads to, signalsandresultsarepresentedinFig.5a. Wethenkeep e tthesetisnecviedreanltinpturlusdeearmsipzleist,udasedceopniscttaendti(n8.F6ig±.05.b5.NE)xpaenrd- u¨n ≃ mκi h(un−1−un)3+/2−(un−un+1)3+/2i, (2) i iments show that the frequency of the intruder nonlinearly increases with the incident solitary wave strength ω2 ≃ 32mκi h(un−1−un)1+/2+(un−un+1)1+/2i, (3) and depends on the intruder size. i The physical behavior of solitary waves in chains of where the first equation, at leading order, provides the equal beads and implications for the existence of local- displacementofthe intruderatslowtime scale. The sec- ized modes is summarized here. Under elastic defor- ond equation, at next order, determines the angular fre- mation, the energy stored at the contact between two quency of the oscillating intruder. We then introduce elastic bodies submitted to an axial compression cor- slowly varying forces at the contacts of the intruder, rδesipsotnhdes otoverHlaeprtzdepfootremnatitaiol,nUbHetw=ee(n2/b5o)κdδie5s/,2,κw−1her=e F− = κi(un−1 − un)3+/2 and F+ = κi(un − un+1)3+/2. (θ + θ′)(R−1 + R′−1)1/2 and θ = 3(1 ν2)/(4Y) are Noticing that they almost behave in phase (F+ ≃ F−, constants, and where R and R′ are res−pective radii of sceeeleFraigti.o2nb3o)s,citlhlaetfiesrsatreoquunadtieoqnuiinlidbirciautmespthosaittiionntr,uu¨dera0c-. n curvature at the contact. The force felt at the interface ≃ The second equation provides the maximum oscillation derives from Hertz potential, F = ∂ U = κδ3/2. In- H δ H + frequency f = max[ω/2π] as a function of the am- m dex + indicates that Hertz force is zero when the beads plitude of the slow time force at the intruder contact, loosecontact(no tensile force). The dynamicsofa chain Fm max[F+] max[F−]: of beads is thus described by the following system of N ≃ ≃ coupled nonlinear equations, 31/2κ1/3F1/6 (R/R )4/3F1/6 f i m C i m (4) mu¨n =κh(un−1−un)3+/2−(un−un+1)3+/2i, (1) m ≃ 2π m1i/2 ≃ (1+Ri/R)1/6 where m and un are the mass and the position of bead where C = (3/4π√πρ)/(2θR4)1/3. Eq. 4 shows that n, respectively. Considering long-wavelength perturba- oscillation frequency tends to zero when the load van- tions, such that the strain ψ = ( ∂xu) (δ/2R) ishes: oscillationsstopsassoonasthesolitarywaveleaves − ≃ ≃ (u/λ) 1, a continuous equation can be derived from the intruder. Using the characteristics of our beads, ≪ Eq.1,whichadmitsanexactsolution[8]intheformofa we find a theoretical estimation, C = 2640 Hz/N1/6. t purelycompressiveandperiodictravelingwave,ψ(x,t)= Matching Eq. 4 to experiments and simulations shown ψ cos4[(x vt)/(R√10)]. Wavespeed,v ψ1/4,nonlin- in Fig. 5, we find C = 2510 151 Hz/N1/6 and C = m m e n early depen−ds on maximum strain. Infini∝tely small per- 2531 52Hz/N1/6,respectivel±y. Theagreementbetween ± 4 characteristic distance that does not depend on static ncy [kHz]30 (a) simulations esximpeurliamtieonntss loaIdnaconndcwluhsiiochn,iswsemhaalvleerrethpaonrteadsoinbgsleervraatdiiounss,oxfce≪nerRg.y ue20 experiments exp. fit localizationin strongly nonlinear chains of elastic beads. q Fre experiments Localization occurs when a light intruder oscillates at a 10 exp. fit (b) timescalemuchshorterthanthedurationoftheincident 0 2 4 6 8 10 12 0.2 0.3 0.4 0.5 0.6 solitarywave. Oscillationsareexcitedbythe localstrain LF force amplitude [N] Radius ratio Ri/R gradient and enhanced by the presence of gap at the in- FIG.5: Inset(a)showsfrequencyversusforce’senvelopeam- truder contact. Oscillations are unable to radiate linear plitude, for an Ri = 2.5 mm in radius intruder. Triangles acoustic waves as they are forbidden in the sonic vac- and circles correspond to experimentalfrequency detected in uum limit. Such nonlinear localization effects are likely the tail of the incident pulse and in the reflected pulse, re- to appear in polydisperse 3D granular assemblies. For spectively. Inset(b)shows frequencyversusintruder’sradius instance,asignificantnumberofweaklyloadedlightpar- measured in the reflected signal, whose envelope amplitude was fixed to Fm = 8.7±0.6 N. Dashed lines correspond to ticles mightplayanimportantrole onsoundtrappingat numericalsimulationsandstraightlinescorrespondstoEq.4. high frequency. This work was supported by Conicyt under Fondap experiments, simulations and theory shown in Fig. 5 is researchprogram No 11980002. satisfactory,consideringthatnoadjustableparametersis used between experiments and simulations. It should be pointed out that weak delayed solitary wave trains, responsible for nonlinear leak of oscillating energy from the intruder to surrounding [16], were de- [1] I. M. Lifshitz, J. Phys. USSR 7, 215 (1943); ibid. 7, 249 (1943); ibid.8, 89 (1944). tected inlong durationacquisitions ofthe force recorded [2] E. W. Montroll and R. B. Potts, Phys. Rev. 100, 525 few beads before or after the intruder (not shown in this (1955); A.A.Maradudinetal.,Theory of lattice dynam- letter). This attenuation mechanism should lead to an ics in the harmonic approximation, vol. 3 of Solid State exponential decay of the oscillating amplitude within a Physics (Academic Press, New York,1971). characteristictime much longer than the duration of the [3] A. F. Vakakis et al., Normal modes and localization in low frequency force envelope, τ /τ m/m 1, nonlinearsystems (JohnWiley&Sons,NewYork,1996). provided the mass of the intrudleearkisLliFgh∼terpthan ai b≫ead [4] A. J. Sievers, Phys. Rev.Lett. 13, 310 (1964). [5] V. F. Nesterenko, Dynamics of heterogeneous materials of the chain, m m. i ≪ (Springer-Verlag, New York,2001). Eq. 4 can also be obtained by considering the low fre- [6] C.Coste,E.Falcon,andS.Fauve,Phys.Rev.E56,6104 quency force envelope amplitude, of the order of F , m (1997). as a static load at the time scale of fast oscillations. [7] S.Job,F.Melo,A.Sokolow,andS.Sen,Phys.Rev.Lett. The dynamics of a loaded monodisperse chain of beads 94, 178002 (2005). of mass m behaves according to linearized Hertz poten- [8] V. F. Nesterenko, J. Appl. Mech. Tech. Phys. 24, 733 tial around static equilibrium; the intergrain stiffness (1984); A. N. Lazaridi and V. F. Nesterenko, ibid. 26, reads k ≃ (3/2)κ2/3F1m/3. Small perturbations propa- [9] T40.5P(ö1s9c8h5e)l.and N. V. Brilliantov, Phys. Rev. E 63, 021 gate as acoustic waves according to the dispersion re- 505 (2001). lation ω = (2 k/m) sin(qR). Forcing a single parti- [10] R.L.DoneyandS.Sen,Phys.Rev.E72,041304(2005). p | | cle to move at an angular frequency ω above the cut- [11] F.Melo,S.Job,F.Santibanez,andF.Tapia,Phys.Rev. off ω = 2 k/m generates an evanescent wave since E 73, 041305 (2006). c p −1 [12] S.Job,F.Melo,A.Sokolow,andS.Sen,GranularMatter the wave number is qR = π/2 jcosh (ω/ω ). The group velocity vg = (∂ [q]/∂ω)−−1 is zero and tche per- [13] J1.0H, 1o3ng(,20P0h7y)s..Rev.Lett. 94, 108001 (2005). ℜ turbation remains localized within a characteristic dis- [14] R. L. Doney and S. Sen, Phys. Rev. Lett. 97, 155502 tance xc = ( [q])−1 = R/cosh−1(ω/ωc). Localization (2006). ℑ achieves when replacing a single particle by a light in- [15] M. A.Porter, C. Daraio, E. B.Herbold,I.Szelengowicz, truder with mass m m. Roughly estimating the an- andP.G.Kevrekidis,Phys.Rev.E77,015601(R)(2008). i ≪ [16] E. Hascoet and H. J. Hermann, Eur. Phys. J. B 4, 183 gular frequency of the intruder from the free oscillation (2000). frequency between two heavy non-oscillating neighbors, [17] See for instance, http://www.marteau-lemarie.fr/. ωi ≃p2ki/mi where ki ≃(3/2)κ2i/3F1m/3, leads to Eq. 4 [18] A. Chatterjee, Phys. Rev.E 59, 5912 (1999). expression. Thefrequencyofoscillationsexceedsthecut- [19] S.Job,F.Santibanez,F.Tapia,andF.Melo,Ultrasonics off, ω ω and the energy remains localized within a 48, 506 (2008). i c ≫