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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 72 (2014) 161–173

http://d0022-50

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journal homepage: www.elsevier.com/locate/jmps

High-amplitude elastic solitary wave propagation in 1-Dgranular chains with preconditioned beads: Experimentsand theoretical analysis

Erheng Wang a, Mohith Manjunath a, Amnaya P. Awasthi a, Raj Kumar Pal b,Philippe H. Geubelle a, John Lambros a,n

a Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104S. Wright St, 306 Talbot Lab,Urbana, IL 61801, USAb Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206W Green St, Urbana, IL 61801, USA

a r t i c l e i n f o

Article history:Received 8 December 2013Received in revised form31 July 2014Accepted 3 August 2014Available online 14 August 2014

Keywords:Stress concentrationsContact mechanicsGranular materialStress wavesKolsky bar

x.doi.org/10.1016/j.jmps.2014.08.00296/& 2014 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ1 217 333 2242.ail address: lambros@illinois.edu (J. Lambros

a b s t r a c t

Elastic solitary waves resulting from Hertzian contact in one-dimensional (1-D) granularchains have demonstrated promising properties for wave tailoring such as amplitude-dependent wave speed and acoustic band gap zones. However, as load increases, plasticityor other material nonlinearities significantly affect the contact behavior between particlesand hence alter the elastic solitary wave formation. This restricts the possible exploitationof solitary wave properties to relatively low load levels (up to a few hundred Newtons).In this work, a method, which we term preconditioning, based on contact pre-yielding isimplemented to increase the contact force elastic limit of metallic beads in contact andconsequently enhance the ability of 1-D granular chains to sustain high-amplitude elasticsolitary waves. Theoretical analyses of single particle deformation and of wave propaga-tion in a 1-D chain under different preconditioning levels are presented, while acomplementary experimental setup was developed to demonstrate such behavior inpractice. The experimental results show that 1-D granular chains with preconditionedbeads can sustain high amplitude (up to several kN peak force) solitary waves. The solitarywave speed is affected by both the wave amplitude and the preconditioning level, whilethe wave spatial wavelength is still close to 5 times the preconditioned bead size.Comparison between the theoretical and experimental results shows that the currenttheory can capture the effect of preconditioning level on the solitary wave speed.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Elastic stress wave propagation in one-dimensional (1-D) granular chains has demonstrated novel characteristicsstemming from the strong non-linear interaction behavior between granular particles in contact, i.e. the Hertzian contact(Nesterenko, 1983, 2001; Shukla et al., 1993; Zhu et al., 1997; Coste et al., 1997; Daraio et al., 2005, 2006; Jayaprakash et al.,2011; Awasthi et al., 2012; Pal et al., 2014). The non-linearity of the Hertzian contact response produces a new type ofsolitary wave with strongly non-linear features, quite different from weakly nonlinear solitary waves originating from the

).

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173162

Korteweg–de Vries equation (Korteweg and de Vries, 1895). For example, in a 1-D granular chain, the spatial width of such asolitary wave does not depend on the loading duration while its wave speed has a nonlinear dependence on the loadingamplitude. These features have been utilized to tailor the propagation of such strongly nonlinear solitary waves in specificmicroscale and macroscale designs to achieve desired goals such as pulse energy trapping and shock disintegration(Nesterenko et al., 1995; Coste et al., 1997; Molinari and Daraio, 2009; Ngo et al., 2012).

Because of the significant stress concentrations present at the contact region between granules, experiments can onlyhave a limited maximum load level imposed by the need to maintain the elastic (Hertzian) nature of particle-to-particlecontact. For example, the maximum load level for a contact point between brass alloy 260 beads with 10 mm diameter isabout 5 N and between stainless steel 440 C beads with 25.4 mm diameter is about 500 N based on Hertzian contact theoryand the von Mises yield criterion (Johnson, 1985). Any realistic dynamic loading application with substantial loading overthe elastic limit such as blast or ballistic loading with a multiple kN loading level will cause plasticity, or other materialnonlinearity, in the contact region. Since the appearance of plasticity significantly affects contact behavior between ductileparticles (Li et al., 2009; Wang et al., 2013a,b; Pal et al., 2013), it also influences the formation of solitary waves. On et al.(2014) have shown the existence of a solitary-like elasto-plastic wave in a granular chain with ductile particles. This type ofwave is different from the regular elastic solitary waves because of its diminishing amplitude, and the dynamics of theformation and propagation of such waves is not fully understood yet. In order to extend the applicability of the strongnonlinearity of solitary waves to much larger loading range, a granular system, which can retain elastic conditions underhigher loading levels is necessary.

From the classic Hertzian contact theory, plasticity can be thought to initiate when the stress in the contacting bodysatisfies the von Mises yield criterion (Johnson, 1985). The elastic contact force limit (critical contact force for yieldinitiation) Fy is then given by

Fy ¼ 16

Rn

En

� �2

ð1:6πσyÞ3; ð1Þ

where σy is the material yield strength. En and Rn are the equivalent Young's modulus and relative radius, respectively,defined as

1En

¼ ð1�v21ÞE1

þð1�v22ÞE2

; ð2aÞ

1Rn

¼ 1R1

þ 1R2

; ð2bÞ

where E is the Young's modulus, R is the radius, and subscripts 1 and 2 refer to the two different contacting particles. It isclear from Eq. (1) that if we want to increase the elastic contact force limit, we can either increase the contact radius (whichis the same as the particle radius for spherical particles), decrease the Young's modulus or increase the yield strength of thematerial. Decreasing Young's modulus is neither easily achieved, since we can only choose from a limited set of existingmaterials, nor necessarily desirable based on static structural considerations. Therefore increasing contact radius and yieldstrength of the material simultaneously is the approach followed in this work.

Pre-yielding or strain hardening has been widely used for increasing the yield strength of ductile materials in applicationsuch as metal rolling or shot peening (Ray et al., 1994; Kim et al., 2008). Here we introduce a similar method, defined aspreconditioning, in which an individual contact point is pre-yielded to a certain load level and then unloaded. Consequently,the contact point will remain elastic upon reloading as long as the load level remains below the original preconditioningamount. The goal of this work is to understand how preconditioning affects solitary wave propagation in 1-D granularchains and how to control the preconditioning process. We focus on three specific tasks: (i) Implementing thepreconditioning treatment and quantitatively controlling the process; (ii) Experimentally and theoretically proving thatthe 1-D granular chain with preconditioned beads can sustain high amplitude elastic solitary waves; and (iii) Performingparametric studies to evaluate elastic solitary wave propagation in 1-D granular chains with preconditioned beads.

2. Theoretical considerations

2.1. Elasto-plastic contact behavior

In the present work, we only consider contact between identical spherical metallic particles that comprise a repeatingunit in a granular chain and each contact point in the chain is preconditioned identically. A sketch of the contact force vs.displacement (between the centers of the two particles) for an identical spherical bead pair in contact is shown in Fig. 1a.Sketches of the contact point are also shown in the figure for three points during the contact loading and unloading process.The elastic region is represented by OA in Fig. 1a. When the load is increased beyond point A, plasticity sets in and isrepresented by the line AB. Since irreversible plastic deformation occurs in region AB, the unloading behavior follows adifferent path, line BC, which corresponds to elastic unloading.

Fig. 1. Elasto-plastic contact response and geometry of the yielded bead. (a) Elasto-plastic net contact force vs. displacement response, and (b) unloadedparticle shape after plastic deformation.

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173 163

For the elastic region the behavior can be described by Hertzian contact as

F ¼ 43En

ffiffiffiffiffiffiffiffiffiffiRnδ3

pð3Þ

where F is the contact force and δ is the displacement between the particle centers. However, as shown in Nesterenko(2001), Daraio et al. (2006), Porter et al. (2009) and Wang et al. (2013a) among others, the yield force in experiments can behigher than the value predicted by Eq. (1) using a simple von Mises yield criterion. It has been observed that if we rewrite Fyin Eq. (1) as

Fy ¼16

Rn

En

� �2

ðCπσyÞ3; ð4Þ

where C is a constant, then the yield force can be realistically evaluated by increasing the value of C (originally 1.6 in Eq. (1)).For example, C¼4.7 for brass alloy 260 (Wang et al., 2013a) and C¼9.14 for stainless steel (grade 316) (Stevens and Hrenya,2005).

The Thornton (1997) model, as modified in Stevens and Hrenya (2005) and Wang et al. (2013a), can be used to describethe subsequent elasto-plastic contact behavior. According to this model the elasto-plastic loading region (line AB in Fig. 1a)in the contact behavior is described by

F ¼ FyþCπσyRnðδ�δyÞ; ð5Þ

where δy is the contact displacement at yield. For the elastic unloading region (line BC in Fig. 1a), the contact behavior is stillassumed to follow Hertzian contact given by

F ¼ 43En

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRn

pðδ�δrÞ3q

ð6Þ

with the deformed relative radius Rpn¼Rp/2 (Rp is shown schematically in Fig. 1b) now replacing the undeformed relative

radius Rn¼R/2 (R is the radius of the original bead). Clearly Rp4R and therefore a consequence of preconditioning is that itincreases both the particle contact radius and the local yield strength for any subsequent contact.

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173164

The geometrical parameters shown in Fig. 1b after plastic deformation can be explicitly calculated using the modifiedThornton model (i.e., based on the appropriate C value in Eq. (5)). The contact radius after plastic deformation, Rp, can beexpressed as

Rp ¼ 2Rn

p ¼8En

3Fmax

2FmaxþFy2Cπσy

� �3=2

; ð7Þ

where Fmax is the maximum normal net contact force schematically shown in Fig. 1a, i.e., the preconditioning level.The residual plastic displacement δr can be calculated as

δr ¼ δmax� 3Fmax

4EnffiffiffiffiffiRn

p

q0B@

1CA

2=3

; ð8Þ

where δmax is the maximum normal contact displacement. Therefore, the final particle size_R after unloading, which is half

of the particle dimension along the loading direction in a preconditioned chain, can be calculated as

_R ¼ R�δr

2¼ R�δmax

2þ12

3Fmax

4EnffiffiffiffiffiRn

p

q0B@

1CA

2=3

ð9Þ

2.2. Elastic solitary wave propagation in a 1-D preconditioned granular chain

Following the analysis process for a “weakly compressed” granular chain (Nesterenko, 2001; Coste et al., 1997), theequation of motion for a preconditioned granular chain is

€ui ¼ A⟨ui�1�ui⟩3=2�A⟨ui�uiþ1⟩

3=2; ð10Þwhere ⟨X⟩¼ X if XZ0 and zero otherwise, and ui is the displacement of ith particle in the chain. A is a coefficient expressedas

A¼ 43En

mRn1=2

p ¼ En

πR3ρ

Rp

2

� �1=2

; ð11Þ

where m is the mass of the bead and ρ the material density. It can be seen that the equation of motion for a preconditionedgranular chain (Eq. (10)) is similar to that of a non-preconditioned granular chain (Eq. 1.12 in Nesterenko (2001)), except forthe expression of the coefficient A, as A for a preconditioned granular chain contains the deformed contact radius Rpseparately from the original contact radius R.

Using a long wavelength approximation (Nesterenko, 2001), we can obtain the wave equation from the equation ofmotion (the partial differential form of Eq. (10)):

utt ¼ Aa5=232ð�uxÞ1=2uxxþ

a2

8ð�uxÞ1=2uxxxx�

a2

8uxxuxxx

ð�uxÞ1=2� a2

64ðuxxÞ3

ð�uxÞ3=2

( ); ð12Þ

where u is the displacement, x is directed along the 1-D chain and a is the particle dimension in the x direction. The particledimension a of a preconditioned granular chain is a¼ 2

_R, which as was pointed out differs from the non-preconditioned

value of 2R.Again, following Nesterenko's procedure (Nesterenko, 2001), the wave speed Vs can then be solved from Eq. (12) as

Vs ¼ 2ffiffiffi5

p ðAa5=2Þ1=2ðξmÞ1=4; ð13Þ

where ξm is the maximum strain and can be expressed by the maximum displacement δm between centers of two particlesand the particle size a as

ξm ¼ j�uxjmax �δma

¼ δm

2_R: ð14Þ

The maximum displacement δm can be calculated from the Hertzian contact law (Eq. (3))

δm ¼ 3Fm4EnRn1=2

� �2=3

¼ 3Fm4EnðRp=2Þ1=2

" #2=3: ð15Þ

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173 165

Substituting ξm and A in Eq. (13) with Eqs. (11), (14) and (15), we obtain a new expression for the wave speed in apreconditioned 1-D granular chain as

Vs �4ffiffiffi5

p 43

� �1=3 En2

m3

!1=6_R

Rp

2

� �1=6

F1=6m : ð16Þ

Thus, the (solitary) wave speed in a preconditioned granular chain is governed by the preconditioned contact radius,Rp, the particle radius after unloading,

_R and the particle mass, m, which is related to the original particle radius, R.

Following Nesterenko's procedure to solve the (solitary) wave spatial length (Nesterenko, 2001), we find that the wavespatial length is only affected by the parameter a given the form of the wave equation, Eq. (12), while the coefficient A doesnot affect the spatial wavelength. We can therefore calculate the solitary wave spatial length in a preconditioned granularchain as

Ls ¼ 5affiffiffiffiffiffi10

p π � 5a; ð17Þ

but now with a¼ 2_R.

3. Experimental methodology

3.1. Preconditioning treatment

The (initially) spherical beads used in this study were made of brass alloy 260 and had diameter of 9.53 mm (3/8 in).Brass alloy 260 has density of 8500 kg/m3, Young's modulus 115 GPa, Poisson ratio 0.3 and yield strength of 550 MPa basedon the manufacturer's data (McMaster-Carr). For such beads the elastic limit contact force is only about 5 N as calculatedfrom Eq. (1). To introduce preconditioning in the beads, individual beads were placed in an Instron servohydraulic machineand statically compressed to predetermined loads. Fig. 2 shows top-view microscope images of the yielded area for differentpreconditioning levels (top row) and corresponding surface profiles measured by a Dektak 3030 profilometer (bottom row).The yield area is circular and clearly has a radius of curvature upon unloading. With reference to Fig. 1b, the bead originalradius R, projected radius of the contact area r, and particle size

_R, can be measured after preconditioning. From geometric

considerations, we can easily show that the contact radius, Rp, after plastic deformation can then be calculated by

Rp ¼ð_R�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2�r2

pÞ2þr2

2ð_R�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2�r2

pÞ

ð18Þ

Fig. 2. Microscope images (top) and surface profiles (bottom) of preconditioned contact areas at different preconditioning load levels: (a) 6 kN, (b) 9 kN,and (c) 15 kN.

Fig. 3. Contact radius after deformation Rp at different preconditioning force levels.

Fig. 4. Deformed particle size_R at different preconditioning force levels.

Fig. 5. Experimental setup for bead impact experiment. Top: sketch, Bottom: photograph of actual setup.

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173166

Experimentally measured results after preconditioning are compared to the Thornton model prediction in Figs. 3 and 4.Fig. 3 shows the contact radius after deformation compared to Eq. (7), and Fig. 4 shows the final particle size compared toEq. (9), both of which compare very well with the appropriate value of C, which can be determined by a series of contactexperiments (Wang et al., 2013a).

3.2. Experimental setup and calibration

The preconditioned beads were then used to form a 1-D chain with the preconditioned areas contacting each other.A schematic and a photograph of the bead shooting setup used here are shown in Fig. 5. The 1-D preconditioned bead chain

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173 167

was held in a clear plastic tube to maintain one-dimensionality during the experiments. A 12.7 mm diameter C350 maragingsteel transmitter bar was in contact with the end of the bead chain to both support the chain and to provide a measure offorce transmitted through it. Following the approach adopted in Nesterenko (1983) and Nesterenko et al. (1995), a flat discpiezoelectric sensor was embedded into specific beads to measure the force profile within the chain. The piezoelectricsensor was obtained from Steiner & Martins Inc. and had a diameter of 7 mm and a thickness of 200 μm. In a typicalexperiment, a single bead is propelled by a gas gun through the tube supporting the 1-D preconditioned array, as shown inFig. 5, and impacts the granular chain thus generating a pulse through it. To synchronize all signals, the data acquisitionsystemwas triggered from the piezoelectric sensor, although a bead impact velocity sensor was also mounted directly aheadof the 1-D array to measure impact velocity.

Beads embedded with piezoelectric sensors have been widely used for measuring elastic solitary wave propagation in a1-D non-preconditioned bead chains. If the sensor mass is assumed to be much smaller than the half bead mass, the sensorforce can be compared with the average value of the contact forces on either side of the bead (Daraio et al., 2005). However,there are two main issues that need additional consideration in the present experiments: precise timing and level ofmeasured force profiles. Forces F1(t) and F2(t) applied on the left and right side of the bead with the embedded piezoelectricsensor, respectively, initiate at a slightly different time because of wave propagation through the bead. This time differencecan affect the average force magnitude and force profiles considerably. Hence, comparison between the average forces from,say, a numerical mass-spring model simulation that ignores wave propagation effects within the beads and an experimentalmeasurement that includes them is not directly feasible. However, correctly reconstructing the contact force profile from theembedded sensor signal is possible, as will be shown below. The second issue pertains to the load level measured by theembedded sensor. Previous studies have been limited to low load levels of several Newtons to hundreds of Newtons(Nesterenko, 1983, 2001; Daraio et al., 2005) and use a calibration process based on conversation of momentum ofimpacting beads (Daraio, 2006). However, this method cannot be extended directly to the high load levels used in thepresent study (kN range) as it is an indirect measurement subject to increasing error as load increases. Therefore, acalibration and signal reconstruction process of the piezoelectric sensor readings at a high load levels has to be devised.

The calibration setup used for the sensor bead is similar to the experimental setup presented earlier, and is shown inFig. 6. The preconditioned sensor bead is placed in contact with the transmitter bar and two non-preconditioned beads areplaced before the sensor bead to serve as pulse shapers. Assuming 1-D wave propagation in the transmitter bar, as is thecase in standard split-Hopkinson pressure bar experiments (Subhash and Ravichandran, 2000), the strain gauge signalrecorded in the transmitter bar represents the contact force between the sensor bead and the bar, i.e., force F2. Since thetransmitter bar measures the force on the right side of the bead only, whereas the piezoelectric sensor records the averageforce from the left and right sides, i.e., (F1þF2)/2, the sensor signal cannot be calibrated directly to the bar strain gaugesignal. However, the peak value of the sensor signal can be calibrated to determine the peak contact force, and thus thesensor sensitivity.

Fig. 7a shows a typical force signal measured from the transmitter bar strain gauge and peak-force-calibrated sensor signal on acommon time scale. Except for the peak force, the calibrated sensor signal does not overlap the measured contact force profile intime. Schematic representations of the contact forces on either side of the sensor bead, F1(t) and F2(t), and the average force profilethat would bemeasured by the sensor are shown Fig. 7b. Generally, we can only obtain the average force profile (red curve) duringthe experiments although we desire to obtain the actual contact force between particles (i.e., blue or black curves). The goal here isto obtain the real contact force profile (blue and black lines) using only the measured average force profile. If the arrival timedifference between the contact forces (blue and black line) is known, the average force profile (red line) can be separated startingfrom its initial time into several segments with time interval tstep, which is same as the arrival time difference seen in Fig. 7b. Thus,in the first segment, since the wave has not arrived at the right hand side of the sensor bead, the contact force there should bezero, i.e., F12ðtÞ ¼ 0, where the superscript denotes the segment sequence. From the average force, the left contact profile in the firstsegment can be calculated as F11ðtÞ ¼ 2F1ðtÞ�F12ðtÞ ¼ 2F1ðtÞ, where F1(t) is the average force in the first segment. As mentionedabove, the left and right contact forces should have same profiles if the sensor bead remains elastic, as is the case here. Thefollowing relation can then be obtained:

Fnþ12 ðtÞ ¼ Fn1ðtÞ; ð19Þ

where n is an integer larger than 0 representing the segment sequence in Fig. 7b. Therefore, the contact force profile, F1, in othersegments can also be calculated as

Fnþ11 ðtÞ ¼ 2Fnþ1ðtÞ�Fnþ1

2 ðtÞ ¼ 2Fnþ1ðtÞ�Fn1ðtÞ: ð20Þ

Fig. 6. Bead sensor calibration setup.

Fig. 7. (a) Comparison between the bar signal and sensor signal used to find detector sensitivity; (b) Contact force reconstruction process; (c) Contact forceprofiles measured from the bar and reconstructed from the sensor.

Fig. 8. Experimental verification of high amplitude solitary wave from experiment with two embedded sensors.

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173168

Repeating this step (Eqs. (19) and (20)) throughout all of segments and calibrating for the peak force value (done earlier) yields anaccurate measurement of contact force.

The key parameter, the arrival time difference of the contact forces or the time interval tstep, is the same as the pulselength difference between the contact force profile (blue line) and the average force profile (red line) as shown in Fig. 7b.Therefore, it can be easily obtained experimentally using the measured strain gauge and piezo-electric signals in Fig. 7a.Fig. 7c shows the measured force F2 from the transmitter bar and the reconstructed F2 from the piezoelectric sensor usingthe procedure described above. It can be seen that the reconstructed contact force agrees very well with the one measuredby the bar, thus validating both the signal reconstruction and calibration process for the contact force.

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173 169

4. Results and discussion

4.1. High-amplitude solitary waves in a preconditioned bead chain

One of the main goals of this work is to generate and sustain a high-amplitude solitary wave in an elastic granular chain.As in many prior works, we use a single bead impact to load the granular chain. However, the impact speed used here, andconsequently the peak force generated, are much higher than previous studies which intentionally kept them low in orderto maintain elastic contact between the (non-preconditioned) beads. To observe the formation of high amplitude solitarywaves, two piezoelectric sensors were placed in a 15 kN preconditioned bead chain at bead locations 3 and 8 of a 12 beadchain. Fig. 8 shows measured signals from the two sensors from a typical experiment. The force profiles of the primary pulsemeasured by both sensors are almost identical, implying an elastic wave is propagating along the chain. The wave amplitudeis about 8 kN, well below the preconditioning level of 15 kN. The arrival time difference between the two sensors is about27 μs, and since the distance between them is 41 mm, the wave speed is calculated at about 1500 m/s, which is much lessthan elastic dilatational wave speed of brass 260 (�3700 m/s). The pulse length is �25 μs, which is close to the propagatingtime (27 μs) from sensor 1 to sensor 2, i.e., a wavelength of 5 beads. This further supports that the wave inside thispreconditioned bead chain is indeed a solitary wave.

4.2. Comparison with non-preconditioned chain

Since the sensor beads with an embedded piezoelectric disk are not exactly the same as regular beads in the chain, theyeffectively act as intruders in the granular chain (although their effect is not very pronounced as seen in the signals of Fig. 8).However, in order to minimize the effect of two sensor beads, we used the setup shown in Fig. 5 with one sensor bead onlyand compared that with the force recorded from the transmitter bar. Both a chain with 15 kN preconditioned beads and onewith non-preconditioned beads were subjected to impact loading. Since the wave in a non-preconditioned bead chaindissipates with propagation distance, slightly different geometries were used in order to obtain similar initial pulse profilesin both cases. Thirteen 15 kN preconditioned beads were used in the preconditioned chain case and the sensor was locatedat the 6th bead from the impact side, while 9 beads were used in the non-preconditioned case and the sensor was located atthe 2nd bead from the impact side. For both cases, however, the distance between the sensor location and the chain-barinterface was the same.

The measured contact force profiles from the bead sensor and the transmitted force from the bar are plotted in Fig. 9 forboth cases. The incident pulses, the reflected pulses at the bead chain-transmitter bar interface and the transmitted pulsesin the transmitter bar are pointed out in the figure. The incident load level is about 3.2 kN, significantly lower than thepreconditioning level (15 kN) but much higher than the elastic contact force limit for non-preconditioned beads. Althoughthe incident contact force profiles for the preconditioned bead chain and non-preconditioned bead chains are almostidentical, the reflected and transmitted profiles are very different. For the preconditioned bead chain, the transmitted forcealso has a single-peak shape and a pulse length that is similar to the incident profile, demonstrating a behavior typical ofelastic solitary wave propagation. The transmitted pulse for the non-preconditioned bead chain does not have a one-peakshape, it is of much lower amplitude, and the pulse length is highly extended to over 100 μs, which is over twice the incidentpulse length. This is characteristic of plastic waves in granular chains undergoing yielding (Pal et al., 2013; On et al., 2014).We also note that a much larger amount of energy was reflected back into the bead chain for the non-preconditioned beadchain than that for the preconditioned one. This observation is associated with the fact that contact area between the barand the last bead for a preconditioned bead chain can be much larger than that for a non-preconditioned one, which reducesthe wave impedance mismatch between the bar and bead and consequently reduces the reflected pulse.

Fig. 9. Wave profiles in 1-D bead chains with 15 kN pre-conditioned beads (labeled as 15 kN) and with non-preconditioned beads (labeled as 0 kN).

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173170

4.3. Effect of preconditioning level on wave propagation

The results of Fig. 9 for the 15 kN preconditioned beads and the non-preconditioned beads imply that the wave speeddepends on preconditioning level, even for the same amplitude of incident wave. In order to conclusively prove thisassertion, which is also expected from the results of Eq. (16), four different granular chains with non-preconditioned, 6 kNpreconditioned, 9 kN preconditioned and 15 kN preconditioned beads were subjected to single-bead impact. Eachexperiment had only one bead with an embedded sensor. The force amplitude of the wave pulse was measured directlyby this embedded sensor. The wave propagation time in the chain from the sensor bead to the chain-transmitter barinterface was obtained by subtracting the wave propagation time in the bar (from the chain-transmitter bar interface to thestrain gauge) from the wave arrival time difference at the embedded sensor and the strain gauge. The average wave speedwas then calculated by dividing the distance between the sensor bead and the chain-transmitter bar interface by the actualpropagation time.

The wave speed vs. wave amplitude experimental data are presented in Fig. 10. The horizontal errors bar on measuredforces is based on the calibrated sensitivity of the bead sensor discussed above, while the vertical error bars on measuredspeed are based on the choice of arrival times of each pulse. Generally, there will be a 75 μs time difference for themeasured wave arrival times. From Fig. 10 we observe that the wave speed is affected by both the wave amplitude, asexpected for elastic solitary waves (Nesterenko, 2001), and the preconditioning level. At the same wave amplitude, solitarywave speed increases with increasing preconditioning level. Since wave speed is only affected by wave amplitude in a non-preconditioned chain, preconditioned bead chains enrich our ability to control and tune solitary wave propagation ingranular media.

Fig. 10 also includes the theoretical predictions of wave speed from Eq. (16). The predicted values do not agree preciselywith the experimental results for each individual preconditioning level, possibly for three reasons: Firstly, the precondi-tioned beads were treated individually which may result in preconditioned shape and size variability. Secondly, and moreimportantly, alignment of preconditioned beads in a 1-D chain so that their pre-yielded areas are in contact is donemanually and can contain misalignments. Thirdly, the model results are somewhat sensitive to the value of δr, that cancontain experimental error (Awasthi et al., 2014). However, the theoretical predictions do capture the general trend of the

Fig. 10. Effect of preconditioning load level on the average wave speed dependence on wave amplitude.

Fig. 11. Effect of preconditioning load level on the average wavelength.

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173 171

experimental results. Another point to note specifically for the non-preconditioned case (i.e., 0 kN and black line in Fig. 10) isthat in the model we assume the chain remains elastic even though the wave amplitude exceeds the elastic contact forcelimit. Therefore, a direct comparison between the measurements, which are definitely plastic for no preconditioning, andthe model, which is elastic, is not valid for the non-preconditioned case.

Recall that we can also measure the pulse duration and wave speed of the solitary wave in the preconditioned beadchain, and can therefore extract its spatial wavelength. Fig. 11 compares the experimental results of the solitary wave lengthwith the prediction from Eq. (17). The experimental results agree reasonably well with the prediction, which verifies theearlier conclusion that the solitary wave spatial length is still about five times of the bead size after preconditioning.

4.4. Theoretical parametric study

Since Eq. (16) is able to capture the trend of how solitary wave speed is affected by preconditioning level and waveamplitude, we perform several parametric studies to investigate in more detail some effects on the solitary wave speed. Alltheoretical predictions are based on the physical parameters for brass alloy 260. From the discussion in Section 2, we knowthat some parameters in Eq. (16) can be related to preconditioning level, namely, Rp and

_R, which can be expressed in terms

of Fmax by Eqs. (7) and (9) respectively. Thus, substituting for Rp and_R in Eq. (16) and using the value of Fy from Eq. (4) gives

Vs ¼ 1:97 4En3

3m3Fmax

� �1=6 2Fmax þ 124

REn

� �2ðCπσyÞ3

2Cπσy

!1=4

F1=6m

� R�Fmax� 1

24REn

� �2ðCπσyÞ3

CπσyRþ3Fmax

8En

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Cπσy

2Fmax� 124

REn

� �2ðCπσyÞ3

vuut264

375: ð21Þ

This expression points to three adjustable parameters that affect solitary wave speed (for a specific material): thepreconditioning level Fmax, the wave amplitude Fm and the particle radius R.

If the bead physical parameters (size and density) and the wave amplitude Fm are fixed, the solitary wave speed Vs is onlyaffected by preconditioning level, Fmax. Fig. 12 shows that the solitary wave speed Vs changes with Fmax as obtained from Eq.(21) at different wave amplitudes. We observe that the solitary wave speed increases with the preconditioning level at agiven wave amplitude, which has also been observed in Fig. 10. Additionally, the influence of preconditioning level on wavespeed diminishes for larger load levels.

The Vs�R�Fmax relation for a fixed wave amplitude Fm¼1 kN is presented in Fig. 13a, showing that the solitary wavespeed in a preconditioned chain with smaller size beads varies more significantly with preconditioning level possiblybecause at the same preconditioning level a smaller bead would exhibit more substantial deformation than a larger bead.Fig. 13b shows a slice of the 3-D plot in Fig. 13a for three different preconditioning levels. Beyond a radius of about 4 mm, theinfluence of preconditioning level changes from increasing to decreasing – although, for radii more than 4 mm, thedifference with preconditioning level is small. Note that in our experiments R is about 4.5 mm. Therefore, for smaller beadsizes, we would expect an even more pronounced effect of preconditioning than what is seen in the current results. Anotherinteresting feature is that the Vs�R relation is not monotonic. There is a peak value of Vs for a certain R at eachpreconditioned level. These phenomena could enhance our ability to control the propagation of the solitary wave by offeringmore avenues of controlling solitary wave speed.

The Vs�R�Fm relation at a fixed preconditioning level Fmax¼9 kN is shown in Fig. 14a, with Fig. 14b presenting similarslices at different wave amplitudes. The results again show that the Vs–R relation is not monotonic and shows a maximum ata given radius. Vs approaches zero when R approaches zero, especially at large preconditioning level Fmax. Recall, however,

Fig. 12. Effect of wave amplitude on the wave speed.

Fig. 13. Vs–R–Fmax relation for a fixed wave amplitude: (a) 3-D plot of relationship, (b) Effect of sphere radius on the wave speed at the fixed waveamplitude and different preconditioning levels.

Fig. 14. Vs–R–Fm relation for a fixed preconditioning load level: (a) 3-D plot of relationship, and (b) effect of sphere radius on the wave speed at the fixedpreconditioning level and different wave amplitudes.

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173172

that this analysis is based on the modified Thornton model. According to the simulation results by Pal et al. (2013), when themaximum contact displacement δmax during the preconditioning treatment is larger than 10% of the bead original radius R,the contact behavior deviates from the model prediction because of the bead shape distortion. Therefore, practically, therewill be a force limitation for the above analysis.

5. Conclusions

In this paper, we introduced and implemented a preconditioning method based on pre-yielded spherical particles toincrease the elastic limit force at the contact location in 1-D ordered granular chains. This preconditioning method isefficient because it can increase not only the yield strength of the material at the contact point but also increase the contactradius, both of which affect the contact yield force. After preconditioning treatment, the contact point can elasticallywithstand several kN of contact force (up to the preconditioning level) without any additional plastic deformation. Atheoretical analysis has shown that the wave propagation behavior in a 1-D granular chain with preconditioned beads iscontrolled by the preconditioning load level. A bead shooting setup was implemented to study wave propagationexperimentally in a 1-D granular chain with preconditioned beads. The experimental results prove that such preconditionedbead chains can sustain high amplitude elastic solitary waves. The measured solitary wave speed depends not only on thewave amplitude, as expected for elastic solitary waves, but also on the preconditioning level. Further analysis shows thatthe current theoretical model can capture the trend of solitary wave speed on wave amplitude and preconditioning level.The propagation of the solitary wave in a preconditioned bead chain is also sensitive to the original size of the bead.The preconditioning level affects the solitary wave speed more efficiently in a preconditioned chain with smaller beads.The theoretical analysis also reveals that there is a peak solitary wave speed at a fixed preconditioning level and waveamplitude.

E. Wang et al. / J. Mech. Phys. Solids 72 (2014) 161–173 173

Acknowledgments

This work was funded by the US Army Research Office (ARO) MURI grant W911NF-09-1-0436. Dr. David Stepp is thegrant monitor. The authors also thank Mr. Owen Kingstedt for his help in obtaining the yield surface profiles.

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