Spatial patterns of sound propagation in sand

Experimental measurements of the granular density of modes via impact

The jamming transition is an important feature of granular materials, with prior work showing an excess of low-frequency modes in the granular analog to the density of states, the granular density of modes. In this work, we present an experimental method for acoustically measuring the granular density of modes using a single impact event to excite vibrational modes in an experimental, three-dimensional, granular material. We test three different granular materials, all of which are composed of spherical beads. The first two systems are monodisperse collections of either 6 mm or 8 mm diameter beads. The third system is a bidisperse mixture of the previous two bead sizes. During data collection, the particles are confined to a box; on top of this box, and resting on the granular material, is a light, rigid sheet onto which pressure can be applied to the system. To excite the material, a steel impactor ball is dropped on top of the system. The response of the granular material to the impact pulse is recorded by piezoelectric sensors buried throughout the material, and the density of modes is computed from the spectrum of the velocity autocorrelation of these sensors. Our measurements of the density of modes show more low-frequency modes at low pressure, consistent with previous experimental and numerical results, as well as several low-frequency peaks in the density of modes that shift with applied pressure. Our method represents an experimentally simple technique for investigating the granular density of modes and may increase the accessibility and number of such measurements.

Thermal shakedown in granular materials with irregular particle shapes

Granular materials with irregular particle shapes undergo a myriad of temperature variations in natural and engineered systems. However, the impacts of cyclic temperature variations on the mechanics of granular materials remain poorly understood. Specifically, little is known about the response of granular materials to cyclic temperature variations as a function of the following central variables: particle shape, applied stress level, relative density, and temperature amplitude. This paper presents advanced laboratory experiments to explore the impacts of cyclic temperature variations on the mechanics of granular materials, with a focus on sands. The results show that cyclic temperature variations applied to sands induce thermal shakedown: the accumulation of irreversible bulk deformations due to microstructural rearrangements caused by thermal expansions and contractions of the constituting particles. The deformation of sands caused by thermal shakedown strongly depends on particle shape, stress level, relative density, and temperature amplitude. This deformation is limited for individual thermal cycles but accumulates and becomes significant for multiple thermal cycles, leading to substantial compaction in sands and other granular materials, which can affect various natural and engineered systems.

The Effect of Very Cohesive Ultra-Fine Particles in Mixtures on Compression, Consolidation, and Fluidization

This paper focuses on the effect of ultra-fine ( d < 10 µm) powders in mixtures with fine ( d < 100 µm) bulk material on compression processes and also evaluates the re-fluidization behavior of the compressed bed (history effect). Achieving this goal, different mixtures of fine and ultra-fine Ground-Carbonate-Calcium were compressed at three pressure levels. The results show that by increasing the applied pressure, the compressibility decreases due to change in compaction regime. Subsequently, for the higher pressure, the slope of packing density versus applied stress curves is noticeably different. However, this slope does not depend on the size distribution of mixtures, but on the type of material. Comparing fluidization and re-fluidization curves (bed pressure drop vs. gas velocity) shows an increase in the maximum bed pressure drop ( Δ P p e a k ) for re-fluidization. By increasing the portion of ultra-fine particles in the binary mixture, Δ P p e a k increases in a non-linear manner. Furthermore, the incipient fluidization point moves to a higher gas velocity. After compression, the peak of the bed pressure drop in the re-fluidization test happens at a lower gas velocity than in the initial fluidization test. Thus, the slope of the loading curve is much larger for re-fluidization. The opposite is observed for the unloading curves.

A nature inspired modularity function for unsupervised learning involving spatially embedded networks

The quality of network clustering is often measured in terms of a commonly used metric known as "modularity". Modularity compares the clusters found in a network to those present in a random graph (a "null model"). Unfortunately, modularity is somewhat ill suited for studying spatially embedded networks, since a random graph contains no basic geometrical notions. Regardless of their distance, the null model assigns a nonzero probability for an edge to appear between any pair of nodes. Here, we propose a variant of modularity that does not rely on the use of a null model. To demonstrate the essentials of our method, we analyze networks generated from granular ensemble. We show that our method performs better than the most commonly used Newman-Girvan (NG) modularity in detecting the best (physically transparent) partitions in those systems. Our measure further properly detects hierarchical structures, whenever these are present.

Evolution of network architecture in a granular material under compression

As a granular material is compressed, the particles and forces within the system arrange to form complex and heterogeneous collective structures. Force chains are a prime example of such structures, and are thought to constrain bulk properties such as mechanical stability and acoustic transmission. However, capturing and characterizing the evolving nature of the intrinsic inhomogeneity and mesoscale architecture of granular systems can be challenging. A growing body of work has shown that graph theoretic approaches may provide a useful foundation for tackling these problems. Here, we extend the current approaches by utilizing multilayer networks as a framework for directly quantifying the progression of mesoscale architecture in a compressed granular system. We examine a quasi-two-dimensional aggregate of photoelastic disks, subject to biaxial compressions through a series of small, quasistatic steps. Treating particles as network nodes and interparticle forces as network edges, we construct a multilayer network for the system by linking together the series of static force networks that exist at each strain step. We then extract the inherent mesoscale structure from the system by using a generalization of community detection methods to multilayer networks, and we define quantitative measures to characterize the changes in this structure throughout the compression process. We separately consider the network of normal and tangential forces, and find that they display a different progression throughout compression. To test the sensitivity of the network model to particle properties, we examine whether the method can distinguish a subsystem of low-friction particles within a bath of higher-friction particles. We find that this can be achieved by considering the network of tangential forces, and that the community structure is better able to separate the subsystem than a purely local measure of interparticle forces alone. The results discussed throughout this study suggest that these network science techniques may provide a direct way to compare and classify data from systems under different external conditions or with different physical makeup.

Topological and geometric measurements of force-chain structure

Developing quantitative methods for characterizing structural properties of force chains in densely packed granular media is an important step toward understanding or predicting large-scale physical properties of a packing. A promising framework in which to develop such methods is network science, which can be used to translate particle locations and force contacts into a graph in which particles are represented by nodes and forces between particles are represented by weighted edges. Recent work applying network-based community-detection techniques to extract force chains opens the door to developing statistics of force-chain structure, with the goal of identifying geometric and topological differences across packings, and providing a foundation on which to build predictions of bulk material properties from mesoscale network features. Here we discuss a trio of related but fundamentally distinct measurements of the mesoscale structure of force chains in two-dimensional (2D) packings, including a statistic derived using tools from algebraic topology, which together provide a tool set for the analysis of force chain architecture. We demonstrate the utility of this tool set by detecting variations in force-chain architecture with pressure. Collectively, these techniques can be generalized to 3D packings, and to the assessment of continuous deformations of packings under stress or strain.

Ultrasonic probing of cohesive granular media at very low consolidation

In this work, we study sound propagation in different granular media made of magnetite particles, steel particles, and glass beads consolidated under their own weight. We used two ultrasonic non invasive methods, which are found in good agreement, to estimate the longitudinal wave velocity Vp. In the first one, the velocity is determined through the resonance peaks of the powder slab. The second method determines the velocity with the time of flight of a short pulse traveling through the medium. Magnetite and steel samples show dispersive properties below a frequency ∼ 10kHz. The increase of attractive forces between particles lead to two competing effects; on one hand the solid fraction decreases and on the other hand the contacts are stiffer. A decrease in solid fraction implies that the number of contacts decreases leading to a decrease in the wave propagation velocity. On the other hand stronger contacts on average lead to an increase of sound velocity. In our experiments an increase in powder cohesion (caused by a decrease of the bead diameter) leads to an increase in sound velocity even though the solid fraction decreases. The same behavior is found in glass beads. These results show the predominant role of cohesive forces on sound propagation in loosely consolidated dry fine powders as well as the accuracy of ultrasonic methods for powder characterization.

Influence of network topology on sound propagation in granular materials

Granular media, whose features range from the particle scale to the force-chain scale and the bulk scale, are usually modeled as either particulate or continuum materials. In contrast with each of these approaches, network representations are natural for the simultaneous examination of microscopic, mesoscopic, and macroscopic features. In this paper, we treat granular materials as spatially embedded networks in which the nodes (particles) are connected by weighted edges obtained from contact forces. We test a variety of network measures to determine their utility in helping to describe sound propagation in granular networks and find that network diagnostics can be used to probe particle-, curve-, domain-, and system-scale structures in granular media. In particular, diagnostics of mesoscale network structure are reproducible across experiments, are correlated with sound propagation in this medium, and can be used to identify potentially interesting size scales. We also demonstrate that the sensitivity of network diagnostics depends on the phase of sound propagation. In the injection phase, the signal propagates systemically, as indicated by correlations with the network diagnostic of global efficiency. In the scattering phase, however, the signal is better predicted by mesoscale community structure, suggesting that the acoustic signal scatters over local geographic neighborhoods. Collectively, our results demonstrate how the force network of a granular system is imprinted on transmitted waves.

High-speed imaging of traveling waves in a granular material during silo discharge

We report experimental observations of sound waves in a granular material during resonant silo discharge called silo music. The grain motion was tracked by high-speed imaging while the resonance of the silo was detected by accelerometers and acoustic methods. The grains do not oscillate in phase at neighboring vertical locations, but information propagates upward in this system in the form of sound waves. We show that the wave velocity is not constant throughout the silo but considerably increases toward the lower end of the system, suggesting increased pressure in this region, where the flow changes from cylindrical to converging flow. In the upper part of the silo the wave velocity matches the sound velocity measured in the same material when standing (in the absence of flow). Grain oscillations show a stick-slip character only in the upper part of the silo.

Thermal expansion effects and heat conduction in granular materials

In this paper, we report results and analysis on a simulation study of the effects of thermal expansion in granular systems. We show that these effects impact the force distribution inside a two-dimensional system of disks that are subject to thermal heating under two different boundary conditions. A significant increase in the average force is observed for steel particles confined within a box with fixed walls at temperature rises of 50 degrees C and 100 degrees C, respectively. As previously noted in the literature, thermal expansion also induces compaction. The results show that a systematic and controllable increase in granular packing can be induced by simply raising and then lowering the temperature, without the input of mechanical energy in agreement with previous experimental observations. We find that the evolution of the packing fraction is well described by a fractional relaxation model, which follows the Mittag-Leffler law.

Packing grains by thermal cycling

A long-standing problem in managing the behaviour of a collection of solid grains concerns the nature of the grain packing, a property that is typically controlled by how the grains are poured or shaken. Here we show that a systematic and controllable increase in granular packing can be induced by simply raising and then lowering the temperature, without the input of mechanical energy. This thermal processing may have important practical implications for the handling and storage of granular materials.

Pressure wave propagation in a shaken granular bed

Pressure waves in a granular material travel through particle contact points and are primarily transmitted by the "force chains" that carry most of the load in a granular medium. However, these force chains tend to be fragile and ephemeral and can be disrupted by very minor perturbations including the waves themselves. External vibration also disrupts the force chains and therefore also changes the wave propagation characteristics. In this paper we study the effects of vibration on wave propagation in a shaken granular bed.

Elastic wave propagation in confined granular systems

We present numerical simulations of acoustic wave propagation in confined granular systems consisting of particles interacting with the three-dimensional Hertz-Mindlin force law. The response to a short mechanical excitation on one side of the system is found to be a propagating coherent wave front followed by random oscillations made of multiply scattered waves. We find that the coherent wave front is insensitive to details of the packing: force chains do not play an important role in determining this wave front. The coherent wave propagates linearly in time, and its amplitude and width depend as a power law on distance, while its velocity is roughly compatible with the predictions of macroscopic elasticity. As there is at present no theory for the broadening and decay of the coherent wave, we numerically and analytically study pulse propagation in a one-dimensional chain of identical elastic balls. The results for the broadening and decay exponents of this system differ significantly from those of the random packings. In all our simulations, the speed of the coherent wave front scales with pressure as p1/6; we compare this result with experimental data on various granular systems where deviations from the p1/6 behavior are seen. We briefly discuss the eigenmodes of the system and effects of damping are investigated as well.

Codalike multiple scattering of elastic waves in dense granular media

We study the multiple scattering of short-wavelength ultrasound through the force networks in dry and wet glass bead packings under stress. Over long distance scales, the diffusion approximation is shown to describe adequately the transport of elastic waves dominated by shear waves. The recovered transport mean path reveals a short-range correlation of the force chains. Also we observe the drastic effect of wetting liquids on the energy dissipation in the granular medium. The relevance of these experimental findings for the seismological applications is discussed.

Contact dynamics in a gently vibrated granular pile

We use multispeckle diffusive wave spectroscopy to probe the micron-scale dynamics of a water-saturated granular pile submitted to discrete gentle taps. The typical time scale between plastic events is found to increase dramatically with the number of applied taps. Furthermore, this microscopic dynamics weakly depends on the solid fraction of the sample. This process is largely analogous to the aging phenomenon observed in thermal glassy systems. We propose a heuristic model where this slowing-down mechanism is associated with a slow evolution of the distribution of the contact forces between particles. This model accounts for the main features of the observed dynamics.

Characterization of soliton damping in the granular chain under gravity

A soliton created in the horizontal granular chain damps due to gravity in the vertical chain. We show that there are two types of propagating modes, quasisolitary and oscillatory, in the vertical chain, depending on the strength of impulse. We find that the type of damping is a power law in depth or time. We also find that the absolute value of the exponent of the power law decreases as the strength of the initial impulse increases in the quasisolitary regime. In the oscillatory regime, however, in which the initial impulse is weak, the power-law exponent is independent of the strength of the initial impulse. We show that the power-law damping is caused by the gravitation which results in the change of the force constant at each contact.

Force Fluctuations in Bead Packs

Experimental observations and numerical simulations of the large force inhomogeneities present in stationary bead packs are presented. Forces much larger than the mean occurred but were exponentially rare. An exactly soluble model reproduced many aspects of the experiments and simulations. In this model, the fluctuations in the force distribution arise because of variations in the contact angles and the constraints imposed by the force balance on each bead in the pile.