Nonlinear acoustics in granular assemblies

Transcription

1 Granular Matter 3, c Springer-Verlag 2001 Nonlinear acoustics in granular assemblies Surajit Sen, Marian Manciu, Robert S. Sinkovits, Alan J. Hurd Abstract We present a brief review of the problem of acoustic propagation in granular media and discuss recent progress on nonlinear acoustics in granular media. The presentation emphasizes the solitary wave like properties of impulse propagation in granular media at vanishingly small loading conditions and discusses the possible spectroscopic applications of nonlinear impulse acoustics in the detection of buried inclusions. Keywords Nonlinear acoustics, granular materials, solitary waves 1 Introduction to impulse propagation in granular beds We start with a brief mention of experiments carried out by several workers nearly forty years ago [1 4]. These experiments have established that (i) sound travels through granular media and (ii) that sound velocity increases as sound travels vertically downward into dry, moist and saturated sand beds. If c denotes the sound velocity and z denotes the vertical depth, as measured from the surface of the granular bed, then one finds that for weak perturbations, c z α, where α is typically a number between 1/6 and 1/4 (see Fig. 1). To understand the origins of the stated scaling law, we note that the arrangement of grains in a sand bed, which is disordered at shallow depths, will become increasingly more compact with depth. Thus, it should become easier for sound to travel from one grain to another as depth increases, i.e., sound velocity should increase with depth. The nature of horizontal sound propagation on the surface and near the surface of granular beds is known to have behavior that is characteristic of transport problems in disordered systems [5]. We shall not address the problem of horizontal propagation in this work. Received: 30 March 2000 S. Sen (&), M. Manciu Department of Physics, State University of New York at Buffalo, Buffalo, New York 14260, USA R. S. Sinkovits P.O. Box 85608, San Diego Supercomputer Center, San Diego, California 92186, USA A. J. Hurd Department 1841, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA As we begin a discussion of impulse propagation in granular beds, we remind ourselves about the motivations for such studies. It is well known that there is a global crisis with regard to humanitarian demining of metallic and plastic anti-personnel and anti-tank land mines [6]. These mines are typically buried at depths that are less than about 30 cm. The mines, which are encased in plastic, have very little metal in them and are therefore effectively invisible (typically because of the presence of other metallic objects in soil) to most metal detectors. The consensus in the humanitarian demining community is that several inexpensive technologies are concurrently needed [7 10]. Examples of materials science applications of impulse acoustics concern the detection of inherent density inhomogeneities in granular ceramic compacts that can lead to low-grade ceramics [11] and in protecting material surfaces by coating them with thermal sprays of granular materials at high temperature [12]. High-power impulse acoustics may be useful for detecting buried underground structures at large depths. 2 From linear to non-linear acoustics Returning to the problem of sound propagation in granular beds, we consider a simple theoretical derivation of the c z α scaling law. We assume that the individual sand grains are spheres of radius R and can be regarded as elastic objects, which repel one another upon contact. If d is the distance between the centers of two adjacent grains in contact and if d is less than 2R, then according to Hertz [13], the repulsion between the grains is V(δ) δ n, where n = 5/2 for spherical grains and δ 2R d 0. The effective spring constant between two grains in contact therefore is k = df/dδ δ n 2, where f denotes the force between the two grains in contact and f δ n 1. Thus, k f/δ, i.e., as f 1 1/(n 1). Sound velocity c (k) 1/2 and hence as f (1 [1/(n 1)])/2.Iff z and n = 5/2, then c z 1/6 [14]. The quantity n depends upon the contact geometry between the grains and turns out to be nearly a constant for grains in a typical bed of grains [15]. To acquire a deeper understanding of the dynamics, one can carry out particle dynamical simulations. In these simulations, one can probe the propagation of an impulse initiated at the surface of a granular bed. For simplicity, we can start with a vertical 1D stack of grains, which is subjected to gravitational loading [16 17]. We first consider weak impulses, i.e. impulses that start off by compressing the surface grain by say 10 6 of the grain

2 34 Fig. 1. Scaling of sound speed for various magnitudes of initial pertubation starting from weak (circles), to intermediate (boxes) to strong (triangles). Observe the deviation from elasticity theory based behaviour at shallow depths for large amplitude perturbations diameter or less [18]. Particle dynamical simulations for such impulses in 1D columns and 2D beds show that the sound velocity scaling behavior predicted above is recovered at depths of about a few hundred grains or less (Fig. 1). With increasing magnitude of the impulse, the scaling law is no longer recovered at modest depths. This behavior is evident upon observing the data in Fig. 1(squares and triangles). It turns out that the deviation from the scaling behavior is quite dramatic for large amplitude impulses. The reasons for this deviation are (i) presence of void space near the surface due to non-compact packing and (ii) nonlinear effects associated with impulse propagation through a system of grains that repel nonlinearly upon compression. For strong enough impulses, significant compaction occurs as the impulse is generated. To probe impulse propagation at shallow depths, we calculated the energy transport as functions of time and space in pristine and slightly disordered 2D beds. Fig. 2 shows the propagation of an impulse in a 2D bed with 3.3% void fraction. The impulse suffers very little spatio-temporal dispersion. Thus, for purposes of studying impulse propagation and backscattering, it is reasonable to start with 2D and 3D beds without disorder. The effects of positional disorder and associated local grain dynamics and stress fluctuations near the surface of a bed can be incorporated in the next step. We expect that the stress fluctuations would make the acoustic signals more noisy (see the left edge of the left panel in Fig. 2) and that such noise can be partially filtered out, if necessary, to clean up the signal. Following the publication of our work [16], we learned about the studies of Miller [19] and Nesterenko [20]. In a series of theoretical and experimental studies, Nesterenko and coworkers [21 23] had shown that impulse travels as a solitary wave through a chain of Hertzian grains in mutual contact. Our studies and those of Coste et al. [24 26] have confirmed the existence of solitary waves in chains of elastic beads. Our discussions above have not directly addressed the issue of dissipation in granular media. This is an impor- Fig. 2. Propagation of an impulse through a 2D system of grains with3.3% void fractions. Snapshots are taken at equal intervals of time with the earliest time data at the bottom. Regardless of the random motion of grains near the surface, the impulse travels as a weakly dispersive bundle of energy tant topic because there is always a certain amount of energy loss associated with the loading and unloading of macroscopic grains during energy transport. Dissipation has important effects on the amplitude of a propagating impulse in granular beds. However, the width of a weakly dispersive or non-dispersive pulse appears to be unaffected by dissipation [27]. Unfortunately, our knowledge of the nature of dissipation, via restitution and via friction, is limited at the present time [29]. Therefore, in modeling granular media, one is reliant upon constructing simplistic models of dissipation which must then be validated by available experimental data on appropriate systems, whenever that is possible. The simplest treatment of dissipation, along the lines suggested by Walton and Braun [29] involves assuming that the ratio of the forces associated with unloading and loading a granular contact, F unloading /F loading w. A study of impulse propagation under such loading conditions reveals that the amplitude of the propagating pulse should decay approximately exponentially in space and time [27]. Fig. 3 presents typical data that shows such decay. Our collaborators, Naughton et al., have experimentally verified this result. In addition, we have demonstrated that besides restitution, polydispersity of grains can have a significant effect in attenuating the intensity of a propagating pulse [30]. It is also known that instead of sending an impulse, if one sends sound waves of fixed frequency, then the nature of attenuation suffered by the propagating signal has a strong dependence on frequency. Experimental studies of Rogers and Don [8] suggest that signals at frequencies above 20 khz are severely attenuated in soil. Available experiments suggest that signal attenuation is modest at 500 Hz. The nature of attenuation is also dependent upon the sample material, moisture content, compaction and other physical properties. The detail of attenuation suffered by acoustic and shock pulses is not firmly understood. 3 Impulse propagation state of the art Studies on sound propagation in granular media have been motivated by issues concerning the stability of grain

3 35 Fig. 3. Exponential-like decay in space of the total energy of the leading pulse due to various restitutions silos and were pioneered by Janssen in 1895 [31]. His work continues to inspire researchers more than a century later [32]. Recently, Boutreux, Raphael and de Gennes [33] have extended Janssen s approach to study the propagation of an acoustic pulse in a 3D cylinder containing a granular medium in the continuum approximation. In recent years, it has been recognized that the nonlinearity of Hertzian contacts between grains plays an important role in determining the characteristics of impulse propagation in granular media and that the use of a purely acoustic approximation to describe impulse propagation may lead to inaccurate results [16 28]. Much work remains to be done to understand impulse propagation in 3D dry, wet and saturated grains. We summarize the recent developments below D granular systems Chains without gravitational loading As mentioned in Section 1, the non-linear nature of shock propagation in granular systems was first recognized by Nesterenko in 1983 [20]. In the original analysis [20], which invoked the long wavelength approximation, Nesterenko demonstrated that the non-linear Hertzian force law results in the impulse propagating as a solitary wave through the chain. The dynamical problem is briefly summarized below. The equation of motion of a spherical grain, in a chain of spherical grains in contact, labeled i at location u i and moving with acceleration d 2 u i /dt 2 can be written as md 2 u i /dt 2 =A(δ 0 u i +u i 1 ) 3/2 A(δ 0 u i+1 +u i ) 3/2 +mg, (1) where A = Y(2R) 1/2 /[3(1 σ 2 )], with Y, σ, m,gandr denoting, respectively, the Young s modulus and the Poisson s ratio of the grain material, the mass of the grains, gravitational acceleration and the radius of each grain (assumed to be homogeneous for the sake of simplicity). We setg=0fornow.thequantity δ 0 gives the distance of closest approach between the grains in the absence of the pulse at time t = 0 and is a parameter that describes the ambient loading of the chain. When δ 0 u i 1 u i, i.e., when the grain compression is small compared to the compression due to loading, one obtains acoustic pulse propagation through the chain [18]. We have recently studied this problem [27] and reported that the amplitude of the maximum velocity of the propagating pulse decays with depth z (measured from the surface) as z 1/2. The envelope of the velocity function associated with the propagating perturbation as experienced by a typical grain decays at large times as t 1.4. The tail is measured in such a way that the behavior of the leading edge of the pulse does not affect the measurement of decay of the tail [27]. To our knowledge, there is no analytic understanding of the relaxation processes associated with the response of grains to a perturbation in a uniformly loaded chain. The most interesting regime occurs when δ 0 0. In this limit [20], one finds that a perturbation initiated at one end of a chain leads to the formation of a solitary wave, at some finite distance of about 10 grain diameters from the perturbed edge. The solitary wave is about 5-grain diameters (Fig. 4) wide for spherical grains. The width of the solitary wave is sensitive to the shape of the grains in the chain [27]. Further, these solitary waves behave in ways that are different compared to continuum solitary waves that have been well studied by mathematicians and physicists [34]. In particular, when two identical solitary waves traveling in opposite directions meet, they very nearly annihilate each other instead of exact annihilation (Fig. 5). The point of intersection typically spawns a hierarchy of secondary solitary waves of same width but of magnitudes that are 10 4 times (or less) of that of the original solitary waves [34]. Our results are qualitatively consistent with related experimental observations made by Nesterenko and coworkers [22, 23]. The process of formation of secondary solitary waves is poorly understood [27] and requires careful experimental study.

4 36 Fig. 4. The displacement, velocity and acceleration of grains (in arbitrary units) as a solitary wave passes through them in a chain of grains Chains with gravitational loading A significant amount of experimental and some theoretical analyses using dynamic photoelasticity and strain-gage techniques and elasticity theory has been carried out by Shukla and collaborators on the problem of propagation of acoustic impulses in granular chains [36 37]. In many of these experiments, the grains were disks of the polyester material Homalite 100. The studies were carried out for various loading magnitudes (i.e., for various δ 0 s). When the granular chain is held vertically, the bottom grain is most compressed due to gravitational loading and the surface grain suffers zero compression. We have carried out studies of pulse propagation for such chains with incremental loading. In the limit of a large amplitude impulse, the effects of non-uniform loading are suppressed and the solitary wave propagation problem is realized. The pulse shape changes significantly due to gravitational loading for modest pulse amplitudes (i.e., when the average compression of a grain is less that 10 3 of granular diameter). For a system in a cylindrical container, this limit leads to the results of Boutreux et al. [33] and of Hill and Knopoff [32]. Fig. 6 depicts the dispersion of a propagating impulse for various initial impulse magnitudes in the presence of gravitational loading. The bottom right panel of Fig. 6 demonstrates that the solitary waves are realized for sufficiently large impulses. Recent work [26,35] suggests that such harmonic approximations fail to provide correct information about the displacement function of the grains, which is strongly sensitive to the nonlinearity of the forces. These studies demonstrate that gravitational compaction is likely to play an important role in determining the long time dynamical behavior of the system in low amplitude impulse propagation Backscattering of an impulse in 1D chains Fig. 7 presents data in which a light and a heavy impurity are placed in the center of a chain of grains. The system is subjected to gravitational loading. A propagating weak perturbation is shown to backscatter off the impurity. The backscattered pulse, as seen at the surface reveals information about the depth at which the buried impurity is located and about its density contrast with respect to the grains in the host medium. The information about the depth can be inferred from the transit time of the impulse. The information on the density contrast can be obtained by studying the phase of the leading edge of the backscattered pulse. Backscattering off a lighter impurity yields a negative acceleration of the leading edge (Fig. 7). The opposite is true for backscattering off a heavy impurity. In Fig. 8 we present typical backscattering data obtained for a heavy impurity. The grains in the chain are not loaded and have a uniform random distribution of masses (typical variation of 15% in mass distribution has been probed). The studies reveal that impulse backscattering can be a potentially useful probe for detecting buried objects with a density constrast in the presence of gravity and polydispersity. Fig. 5. (Left) Crossing of two identical solitary waves in a chain of elastic grains (only half of a chain is shown). (Right) Spawning of secondary solitary waves from grains adjacent to the point of intersection

5 37 Fig. 6. Data showing the dispersion of an impulse in the presence of gravitational loading for progressively larger impulse amplitudes. In these calculations, we used parameters that are appropriate for sand grains. Bottom right panel shows that for sufficiently strong impulses, there is no dissipation and solitary wave propagation is recovered Fig. 7. Backscattering of an acoustic impulse from an impurity in a gravitationally loaded chain (left [right] panel has a light [heavy] impurity). Acceleration and time are in arbitrary units 3.2 Impulse propagation in 3D beds One often turns to Biot theory [38] for a qualitative, elastic theory based analysis of sound propagation in dry, moist and saturated granular media. At present, there is considerable interest in the study of impulse propagation in dry and wet granular beds in 2D and 3D [8, 14, 16 18, 39 41]. The bulk of the available literature does not treat the dynamical behavior of the granular beds. The study of impulse propagation in wet and saturated beds is a challenging topic and there appears to be few controlled experiments on shock propagation in wet soil. Theoretical analyses of the dynamics of shock propagation in wet soil remains to be carried out. In this brief review, we shall mostly concern ourselves with impulse propagation in nominally dry granular beds. Shukla and collaborators have studied the propagation of long wavelength acoustic perturbations in 2D arrays of disks [36 37]. Their disks were made of the well-characterized polyester material Homalite 100. The experiments reveal the details of propagation of a compression wave through individual grains as well as through the granular fabric. These studies have established the validity of

6 38 Fig. 8. Backscattering of an impulse as it propagates through a chain of beads with randomly distributed masses. Kinetic energy of grains in arbitrary units) is shown along the vertical axis time and grain position being shown along the two planar axes earlier theoretical analyses of several workers and of simulations done by Sinkovits and Sen in 1995 [16]. Shukla et al. has also investigated sound propagation in granular compacts with voids and inclusions [42]. The research has set the stage for studies on the impulse propagation problem in which the size of the perturbation is comparable to the grain size, i.e., a limit in which the acoustic propagation picture breaks down. In this limit, one is drawn to the problem of impulse propagation where the impulse is of significant enough magnitude that non-linear dynamical effects become important. As suggested in the works of Nesterenko [20 21] and demonstrated by our simulations [16 18], strong perturbations have solitary wave like characteristics in 2D and 3D systems. Our preliminary calculations show that spatially localized impulses imparted at a surface disperse slowly in 2D and 3D systems. Given the slow dispersion, it is likely that the impulses can be backscattered off buried objects and can hence be used as a tool for acoustically viewing buried backscatterers. Fig. 9 presents a preliminary calculation in which we show the backscattering of an impulse from a buried 2D plate that is embedded in a 3D bed. As mentioned in Section 1, impulse backscattering probes may prove to be useful in detecting buried non-metallic objects such as plastic anti-personnel land mines as well as metallic ones, buried structures, deposits and other objects. It turns out that Rogers and Don [8] have already explored some of these issues and have obtained positive results. Recently, Naughton, Shelton, Sen and Manciu [10] have also explored acoustic backscattering at lower frequencies. Our preliminary results suggest that the specific frequency window in the low frequency (typically below 1KHz) is dependent upon the properties of the granular bed itself. In closing, our studies suggest that impulses propagate as solitary waves in 1D chains of elastic grains, exhibit approximately exponential decay in amplitude due to restitution and weak polydispersity and show solitary wave like propagation in 2D and 3D beds. Better understanding Fig. 9. Backscattering of an impulse from a buried square in a 3D granular bed made out of a hexagonal close packed array of Hertzian grains. The buried object possesses a heavier mass in this case. Acceleration and time are in arbitrary units. Polydispersity and gravity make the return signals noisier but do not obliterate the signal. The backscattered signals can be easily detected using sensitive microelectromechanical sensors, as shown in Ref. [10] of the backscattering process may allow one to decipher information about the geometric and material characteristics of the inclusions in granular assemblies. We acknowledge many helpful discussions with Vitali Nesterenko, Felicia Manciu, Vicky Tehan, Mike Naughton, Rick Shelton, Jongbae Hong and Ernie Cespedes. The research reported here has been partially supported by Sandia National Laboratories, U.S. Army and SUNY-Buffalo. References 1. B. O. Hardin, Ph.D. dissertation, University of Florida, (1961) 2. B. O. Hardin & F. E. Richart, Jr., J. of Soil Mech. and Found. Div., Proc. ASCE, 89, No. SM 1, (1963), p B. O. Hardin & W. L Black, J. of Soil Mech. and Found. Div., Proc. ASCE, 92, No. SM 2, (1968), p F. E. Richart, Jr., R. D. Woods, & J. R. Hall, Jr., Vibrations of Soils and Foundations, (Prentice Hall, Englewood Cliffs, NJ, 1970) and references therein 5. C.-h. Liu & S. R. Nagel, Phys. Rev B, 48 (1993), p See and C. Wu, Science News, 153 (1998), p Brig. P. M. Blagden, in Detection of Abandoned Land Mines, IEE Conf. Pub. No., 458 (1998), p A. J. Rogers & C. G. Don, Acoustics Australia, 22 (1994), p S. Sen & M. J. Naughton, U.S. Army Research Report for Contract No. DACA39 97-K-0026, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss., (July, 1998) 10. M. J. Naughton, R. S. Shelton, S. Sen, & M. Manciu, in Detection of Abandoned Land Mines, IEE Conf. Pub. No., 458 (1998), p. 249

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