Spatiotemporally Resolved Acoustics in a Photoelastic Granular Material

Eli T. Owens , Stéphanie Couvreur†, and Karen E. Daniels

Department of Physics, North Carolina State University, Raleigh, NC, USA 27695 †Département de Physique, École Normale Supérieure, 24 rue Lhomond, 75 005 Paris, France

Abstract. The effect of the force chain network on sound propagation in a granular material is poorly understood. To quantitatively study these effects, we perform acoustics experiments in a two dimensional photoelastic granular material in which force chains are visible. We send acoustic pulses into the material from a point source and measure the effects of this pulse via two methods: accelerometers within individual grains and movies which produce spatiotemporally resolved measurements of the acoustic propagation.

Keywords: force chains, piezoelectric, photoelastic PACS: 43.25.-x, 61.43.Gt, 81.05.Rm

INTRODUCTION

Sound propagation in granular materials differs in sig­nificant ways from propagation in ordinary elastic ma­terials. In particular, the speed of sound is no longer specified by the elastic modulus and density via the re­lation c = ^Ejp and is instead observed to depend strongly on the particular velocity measured [1]. Fur­thermore, the observed speed of sound is proportional to p1/4, rather than the p1l6 predicted from effective elas­tic theory for Hertzian contacts [2, 3]. In granular mate­rials, unlike conventional elastic materials, stress trans­mission is manifested as force chains in which larger-than-average forces propagate through the material in a branching structure. The presence of these force chains means that any assumption of homogeneity, used in cre­ating an effective elastic theory, is violated when the wavelength of the acoustic wave is similar to the parti­cle or force chain length scales. Thus, some of the pecu­liar features of acoustic transmission in granular materi­als are likely to be due to the force chains.

We seek to understand what role the force chains play in the propagation of acoustic waves. Two sets of prior experiments [4, 5] have inferred their importance even though the chains themselves were not visible. Liu and Nagel [4] placed a resistor within a granular pack and disturbed the configuration by thermally expanding the resistor. This small disturbance was observed to create large variations in the tail of the transmitted sound pulse, likely due to having altered the force chain network. Similarly, Hostler and Brennen [5] argue, based on ex­periments in three-dimensional systems, that the specific arrangement of force chains affects sound propagation. Their system was either prepared in a loose, unconsol­idated state, or shaken to consolidate the system. The

brass weight

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FIGURE 1. (a) Schematic of the apparatus and (b) image of plain photoelastic particle (11 mm diameter) and one embedded with a piezoelectric accelerometer.

unconsolidated pack showed wider variation in the re­ceived signal than the more consistent consolidated pack, which may be attributed to the unconsolidated system’s ability to rearrange its force chain network more easily. In contrast, numerical simulations by Somfai et al. [6] showed no correspondence between sounds wave inten­sity on and off a fixed network of force chains.

One of the difficulties in establishing the role force chains play in acoustics is the inability to see the force chains in ordinary three-dimensional granular materials. Therefore, we perform sound propagation experiments in a two dimensional system of photoelastic disks, giving

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us detailed information about the force chain network. Since the force chains are optically visible, we can ob­serve the sound traveling through the material via a high speed digital camera. These movies suggest that sound does in fact primarily travel along the force chains. We also employ piezoelectric sensors embedded inside par­ticles to obtain more precise information about the speed of sound and the shape of the wave as it travels through the material. This is a novel approach to the problem, and will ultimately allow us to combine these two forms of data collection to characterize how small changes in the force chain network are manifested in the acoustic response.

EXPERIMENT

Our apparatus consists of a vertical, two dimensional ar­ray of photoelastic disks, as shown in Figure 1. The parti­cles are confined in a single layer between two sheets of Plexiglas, and are kept at constant pressure by a brass weight resting on the upper surface; typical pressures range from 1 kPa to 15 kPa. The width of the cell is 30 cm, and the height is set by the number of particles in the system (up to 60 cm for 1500 particles). The particles are cut from Vishay PhotoStress material, a well-calibrated photoelastic polymer. We use a bi-disperse mixture of circular PSM-4 particles with diameters 9 mm and 11 mm, in equal concentrations.

Sound waves are produced by a voice coil currently af­fixed to the bottom wall of the system. The driver sends wave packets consisting of three sine waves with fre­quencies ranging from 10 Hz to 120 Hz, with a maximum acceleration of 40g. The head of the driver is approxi­mately two particle diameters wide, providing a point­like source.

We observe the acoustic wave propagation through the pack with two different techniques: visually with a high speed camera, and through particle-scale piezoelec­tric accelerometers. By using photoelastic particles, we can visualize the force chains by viewing them through crossed circular polarizers. We obtain digital movies of the changes in the force chain network during sound propagation using a Phantom V5.2 camera operating at 256 ×256 resolution at 10 kHz. The change in the opti­cal signal is very weak, often less than 1 out of 256 bits and of similar magnitude as the noise. For this reason, we transmit the pulse one hundred times and take the av­erage over all the pulses to obtain a single movie. This technique has the additional advantage of removing fluc­tuations in the intensity of the halogen bulb. By examin­ing differences in the images between successive frames, we are able to resolve the spatiotemporal dynamics of the propagation process.

FIGURE 2. A closeup of the force chains and accelerometers for the data presented in Figure 3, under 4 kPa of pressure. The force chains are the bright lines, made visible by crossed cir­cular polarizers. Each of the sensor particles (1,2,3) is marked with a circle.

1

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FIGURE 3. The signal resulting from two 100 Hz sinusoidal pulses, recorded both from a piezoelectric accelerometer at­tached to the driver (reduced in magnitude by 50%) and at three pizeo-embedded sensors (1,2,3 in Figure 2). Offsets between signals are provided for clarity.

The second method of studying sound propagation is with piezoelectric sensors embedded in select parti­cles. Piezoelectric sensors were chosen because they are lightweight, produce a voltage proportional to the ap­plied stress (instantaneous acceleration), and have an ex­cellent frequency response. Because piezoelectric ma­terials are extremely sensitive and require no external power source, they are small and lightweight enough to place within single particles. We use 2 mm thick lead zirconate titanate piezoelectric ceramics, cut to 4 mm × 4 mm squares, adding about 0.25 grams (40%) to each particle’s mass.

During the propagation of an acoustic wave, we record

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CD >

base image t = 0.0 ms 0.1 ms 0.2 ms

0.3 ms 0.4 ms 0.5 ms 0.6 ms

FIGURE 4. High-speed changes in the force chain network, as compared to a base image, obtained during the initial (compres-sive) portion of the pulse. Each subsequent image is measured as the difference between an averaged image (from 100 pulses) and the base image. Data is collected in a horizontal granular aggregate at constant volume.

the time-varying voltage produced by each of four sen­sor particles via a computer running data acquisition soft­ware. The signal from each of three piezo-embedded par­ticles within the pack records both when the acoustic signal arrives (time-of-flight velocity) and the resulting acceleration as the wave passed through. The driver is also fitted with a piezoelectric sensor so that we know the acoustic input to the system. Due to slight differ­ences in the individual piezoelectric sensors, we calibrate each one by vibrating it at 100 Hz and rescaling the re­sults among sensors by the standard deviation of their response.

RESULTS

Using both the piezoelectric sensors and the high speed camera, we can both observe how the acoustic wave moves through the material and obtain quantitative mea­surements of the acoustic response of the individual par­ticles fitted with piezoelectric sensors. Figure 2 shows a closeup image of a force chain network with three piezo-embedded particles marked within the pack, and the driver visible at the bottom. The side walls and brass weight are both outside the field of view. The brightness of a particle, along with the presence of multiple black fringes, gives the magnitude of the force on a particular particle [7]. Due to the magnitude of the applied pres­sure, all three sensors are located on force chains, but #3 is connected by the least direct path.

When a sound wave is produced by the driver, it trav­els through the material where its effects can be observed with the piezo-embedded particles. For the pack shown in Figure 2, a signal sent from the driver takes approxi­mately 3 ms to reach the sensors, as shown in Figure 3. This provides a time of flight velocity of approximately 20 m/s, which is, as expected, significantly slower than the bulk speed of approximately 200 m/s calculated from the density and elastic modulus. Notice that the differ­ent sensors record signals with different magnitudes and shapes; future work will provide quantitative results on how the speed and amplitude of the signals depends on properties of the force chains between the driver and the particle.

To visualize the spatiotemporal dynamics of the force chain network, we examine changes in the network using high-speed imaging. Figure 4 shows the increased load on the particles during the initial propagation on a single square wave pulse of width 0.5 ms which begins at t = 0. Each image was produced by taking the difference between a single frame of an averaged movie and a base image averaged over the whole movie. Where a particle is white, the load on the force chain increased; where a particle is dark, it decreased. For the few particles directly in front of the driver head, the presence of a dark band in the middle of the particle represents an increase in force as a dark fringe (see Figure 2) forms within the central portion particle.

As can be seen in Figure 4, the changes in stress provided by the propagating wave fall primarily on the

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existing network of force chains. In rarer cases, we also observe that the transient wave can close a contact and create a new force chain. We find that changes in the force chain structure are typically visible a maximum of

10d from the driver, due to attenuation of the signal.

DISCUSSION

In this paper, we have presented the development of new tools for the investigation of the response of a granular material to an acoustic source. We are able to quantita­tively measure particle-scale sound propagation in a sys­tem for which we know both the locations of force chains and how they change during propagation. This will allow for the investigation of a number of unanswered ques­tions surrounding sound propagation in granular mate­rials. For example, we will be able to examine possible differences in propagation of sound along paths primar­ily on and off the force chain network, and the sensitiv­ity of propagation to small changes in that network. A second line of investigation involves changes in the role that force chains play when the wavelength of the sound wave is close to either the particle size or the force chain length scale. This can be studied through the dispersion relation in our material, as we can observe changes in fre­quency, wavelength, and amplitude of acoustic signals. These sorts of studies will ultimately allow us to charac­terize force chains by their acoustic response.

ACKNOWLEDGMENTS

This work has been supported by North Carolina State University. The authors thank C. Chafin for preliminary work on an earlier incarnation of the apparatus.

REFERENCES

1. C. H. Liu, and S. R. Nagel, Physical Review Letters 68, 2301–2304 (1992).

2. J. D. Goddard, Proceedings Of The Royal Society Of London Series A-Mathematical Physical And Engineering Sciences 430, 105–131 (1990).

3. H. A. Makse, N. Gland, D. L. Johnson, and L. Schwartz, Physical Review E 70, 061302 (2004).

4. C. H. Liu, and S. R. Nagel, Journal of Physics – Condensed Matter pp. A433–A436 (1994).

5. S. R. Hostler, and C. E. Brennen, Physical Review E 72, 031303 (2005).

6. E. Somfai, J. N. Roux, J. H. Snoeijer, M. van Hecke, and W. van Saarloos, Physical Review E 72, 021301 (2005).

7. M. Frocht, Photoelasticity, John Wiley & Sons (1941).

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