J. Appl. Phys. 125, 145104 (2019); https://doi.org/10.1063/1.5081035 125, 145104

© 2019 Author(s).

Weak-shock wave propagation in polymer-based particulate compositesCite as: J. Appl. Phys. 125, 145104 (2019); https://doi.org/10.1063/1.5081035Submitted: 13 November 2018 . Accepted: 21 March 2019 . Published Online: 09 April 2019

S. Ravindran, A. Tessema , A. Kidane , and J. Jordan

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Weak-shock wave propagation in polymer-basedparticulate composites

Cite as: J. Appl. Phys. 125, 145104 (2019); doi: 10.1063/1.5081035

View Online Export Citation CrossMarkSubmitted: 13 November 2018 · Accepted: 21 March 2019 ·Published Online: 9 April 2019

S. Ravindran,1 A. Tessema,1 A. Kidane,1,a) and J. Jordan2

AFFILIATIONS

1Department of Mechanical Engineering, University of South Carolina, 300 Main Street, Columbia, South Carolina 29208, USA2Shock and Detonation Physics, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

a)Electronic mail: Kidanea@email.sc.edu

ABSTRACT

Shock waves are common in polymer-based particulate composites that are subjected to intermediate to high-velocity impact loading.However, quantitative information on the spatial variation of stress, particle velocities, and energy dissipation during the formation andpropagation of weak-shock waves is limited. In this paper, a systematic experimental study is conducted to understand the characteristics ofweak-shocks in polymer-bonded particulate composites. Specimens made of polymer-bonded sugar are subjected to a projectile impactloading, at varying velocities, using a modified Hopkinson pressure bar apparatus. Full-field displacement and strains of the deformedsamples are obtained with the help of an ultrahigh-speed imaging and digital image correlation technique. Using the full-field displacementdata, the shock wave velocity, shock front thickness, and the full-field stress fields are calculated. From the spatial stress field and the strainrate data, the spatial energy dissipation profile is also estimated. The effect of impact velocity on the spatial stress profile, shock wavevelocity, and energy dissipation are discussed.

Published under license by AIP Publishing. https://doi.org/10.1063/1.5081035

I. INTRODUCTION

Polymer-bonded explosives (PBXs) are a class of particulatecomposites with a high solid volume fraction. They typically contain80%–95% explosive crystals such as 1,3,5,7-tetranitro-1,3,5,7-tetrazo-cane (HMX), 1,3,5-trinitro-1,3,5-triazinane (RDX), 2,2-bis[(nitrooxy)methyl] propane-1,3-diyl dinitrate (PETN), and so on and 5%–20%of the soft polymer binder. The soft polymer binder is used to reduceshock sensitivity and to improve castability and machinability. Thesematerials are highly heterogeneous due to a substantial mismatch inthe material properties of the constituents and an unavoidable poros-ity in the material at multiple length scales.

These materials can be subjected to a wide range of loadingconditions from low mechanical insults during handling and trans-portation to strong impacts during an accident. It has been seenthat weak insults with an impact velocity of around 40–150 m/s cancause the material to have a deflagration (low-speed combustion)to detonation (rapid combustion) transition (DDT).1–4 Severalexperimental and numerical studies have been performed to under-stand the deformation mechanisms that cause such reactions.1,3,5–18

It has been found that bulk load transfers in heterogeneouscomposites, such as PBXs, take place through particle to particle

contacts, which lead to stress–strain localization in the area of thecontact surfaces, even for weak impact loading conditions.15,16,19

Also, the plastic deformation of the binder, void collapse, debond-ing of the crystals, and frictional heating of the failure surface ofthe crystals can cause the dissipation of the energy in localizedregions in the material.5,7,15,19,20 These local energy dissipationmechanisms produce small regions of high temperature calledhotspots, which eventually trigger a reaction. Recently it has beenshown, using the ultrasonication-based loading, that the hotspotscan be formed due to the friction between the binder and thecrystals.21 In addition, crystals that are contacting each other areseen to generate higher temperatures compared with crystals thatare separated by a thin polymer binder, which indicates that fric-tional heating plays a dominant role in the hot spot formation.22

On the other hand, numerical studies in PBXs have shown apropagation of a compaction type wave at low impact velocities(50–200 m/s).23 Though several experimental-based investigationsare available to quantify the shock propagation nature in PBX atdifferent loading regimes,24–26 no attempt has been made tomeasure the spatial stress profile, which is essential in estimatingthe spatial energy dissipation during shock wave propagation.

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J. Appl. Phys. 125, 145104 (2019); doi: 10.1063/1.5081035 125, 145104-1

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Recently, digital image correlation (DIC)-based high strain rateexperiments have been developed, which allow for full-field strainand displacement measurement at a high spatial resolution underdynamic loading conditions.12,13,27,28 It is a promising method tomeasure the local strain field and particle displacement profileduring the propagation of the weak shocks.29

In this paper, a full-field deformation measurement techniqueat a high temporal resolution is used to investigate the weak-shockpropagation in polymer-bonded explosives. A detailed discussionof the weak-shock wave properties such as wave velocity, spatialstress profile, energy dissipated, and the wave thickness at threeimpact velocities is presented.

II. MATERIALS AND METHODS

A. Material preparation

Polymer-bonded sugar (PBS), a mechanical simulant ofHMX-based PBX, is used in this study. Sugar crystals have a mono-clinic crystal structure30 and morphological characteristics similar tothat of HMX, and they are a suitable mechanical surrogate for HMX.Polymer-bonded sugar has been extensively used as an inert simulantto investigate PBX under different loading conditions.11,12,14,18,31–33

The polymer-bonded sugar specimens used in this study wereprepared by cold pressing sieved sugar crystals with the particle sizedistribution shown in Fig. 1 and a plasticized hydroxyl-terminatedpolybutadiene (HTPB) mixture. The formulation composed of87.5/9.0/2.4/1.1 wt. % of sugar/hydroxyl-terminated polybutadiene(HTPB)/di-octyl sebacate (DOS)/toluene diisocyanate (TDI), whereHTPB is the monomer, DOS is the plasticizer, and TDI is the curingagent. The HTPB is mixed with the plasticizer (DOS) and curingagent (TDI); then, the sugar crystals are added to the mixture. Themixture is stirred to ensure thorough mixing and then placed in a

vacuum oven for 6–7 h at 80 °C for partial curing. The partially curedgluey mixture is pressed at 30MPa pressure in a cylindrical mold ofbore diameter 25mm to produce cylindrical billets. These billets arethen completely cured at 80 °C for 120 h and cooled slowly to roomtemperature. Samples for the experiments are machined from thecylindrical billets using a milling machine. The extracted samples aremechanically dry polished with abrasive grinding paper up to a gritof 1200 to have a flat and smooth surface appropriate for DICimaging. The density of the pressed PBS samples is 1.34 g/cm3.The theoretical maximum density (ρTMD) of the sample is calcu-lated to be 1.47 g/cm3, so the porosity present in the pressed sample

is about 9% w ¼ 1� ρbulkρTMD

� �100%

h i.

Quasistatic uniaxial compression experiments were performed,at a strain rate of 1.8 × 10−3 s−1, to obtain the elastic properties ofthe material and the bulk sound velocity in the pressed samples.The elastic modulus and Poisson’s ratio were found to be 59MPaand 0.46, respectively. From the elastic properties, the bulk soundvelocity was calculated using the well-known elastic wave equa-tions34 and found to be 428 m/s. Previously, using a pulse-echotechnique, the sound velocity in a cast polymer-bonded sugarsample of similar composition to the material used in this studywas measured to be 420 m/s.35

B. Experimental setup

A complete schematic representation of the experimental setupused in this study is shown in Fig. 2. The direct impact Hopkinsonpressure bar was modified for a projectile impact loading application.It consists of a gas gun, impactor, a fixed transmitter bar, and animaging system. The gas gun is composed of two main parts: a pres-sure chamber and a launching tube. The gas gun uses inert heliumfor launching the projectile. In this setup, a transmitter bar ismounted on linear bearings aligned precisely with the launchingtube to have a plane impact on the specimen. It should be noted thatthe transmitter bar is 1.5m long and fixed at one end. One side ofthe specimen is attached to the transmitter bar with the help of athin layer of lithium grease. The other side of the specimen is freeand close to the barrel of the launcher facing the impactor. In con-trast to typical split Hopkinson pressures bars, there is no incidentbar in this setup; instead, the impactor directly contacts the specimenupon exiting the launching tube. In this study, polycarbonate wasused for both the transmitter bar and the impactor. The soundvelocity and density of the impactor and transmitter bar are 1437m/sand 1.20 g/cm3, respectively. The time required for the completedeformation of the material was first measured by performing pre-liminary experiments on similar material. In light of this, the lengthof the impactor (88mm) was chosen to avoid reloading the sampleduring the shock wave propagation. The time required for the back-ward traveling wave, formed in the projectile upon impact, to reachback the impact face of the specimen, after the reflection from thefree end, was always kept higher than the total time required forthe compaction wave to reach the supported end of the sample. Theexperiments were conducted at three impact velocities: 53.1 ± 2.5m/s(∼55m/s), 74.3 ± 1.8m/s (∼75m/s), and 91 ± 3.2m/s (∼94m/s). Tocheck the repeatability of the experiment, three experiments wereconducted at 53.1 ± 2.5m/s and two experiments were conducted at74.3 ± 1.8m/s.

FIG. 1. Particle size distribution in the specimen.

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An ultrahigh-speed camera, HPVX-2 (Shimadzu Inc.),equipped with a 100mm Tokina lens, was employed to observe thefull-field macroscale deformation in situ at a framing rate of 2 × 106

frames/s. The camera can capture 128 images at all framing speedswith a fixed resolution of 400 × 250 pixels2. A metal arc lamp(Lumen 200) is utilized to illuminate the samples, which providesenough light for acquiring images at 2 × 106 frames/s. The arc lampis a high-intensity continuous illumination system and does notrequire the complicated triggering that is needed for flashlamps com-monly used in high-speed photography. The specimen is illuminatedright before the experiment to avoid any heating of the sample.

To facilitate the DIC measurement, a high contrast, random,and isotropic speckle pattern must be applied on the surface ofthe specimen. The size of the speckles is carefully selected by con-sidering the image resolution of the optical system used in theexperimental setup. As a rule of thumb, for good displacement reso-lution and accuracy, every speckle has to be sampled by at least3–5 pixels.36 The image resolution for the optical setup in this exper-iment is 74 μm/pixel and, therefore, a speckle size of 240–400 μm isrequired. These speckles are obtained using an airbrush and flatpaint. First, a thin layer of white paint is applied on the surface ofthe specimen, and, after complete drying of the white paint, blackpaint is sprayed on the top of the white layer using an airbrush.The specimen dimensions and the speckle pattern for the macroscaleDIC measurement are shown in Fig. 2; a bell-shaped intensity curvewas obtained, which is ideal for DIC strain measurements.

The images acquired in situ during the deformation process ofthe specimen were imported into Vic 2D software (CorrelatedSolution Inc.) for postprocessing. During image processing, an areaof interest (AOI) slightly smaller than the entire speckled region wasselected by considering the subset size and speckle quality at theedges of the sample. Due to the inherent nature of the DIC algo-rithm, the displacement calculation is made at the center of thesubset. The subset size chosen in this study was 15 × 15 pixel2, whichomits approximately 8 pixels near the edges of the sample. Also, dueto the insufficient contrast of the speckles close to the impact faceregion, an additional 4 pixels were excluded from the AOI during

DIC postprocessing. Therefore, the displacement and strain fieldsobtained are 12 pixels (X1 = 0.894mm) away from the impact face.As mentioned above, the resolution of the macroscale experiment is74 μm/pixel, and the average speckle size is 300 μm. Hence, a subsetsize of 15 × 15 pixel2 (1.11 × 1.11mm2) is used considering aminimum of 3 × 3 speckles in one subset. In order to have sufficientdata points and at the same time to reduce the noise, a step size of3 pixels and a filter size of 9 pixels were selected. The strain calcula-tion was done with the exhaustive search mode, which enables thehighest amount of data recovery when the correlation fails at theexpense of processing time. A higher order interpolation function(Optimized 8-tap) is used to convert discrete digital data points intocontinuous data. The correlation criterion is chosen to be zero nor-malized, which is insensitive to the scaling of light intensity. Thearea of interest (AOI) is shown in Fig. 3(b) and the completeimaging and postprocessing parameters are listed in Table I.

C. Density and spatial stress field calculation

The density field can be calculated by using the full-field dis-placement field obtained from the experiment. The density of thematerial, ρ, and the initial density of the specimen, ρo, are related bythe following equation:37

Jρ ¼ ρo, (1)

where J is the Jacobian at a point at any time t and is calculated byJ = detF. F is the deformation gradient and is calculated from the dis-placement field (d) obtained from DIC,

F ¼

1þ @d1@X1

@d1@X2

@d1@X3

@d2@X1

1þ @d2@X2

@d2@X3

@d3@X1

@d3@X2

1þ @d3@X3

26666664

37777775, (2)

FIG. 2. Schematic of the completeexperimental setup. Image of thespeckled sample shows the field ofview (FOV), projectile, and the supportend before the impact.

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J ¼ det F � (1þ ε11)(1þ ε22)(1þ ε33), (3)

where ϵ is the strain and d is the displacement. The compressionstress before the wave reaches the supported end of the specimen canbe calculated using the acceleration field obtained from the full-fielddisplacement. A brief description of the method is outlined below.Figure 4 shows a depiction of the direct impact experiment indicat-ing that the wave propagates along the loading direction. The DICmeasurement is performed on the surface abcd.

The linear momentum conservation equation in the Lagrangiandescription, when neglecting the body force, can be written as

r � P ¼ ρo@u@t

jX¼Constant, (4)

where P is the first Piola Kirchoff’s stress and @u@t is the Lagrangian

acceleration.The expansion of Eq. (4) gives

@P11(X1, t)@X1

þ @P12(X1, t)@X2

þ @P13(X1, t)@X3

¼ ρ0@u1@t

: (5)

In order to calculate the stress at a point in the specimen, Eq. (5) isintegrated over a differential area dX2dX3

ðA

@P11(X1, t)@X1

þ @P12(X1, t)@X2

þ @P13(X1, t)@X3

� �dX2dX3

¼ðA

ρ0@u1@t

dX2dX3: (6)

By neglecting the shear gradient terms, Eq. (6) can be reduced to

@P11(X1, t)@X1

¼ ρ0@u1@t

: (7)

By integrating Eq. (7), the first Piola Kirchoff’s stress, P11(X1, t),can be determined by

P11(X1, t) ¼ðX1

0

ρ0@u@t

dX1: (8)

In order to calculate the true stress (Cauchy’s stress), P11(X1, t) hasto be transformed into the deformed coordinates incorporating the

FIG. 4. (a) Depiction of wave propagation and a horizontal section, (b) lateralrelief wave on face abcd and feba, and (c) horizontal section with a small stripmarked at X2 = h/2 and X3 = b/2.

FIG. 3. (a) Specimen geometry anddimension, (b) typical speckle pattern,and (c) grayscale intensity of thespeckle pattern.

TABLE I. The complete imaging setup camera and postprocessing parameters.

Imaging parameter Postprocessing

Imaging lens: 100 mm TikonaIllumination: Lumen 200 PRO metalarc lamp

Imaging framing rate: 2 × 106

Resolution: 400 × 250 pixel2

Field of view: 29.6 × 18.5 mm2

Length to pixel ratio: 74.0 μm/pixel

Subset size: 15 × 15 pixel2

Step size: 3 pixelsFilter size: 9

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change in the volume. The true stress can be given as

σ ¼ J�1FTP: (9)

By substituting Eq. (2) and Eq. (3) into Eq. (9), the true axial stresscan be written as

σ11(X1, t) ¼ P11(X1, t)(1þ ε22(X1, t))(1þ ε33(X1, t))

: (10)

The wavefront propagates from the impact end to the supportedend, as shown in Fig. 4(a). At the same time, a tensile lateral reliefwave is initiated at the edges of surface abcd and propagates towardthe center of the specimen, as shown in Fig. 4(b). A similar obser-vation can be seen on surface feba. The component ε22 will be zeroin the horizontal line on surface abcd until the tensile wave reachesthe center. A similar observation can be made for ε33 on surfacefeba. Therefore, the surface of the specimen is not in the uniaxialstrain condition at any time t. However, the material points alongline AB at the center of plane X2= h/2 can be regarded as underthe uniaxial strain condition until the lateral relief from both theedges reaches the center, as seen in Fig. 4(c).

In the case of the uniaxial strain condition, Eq. (10) can bereduced to

σ11(X1, t) ¼ P11(X1, t): (11)

In this study, the measurements were performed on the surface ofthe sample. For the calculation of the stress, based on the assump-tion of the uniaxial strain condition, to be valid, stress correctionfor the out of plane deformation is essential. The correction is per-formed by calculating stress at the top edge, ab, (see Fig. 4) of thesample, which is affected by the lateral deformation in the X3 andX2 directions. It is important to note that there is no lateral confi-nement in any direction on the top edge of the rectangular sample,whereas the material points along the line joining PQ are affectedonly by the lateral deformation in the X3 direction. The effect oflateral deformation in the X2 direction can be ignored due to thelateral inertia confinement in the X2 direction until the tensile wavereaches the centerline PQ, as discussed above. The differencebetween the stress calculated along the top edge ab and the center-line PQ, before the lateral relief arrival, gives the error stress con-tributed by the lateral deformation in one direction; see Eq. (12).

σ11(error) ¼ σ11(PQ)� σ11(ab): (12)

The stress inside the material along line AB can be estimated byadding the error due to out of plane deformation to the stress cal-culated in the material points along PQ, as shown in Eq. (13):

σ11(AB) ¼ σ11(PQ)þ σ11(error): (13)

Equation (13) is valid since the midline AB within the material isfree from lateral deformation due to lateral confinements in all sides.

Energy dissipation is estimated from the plastic work rate cal-culated from the Cauchy’s stress using the method described aboveand the strain rate obtained from the experiment. The plastic workrate can be obtained as the product of Cauchy’s stress and theplastic strain rate as

W:

p¼ σ ij : ε

:

ij: (14)

The energy dissipated per unit volume was computed by integrat-ing Eq. (14) over time.

D. Measurement performance, data smoothing, anduncertainty propagation

To evaluate the performance of the displacement measurement,the images that are captured before the projectile impacts the speci-men were processed using the Vic2D software. The postprocessingparameters are described in Table I. For an ideal system beforeimpact, the strain and displacement should read zero everywhere inthe field of view. However, a nonzero displacement and strain will begenerated in the field due to sources of error such as camera sensornoise, poor speckling, lighting, and improper subset selection.

In order to quantify the error associated with the experiment,the mean value of the axial strain and its standard deviation (SD) areplotted for ten images of the undeformed specimen, as depicted inFig. 5(a). The mean strain, mainly due to bias, remained very small,less than 0.0015%. The standard deviation is also minimal, less than0.060%, compared with the large deformation expected in theimpact experiments. The displacement field indicates a standarddeviation of about 0.7 μm, as shown in Fig. 5(b). The uncertainty of0.7 μm is negligible compared with the expected displacement of1–2mm. However, the first and second derivative of the displace-ment field used to calculate the acceleration causes a significant errorin the stress calculation. Therefore, temporal and spatial datasmoothing were performed using weighted nonlinear smoothing.The smoothed data show a lower SD (∼0.25 μm) compared with theunsmoothed data. The velocity and the acceleration were calculatedfrom the smoothed displacement data and plotted in Figs. 5(c) and5(d), which show that the uncertainty in the velocity and the acceler-ation are 0.08 m/s and 1.75 × 104 m/s2, respectively; this value is neg-ligible compared with the expected particle velocity (∼45–70m/s)and the acceleration (∼106 m/s2) in the experiment.

The acceleration and displacement data are used to calculatethe Cauchy stress in the material. The propagated uncertainty inthe stress calculation was calculated based on the uncertaintyvalues of the acceleration and displacement. In the true stress calcu-lation, the variables that are required are the acceleration, Jacobian,and deformation gradient.

The following equation can be used to calculate the true stressuncertainty:

δσ

σ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiδJJ

� �2

þ δaa

� �2

þ δFF

� �2s

: (15)

Uncertainty in the Jacobian, acceleration, and deformation

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gradients is calculated from the displacement uncertainty in themeasurement system.

III. RESULTS AND DISCUSSION

All the discussions are limited to the time duration of thepropagation of the wave from the impact end to the support end ofthe sample. The term “impact face” corresponds to a location0.894 mm away from the plane of contact between the projectileand the specimen. Due to similarity between the results, the fullanalysis of the 55 m/s experiment is presented, and the highervelocity results are included for comparison as needed.

A. Axial stress wave and lateral relief

The distribution of true full-field axial strain along the lengthof the specimen at time t = 0, 5, 10, 15, 20, 25, and 35 μs is shownin Fig. 6(a). From the figure, it is evident that a compression wavewas formed and propagated from the impact end to the supportend of the specimen. The compressive axial strain behind the wave-front, is significantly high, whereas ahead of the wavefront, theaxial strain is zero, indicating an undisturbed region. The strain-time plot at point A shown in Fig. 6(c) indicates that the axialstrain linearly increases with time until t = 8–9 μs. After t = 8–9 μs,the axial strain is approximately constant, which indicates compac-tion type waves that are commonly seen in the granular explosives.It is very important to note that in granular materials the

compaction waves are formed by the distortion of the crystals whenthe local stress exceeds the yield strength of the crystal.38,39 In thisstudy, as discussed later in Sec. III E, the maximum average stressbehind the compaction front is about 25MPa for the lowest veloc-ity impact loading (55 m/s), which is far from the yield stress(110MPa) of the sugar crystal. The plastic deformation of the crys-tals may be playing only a minor role in the compaction wave for-mation and propagation. In contrast, the soft binder can undergolarge deformation locally,8 which can result in the closing of voidsby squeezing out the pores present in the material. The deforma-tion of the binder can be either plastic or viscoelastic depending onthe strength of local heterogeneity in the deformation. As shown inFig. 6, it is possible that maximum compaction, i.e., quasisteadycondition, is achieved in the time scale considered. However, thequasisteady condition could be transient if viscoelastic deformationdominates the compaction wave formation in the material. In thatcase, the deformation of the binder can relax once the compactionwaves pass through the material. However, during the time scaleconsidered in this study, such relaxation was not observed.

The axial strain field in Fig. 6(a) ascertains two points: (1) thewavefront has a finite thickness that changes with time and (2) thewavefront is planar at the beginning but becomes nonplanar as itpropagates across the specimen. The wide compaction type bandobserved indicates a weak-shock nature, as was expected since alow impact velocity was applied to the specimen. On the otherhand, the nonplanar wavefront observed at later times could be

FIG. 5. Mean and the standard devia-tion (SD) in the specimen in an unde-formed condition: (a) strain, (b)displacement, (c) axial material velocity,and (d) axial acceleration.

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associated with the arrival of a lateral relief wave from the twoedges of the sample. In order to investigate the formation of thenonplanar front, lateral strain (ε22) is plotted in Fig. 6(b). Thelateral strain field shows a tensile strain expanding from the leftcorners of the area of interest. This lateral deformation generates adiagonally propagating expansion stress wave (lateral relief ) towardthe center of the specimen at a velocity close to the sound velocityof the material in the compressed region. A quiver plot is shown inFig. 6(b), which clearly shows the propagation of the lateral reliefwave. The axial and lateral strain at point A (X2 = h/2 = 8.65 mm) isplotted in Fig. 6(c) to find the time at which the lateral reliefreaches the center of the specimen. Lateral strain ε22 is close tozero for a period of 16 μs after the impact. At time t = 16 μs, anonzero lateral strain is observed, and it increases linearly withtime, as shown in Fig. 6(c). Later, these two tensile waves mergeand propagate along the loading direction of the sample, as shownin Fig. 6(b). A similar observation can also be expected on thesurface aefg, where ε33 will be zero at the center along the materialpoints in line RS, until t = 16 μs, as shown in Fig. 4(b). Therefore,until the relief wave, due to transverse deformation in X2 and X3,reaches the line AB, center plane X2= h/2, the deformation alongline AB will be under the uniaxial strain condition.

Figure 6(d) shows the axial strain field at 10 μs for the threeimpact velocities (55, 75, and 94 m/s) considered in this study. Itshows that the wavefront position is farther from the impact end athigher impact velocities, indicating a higher wave speed. The widthof the shock front does not appear to be changing significantlywith the impact velocity range considered in this study.

B. Wave velocity and spatial density

The X1-t diagram in Fig. 7(a) shows the location of the wave-front at different times. A linear fit is obtained for the wavefront

location with time, indicating a constant wave velocity during propa-gation. The wavefront velocity of 506m/s is obtained by calculatingthe slope of the linear fit. The wave velocity estimated is 1.18 timesthe sound speed in the material, indicating a weak-shock formationin the material. The estimated shock wave velocity is higher thanporous explosives without any binder under similar impact veloci-ties.40 The difference is due to the binder that provides more pathsfor force transfer between crystals compared with porous explosivesin which the sole mechanism of force transfer is crystal-to-crystalcontact.17 Also, since the porosity in the PBS is less than the porousexplosives, it is expected to have higher shock wave velocity. In orderto confirm the repeatability of the experimental measurements, threeexperiments were conducted at 53.1 ± 2.5m/s, and two experimentswere conducted at 74.3 ± 1.8m/s. Figure 7(b) shows the shock veloc-ity estimated for three different impact velocities. The shock wavevelocities corresponding to the impact velocities of 74.3 ± 1.8m/sand 91 ± 3.2m/s are 706.5 ± 80.1m/s and 825.3 ± 42.3m/s,respectively.

The instantaneous density of the material is calculated usingEq. (2). The variation of the density along the line PQ at differentlocations is plotted in Fig. 7(c). A maximum density of 1.425 g/cm3

is observed in the compacted region behind the front. For an idealshock (uniaxial strain) condition, a constant density is expectedbehind the shock front. However, the unconfined experiments con-sidered in this work have transverse strains in the X3 and X2 direc-tions, and, subsequently, the density in the region behind the shockfront is not constant. The density profile shows a lower densityregion close to the impact end due to the high transverse strain inthe X3 direction. As discussed above, the transverse strain in the X2

direction is negligible until t = 16 μs. On the other hand, thedensity calculated along the line material points AB where the uni-axial strain condition prevails gives a maximum density behind theshock front of 1.46 g/cm3. The density behind the front is close to

FIG. 6. (a) Axial strain field at t = 0, 5,10, 15, 20, 25, 30, and 35 μs, propaga-tion of the compressive wave from theimpact end and to support end is mani-fested, (b) transverse strain field att = 0, 10, 20, and 30 μs, lateral reliefwave from both the left corners isshown in Quiver vector plot, and (c)axial and transverse strain at a locationA with time. The figure inside showsthe point at which the strains areextracted, and (d) axial strain field att = 10 μs for the impact velocities55 m/s, 75 m/s, and 94 m/s.

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the theoretical maximum density of the sample and is nearlyconstant.

C. Full-field particle velocity

The contour plot of the full-field particle velocity for the55 m/s impact velocity is shown in Fig. 8(a). The particle velocity

profile is qualitatively very similar to the axial strain field shown inSec. III A [see Figs. 6(a) and 8(a)]. The particle velocity is zeroahead of the shock front and become constant on the region of thematerial fully compacted behind the shock front. The particlevelocity along the centerline PQ at four different Lagrangian loca-tions, X1 = 0, 2, 4, and 6 mm, from the impact end, is shown inFig. 8(b). The gray lines and solid lines in the figure show the

FIG. 8. (a) Contour plot of axial parti-cle velocity with time at t = 0, 4, 8, 12,16, and 20 μs, (b) axial particle veloci-ties at four different Lagrangian loca-tions with time, and (c) particle velocityalong Lagrangian axial coordinates att = 4, 8, 12, and 16 μs. Uncertainty inthe velocity is 3.1 m/s. (d) Particlevelocity for three impact velocities.

FIG. 7. (a) X1-t diagram for the impact velocity 55 m/s and slope of the linear fit gives the shock velocity Us = 506 m/s, (b) shock velocity for the three impact velocities,53.1 ± 2.6 m/s, 74.3 ± 1.8 m/s, and 91 ± 3.1 m/s, and (c) spatial density variation at t = 8 μs and 16 μs at an impact velocity of 55 m/s.

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velocity profile obtained from the unfiltered and filtered data,respectively. A piecewise spatial smoothing with a window size of10% of the total time scale was used for smoothing the particle dis-placement and to obtain the velocity. The total time duration of thedata is 20 μs, and hence the displacement was smoothed with every2 μs time interval. As shown in Fig. 8(b), the smoothed velocityprofile passes through the mean position of the oscillating particlevelocity profile. Note that the impactor has a significantly highermechanical impedance compared with the polymer-bonded sugarsamples. Therefore, the particle velocity in the sample is expectedto be close to the impact velocity of the projectile. The particlevelocity at the impact face behind the shock wave front increasesfrom 0 m/s to 41 m/s in 8 μs. Interestingly, the particle velocitybehind the shock front after t = 8 μs is nearly constant, indicatingpossible attainment of a quasisteady state. The rise time at theimpact end (X1 = 0) is close to 8 μs, whereas at X1 = 6 mm, the risetime increases to 10 μs. Also, as the shock wave travels across thespecimen, the peak particle velocity decreases from 41 m/s at X1 = 0to 37 m/s at X1 = 6 mm, indicating a reduction in shock strength.The decrease in the wave strength is an indication of energy dissi-pation in the material. In order to see the shock wave profile, theparticle velocity as a function of position (X1) is plotted as shown inFig. 8(c). Interestingly, the wave profile is qualitatively very similar tothe weak-shock profile seen in 2D numerical simulation of pistoncompressed disordered brittle spheres.41 As shown in Fig. 8(c), theparticle velocity, from unsmoothed data, indicates a possible forma-tion of elastic precursor ahead of the shock wave that weakens withtime. This was also previously seen in a 2D numerical simulation ofporous explosives; where the weak elastic precursor weakens as itpropagates.42 The particle velocities obtained for the three differentimpact velocities are shown in Fig. 8(d).

D. Shock thickness

In the case of an ideal strong shock front, the gradient acrossthe shock wave front will be infinite. Nevertheless, a real shockwave will have a finite rise time corresponding to material defor-mation, and the slope of the shock wave front will vary with thestrength of the shock wave. Across the shock wave front, disconti-nuity in strain, stress, and material velocity is expected. It is

important to note here that, in the case of weak-shock waves, thefront will have a finite thickness, and, therefore, the gradientacross the shock wave front will not be sharp compared with astrong shock, as shown in Fig. 9(a). Hence, in order to understandthe process of shock formation under weak-shock condition char-acteristics, a complete analysis of the shock wave thickness isrequired. Shock front wave thickness is an important quantitythat determines the gradient of stress, strain, or velocity across thewavefront. The shock wave thickness, δ, can be defined as shownin Eq. (16):

Shockwave thickness, δ ¼ u1

max@u1@x1

� � , (16)

where u1 is the particle velocity behind the shock front. Since thematerial velocity is directly calculated from DIC, the shockwavethickness can be easily calculated from Eq. (16).

The particle velocity, as well as its gradient at two differentinstances across the length of the specimen for an impact velocityof 55 m/s, is shown in Fig. 9(a). A circular marker indicates theaxial location of the maximum velocity gradient. The gradient ofthe velocity shows a sharper peak at t = 8 μs compared with thevelocity gradient peak at t = 16 μs. The decrease in the velocity isattributed to the dispersion or dissipation of the shock wave as ittravels across the specimen. The shock wave thickness is calculatedusing Eq. (16) and plotted in Fig. 9(b). Inside the shock front, thepores are collapsed, and the crystals reorient themselves and mayeven fracture due to the stress concentrations as a result of forcechain formation. The shock wave thickness is close to 3.4–3.8 mm(6–7 crystals) and it is nearly constant for a period of t = 8–12 μs;however, after t = 12 μs, the shock thickness gradually increasesto ∼4.1–4.5 mm (8–9 crystals). Comparatively, the 1D numericalsimulation of the compaction wave in porous high meltingexplosives under constrained experiments shows a constant wavethickness of 5.1 mm.43

A dissipative compressive front will always tend to broaden;however, the stress–strain nonlinearity effect steepens the wavefront.44

FIG. 9. (a) The particle velocity and itsgradient along the Lagrangian axialcoordinates at an impact velocity of 55m/s and (b) shock wave thicknessevolution.

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In the case of a steady shock wave, there is a balance between thebroadening due to dissipation and steepening due to nonlinearity.Therefore, in this study, a nearly constant shock thickness right afterthe attainment of quasisteady condition would indicate a balanceof dissipation and nonlinear steepening. However, at later times(>12 μs), the increased broadening is attributed to either dissipationbecoming dominant or the transformation of the planar front to anonplanar front due to lateral wave release. Shock experiments ongranular materials at a very high impact velocity show a similarincrease in thickness of the front with time.45 In order to distinguishthe shock front widening mechanisms, it would be ideal to perform aconstrained experiment in which the effect of relief wave is minimal.However, this is outside the scope of this work and could be a subjectfor future research.

E. Axial stress from full-field acceleration and energydissipation

The acceleration is calculated by numerically differentiatingthe particle displacement as a function of time. Note that the

fluctuation in the particle velocity from the unfiltered data wasseen to be high. Therefore, smoothing the displacement data wasinevitable to obtain accurate acceleration. As discussed earlier,the temporal smoothing window size was 2 μs., and this couldhave possibly smoothed out the small-scale features in the accelera-tion. At a 2 μs window, the acceleration uncertainty is close to0.8 × 106 m/s2. Therefore, any fluctuation below the uncertainty isignored in the analysis. The acceleration evolution with time atdifferent axial Lagrangian locations for the impact velocity of 55 m/s is plotted in Fig. 10(a). The acceleration at any Lagrangian loca-tion increases to a peak value, when the shock wave arrives, and itgradually drops to zero, when it departs, as shown in Fig. 10(a).The acceleration profile shown in Fig. 10(b) has a typical structurefor a stress wave in materials. The acceleration drop behindthe peak indicates the attainment of constant velocity behindthe shock front. In addition, the magnitude of the accelerationpeak drops and the acceleration profile widens as the wave traversesacross the sample [see Figs. 10(a) and 10(b)]. For instance, consid-ering t = 8 μs and 16 μs, two important points need to be noted:(1) a decrease in the peak value of acceleration and (2) widening of

FIG. 10. (a) Axial acceleration vs timeat four different Lagrangian axial loca-tions, (b) spatial variation of axialacceleration at different times t = 2, 4,8, 12, and 16 μs, (c) spatial axialstress variation at different times t = 4,8, 12, and 16 μs, and (d) spatial axialstress along different Lagrangian axiallocations for three experiments per-formed at impact velocity 53 ± 2.6 m/s.

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the pulse with time. The drop in peak acceleration at t = 8 μs is8.48 × 105m/s2, which is 15% of the peak acceleration. This couldbe due to energy dissipation or/and possible dispersion of theshock wave, as discussed in Secs. III A–III D.

The axial stress is calculated using the inertia stress analysisdiscussed above and used to quantify the energy dissipated. Theaxial stresses at different Lagrangian locations along the specimenat different times t = 4, 8, 12, and 16 μs for the impact velocity of55 m/s are shown in Fig. 10(c). Behind the shock front, the stress isapproximately constant (close to 25MPa), and ahead of the front,the stress is zero. Uncertainty in the stress is close to 3MPa; there-fore, fluctuations of the stress of amplitude 3MPa and below arenot resolved in this study. The stress profile is very similar to theaverage compaction wave profile observed in the granular materialsand polymer-bonded particulate composites.42,46 The repeatabilityof the stress estimation was confirmed by running two moreexperiments at impact velocities close to 55 m/s, as presented inFig. 10(d). The axial stress profiles are identical, and the stress vari-ation behind the front was seen to be within 10%.

Stress profiles for three different impact velocities consideredin this study are shown in Fig. 11(a). A steeper stress profile wasobtained at higher impact velocities. The stress behind the shock

front for impact velocities 55, 75, and 94 m/s are approximately 25,48, and 72MPa, respectively. Figure 11(b) shows the comparison ofaxial stress based on the Rankine–Hugoniot equations (ρ0 u1Us)and the experimentally determined value. The table in Fig. 11(b).gives a comparison between uncorrected and corrected stress valuesfor the three different impact velocities. The corrected shock stressvalues for 55, 75, and 94 m/s are 25.4, 46.0, and 72.8 MPa, respec-tively, which are very close to the Rankine-Hugoniot Relations.Uncorrected stresses are low compared with the corrected stressdue to the release of stress due to lateral deformation in the X3

direction. In addition, the error associated with the measurement isslightly increased with impact velocity, mainly due to significantout of plane deformation at higher impact velocities.

Figure 11(c) shows the energy dissipation per unit volume forthe impact velocity of 55 m/s, calculated by integrating Eq. (14)with respect to time. Energy dissipation in the material can bedirectly connected to the temperature evolution in the material.The energy dissipation profile shows an increase in dissipationwith time, which indicates the temperature increase during thewave propagation. Energy dissipation for three impact velocities(55, 75, and 94 m/s) at t = 8 μs is shown in Fig. 11(d). The structureof the dissipation profiles for all the impact velocities are nearly

FIG. 11. (a) Axial stress along differentLagrangian axial coordinates for thethree impact velocities, 55 m/s, 75 m/s,and 94 m/s and (b) the comparison ofaxial stress based on shock theorywith the experimentally determinedvalue. (c) Energy dissipation at t = 4, 6,8, and 10 μs for impact velocity 55 m/sand (d) energy dissipation at t = 8 μsfor the impact velocities 55 m/s, 75 m/s,and 94 m/s.

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identical, and the energy dissipated was seen to significantlyincrease with impact velocity. The energy dissipated near theimpact face of the specimen, for impact velocity 55, 75, and94 m/s, is close to 0.9, 2.9, and 10.23MJ/m3, respectively. Theincrease in dissipated work with impact velocity indicates a rise intemperature in the PBX material at a higher impact velocity thatcould cause hotspot formation and a subsequent sensitization ofthe explosives.

The possible energy dissipation mechanisms in PBS are docu-mented as the fracture of crystals, plastic deformation of thebinder, and frictional relative movement of the crystals.5,13,15,16,47

However, a detailed understanding of grain-scale mechanisms canbe achieved by performing high spatial and temporal resolutionexperiments. The present study is limited to macroscale investiga-tion, but a mesoscale study could reveal the mechanisms associatedwith the energy dissipation.

IV. CONCLUSION

An optical-based experimental method was developed to inves-tigate the weak-shock wave properties of PBS subjected to impactloading. Specimens made of polymer-bonded sugar were subjectedto projectile impact, and the propagation of the weak-shock wavewas captured with the help of high-speed optical imaging. A quanti-tative analysis of the shock wave structure, including shock wavevelocity, shock front thickness, and full-field strain and stress, wasperformed with the help of digital image correlation. The shockwave stress in the material was estimated based on the full-fielddeformation measured using DIC and validated using Rankine–Hugoniot equations’ calculation. The energy dissipated as a functionof time was also calculated based on the stress–strain relationobtained in the experiment. The key findings are summarized asfollows:

• For the material considered in this study, the shock wave velocityis close to 506 m/s at a projectile impact velocity of 55 m/s. Theshock wave velocity was slightly higher than the sound speed inthe material. The wave structure indicates a weak-shock with awide shock front.

• Compaction waves can be generated in PBS at stresses much lessthan the plastic deformation of the crystals. In this case, thecompaction is generated mainly due to viscoelastic or plasticdeformation of the binder.

• The shock wave front thickness was experimentally estimated fromthe material velocity. It nearly remains constant (3.1–3.4 mm) for ashort period right after the attainment of a quasisteady condition,and it gradually increases as it propagates (4.1–4.6mm). Thewidening of the shock front may be due to energy dissipation inthe material or the evolution of a nonplanar front as a result of alateral relief wave.

• The spatial energy dissipation profile is calculated from thestress estimated from the displacement field and the axial strainrate. It was found that the amount of energy dissipated ishighly related to the impact velocity. Higher dissipation isobserved near the impact end, pointing to the possibility ofhigher temperature near the impact surface during wavepropagation.

ACKNOWLEDGMENTS

The financial support of the Air Force Office of ScientificResearch (AFOSR) (Grant Nos. FA9550-14-1-0209 and FA9550-16-1-0623) is gratefully acknowledged.

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