Frequency- and Amplitude-Dependent Transmission of StressWaves in Curved One-Dimensional Granular CrystalsComposed of Diatomic Particles

Jinkyu Yang & Chiara Daraio

Received: 19 January 2012 /Accepted: 25 June 2012 /Published online: 31 July 2012# Society for Experimental Mechanics 2012

Abstract We study the stress wave propagation in curvedchains of particles (granular crystals) confined by bent elasticguides. We report the frequency- and amplitude-dependentfiltering of transmitted waves in relation to various impactconditions and geometrical configurations. The granular crys-tals studied consist of alternating cylindrical and sphericalparticles pre-compressed with variable static loads. First, weexcite the granular crystals with small-amplitude, broadbandperturbations using a piezoelectric actuator to generate oscil-latory elastic waves. We find that the linear frequency spec-trum of the transmitted waves creates pass- and stop-bands inagreement with the theoretical dispersion relation, demon-strating the frequency-dependent filtering of input excitationsthrough the diatomic granular crystals. Next, we excite high-amplitude nonlinear pulses in the crystals using strikerimpacts. Experimental tests verify the formation and propa-gation of highly nonlinear solitary waves that exhibitamplitude-dependent attenuation. We show that the wavepropagation can be easily tuned by manipulating the pre-compression imposed to the chain or by varying the initialcurvature of the granular chains. We use a combined discreteelement (DE) and finite element (FE) numerical model tosimulate the propagation of both dispersive linear waves andcompactly-supported highly nonlinear waves.We find that thetunable, frequency- and amplitude-dependent filtering of theincoming signals results from the close interplay between thegranular particles and the soft elastic media. The findings in

this study suggest that hybrid structures composed of granularparticles and linear elastic media can be employed as newpassive acoustic filtering materials that selectively transmit ormitigate excitations in a desired range of frequencies andamplitudes.

Keywords Granular crystals . Phononic crystals . Diatomicchains .Highly nonlinear solitarywaves . Impactmitigation .

Acoustic filtering

PACS numbers . 43.58.Kr . 45.70.-n . 46.40.Cd . 43.20.Ks

Introduction

Granular particles in contacts, arranged in ordered lattices (i.e.,granular crystals), are efficient media for the transmission ofnonlinear stress waves [1, 2]. One-dimensional (1D) chains ofelastic spheresunderzeroorweakstaticcompressionhavebeenshown to support the formation and propagation of compactnonlinear waves in the form of solitary waves [1–6]. Thesesolitary waves derive from the nonlinear (Hertzian) contact

interaction between particles (i.e., F / d3=2 , where F is thecompressive force and δ is the approach between the particles)and a zero tensile response [1, 2]. Such formation of stablenonlinear waves in ordered 1D systems is in contrast to thehighlydispersiveanddissipative stresspropagationobservedindisordered granular media [7].

Highly nonlinear solitary waves in granular crystals exhibitunique physical properties compared to the conventionalstress waves in linear elastic media. First, solitary waves haveconstant spatial wavelength, confined to approximately fiveparticle diameters [2, 8]. Second, solitary waves have ex-tremely slow propagation speed, typically an order of magni-tude slower than that of dilatational waves within particles [2,6]. Furthermore, the propagation speed of solitary waves isamplitude-dependent. For a homogeneous granular chain, the

J. Yang :C. Daraio (*)Engineering and Applied Science,California Institute of Technology,1200 E. California Blvd.,Pasadena, CA 91125, USAe-mail: daraio@caltech.edu

J. YangMechanical Engineering, University of South Carolina,300 Main Street, A224,Columbia, SC 29208, USA

Experimental Mechanics (2013) 53:469–483DOI 10.1007/s11340-012-9652-y

wave speed can be expressed as Vs / F1=6m (Vs and Fm are the

speed of solitary waves and their maximum force amplitude,respectively) [2]. Lastly, solitary waves have unique interac-tion properties, allowing the formation of single solitarypulses, trains, and shock-like signals [8–13]. By leveragingthese characteristics, granular crystals have been proposed forseveral engineering applications, such as nonlinear lenses foracoustic imaging [14, 15], impact mitigation devices [16–20],and actuators and sensors for nondestructive evaluations[21–23].

Previous studies have shown that granular crystals canalso propagate linear elastic waves when they are excited bysmall-magnitude dynamic forces, relative to the static pre-compression applied to the system [2, 24, 25]. In this case,diatomic granular crystals (i.e., periodic chains with twoparticles per unit cell) support the formation of a frequencyband structure with allowable and forbidden frequencybands of signal propagation (i.e., pass- and stop-bands)[24]. Since stress waves in forbidden frequency bands areprohibited to transmit, such granular crystals can function aspassive acoustic filter structures. The selection of materialproperties, geometry, and boundary conditions of the gran-ular crystals allows controlling the number and location offrequency pass-/stop-bands. Herbold et al. investigated thetunable frequency band-gaps in diatomic granular crystals invarious combinations of dynamic and static forces appliedto the chain [24]. Boechler et al. controlled the frequencyband structures of a granular crystal composed of three-particle unit cells by simply manipulating the particle geometryand pre-compression [25].

This study investigates the comprehensive mechanismof wave formation and transmission in a diatomic gran-ular crystal under various geometrical configurations andimpact conditions, encompassing linear to highly nonlin-ear dynamical regimes. While previous studies focusedmainly on a straight, 1D chain of particles, here weconsider a curved granular system guided by soft elasticmedia to account for the coupling mechanism betweenthe granular particles and their surrounding elastic media.By using such a hybrid structure, we demonstrate exper-imentally the amplitude- and frequency-dependent filter-ing of compressive waves. First, we report the formationof acoustic band structures in the coupled granular andelastic media in the linear regime. We show that theacoustic band structure can be tuned in situ by manipu-lating the applied static pre-compression or the initialcurvature imposed on the elastic guides. Then, we in-crease the amplitude of the excited dynamic disturbancesrelative to the static pre-compression. As the dynamicamplitude increases, we observe the disappearance of thelinear acoustic band structure and the generation andpropagation of highly nonlinear solitary waves. We showexperimentally that the suppression of frequency band

structures is critically dependent on the level of nonlin-earity in the system, represented by the ratio of dynamicforce magnitudes to the static pre-compression. We alsodemonstrate that the transmission efficiency of solitarywaves can be controlled by the curvature initially im-posed to the diatomic granular system. The frequency-and amplitude-dependent behavior of the granular systemis verified by numerical simulations. We use a numericalmodel that integrates discrete element (DE) and finiteelement (FE) methods to simulate the propagation ofboth dispersive linear waves and compactly-supportedhighly nonlinear waves [26]. The experimental and numericalresults in this study demonstrate that the studied systems canbe excellent candidates for creating a new class of engineeredmaterials, which can transmit selected ranges of amplitudesand frequencies from external excitations.

The rest of the manuscript is structured as follows: Wedescribe the experimental setup in Section II. We discuss theformation of frequency band structures and the propagationof highly nonlinear solitary waves in Section III. In SectionIII, we also describe the numerical approach based on acombined DE and FE model. Section IV compares analyt-ical, numerical, and experimental results, focusing on theeffect of pre-compression, chain curvature, and strikermasses on the propagation of stress waves through thegranular system. Lastly, in Section V, we conclude themanuscript with a summary.

Experimental Setup

The overall configuration of the experimental setup isshown in Fig. 1. The test setup consists of two parts: agranular crystal composed of diatomic unit cells and anapparatus for applying external excitations to the topparticle of the granular crystal. The granular crystal consistsof an array of alternating spheres and cylinders made fromstainless steel (type 440 C, McMaster-Carr). The sphericalelements have a radius Rs09.53 mm, mass ms028.2 g, elasticmodulus Es0200 GPa, and Poisson’s ratio vs00.28. Thecylindrical elements have a radius Rc09.53 mm, aheight hc019.1 mm, and mass mc042.4 g, with materialproperties identical to those of the spherical elements.The granular crystal tested is 21 particles long with 10repeating unit cells (N010) and one additional sphericalparticle (i021) positioned at the bottom of the chain(see Fig. 1).

In this study, we consider two different configurationsof the granular crystal: a straight chain (Figs. 1(a)) and acurved chain (Figs. 1(b)), to investigate the effect ofchain curvature on stress transmission. The straight chainserves as a reference system by restricting the lateralmotions of granular particles using rigid guides made

470 Exp Mech (2013) 53:469–483

of hardened precision steel shafts. In this straight system,the propagation of stress waves is governed primarily bythe axial interactions between the granular particles ex-cited by an impact. In the curved diatomic chain, theparticles are guided by four polytetrafluoroethylene(PTFE) rods, which can flex upon the vertical impactapplied to the granular chains. As a result, the transmis-sion of waves along the granular crystals becomesaffected by the coupling mechanism between the gran-ules and the soft PTFE guides in the lateral direction. Inthis study the PTFE material is chosen for its flexibilityand effectiveness in reducing friction during these inter-actions. The PTFE rods have an outer radius RG06.35 mm,density ρG04,302 kg/m3, elastic modulus EG00.46 GPa,and Poisson’s ratio vG00.46. The initial curvature of thechain is determined by the bent PTFE rods, which areheld by the upper and lower stainless steel plates(Fig. 1(b)). We represent the curvature of the chain byΔ, which measures the offset of the chain from itscenterline. In this study, we test six different configurationsof curved chains (Δ0[22.4, 35.2, 58.9, 66.2, 73.3, 85.2] mm).A customized linkage structure provides vertical androtational degrees of freedom of the upper plate uponthe external impact.

To vary the profiles and magnitudes of external exci-tations applied to the granular chains, we use two differ-ent apparatuses: a piezoelectric actuator (Fig. 1(c)) andimpact strikers (Fig. 1 (d)). The piezoelectric actuator isused to apply small-amplitude disturbances to the granu-lar chain over a wide range of frequencies. In this study,we use a commercial piezoelectric actuator (PhysikInstrumente P-212) powered by an external voltage am-plifier (Piezo Systems Linear Amplifier EPA-104). Wecondition its output with white-noise signals (1-secondduration) generated by MATLAB. The piezoelectric ac-tuator is mounted to a guided support plate to make adirect contact with the top sphere of the chain. To im-pose pre-compression to the granular crystal, we place aring-shaped weight on top of this support plate. In thisstudy, we vary the amount of pre-compression by twoorders of magnitude from 4.7 N (weight of the supportplate) to 306.9 N.

Using a piezoelectric actuator, the maximum amplitudeof the excitation forces is limited by the stroke length of thetransducer and by the power available from the externalamplifier. To investigate the nonlinear responses of thegranular chain using large disturbances beyond the limitsof the piezoelectric transducer, we excite dynamic impacts

External excitations

i = 1

i = 2N+1

i = 2N

i = 2

i = 3

x

y

Amp.

Piezoactuator

WeightSupport

plate

Guide rods

Striker

Switch

Solenoid

(d)

(c)

i = 2N-1

x

y

. . .. . .

i = 1

i = 2

i = 3

i = 2N+1

External excitations

Force sensor

Upper plate

Lower plate

(a) (b)

i = 2N

i = 2n

i = 2n+1

i = 2n-1

wn-1

12,2 nnF

12n,2nn̂

lin,2P

wnun

nn 2,12Fr

in,2P

Fig. 1 Schematic of experimental setup. (a) A straight chain constrained by stainless steel guides. (b) A curved chain guided by bent PTFE guides.Inset shows a magnified view of particles under the interaction with linear guides. (c) A piezoelectric actuator to apply broadband excitations to thegranular chain. (d) Striker impacts using cylindrical masses. A solenoid is used to release the striker from a designated drop height

Exp Mech (2013) 53:469–483 471

via striker drop (Fig. 1(d)). We use two different types ofstrikers: a spherical striker made of aluminum 2017-T4(19.05-mm diameter) and a group of cylindrical strikers(440 C stainless steel) in various lengths. Using these strikerimpacts, we can achieve a wide range of compressive forcesfrom ~50 N (aluminum sphere drop from 3.2 mm) to ap-proximately 1,000 N (679 g cylinder drop from 10 mm). Tocontrol the amplitude of dynamic forces, we employ 13different cylindrical strikers with a 9.53-mm radius andlength L0[6.35, 9.53, 12.7, 15.9, 19.1, 22.2, 25.4, 50.8,102, 152, 203, 254, 305] mm. This corresponds to differentstrikers’ mass M0[14.1, 21.2, 28.2, 35.3, 42.3, 49.4, 56.5,113, 226, 339, 452, 565, 678] g. To impact the top sphere ofthe chain, we release spherical and cylindrical strikers from3.2-mm and 10-mm drop heights respectively using a DC-powered solenoid [27]. We apply pre-compression to thechain by resting a weight on the top of a guided supportplate, which has a central hole to expose partially the topsphere for striker impacts. The ring-shaped weight is cutopen to allow the access of the solenoid in the lateraldirection.

We measure the transmitted waves through the gran-ular chain using a commercial force sensor (Piezo-tronics, PCB-C04) positioned at the bottom of thegranular crystal. We mount the force sensor on a mas-sive block that simulates a rigid wall. The physicalstiffness of the force sensor is 1.05×109 N/m, whichis much higher than the contact stiffness between thediatomic particles over the range of pre-compressionforces considered in this study (the maximum contactstiffness calculated is 5.96×107 N/m under static pre-compression 306.9 N for the straight chain). The sen-sor’s cap is made of hardened 17-4 PH (H900) stainlesssteel, which is assumed to have identical density, elasticmodulus, and Poisson’s ratio to those of the beadscomposing the chain. The sensor is connected to a dataacquisition board (National Instrument PCI-6115) tocollect signals at a sampling frequency up to 10 MHz.For the statistical treatment of signals, we average themeasurements over five acquisitions, and the acquiredsignals are processed in MATLAB.

Theoretical Background and Numerical Approach

Previous studies have shown that a 1D granular crystalcan be modeled as a chain of point masses connectedby nonlinear springs based on Hertzian law [2, 5]. Thenumerical approach using this discrete element (DE)model is valid if the particles’ interactions are restrictedto small displacements and if the transit times of thestress waves in the granular crystal are much longerthan the oscillation period of elastic waves within the

particles [2]. Using this DE model, we can express theequations of motion for the diatomic particles in acurved chain as:

ms�u�n ¼ F2n�1;2n þ F2n;2nþ1 þ Pl

2n þ Pr2n þ msg

mc�w�n ¼ F2n;2nþ1 þ F2nþ1;2ðnþ1Þ þ Pl

2nþ1 þ PR2nþ1þmcg;

ð1Þwhere ms and mc are the masses of a sphere and acylinder, un and wn are the position vectors to theircenters in the n-th unit cell, and g is the gravitationalconstant (see the inset in Fig. 1(b)). The axial forceF2n,2n+1 exerted on the 2n-th particle by the neighboring(2n+1)-th particle can be expressed as:

F2n;2nþ1 ¼ A D� un � wnj j½ �kþbn2n;2nþ1; ð2Þaccording to theHertziancontact law.Here theunit cell length isD, and thebracket [x]+ takesonlypositivevaluesand returns0 ifx<0, implying no tensile strength between particles. The unitnormal vector bn2n;2nþ1 is defined as bn2n;2nþ1 ¼ un � wnð Þ=un � wnj j . Given the configurations of the sphere-cylindercontact, the Hertzian factor is k03/2 and the coefficient A can

be derived as A ¼ 4ffiffiffiffiRs

p3

1�v2sEs

þ 1�v2cEc

� ��1

. Here, R, E, and v

represent the radius, elastic modulus, and Poisson’s ratioof particles with the subscript s and c denoting sphericaland cylindrical elements [28]. Note that the particle dis-tance D and the Hertzian coefficient A in (equation (2))need to be modified at the end of the granular chain, suchas the interface with the elastic half space and the striker,to account for the bounding medium’s material and geo-metrical condition [27]. The lateral interactions with the

right- and left-hand guiding media, Pr2n and P

l2n can be also

expressed using the modified Hertzian contact law. Forthe sake of brevity, we do not include the expressions ofthe lateral contact forces in this manuscript. The completedescriptions of the axial and lateral forces including dis-sipative terms can be found in [26].

We simplify the equations of particles’motion into a one-dimensional form for theoretical considerations, while wetackle the two-dimensional (equation (1)) for numericalintegrations. After neglecting the lateral forces and the grav-itational effect, (equation (1)) becomes:

ms�u�n ¼ A wn�1 � unð Þkþ�A un � wnð Þkþ

mc�w�n ¼ A un � wnð Þkþ�A wn � unþ1ð Þkþ:

ð3Þ

Here, w and u are the displacements of cylindrical andspherical particles from their equilibrium positions (Fig. 2(a)), which should not be confused with the location vectorsof particles in (equation (1)). Depending on the level ofnonlinearity in the granular system, (equation (3)) can beapproximated into different mathematical forms, therebyproducing distinctive sets of mathematical solutions. Phys-ically, these solutions correspond to different types of stress

472 Exp Mech (2013) 53:469–483

waves ranging from linear elastic waves to highly nonlinearsolitary waves that can be generated and propagated in adiatomic chain.

Highly Nonlinear Regime

In a highly nonlinear regime, the interactions of granu-lar particles are governed by a full range of Hertziannonlinear contact (Fig. 2(b)). In this configuration, theamplitude of dynamic disturbances (Fd) applied to thegranular chain is much larger than that of the static pre-compression (F0). Given such a high-level nonlinearity,Porter et al. derived a closed-form solution of (equation(3)) using the long-wave approximation to describewave dynamics in a general one-dimensional diatomicchain [29]. We briefly introduce the analytical processof the closed-form solution derivation, while details canbe found in [29, 30].

Under the assumption of acoustic-mode excitations(i.e., in-phase oscillations of diatomic particles in a unitcell), the displacement of a cylindrical particle can beapproximated in terms of the neighboring spherical par-ticle’s motion by a Taylor expansion up to the fourthorder [29, 30]:

w ffi uþ b1Dux þ b2D2 uxx þ b3D

3 uxxx þ b4D4 uxxxx: ð4Þ

Here, D denotes the distance between two particles in aunit cell, and the coefficients are found to be b101,b20ms/(ms+mc), b30(2ms – mc)/3(ms+mc), and b40ms

(2ms2 – msmc+mc

2)/3(ms+mc)3 [30]. After plugging this

into (equation (3)) and incorporating terms, it is possi-ble to obtain a differential equation in the followingform [29, 30]:

utt ¼ uk�1x utt þ Guk�3

x u3xx þ Huk�2x uxxuxxc þ Iuk�1

x uxxxx;

ð5Þwhere t ¼ t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kDkþ1=ðms þ mcÞ

pis a rescaled time. The

constants in (equation (5)) are G0D2(2–3k+k2)ms2/6

(ms+mc)2, H0D2(k–1)(2ms–mc)/3(ms+mc), and I0D2

(ms2 – msmc+mc

2)/3(ms+mc)2 [29, 30]. This equation

is fundamentally similar to the differential equation thatNesterenko derived to describe the dynamics of a uniformspherical chain [2].

After several algebraic steps, an exact solution of(equation (5)) can be obtained in the highly nonlinearregime, i.e., when the pre-compression applied to thegranular chain is significantly smaller than the dynamic

F

d00F

F

ddF

dF

d

0F

2/3dd AFun-1

wn-1

un

wn

un+1

wn+1

F0 +Fd

ms

mc

D

(a) (b)

(c)

Fig. 2 (a) Schematic of a one-dimensional granular chaincomposed of spherical and cy-lindrical elements. (b) The force-displacement diagram of Hert-zian nonlinear contact. A high-lighted region in gray colorrepresents a fully nonlinear re-gime under large disturbancesrelative to the static pre-compression. (c) Force-displacement diagram under thelinear approximation. Givensmall disturbances compared topre-compression, the force-displacement relationship can belinearized as shown with adashed line in the highlightedregion

Exp Mech (2013) 53:469–483 473

disturbances (Fd >> F0). The trigonometric solution canbe expressed as [29, 30]:

ux ¼ Bcos2=ðk�1ÞðaxÞ; � p2� ax � p

2; ð6Þ

where ξ is a rescaled strain in the homogenized granularsystem, B is a parameter dependent on the wave speedand material properties, and α is a coefficient deter-mined by mass-ratios (ms/mc) of the diatomic granularsystem. The solution of the nonlinear equation (3) is asingle arch of the periodic profile as represented in(equation (6)), implying a finite-width wavelength ofthe nonlinear waves formed in the granular system[30]. For a monodispersed chain composed of sphericalparticles, Nesterenko found that the spatial width of thepropagating waves is equivalent to approximately five-particle diameters [2]. For a diatomic granular chain, theshape and propagating properties of nonlinear wavesdepend on the combination of the materials and mass-ratios of particles that compose the chain [29, 30]. Newfamilies of solitary waves in dimer chains have alsobeen presented in [31].

Linear Regime

Now we examine the propagation of acoustic waves when agranular chain is strongly compressed relative to the ampli-tude of the dynamic disturbances (Fd << F0) (Fig. 2(c)). Inthis case, we can separate the displacements of particles intostatic and dynamic components as un ¼ un;sta þ un;dyn andwn ¼ wn;sta þ wn;dyn , where subscript ‘sta’ and ‘dyn’ denotestatic and dynamic elements. We can plug these into (equation(3)) and expand it using the Taylor series under the assump-tion that the static displacement is much larger than thedynamic one. Neglecting the dynamic displacement terms ofthe second and higher order, we obtain a linearized equation ofmotion as [24]:

ms�u�n;dyn ¼ b wn�1;dyn � un;dyn

� �� b un;dyn � wn;dyn

� �mc

�w�n;dyn ¼ b un;dyn � wn;dyn

� �� b wn;dyn � unþ1;dyn

� �� ð7Þ

Here, the static displacement components are cancelledout due to the force equilibrium and thus, the equationof motion is described solely in terms of dynamicdisturbances. However, it should be noted that a line-arized stiffness β is determined by the static compres-sion as:

b ¼ 3

2A2=3F1=3

0 ; ð8Þ

where F0 denotes the static pre-compression force ini-tially applied to the granular crystal. This implies thatlarger pre-compression yields higher stiffness betweenthe adjoined particles.

We analytically solve (equation (7)) under the as-sumption that there is no dissipation in the systemand that the chain is infinitely long (i.e., there is noloss of energy and no boundary effects). We use theFloquet’s principle in which the particle dynamics in agiven cell is assumed to be fundamentally identical tothat of its neighboring cell based on the periodicity ofthe system [32, 33]. Mathematically, this implies thatthe particles’ motion in the (n+1)-th cell is expressed

as unþ1;dynðtÞ ¼ un;dynðtÞ e�ið4pD=lÞ and wnþ1;dynðtÞ ¼ wn;dyn

ðtÞe�ið4pD=lÞ , where λ is the wavelength of the propa-gating waves and un;dynðtÞ and wn;dynðtÞ are the dynamicdisplacements of particles in a reference (i.e., n-th) cell.After substituting these periodic expressions into (equa-tion (7)) and imposing the condition of non-trivialsolution, we obtain the following characteristic equationdefining the dispersive relation [24]:

msmcw4 � 2bðms þ mcÞw4 þ 2b2 1� cosð4pD=lÞð Þ ¼ 0;

ð9Þwhere ω is the angular frequency. By solving (equation(9)), we obtain the dispersion curve that defines theangular frequencies of the propagating waves as afunction of their wavelengths (see Fig. 3 for F00

00

1

2

3

4

5

6

7

8

Fre

qu

en

cy [k

Hz]

2πD/λ

AcousticOptical

π

f1

f2

f3

f4

Fig. 3 Dispersion diagram of a diatomic granular chain. The lowerand upper curves represent acoustic and optical pass bands respective-ly. A band-gap is formed between two distinctive pass bands. Thecutoff frequencies are [f1, f2 f3, f4]0[0, 4.827, 5.911, 7.632] kHz, giventhe 10 N pre-compression

474 Exp Mech (2013) 53:469–483

10 N). In this dispersion curve, the cutoff frequenciesof this dispersion relationship are found at l ¼ 2D andl ¼ 1:

f1 ¼ 0; f2 ¼ 1

2p

ffiffiffiffiffiffi2bmc

r; f3 ¼ 1

2p

ffiffiffiffiffiffi2bms

r;

f4 ¼ 1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2bðms þ mcÞ

msmc

s:

ð10Þ

It is evident that a 1D granular crystal in the linearregime allows the propagation of waves only in certainfrequency ranges [24]. The allowable frequency bands in[f1, f2] and [f3, f4] are called acoustic and optical bands,respectively. The stress waves whose frequencies belongto the frequency band in [f2, f3] cannot propagate in the granularsystem. These forbidden frequency bands are referred to asband-gaps or stop bands. The cutoff frequencies in (equation(10)) are functions of the particle masses and of the linearizedstiffness β. As described in (equation (8)), β can be altered bysimple manipulation of the pre-compression applied to thegranular system. This means that the acoustic structure ofdiatomic granular crystals can be tuned in situ without chang-ing materials or geometry of unit cell particles [24, 25].

The theoretical aspects of wave propagation in curvedgranular crystals are not as simple as those in a straight chain,due to the complex coupling mechanism between the granularand elastic media. To calculate the particles’ dynamics in acurved chain and their interactions with soft elastic guides, weemploy a numerical method that combines a discrete element(DE) and a finite element (FE) model. The DE model simu-lates the two-dimensional interactions of granular particlesbased on the Cundall [34] and Tsuji [35] models, which arebuilt on the classical Hertz-Mindlin theory [28]. The dynamicsof linear elastic guides are calculated by the FE model thatdiscretizes the linear elastic guides into the Bernoulli-Eulerbeam elements [36]. The combined DE and FE model takesinto account the axial and tangential interactions among par-ticles, including dissipative terms. In this study, we suppressthe dissipative terms among particles to assess the waveattenuation solely contributed by the structural coupling be-tween granules and the linear elastic guides. The details of thecombined DE and FE model can be found in [26].

Results and Discussion

In this section, we present the experimental and numericalresults obtained by varying the static pre-compression, thechain curvatures, and the impact amplitude via differentstriker masses. We compare these results with the theoreticalpredictions and discuss the findings.

Pre-Compression Effect

We begin investigating the effect of pre-compression on thepropagation of stress waves in granular crystals. We excitethe straight granular chain (Fig. 1(a)) using a sphericalaluminum striker and measure the transmitted waves arriv-ing in the force sensor under various pre-compressionsapplied to the chain. Figure 4(a) reports selected forceprofiles that are experimentally measured all under the iden-tical striker impact conditions. To ease graphical illustration,the results are plotted with a vertical shift of 50 N. Here, thepositive components in the y-axis denote compressive force,while the negative components represent tensile forces. Thetime in the x-axis is measured with respect to the trigger ofthe sensor at t00.4 ms, when the force amplitude reaches athreshold level. In Fig. 4(a), the lowest curve shows a force-time history obtained from a straight chain under no pre-compression, in which the dynamic disturbances are domi-nant in comparison to the pre-compression. In this configu-ration, we expect the propagation of highly nonlinearsolitary waves, as evident in the experimental measurementby the presence of a single pulse appearing around a time t~0.5 ms. We also observe small-amplitude fluctuations fol-lowing this impulse, which are likely caused by the oscil-lations of the particles within a unit cell. This differs from aclean, single hump of compressive waves witnessed in amonodispersed granular chain [2].

The nonlinear waves under zero-precompression are attrib-uted to the highly nonlinear interactions between the particles asdiscussed in Section III. As the pre-compression increasesrelative to the dynamic disturbances, the interfacial stiffnessbetween the particles transits from nonlinear to linear relation-ship, which yields an approximately uniform interfacial stiff-ness value between particles. In other words, the granulararchitecture becomes similar to a linear lattice structure, whichsupports the propagation of linear elastic waves. As a result,under the large values of pre-compression, we find that thetransmitted waves are highly oscillatory, showing the propaga-tion of not only compressive forces but also tensile componentsamong the particles (Fig. 4(a)). Note that these tensile forces inthe compressed chain do not imply the separation of particles,since the magnitude of pre-compression is much larger than thedynamic forces (Fd=F0 � 0:1for F00307 N) and all particlesremain compressed in contact with each other. The oscillatorywaves observed in the strongly pre-compressed chains arecaused by the propagation of particles’ vibration (also calledphonon) in linear lattices. This is fundamentally different fromhighly nonlinear solitary waves deriving from the unidirectionalcompressive motions of particles under zero pre-compressionand no tensile strength. This is why uncompressed granularcrystals are sometimes referred to as “sonic vacuum” due totheir incapability of transmitting linear elastic waves [2].

Exp Mech (2013) 53:469–483 475

The numerical results are reported in Fig. 4(b), in which atime t00 ms denotes the moment that the striker collideswith the top particle in the chain. We find that they corrobo-rate the experimental results. However, the amplitude of thenumerical force profiles is larger than that of the experimentalmeasurements. Such a discrepancy stems from the absence ofdissipative effects in the numerical simulations. Dissipativemechanisms in granular crystals have been studied previously,including the authors’workbasedonempiricalLaplacianmod-el [37] and quasi-static model using a Hertz-Mindlin theory[26]. However, these models are primarily focused onthe attenuation of solitary waves in a 1D granular chain,while they are not properly applicable to the stronglycompressed chain. Despite the negligence of the dissi-pative effects, however, we find that the DE-FE modelsuccessfully predicts the transition of wave characteristicsfrom nonlinear to linear modes. From the numericalsimulation results in Fig. 4(b), it is evident that thespeed of the transmitted waves is faster as the amplitudeof the static pre-compression increases. Such amplitude-dependence of wave speed is a unique property of generaltypes of nonlinear waves observed in a variety of physicalsystems [2, 38–41].

It is also worth discussing the effect of gravity, given thechain’s vertical configuration in this study. To investigatethe gravitational effect, we calculate the transmitted waveprofiles with and without the inclusion of gravity, whenthere is no external pre-compression (F000). Comparing

the thick (gravity) and thin (no gravity) curves at the bottomof Fig. 4(b), we find that the gravitational effect is noticeablein terms of wave form and speed. Particularly, the forceprofile under the gravitational effect contains more oscilla-tory components of stress waves compared to that withoutgravity. The offset between the two curves at the beginningof the signals corresponds to the weight of the granularchain that amounts to 7.7 N. However, the amount ofcompressive force induced by the chain weight is yet anorder of magnitude smaller than the dynamic disturbancescaused by the striker drop (Fd=F0 � 11:0). Therefore, thequalitative nature of the propagating waves remains thesame, exhibiting highly nonlinear solitary wave shapes.Figure 4(b) also includes a simulated result of the gravita-tional effect under the maximum pre-compression (F00307).In this strongly compressed granular chain, the effect ofgravity is even more insignificant as comparing the twocurves in the top of Fig. 4(b). This is because the gravity-induced force is very small relative to the externally imposedcompression. Therefore, the oscillatory shapes of the propa-gating waves are similar, and the change of cutoff frequencyof the propagating waves is expected to be negligible, consid-

ering fc / F1=60 deduced from (equations (8) and (10)) (cutoff

frequency shift is ~0.4 % due to gravity when F00307).To investigate the frequency spectrum of the transmitted

waves, we perform a fast Fourier transform (FFT) of the mea-sured time-domain signals. As a result, Figure 5(a) shows anexperimental plot of the power spectral density (PSD) in a

0 1 2 3 4 5Time [ms]

Fo

rce

[N]

0 1 2 3 4 5Time [ms]

Fo

rce

[N]

Sca

le:5

0N

F0 = 0 N

6.4N

14 N

33 N

77 N

166N

307N

Sca

le:1

00N

F0 = 0 N

6.4N

14 N

33 N

77 N

166N

307N

(a) (b)

Without gravityWith gravity

Fig. 4 (a) Experimental and (b) Numerical force profiles of transmitted waves through the diatomic granular chain under various pre-compressions. Toease visualization, the experimental and numerical force profiles are plotted with vertical shifts of 50 N and 100 N, respectively. The force profile positionedat the top shows oscillatory mechanical waves under high pre-compression (307 N), whereas the bottom curve corresponds to compactly-supported solitarywaves under no pre-compression (0 N). The numerical force profiles under F000 and 307 N (top and bottom curves in (b)) also include the simulationresults without gravitational effects (thinner lines) on the top of those under gravity (thicker lines)

476 Exp Mech (2013) 53:469–483

logarithmic scale, as a function of the static preload and fre-quency. Here, the light color regions represent the high ampli-tude ratios of transmitted waves to the input excitations (alsocalled transfer functiongain) (see the color bar inFig. 5(a)).Thedashed lines correspond to the theoretical cutoff frequencies inthe dispersion relation, obtained from (equation (10)). Theabscissa of the figure denotes the level of system nonlinearityquantified by the ratio of the dynamic disturbances to the staticpre-compression (Fd/F0). Since the dynamic disturbances areidentical throughout theexperiments, thevariationsofFd/F0aresolely determined by the pre-compression effect. The ordinaterepresents the frequency components of the transmittedwaves.

We note that frequency band structures are formed withdistinctive pass- and stop-bands, whose frequency ranges varysensitively with the amount of static pre-compression. This isconsistentwith theanalyticalpredictionthatadiatomicgranularcrystal forms acoustic and optical pass-bands and that theircutoff frequencies are governed by the linearized stiffness β,which in turn is a functionof the chain pre-compression [24]. InFig. 5(a), the acoustic and optical frequency bands are denotedas ‘pass-1’ and ‘pass-2’. The stopbandpositionedbetween twopass-bands is denoted as ‘stop-1’, and we conventionally referto the forbidden band located above the optical band as the‘stop-2’ band. We find that our experimental results are inagreement with the theoretical prediction.

Another interesting point is the dependence of transmissiongains (i.e., PSD values) on Fd/F0. In Fig. 5(a), the differencesof PSD values between pass- and stop-bands are obvious inthe lower Fd/F0, while such distinction become less evident inthe higher Fd/F0. This translates into the fact that the classicalfrequency band structure is a characteristic of the linear re-gime, with small-amplitude dynamic disturbances relative tothe pre-compression (i.e., small Fd/F0). In the case of zero orweak pre-compression (rightmost data sets in Fig. 5(a)), weobserve the concentration of PSD values in the low frequency

regime, but it is hard to distinguish pass- and stop-bandswithin this region. Figure 5(b) shows the corresponding nu-merical results based on the DE model.

To analyze the evolution of the frequency band struc-tures, we calculate the average PSD values in the pass- andstop-bands over a range of Fd/F0 tested in this study. Math-ematically, the average PSD can be quantified by:

PSD ¼ 1

fu � fl

Z fu

fl

PSD2df

� �1=2; ð11Þ

where fl and fu are the cutoff frequencies corresponding tothe lower and the upper boundaries of the frequency band.This value represents the amount of energy transmitted inthe given frequency band normalized by the width of thefrequency bands. Figures 6(a) and (b) show the experimen-tal and numerical results of the average PSD values as afunction of the nonlinearity level (Fd/F0). We find that theaverage PSD values in pass-bands are significantly higherthan those in the stop-bands in the linear regime (i.e., lowFd/F0). However, the discrepancies between pass- and stop-bands are reduced as the nonlinearity level is increased (i.e.,high Fd/F0). In particular, the average PSD values of thestop-bands become equivalent to those of the pass-bands atthe maximum nonlinearity level. The experimental and nu-merical results are qualitatively in agreement.

We quantify the prominence of frequency band structuresby calculating the ratios of pass-band PSD values to those ofthe neighboring stop-bands (Fig. 7). Here, the blue curvewith circular marks represents the PSD ratios of the firstpass- and stop-bands (Pass-1/Stop-1), while the green curvewith rectangular marks denotes the PSD ratios of the secondpass- and stop-bands (Pass-2/Stop-2). We find that at a lowFd/F0, the PSD ratio is as high as 103~104 due to theobvious dispersion of propagating waves and the resulting

(a) (b)

PASS-1

STOP-1

PASS-2

STOP-2

PASS-1

STOP-1

PASS-2

STOP-2

Fig. 5 (a) Experimental and (b) numerical results of PSD in a diatomic band structure as a function of Fd/F0. The highlighted zones represent passbands, while dark zones correspond to band gaps. The color bars denote transmission gains in a logarithmic scale. Yellow dotted lines denote cutofffrequencies based on theoretical dispersion relationship

Exp Mech (2013) 53:469–483 477

formation of frequency band structures. However, as the degreeof nonlinearity increases, the PSD ratios decrease to the level of10-1~101, which suggests the disappearance of the frequencyband structures. The experimental and numerical results are inagreement. The observed trend proves that the frequency bandstructure becomes dominant only if the granular systemsupports the propagation of linear elastic waves with intrinsicdispersive characteristics. As a granular chain is excited bystrong excitations and starts to form nonlinear waves, thetransmitted waves no longer contain frequency componentsthat correspond to the acoustic and optical pass bands.

Chain Curvature Effect

In this section, we discuss the effects of the chain’s curva-ture on the formation of frequency band structures. Weexcite the granular crystals using a piezoelectric actuator toconsistently apply small-amplitude, wide-band signals in acontrollable manner. The maximum amplitude of the dy-namic disturbances is limited to 0.5 N, which is orders of

magnitude smaller than the static pre-compression used inthis study [F0056.4 N, Order(Fd/F0)≈10-2]. For numericalsimulations, we use the same order of Fd/F0 to allow theformation of a full-fledged frequency band structure.

Figure 8 reports the frequency spectra of stress wavestransmitted through diatomic chains with four different cur-vatures. The y-axis is in a logarithmic scale, and the signalsare shifted by 102 to ease visualization. The black verticaldashed lines represent analytic predictions of cutoff frequen-cies from (equation (10)) for an equivalent straight chain(Δ00). Given the amplitude of the pre-compression used inexperiments, we obtain cutoff frequencies at f100 kHz, f206.46 kHz, f307.91 kHz, and f4010.2 kHz. From the exper-imental results in Fig. 8(a), we observe the presence of bothacoustic and optical bands depicted by the shaded regions inthe left- and right-hand sides (i.e., ‘pass-1’ and ‘pass-2’).These pass-bands are characterized by the sharp spikesrepresenting the resonant frequencies supported by the finitediatomic chains. In the straight chain, the measured passbands agree well with the theoretically predicted cutofffrequencies (top blue curve and vertical dashed lines in

100

101

10-1

100

101

102

103

104

PS

Dpa

ss /

PS

Dst

op

Pass-1/Stop-1Pass-2/Stop-2

100

101

100

101

102

103

104

Fd/F0 Fd/F0

PS

Dpa

ss /

PS

Dst

op

Pass-1/Stop-1Pass-2/Stop-2

(a) (b)Fig. 7 The average PSD ratiosof pass-bands to their adjacentstop-bands. (a) Experimentalresults. (b) Numerical results

100

101

10-6

10-5

10-4

10-3

10-2

Ave

rage

PS

D

Pass-1Pass-2Stop-1Stop-2

100

101

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Fd/F0 Fd/F0

Ave

rage

PS

DPass-1Pass-2Stop-1Stop-2

(a) (b)Fig. 6 Average PSD values as afunction of nonlinearity levels(Fd/F0). Four curves representthe PSD values calculated inpass- and stop-bands formed in adiatomic granular chain. (a) Ex-perimental results. (b) Numeri-cal results

478 Exp Mech (2013) 53:469–483

Fig. 8(a)). In curved chains, however, we find frequencybands gradually down-shifting as the chain curvatureincreases. In particular, when the chain’s curvature becomes85.2 mm (bottom cyan curve in Fig. 8(a)), the optical passband shifts to the middle of band-gap predicted for a straightchain. The numerical results in Fig. 8(b) corroborate theexperimental results.

To show the evolution of frequency bands as a functionof the chain curvature, we report in Fig. 9 the surface mapscorresponding to the experimental and numerical resultsshown in Fig. 8. We plot the experimental results from sevendifferent curvatures (Δ0[0, 22.4, 35.2, 58.9, 66.2, 73.3,85.2] mm) in Fig. 9(a), and we present the shift of frequencybands obtained numerically for every 5-mm offset from Δ00 mm to 100 mm in Fig. 9(b). Here, the horizontal dashedyellow lines denote the analytic cutoff frequencies of acous-tic and optical pass bands for the straight chain. In Fig. 9(a)

and (b), the upper bright band corresponds to the opticalpass band, while the lower bright band denotes the acousticpass band. The presence of a stop band between the opticaland acoustic bands is evident in both experimental andnumerical results. We can also verify that the frequencybands shift towards a lower frequency zone as the curvedchain exhibits a larger curvature. In particular, the shift ofthe optical pass band is more drastic than that of the acousticband.

We can qualitatively explain the dependence of frequen-cy band structures on the chain’s curvature by assessing thestatic compression between granules in a curved chain:From the force equilibrium when an identical static load isapplied to the granular chains, the compressive force be-tween granules is decreased as the chain’s curvature isincreased. This is because a curved chain dispenses a por-tion of compressive force to the guides through lateral

2 4 6 8 10 12Frequency [kHz]

PS

D

2 4 6 8 10 12Frequency [kHz]

PS

D

(a) (b)

Sca

le: 1

02

Sca

le: 1

02

0 mm0 mm

35.2 mm

66.2 mm

85.2 mm

35.2 mm

66.2 mm

85.2 mm

PASS-1 PASS-2STOP-1 STOP-2 PASS-2 STOP-2PASS-1 STOP-1Fig. . 8 PSD of transmittedwaves for the curved chain withΔ0[0, 35.2, 66.2, 85.2] mm. (a)Experimental results. (b) Nu-merical results. The shadedregions in the left- and right-hand sides represent the acousticand optical pass-bands

(a) (b)

Fig. 9 Surface maps of PSD as a function of diatomic chain curvature represented by chain offset values. (a) Experimental results forΔ0[0, 22.4, 35.2,58.9, 66.2, 73.3, 85.2] mm. (b) Numerical results for the various curvatures. The color bar reports the ratio of transmitted-wave amplitudes to those ofinput excitations in a logarithmic scale. Dashed yellow lines denote cutoff frequencies of acoustic and optical pass bands

Exp Mech (2013) 53:469–483 479

contact, while all compressive loading is imposed on thegranular particles in the case of a straight chain. Thus, despitethe identical pre-compression applied to the chain, the gran-ules in a curved chain are less compressed than those in astraight chain. According to the calculation of the linearizedstiffness in (equation (8)), a reduced amount of compressionbetween the particles produces a smaller stiffness value, whichleads to the decreased cutoff frequencies. Consequently, weobserve a down-shifting of frequency bands as we apply alarger offset to a curved granular chain. This means thatcompared to the straight chain, it is possible to obtain anadditional parameter to control the frequency response of acurved chain by leveraging the coupling mechanism betweenthe granules and the guided structures.

Striker Mass Effect

In this section, we study the combined effects of bothnonlinearity and geometry by impacting curved granularchains with cylindrical strikers. We control the level ofnonlinearity in the system, not by changing the pre-compression (F0), but by generating different dynamic forceamplitudes (Fd) varying the striker masses. The pre-compression in these tests is kept constant at F0056.4 N.Using various combinations of strikers and chain’s curva-tures, we characterize the dynamic responses of the bentgranular crystals focusing on the amplitude-dependent be-havior of the combined granular and linear elastic media.

We first investigate the propagation of stress waves ex-cited by a light mass striker (M014.1 g) with a velocity of0.443 m/s. The temporal force profiles recorded by the forcesensor are reported in Fig. 10(a) for the straight chain (solidblue line) and the curved chain with Δ058.9 mm (dashedred line). For this light striker with a mass smaller than thatof a bead (m028.2 g), we find that both the straight and thecurved chains result in the formation of stress waves con-taining both compressive (positive) and tensile (negative)components. We also observe that the initial pulses arefollowed by small amplitude oscillatory waves. The maxi-mum amplitude of the leading compressive wave is approx-imately 100 N, which is in the same order of the static pre-compression applied to the structure. This corresponds to aweakly-nonlinear regime, characterized by the presence ofleading solitary wave-like pulses followed by oscillatorywaves. Comparing the responses from the straight andcurved chains, we observe that the waves triggered by thelight striker in the curved chain are very similar to those inthe straight chain. This implies the wave transmissionbehaviors of the straight and curved chains are qualitativelysimilar in the weakly nonlinear regime. However, the max-imum force measured in the curved chain is reduced by22.9 % as compared to the straight chain (from 104 N inthe straight chain to 80.2 N in the curved one). This is due tothe loss of axial force components among the granules byinteracting with the linear elastic guides in lateral direction.

We then characterize the wave propagation through thestraight and curved chains excited by a larger mass striker

0.5 1 1.5-50

0

50

100

Time [ms]

For

ce [N

]

Experiment:14.1 g

0 mm58.9 mm

0.5 1 1.5

0

200

400

600

800

1000

Time [ms]

For

ce [N

]

Experiment:676 g

0 mm58.9 mm

0.5 1 1.5-50

0

50

100

150

Time [ms]

For

ce [N

]

Simulation:14.1 g

0 mm58.9 mm

0.5 1 1.5

0

200

400

600

800

1000

1200

Time [ms]

For

ce [N

]

Simulation:676 g

0 mm58.9 mm

(a) (b)

(c) (d)

Fig. 10 Experimental results ofthe force-time profiles for (a)light-mass (14.1 g) and (b)heavy-mass (678 g) impacts,measured by the force sensorlocated at the base of the granu-lar chains. Solid blue lines rep-resent the measurements fromthe straight chain, while dottedred lines denote those from thecurved chain with an initial cur-vature Δ058.9 mm. Numericalresults are plotted for (c) light-mass and (d) heavy-massimpacts

480 Exp Mech (2013) 53:469–483

(M0678 g). In Fig. 10(b), we observe that the maximumforce induced by the striker is in the range of Fd≈1,000 N.Note that according to [28], the maximum contact pressuredeveloped in the system remains below plastic limit evenunder the heaviest striker impact. Therefore, the measuredforce profiles are not significantly affected by plastic effects.The maximum dynamic force is much larger than the staticpre-compression applied to the chain (F0056.4 N). Thisimplies that the chain is now under the highly-nonlinearregime. In particular, we find that the shape of the nonlinearwaves propagating in the chain is totally different from thatof the weakly-nonlinear waves observed in the case of thelight mass impact. For the straight chain, we now observethe generation of solitary wave trains characterized by aleading pulse in large amplitude and the trailing wavesdecaying exponentially. The mechanism behind the forma-tion of the solitary wave trains can be explained by themultiple collisions between the heavy-mass striker and thegranular chain during the impact [8, 16, 42].

It is notable in Fig. 10(b) that the transmitted waves in thestraight and curved chains present a remarkably differentbehavior. We find that the transmitted waves in thecurved chain results in faster decay than that in thestraight chain, such that no significant waves are ob-served after the initial impulse around 0.5 ms. The max-imum amplitude of this initial pulse is also reduced by40.3 %, from 1,027 N in the straight chain to 613 N inthe curved one (see Fig. 10(b)). Considering that the areaunder the force-time curve corresponds to the amount ofimpulse transferred, we find that the transferred impulsein the curved chain is a mere 20.2 % compared to that inthe straight chain. This means that the curved systemscan attenuate large-amplitude impacts more effectivelyrelative to the straight chains. Furthermore, we can ob-serve an amplitude-dependent behavior of the curvedgranular chains, in which the energy absorption by theelastic guides is facilitated under stronger impacts com-pared to the small-amplitude disturbances. Figures 10(c)and (d) show the numerical results of the transmittedwaves under the small- and large-amplitude impact. Wefind that the numerical results agree well with the exper-imental findings.

The amplitude-dependent behavior of the curved granularsystem can be explained by the coupling mechanism be-tween the granules and the surrounding elastic media. Whilea straight chain transmits impact energy without significantlosses, a curved chain dispenses the kinetic energy carriedby the granular particles to the elastic guides through theirlateral interactions. If we apply a small-amplitude impact toa curved granular chain, the amount of energy absorbed bythe elastic guides is relatively small due to the weak pertur-bations experienced by the elastic guides. However, if alarge-amplitude impact is applied to the curved granular

chain, the elastic guides deform significantly during theshort impact event, and a large portion of the kinetic energyis dispensed to the elastic guides due to the structural defor-mation of the elastic guides. This means that the elasticguides present an improved efficiency of energy absorptionunder large dynamic disturbances relative to that of smallexcitations. In [26], such amplitude-dependent behavior in amonodispersed granular chain has been experimentally andnumerically verified using high-speed photography, and thefeasibility of tuning the efficiency of the energy absorptionwas demonstrated by manipulating the initial curvature ofthe granular system.

The frequency spectra of transmitted waves under vari-ous impactors are shown in Fig. 11 as a function of strikermasses and frequency components. Both numerical andexperimental results are based on the FFT of time-domainsignals, which are normalized with respect to their maxi-mum PSD values. For the straight chain, we observe thepresence of the optical and acoustic pass-bands in the lowrange of striker masses (see Fig. 11(a)). Here the yellowdotted lines represent the cutoff frequencies predicted ana-lytically. As the striker mass increases, however, the distinc-tion between pass- and stop-bands becomes less evident.This is consistent with the results observed in the previoussection that discussed the effect of system nonlinearity onfrequency band structures by altering pre-compression. Wealso find that most energy is concentrated in the near-zerofrequency domain when the system is excited by large-amplitude impacts, while the transmitted energy is distributedin the pass-bands when the system is excited by small-amplitude impacts. This means that the transmitted wavesexhibit distinctive energy distribution in the frequencydomain, which is highly dependent on the impact amplitudesimulated by various striker masses. Figure 11(b) shows thefrequency responses of the curved chain (Δ058.9 mm) undervarious impacts. We again confirm the presence of frequencyband structures resulted from the dispersive effect of propa-gating waves under the light-striker impacts. In comparison tothe straight chain, the band structure is down-shifted, which isin agreement with the findings in the previous section. Weperform numerical simulations using the combined DE andFE model, and the results are shown in Figs. 11(c) and (d). Wefind that the numerical results corroborate the experimentalresults satisfactorily both in the straight and curved chains.

The experimental and numerical results in this sectionconfirm the amplitude- and frequency-dependent behaviorof a curved granular chain. We observe an evident frequencyband structure in the linear regime, which can be tuned bymanipulating the initial curvature of the bent chains. We alsowitness the formation of highly nonlinear solitary waves inthe nonlinear regime, whose transmission efficiency is crit-ically determined by the amplitude of external impacts. Thisstudy establishes a foundation for utilizing the combined

Exp Mech (2013) 53:469–483 481

granular and elastic system as a novel structural material forengineering applications.

Conclusions

In this work, we demonstrated the frequency- andamplitude-dependent filtering properties of compressivestress waves transmitted through one-dimensional curvedgranular crystals composed of diatomic particles. Thegranular crystals studied consist of periodic arrays ofalternating spherical and cylindrical particles, which areconstrained by bent elastic guides. When external exci-tations are applied to the chains, the elastic guides canflex and absorb impact energy by converting the par-ticles’ kinetic energy to potential energy via structuraldeformation. Under small-amplitude disturbances rela-tive to the static pre-compression, the combined granu-lar and linear elastic system can transmit and supportlinear elastic waves, leading to the formation of acousticband structures with allowable and forbidden frequencybands. These band structures can be tuned by manipulatingthe amount of the static pre-compression or the chain

curvature initially imposed to the diatomic granular system.Hence, the combined granular and linear elastic media arecapable of filtering a selected range of frequencies containedin external impacts. Upon the excitation by large-amplitudeimpacts, this hybrid system can generate and propagatecompactly-supported nonlinear waves in the form of soli-tary waves. In this case, the amount of energy transmittedthrough the granular system is critically dependent on thewave amplitude, and the combined granular and linearelastic structure performs as an amplitude-dependent filter.We show that the efficiency of the energy transmission canbe tuned via simple manipulation of the curvature of thegranular system.

The findings of this work suggest the use of hybridgranular and soft elastic media in engineering applications,such as tunable protective devices and impact mitigatingstructures that can selectively allow or reject the transmis-sion of external impacts. The characteristics of these newstructures can be tailored by varying material types, geom-etry, and boundary conditions. In particular, by includingmaterials of contrasting properties (e.g., soft vs. rigid ele-ments), these structures can present unique combinations ofmechanical properties unprecedented in other mechanical

(a) (b)

(c) (d)

Fig. 11 Experimental (top row) and numerical (bottom row) surface plots of PSD for straight (left column) and curved (right column) chains. Eachsurface plot depicts the force profiles of all 13 striker impacts after being normalized with respect to their maximum PSD values. The dashed yellowlines denote the band edges of acoustic and optical pass bands predicted by analysis

482 Exp Mech (2013) 53:469–483

systems. This study is limited to the construction of a one-dimensional granular crystal, but the mechanism developedin this study can be extended to a three-dimensional archi-tecture, for example by arraying the granular crystals inelastic matrices. This can lead to the development of tunableand lightweight materials system that can form innovativestructures used in space, civil infrastructure, and biomedicalapplications.

Acknowledgements We acknowledge support from DARPA (Con-tract N. HR0011-10-C-0089, Dr. Jinendra Ranka), the National ScienceFoundation, Grant Number CMMI-0844540 (Career), and the ArmyResearch Office (MURI grant US ARO W911NF-09-1-0436, Dr. DavidStepp).

References

1. Nesterenko VF (1983) Propagation of nonlinear compressionpulses in granular media. J Appl Mech Tech Phys 24:733–743

2. Nesterenko VF (2001) Dynamics of Heterogeneous Materials.Springer-Verlag New York, Inc, New York

3. Coste C, Falcon E, Fauve S (1997) Solitary waves in a chain ofbeads under Hertz contact. Physical Review E 56(5):6104–6117

4. Chatterjee A (1999) Asymptotic solution for solitary waves in achain of elastic spheres. Physical Review E 59(5):5912–5919

5. Sen S, Hong J, Bang J, Avalos E, Doney R (2008) Solitary wavesin the granular chain. Physics Reports 462(2):21–66

6. DaraioC,NesterenkoVF,HerboldEB,JinS(2005)Stronglynonlinearwaves in a chain of Teflon beads. Physical Review E 72(1):016603

7. Jaeger HM, Nagel SR, Behringer RP (1996) Granular solids,liquids, and gases. Rev Mod Phys 68(4):1259–1273

8. Sokolow A, Bittle EG, Sen S (2007) Solitary wave train formationin Hertzian chains. EPL (Europhysics Letters) 77(2):24002

9. Manciu M, Sen S, Hurd AJ (2001) Crossing of identical solitarywaves in a chain of elastic beads. Physical Review E 63(1):016614

10. HerboldEB,NesterenkoVF(2007)Shockwavestructure ina stronglynonlinear lattice with viscous dissipation. Physical Review E 75(2):021304

11. Molinari A, Daraio C (2009) Stationary shocks in periodic highlynonlinear granular chains. Physical Review E 80:056602

12. Avalos E, Sen S (2009) How solitary waves collide in discretegranular alignments. Physical Review E 79(4):046607

13. Santibanez F, Munoz R, Caussarieu A, Job S, Melo F (2011)Experimental evidence of solitary wave interaction in Hertzianchains. Physical Review E 84(2):026604

14. Sen S, Mohan TRK, Donald P, Visco J, Swaminathan S, Sokolow A,Avalos E, Nakagawa M (2005) Using Mechanical Energy as a Probefor the Detection and Imaging of shallow Buried Inclusions in DryGranular Beds. Int J Mod Phys B (Singapore) 19(18):2951–2973

15. Spadoni A, Daraio C (2010) Generation and control of soundbullets with a nonlinear acoustic lens. Proc Natl Acad Sci U S A(107), pp. 7230-7234.

16. Sen S,Manciu FS,ManciuM (2001) Thermalizing an impulse. Phys-ica A: Statistical Mechanics and its Applications 299(3–4):551–558

17. Hong J (2005) Universal power-law decay of the impulse energy ingranular protectors. Physical Review Letters, 94,108001(10)

18. Daraio C, Nesterenko VF, Herbold EB, Jin S (2006) Energytrapping and shock disintegration in a composite granular medium.Phys Rev Lett 96(5):058002

19. Melo F, Job S, Santibanez F, Tapia F (2006) Experimental evi-dence of shock mitigation in a Hertzian tapered chain. PhysicalReview E 73(4):041305

20. Fraternali F, Porter MA, Daraio C (2009) Optimal Design of Com-posite Granular Protectors. Mech Adv Mater Struct 17(1):1–19

21. Khatri D, Rizzo P, Daraio C (2008) Highly nonlinear waves’sensor technology for highway infrastructures. SPIE Smart Struc-tures/NDE, 15th annual international symposium San Diego, CA,6934-6925

22. Yang J, Silvestro C, Sangiorgio S, Borkowski S, Ebramzadeh E,De Nardo L, Daraio C (2012) Nondestructive evaluation of ortho-pedic implant stability in THA using highly nonlinear solitarywaves. Smart Materials and Structures 21:012001

23. Yang J, Sangiorgio S, Silvestro C, De Nardo L, Daraio C,Ebramzadeh E (2012) Site-specific quantification of bonequality using highly nonlinear solitary waves, Journal of Bio-mechanical Engineering (in print).

24. Herbold E, Kim J, Nesterenko V, Wang S, Daraio C (2009) Pulsepropagation in a linear and nonlinear diatomic periodic chain: effectsof acoustic frequency band-gap. Acta Mechanica 205(1):85–103

25. Boechler N, Yang J, Theocharis G, Kevrekidis PG, Daraio C(2011) Tunable vibrational band gaps in one-dimensional diatomicgranular crystals with three-particle unit cells. J Appl Phys 109(7):074906–074907

26. Yang J, Dunatunga S, Daraio C (2012) Amplitude-dependentattenuation of compressive waves in curved granular crystals con-strained by elastic guides. Acta Mechanica 223(3):549–562

27. Yang J, Silvestro C, Khatri D, De Nardo L, Daraio C (2011)Interaction of highly nonlinear solitary waves with linear elasticmedia. Physical Review E 83:046606

28. Johnson KL (1985) Contact mechanics. Cambridge UniversityPress.

29. Porter MADC, Szelengowicz I, Herbold EB, Kevrekidis PG(2009) Highly Nonlinear Solitary Waves in Heterogeneous Peri-odic Granular Media. Physica D 238:666–676

30. Porter MA, Daraio C, Herbold EB, Szelengowicz I, Kevrekidis PG(2008) Highly nonlinear solitary waves in periodic dimer granularchains. Physical Review E 77:015601

31. Jayaprakash KR, Starosvetsky Y, Vakakis AF (2011) New familyof solitary waves in granular dimer chains with no precompression.Physical Review E 83(3):036606

32. Brillouin L (1953) Wave Propagation in Periodic Structures.Dover, New York

33. Phani AS, Fleck NA (2008) Elastic boundary layers in two-dimensional isotropic lattices. J Appl Mech 75(2):021020–021028

34. Cundall PA, Strack ODL (1979) A discrete numerical model forgranular assemblies. Geotechnique 29(1):47–65

35. Tsuji Y, Tanaka T, Ishida T (1992) Lagrangian numerical simula-tion of plug flow of cohesionless particles in a horizontal pipe.Powder Technol 71(3):239–250

36. Gere JM,TimoshenkoSP(1997)Mechanicsofmaterials, PwsPubCo.37. Carretero-González R, Khatri D, Porter MA, Kevrekidis PG, Daraio

C (2009) Dissipative solitary waves in granular crystals. Phys RevLett 102(2):024102

38. Lamb GL (1980) Elements of soliton theory, JohnWiley & Sons Inc.39. Scott A (2003) Nonlinear science: emergence and dynamics of

coherent structures, Oxford University Press.40. Dauxois T, Peyrard M (2006) Physics of solitons, Cambridge

University Press41. Daraio C, Nesterenko VF, Herbold EB, Jin S (2006) Tunability of

solitary wave properties in one-dimensional strongly nonlinearphononic crystals. Phys Rev E 73(2):026610

42. Job S, Melo F, Sokolow A, Sen S (2007) Solitary wave trains ingranular chains: experiments, theory and simulations. Granul Mat-ter 10:13–20

Exp Mech (2013) 53:469–483 483