Partially embedded metabarrier to suppress surface waves in granular mediaa)

The gravity-induced depth-dependent elastic properties of a granular half-space result in multiple dispersive surface modes and demand the consideration of material heterogeneity in metabarrier designs to suppress surface waves. Numerous locally resonant metabarrier configurations have been proposed in the literature to suppress Rayleigh surface waves in homogeneous media, with little focus on extending the designs to a heterogeneous half-space. In this work, a metabarrier comprising partially embedded rod-like resonators to suppress the fundamental dispersive surface wave modes in heterogeneous granular media known as first order PSV (PSV1; where P is the longitudinal mode and SV is the shear-vertical mode) and second order PSV (PSV2) is proposed. The unit-cell dispersion analysis, together with an extensive frequency-domain finite element analysis, reveals preferential hybridization of the PSV1 and PSV2 modes with the longitudinal and flexural resonances of the resonators, respectively. The presence of the cutoff frequency for the longitudinal-resonance hybridized mode facilitates straightforward suppression of the PSV1 mode, while PSV2 mode suppression is possible by tailoring the hybridized flexural resonance modes. These PSV1 and PSV2 bandgaps are realized experimentally in a granular testbed comprising glass beads by embedding 3D-printed resonator rods. Also explored are novel graded metabarriers capable of suppressing both PSV1 and PSV2 modes over a broad frequency range for potential applications in vibration control and seismic isolation.

Contact networks and force transmission in aggregates of hexapod-shaped particles

Hexapods, consisting of three mutually orthogonal arms, have been utilized as a representative nonconvex shape to demonstrate the impact of interlocking on the strength properties of granular materials. Nevertheless, the microstructural characteristics of hexapod packings, which underlie their strength, have remained insufficiently characterized. We use particle dynamics simulations to build isotropically-packed aggregates of hexapods and we analyze the effects of aspect ratio and interparticle friction on the microstructure and force transmission. We find that the packing fraction is an unmonotonic function of aspect ratio due to competition between steric exclusions and interlocking. Interestingly, the contact coordination number declines considerably with friction coefficient, showing the stronger effect of friction on the stability of hexapod packings as compared with sphere packings. The pair distribution functions show that local ordering due to steric exclusions disappears beyond the aspect ratio 3 and the hexapods touch their second neighbors. Remarkably, hexapods of aspect ratio 3 tend to align with their neighbors and form locally ordered structures, implying a contact coordination number which is highly sensitive to the confining pressure. We also show that the probability density function of forces between hexapods is similar to that of sphere packings but with broadening exponential fall-off of strong forces as aspect ratio increases. Finally, the elastic bulk modulus of the aggregates is found to increase considerably with aspect ratio as a consequence of the rapid increase of contact density and the number of contacts with second neighbors.

Acoustic monitoring of compaction in cohesive granular materials

We study the transition from cohesive to noncohesive states of cemented granular materials (synthetic rocks) under oedometric loading, combining simultaneous measurements of ultrasound velocity and acoustic emission (AE: microseosmicity). Our samples are agglomerates made of glass beads bonded with a few percent of cement, either ductile or brittle. These cemented granular samples exhibit an inelastic compaction beyond certain axial stresses likely due to the formation of compaction bands, which is accompanied by a significant decrease of compressional wave velocity. Upon subsequent cyclic unloading-reloading with constant consolidation stress, we found the mechanical and acoustic responses like those in noncohesive granular materials, which can be interpreted within the effective medium theory based on the Digby's bonding model. Moreover, this model allows P-wave velocity measured at vanishing pressure to be interpreted as an indicator of the debonding on the scale of grain contact. During the inelastic compaction, stick-slip-like stress drops were observed in brittle cement-bonded granular samples accompanied by the instantaneous decrease of the P-wave velocity and AEs which display an Omori-like law for foreshocks, i.e., precursors. By contrast, mechanical responses of ductile cement-bonded granular samples are smooth (without visible stick-slip-like stress drops) and mostly aseismic. By applying a cyclic loading-unloading with increasing consolidation stress, we observed a Kaiser-like memory effect in the brittle cement-bonded sample in the weakly damaged state which tends to disappear when the bonds are mostly broken in the noncohesive granular state after large-amplitude loading. In this paper, we show that the macroscopic ductile and brittle behavior of cemented granular media is controlled by the local processes on the scale of the bonds between grains.

Finite amplitude waves in jammed matter

Here we use simulations and theory to show that, close to the jamming point, an arbitrary initial distortion of a granular media induces the formation of forward and backward non-linear finite amplitude waves. There are two regimes in the evolution of these waves (near field and far field). Initially, non-linear interactions between forward and backward waves dominate the propagation, leading to complex early evolution (near field). At longer times, forward and backwards waves cease interacting in the far field, and the propagation enters a new regime. Here the waves acquire a triangular-like profile, and evolve in a self-similar fashion characterized by a power law attenuation, whose exponent is weakly dependent on the initial pressure of the system. The finite amplitude waves gradually become linear waves when the amplitude of the initial distortion decreases, or the confining pressure on the system increases.

The Tides of Enceladus' Porous Core

The inferred density of Enceladus' core, together with evidence of hydrothermal activity within the moon, suggests that the core is porous. Tidal dissipation in an unconsolidated core has been proposed as the main source of Enceladus' geological activity. However, the tidal response of its core has generally been modeled assuming it behaves viscoelastically rather than poroviscoelastically. In this work, we analyze the poroviscoelastic response to better constrain the distribution of tidal dissipation within Enceladus. A poroviscoelastic body has a different tidal response than a viscoelastic one; pressure within the pores alters the stress field and induces a Darcian porous flow. This flow represents an additional pathway for energy dissipation. Using Biot's theory of poroviscoelasticity, we develop a new framework to obtain the tidal response of a spherically symmetric, self-gravitating moon with porous layers and apply it to Enceladus. We show that the boundary conditions at the interface of the core and overlying ocean play a key role in the tidal response. The ocean hinders the development of a large-amplitude Darcian flow, making negligible the Darcian contribution to the dissipation budget. We therefore infer that Enceladus' core can be the source of its geological activity only if it has a low rigidity and a very low viscosity. A future mission to Enceladus could test this hypothesis by measuring the phase lags of tidally induced changes of gravitational potential and surface displacements.

The mechanics of brittle granular materials with coevolving grain size and shape

The influence of particle shape on the mechanics of sand is widely recognized, especially in mineral processing and geomechanics. However, most existing continuum theories for engineering applications do not encompass the morphology of the grains and its evolution during comminution. Similarly, the relatively few engineering models accounting for grain-scale processes tend to idealize particles as spheres, with their diameters considered as the primary and sole geometric descriptor. This paper inspires a new generation of constitutive laws for crushable granular continua with arbitrary, yet evolving, particle morphology. We explore the idea of introducing multiple grain shape descriptors into Continuum Breakage Mechanics (CBM), a theory originally designed to track changes in particle size distributions during confined comminution. We incorporate the influence of these descriptors on the elastic strain energy potential and treat them as dissipative state variables. In analogy with the original CBM, and in light of evidence from extreme fragmentation in nature, the evolution of the additional shape descriptors is postulated to converge towards an attractor. Comparisons with laboratory experiments, discrete element analyses and particle-scale fracture models illustrate the encouraging performance of the theory. The theory provides insights into the feedback among particle shape, compressive yielding and inelastic deformation in crushable granular continua. These results inspire new questions that should guide future research into crushable granular systems using particle-scale imaging and computations.

Resonance ultrasonic spectroscopy applied to normal and tangential contact modes investigation of a constrained metallic sphere

This article reports on the analysis of the vibration modes (central eigenfrequencies) observed on a dry steel sphere clamped between two planes, under static compression and subjected to an acoustic excitation. The experimental technique is based on the analysis of the ultrasonic resonance spectrum (RUS); compression or transverse excitations are used. Two types of vibrations are observed. The first occurring in the low frequency range corresponds to modes of movement conditioned by the contacts; the second corresponds to the internal, toroidal and spheroidal deformation modes. In this article we limit our research to the modes observed in the low frequency regime and our objective is to study the influence of the static force on the position of the peaks observed in the frequency spectra. In the case of a longitudinal excitation, only one mode is observed; the transverse excitation makes it possible to detect two modes. The experimental results are compared to theoretical calculations based on theoretical models of Hertz and Hertz-Mindlin, on a development of the JKR model based on adhesion due to Van der Waals forces. In this work, the mechanical coefficients are considered to be independent of time and we do not study the phenomena of hysteresis of these coefficients.

Elastic waves in particulate glass-rubber mixtures

We investigate the propagation of waves in dense static granular packings made of soft and stiff particles subjected to hydrostatic stress. Physical experiments in a triaxial cell equipped with broadband piezoelectric wave transducers have been performed at ultrasound frequencies. The time of flight is measured in order to study the combined effect of applied stress and rubber content on the elastic properties of the mixtures. The bulk stiffness deduced from the wave speed is nonlinear and non-monotonic with the increasing percentage of rubber with a more prominent effect at higher pressures. Moreover, in the frequency domain, a spectral analysis gives insights on the transition from a glass- to a rubber-dominated regime and the influence of rubber particles on the energy dissipation. Mixtures with rubber content below 30% show enhanced damping properties, associated with slightly higher stiffness and lighter weight.

Physics-based characterization of soft marine sediments using vector sensors

In a 2007 experiment conducted in the northern North Sea, observations of a low-frequency seismo-acoustic wave field with a linear horizontal array of vector sensors located on the seafloor revealed a strong, narrow peak around 38 Hz in the power spectra and a presence of multi-mode horizontally and vertically polarized interface waves with phase speeds between 45 and 350 m/s. Dispersion curves of the interface waves exhibit piece-wise linear dependences between the logarithm of phase speed and logarithm of frequency with distinct slopes at large and small phase speeds, which suggests a seabed with a power-law shear speed dependence in two distinct sediment layers. The power spectrum peak is interpreted as a manifestation of a seismo-acoustic resonance. A simple geoacoustic model with a few free parameters is derived that quantitatively reproduces the key features of the observations. This article's approach to the inverse problem is guided by a theoretical analysis of interface wave dispersion and resonance reflection of compressional waves in soft marine sediments containing two or more layers of different composition. Combining data from various channels of the vector sensors is critical for separating waves of different polarizations and helps to identify various arrivals, check consistency of inversions, and evaluate sediment density.

Stress driven fractionalization of vacancies in regular packings of elastic particles

Elucidating the interplay of defects and stress at the microscopic level is a fundamental physical problem that has a strong connection with materials science. Here, based on the two-dimensional crystal model, we show that the instability mode of vacancies with varying size and morphology conforms to a common scenario. A vacancy under compression is fissioned into a pair of dislocations that glide and vanish at the boundary. This neat process is triggered by the local shear stress around the vacancy. The remarkable fractionalization of vacancies creates rich modes of interaction between vacancies and other topological defects, and provides a new dimension for mechanical engineering of defects in extensive crystalline structures.

Compaction Model for Highly Deformable Particle Assemblies

The compaction behavior of deformable grain assemblies beyond jamming remains bewildering, and existing models that seek to find the relationship between the confining pressure P and solid fraction ϕ end up settling for empirical strategies or fitting parameters. Using a coupled discrete-finite element method, we analyze assemblies of highly deformable frictional grains under compression. We show that the solid fraction evolves nonlinearly from the jamming point and asymptotically tends to unity. Based on the micromechanical definition of the granular stress tensor, we develop a theoretical model, free from ad hoc parameters, correctly mapping the evolution of ϕ with P. Our approach unveils the fundamental features of the compaction process arising from the joint evolution of grain connectivity and the behavior of single representative grains. This theoretical framework also allows us to deduce a bulk modulus equation showing an excellent agreement with our numerical data.

Acoustic waves in granular packings at low confinement pressure

Elastic properties of a granular packing show a nonlinear behavior determined by its discrete structure and nonlinear inter-grain force laws. Acoustic waves show a transition from constant, pressure-dependent sound speed to a shock-wave-like behavior with an amplitude-determined propagation speed. This becomes increasingly visible at low static confinement pressure as the transient regime shifts to lower wave amplitudes for lower static pressure. In microgravity, confinement pressure can be orders of magnitude lower than in a ground-based experiment. In addition, the absence of hydrostatic gradients allows for much more homogeneous and isotropic pressure distribution. We present a novel apparatus for acoustic wave transmission measurements at such low packing pressures. A pressure control loop is implemented by using a microcontroller that monitors static force sensor readings and adjusts the position of a movable wall with a linear-motor until the desired pressure is reached. Measurements of acoustic waves are possible using accelerometers embedded in the granular packing as well as piezos. For excitation, we use a voice-coil-driven wall, with a large variety of signal shapes, frequencies, and amplitudes. This enables experiments in both the linear and strongly nonlinear regimes.

Drastic slowdown of the Rayleigh-like wave in unjammed granular suspensions

We present an experimental investigation of Rayleigh-like wave propagation along the surface of a dense granular suspension. Using an ultrafast ultrasound scanner, we monitor the softening of the shear modulus via the Rayleigh-like wave velocity slowdown in the optically opaque medium as the driving amplitude increases. For such nonlinear behavior two regimes are found when increasingthe driving amplitude progressively: First, we observe a significant shear modulus weakening due to the microslip on the contact level without macroscopic rearrangements of grains. Second, there is a clear macroscopic plastic rearrangement accompanied by a modulus decrease up to 88%. A friction model is proposed to describe the interplay between nonlinear elasticity and plasticity, which highlights the crucial effect of contact slipping before contact breaking or loss. Investigation of this nonlinear Rayleigh-like wave may bridge the gap between two disjoint approaches for describing the dynamics near unjamming: linear elastic soft modes and nonlinear collisional shock.

Ultrasonic investigation of granular materials subjected to compression and crushing

Ultrasonic wave propagation measurement has been used as a suitable technique for studying the granular materials and investigating the soil fabric structure, the grain contact stiffness, frictional strength, and inter-particle contact area. Previous studies have focused on the variations of shear and compressional wave velocities with effective stress and void ratio, and lesser effort has been made in understanding the variation of amplitude and dominant frequency of transmitted compressional waves with deformation of soil packing. In this study, continuous compressional wave transmission measurements during compaction of unconsolidated quartz sand are used to investigate the impact of soil layer deformation on ultrasonic wave properties. The test setup consisted of a loading machine to apply constant loading rate to a sand layer (granular quartz) of 6 mm thickness compressed between two forcing blocks, and an ultrasonic wave measurement system to continuously monitor the soil layer during compression up to 48 MPa normal stress. The variations in compressional wave attributes such as wave velocity, transmitted amplitude, and dominant frequency were studied as a function of the applied normal stress and the measured normal strain as well as void ratio and particle size. An increasing trend was observed for P-wave velocity, transmitted amplitude and dominant frequency with normal stress. In specimen with the largest particle size (D₅₀ = 0.32 mm), the wave velocity, amplitude and dominant frequency were found to increase about 230%, 4700% and 320% as the normal stress reached the value of 48 MPa. The absolute values of transmitted wave amplitude and dominant frequency were greater for specimens with smaller particle sizes while the normalized values indicate an opposite trend. The changes in the transmitted amplitude were linked to the changes in the true contact area between the particles with a transitional point in the slope of normalized amplitude, coinciding with the yield stress of the granular soil layer. The amount of grain crushing as a result of increase in the normal stress was experimentally measured and a linear correlation was found between the degree of grain crushing and the changes in the normalized dominant frequency of compressional waves.

Contact laws between nanoparticles: the elasticity of a nanopowder

Studies of the mechanical contact between nanometer-scale particles provide fundamental insights into the mechanical properties of materials and the validity of contact laws at the nanoscale which are still under debate for contact surfaces approaching atomic dimensions. Using in situ Brillouin light scattering under high pressure, we show that effective medium theories successfully predict the macroscopic sound velocities in nanopowders if one takes into account the cementation of the contacts Our measurements suggest the relevance of the continuum approach and effective medium theories to describe the contact between nanoparticles of diameters as small as 4 nm, i.e. with radii of contact of a few angstroms. In particular, we demonstrate that the mechanical properties of nanopowders strongly depend on the surface state of the nanoparticles. The presence of molecular adsorbates modifies significantly the contact laws.

Dynamic induced softening in frictional granular materials investigated by discrete-element-method simulation

A granular system composed of frictional glass beads is simulated using the discrete element method. The intergrain forces are based on the Hertz contact law in the normal direction with frictional tangential force. The damping due to collision is also accounted for. Systems are loaded at various stresses and their quasistatic elastic moduli are characterized. Each system is subjected to an extensive dynamic testing protocol by measuring the resonant response to a broad range of ac drive amplitudes and frequencies via a set of diagnostic strains. The system, linear at small ac drive amplitudes, has resonance frequencies that shift downward (i.e., modulus softening) with increased ac drive amplitude. Detailed testing shows that the slipping contact ratio does not contribute significantly to this dynamic modulus softening, but the coordination number is strongly correlated to this reduction. This suggests that the softening arises from the extended structural change via break and remake of contacts during the rearrangement of bead positions driven by the ac amplitude.

Synchronized oscillations and acoustic fluidization in confined granular materials

According to the acoustic fluidization hypothesis, elastic waves at a characteristic frequency form inside seismic faults even in the absence of an external perturbation. These waves are able to generate a normal stress which contrasts the confining pressure and promotes failure. Here, we study the mechanisms responsible for this wave activation via numerical simulations of a granular fault model. We observe the particles belonging to the percolating backbone, which sustains the stress, to perform synchronized oscillations over ellipticlike trajectories in the fault plane. These oscillations occur at the characteristic frequency of acoustic fluidization. As the applied shear stress increases, these oscillations become perpendicular to the fault plane just before the system fails, opposing the confining pressure, consistently with the acoustic fluidization scenario. The same change of orientation can be induced by external perturbations at the acoustic fluidization frequency.

Numerical study of one-dimensional compression of granular materials. II. Elastic moduli, stresses, and microstructure

The elastic moduli of a transversely isotropic model granular material, made of slightly polydisperse elastic-frictional spherical beads, in equilibrium along a one-dimensional (oedometric) compression path, as described in the companion paper [M. H. Khalili et al., Phys. Rev. E 95, 032907 (2017)]10.1103/PhysRevE.95.032907, are investigated by numerical simulations. The relations of the five independent moduli to stresses, density, coordination number, fabric and force anisotropies are studied for different internal material states along the oedometric loading path. It is observed that elastic moduli, as in isotropic packs, are primarily determined by the coordination number, with anomalously small shear moduli in poorly coordinated systems, whatever their density. Such states also exhibit faster increasing moduli in compression, and larger off-diagonal moduli and Poisson ratios. Anisotropy affects the longitudinal moduli C_{11} in the axial direction and C_{22} in the transverse directions, and the shear modulus in the transverse plane C_{44}, more than the shear modulus in a plane containing the axial direction C_{55}. The results are compared to available experiments on anisotropic bead packs, revealing, despite likely differences in internal states, a very similar range of stiffness level (linked to coordination), and semiquantitative agreement as regards the influence of anisotropy. Effective medium theory (the Voigt approach) provides quite inaccurate predictions of the moduli. It also significantly underestimates ratios C_{11}/C_{22} (varying between 1 and 2.2) and C_{55}/C_{44} (varying from 1 to 1.6), which characterize elastic anisotropy, except in relatively weakly anisotropic states. The bulk modulus for isotropic compression and the compliance corresponding to stress increments proportional to the previous stress values are the only elastic coefficients to be correctly estimated by available predictive relations. We discuss the influences of fabric and force anisotropies onto elastic anisotropy, showing in particular that the former dominates in sample series that are directly assembled in anisotropic configurations and keep a roughly constant lateral to axial stress ratio under compression.

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.

Mechanical impulse propagation in a three-dimensional packing of spheres confined at constant pressure

Mechanical impulse propagation in granular media depends strongly on the imposed confinement conditions. In this work, the propagation of sound in a granular packing contained by flexible walls that enable confinement under hydrostatic pressure conditions is investigated. This configuration also allows the form of the input impulse to be controlled by means of an instrumented impact pendulum. The main characteristics of mechanical wave propagation are analyzed, and it is found that the wave speed as function of the wave amplitude of the propagating pulse obeys the predictions of the Hertz contact law. Upon increasing the confinement pressure, a continuous transition from nonlinear to linear propagation is observed. Our results show that in the low-confinement regime, the attenuation increases with an increasing impulse amplitude for nonlinear pulses, whereas it is a weak function of the confinement pressure for linear waves.

Softening of stressed granular packings with resonant sound waves

We perform numerical simulations of a two-dimensional bidisperse granular packing subjected to both a static confining pressure and a sinusoidal dynamic forcing applied by a wall on one edge of the packing. We measure the response experienced by a wall on the opposite edge of the packing and obtain the resonant frequency of the packing as the static or dynamic pressures are varied. Under increasing static pressure, the resonant frequency increases, indicating a velocity increase of elastic waves propagating through the packing. In contrast, when the dynamic amplitude is increased for fixed static pressure, the resonant frequency decreases, indicating a decrease in the wave velocity. This occurs both for compressional and for shear dynamic forcing and is in agreement with experimental results. We find that the average contact number Zc at the resonant frequency decreases with increasing dynamic amplitude, indicating that the elastic softening of the packing is associated with a reduced number of grain-grain contacts through which the elastic waves can travel. We image the excitations created in the packing and show that there are localized disturbances or soft spots that become more prevalent with increasing dynamic amplitude. Our results are in agreement with experiments on glass bead packings and earth materials such as sandstone and granite and may be relevant to the decrease in elastic wave velocities that has been observed to occur near fault zones after strong earthquakes, in surficial sediments during strong ground motion, and in structures during earthquake excitation.

Revealing the structure of a granular medium through ballistic sound propagation

We study the propagation of sound through a bidimensional granular medium consisting of photoelastic disks, which are packed into different crystalline and disordered structures. Acoustic sensors placed at the boundaries of the system capture the acoustic signal produced by a local and well-controlled mechanical excitation. By compressing the system, we find that the speed of the ballistic part of the acoustic wave behaves as a power law of the applied force with both exponent and prefactor sensitive to the internal geometry of the contact network. This information, which we are able to link to the force-deformation relation of single grains under different contact geometries, provides enough information to reveal the structure of the granular medium.

Acoustic probing of elastic behavior and damage in weakly cemented granular media

We investigate the elastic behavior and damage of weakly cemented granular media under external load with ultrasound. The cementation controlled experiments are performed by freezing the capillary liquid at the bead contact in a dense glass or polymeric [poly(methyl methacrylate)] bead pack wet by tetradecane of volume fraction ϕ = 0.1%-4%. When the pendular rings are solidified, an abrupt increase by a factor of 2 in the compressional wave velocity is observed. We interpret the data in terms of effective medium models in which the contact stiffnesses are derived by either a bonded contact model [P. J. Digby, J. Appl. Mech. 48, 803 (1981)] or a cemented contact model [J. Dvorkin, A. Nur, and H. Yin, Mech. Mater. 18, 351 (1994)]. The former fails to quantitatively account for the results with a soft cement relative to the grain, whereas the latter considering the mechanical properties of the cement does apply. Moreover, we monitor the irreversible behavior of the cemented granular packs under moderate uniaxial loading (1.3 MPa) with the correlation method of ultrasound scattering. The damage of the cemented materials is accompanied by a compressional wave velocity decrease up to 60%, likely due to the fractures induced at the grain-cement interfaces.

Evolution of grain contacts in a granular sample under creep and stress relaxation

This article deals with the characterization, using an acoustic technique, of the mechanical behavior of a dry dense granular medium under quasistatic loading. Ultrasound propagation through the contact-force network supporting the external load offers a noninvasive probe of the viscoelastic properties of such heterogeneous media. First the response of a glass bead packing is studied in an oedometric configuration during creep and relaxation tests. Quasilogarithmic increases of sound velocities are found in both mechanical tests. A model based on the mechanics of microcontacts between rough grains adequately reproduces our experimental results, especially for the evolution of elastic modulus. Another main experimental finding is that collective grain rearrangements within the packing also play a crucial role at the early stage of creep and relaxation.

Wave propagation in square granular crystals with spherical interstitial intruders

We investigate the propagation and scattering of highly nonlinear waves in granular systems composed of spheres in contact arranged in a square packing, and study how the presence of small and light spherical interstitial defects, also referred to as intruders, affects the wave propagation. The effects of a single defect are investigated experimentally and compared to numerical simulations, showing very good quantitative agreement. Transmitted and scattered waves are formed, whose characteristics depend on the material properties of the defect in relation to the properties of the particles in the lattice. Experiments and numerical simulations reveal that stiffer defects are more efficient at redistributing energy outside the impacted chain and soft defects induce a localization of the energy at the defect. Finally, the effects of the presence of two defects, placed diagonally or aligned in the square packing are also investigated, as well as how their interaction depends on their relative positions.

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.

Effects of weak disorder on stress-wave anisotropy in centered square nonlinear granular crystals

The present study describes wave propagation characteristics in a weakly disordered two-dimensional granular media composed of a square array of spheres accommodating interstitial cylindrical intruders. Previous investigations, performed experimentally as well as numerically, emphasized that wave-front shapes in similar systems are tunable via choice of material combinations. Here, we investigate the effects of statistical variation in the particle diameters and compare the effects of the resulting disorder in experiments and numerical simulations, finding good agreement.

Sonic sands

Many desert sand dunes emit a loud sound with a characteristic tremolo around a well-defined frequency whenever sand is avalanching on their slip face. This phenomenon, called the 'song of dunes', has been successfully reproduced in the lab, on a smaller scale. In all cases, the spontaneous acoustic emission in air is due to a vibration of the sand, itself excited by a granular shear flow. This review presents a complete characterization of the phenomenon-frequency, amplitude, source shape, vibration modes, instability threshold-based on recent studies. The most prominent characteristics of acoustic propagation in weakly compressed granular media are then presented. Finally, this review describes the different mechanisms proposed to explain booming avalanches. Measurements performed to test these theories against data allow one to contrast explanations that must be rejected-sound resonating in a surface layer of the dune, for instance-with those that still need to be confirmed to reach a scientific consensus-amplification of guided elastic waves by friction, in particular.

Protocol dependence of mechanical properties in granular systems

We study the protocol dependence of the mechanical properties of granular media by means of computer simulations. We control a protocol of realizing disk packings in a systematic manner. In 2D, by keeping material properties of the constituents identical, we carry out compaction with various strain rates. The disk packings exhibit the strain rate dependence of the critical packing fraction above which the pressure becomes non-zero. The observed behavior contrasts with the well-studied jamming transitions for frictionless disk packings. We also observe that the elastic moduli of the disk packings depend on the strain rate logarithmically. Our results suggest that there exists a time-dependent state variable to describe macroscopic material properties of disk packings, which depend on its protocol.

Elastic weakening of a dense granular pack by acoustic fluidization: slipping, compaction, and aging

Sound velocity measurements in dense glass bead packs reveal significant softening effect at large amplitudes, due to the frictional nonlinearity at the grain contacts. Beyond a certain amplitude, the sound-matter interaction becomes irreversible, leaving the medium in a weakened and slightly compacted state. A slow recovery of the initial elastic modulus is observed after acoustic perturbation, revealing the plastic creep growth of microcontacts. The cross-correlation function of configuration-specific acoustic speckles highlights the relationship between the macroscopic elastic weakening and the local change of the contact networks, induced by strong sound vibration, in the absence of appreciable grain motion.

Propagation of acoustic waves in a one-dimensional array of noncohesive cylinders

By means of a photoelastic method, we access the visualization of acoustic waves propagating in a one-dimensional array of noncohesive cylinders. As pointed by Nesterenko in the case of spherical grains [V. F. Nesterenko, J. Appl. Mech. Tech. Phys. 24, p. 567 (1983)], the nonlinearity of the contact law between the grains induces a dependence of the wave velocity both on its amplitude and on the confinement force. Our experimental method allows one to access the evolution in time of the internal state of stress of individual grains with excellent accuracy. We show that the velocity of the sound presents two regimes as a function of the confining force. For low forces, the dependence is strongly nonlinear, while it weakens for higher forces. By means of the direct visualization of the contact zone, we show that both micro- and macroscale imperfections of the surface of contact explain the low forces behavior. We test the consistency of our experimental findings results with both the theoretical expectations and with the experimental determination of the force-displacement dependence. We show, moreover, that the main damping process originates in solid friction.

Comparison of hyperelastic models for granular materials

Three recently proposed hyperelastic models for granular materials are compared with experiment data. Though all three are formulated to give elastic moduli that are power law functions of the mean stress, they have rather different dependencies on individual stresses, and generally differ from well established experimental forms. Predicted static stress distributions are in qualitative agreement with experiments, but do not differ greatly from isotropic linear elasticity, and similarly fail to account for variability in experiment data that presumably occurs due to a preparation dependence of granular materials.

Nonlinear Biot waves in porous media with application to unconsolidated granular media

The nonlinear propagation through porous media is investigated in the framework of Biot theory. For illustration, and considering the current interest for the determination of the elastic properties of granular media, the case of nonlinear propagation in "model" granular media (disordered packings of noncohesive elastic beads of the same size embedded in a visco-thermal fluid) is considered. The solutions of linear Biot waves are first obtained, considering the appropriate geometrical and physical parameters of the medium. Then, making use of the method of successive approximations of nonlinear acoustics, the solutions for the second harmonic Biot waves are derived by considering a quadratic nonlinearity in the solid frame constitutive law (which takes its origin from the high nonlinearity of contacts between grains). The propagation in a semi-infinite medium with velocity dispersion, frequency dependent dissipation, and nonlinearity is first analyzed. The case of a granular medium slab with rigid boundaries, often considered in experiments, is then presented. Finally, the importance of mode coupling between solid and fluid waves is evaluated, depending on the actual fluid, the bead diameter, or the applied static stress on the beads. The application of these results to other media supporting Biot waves (porous ceramics, polymer foams, etc.) is straightforward.

Anisotropic nonlinear elasticity in a spherical-bead pack: influence of the fabric anisotropy

Stress-strain measurements and ultrasound propagation experiments in glass bead packs have been simultaneously conducted to characterize the stress-induced anisotropy under uniaxial loading. These measurements realized, respectively, with finite and incremental deformations of the granular assembly, are analyzed within the framework of the effective-medium theory based on the Hertz-Mindlin contact theory. Our work shows that both compressional and shear wave velocities and consequently the incremental elastic moduli agree fairly well with an effective-medium model developed by Johnson [J. Appl. Mech. 65, 380 (1998)] for the oedometric test, but the anisotropic stress ratio resulting from finite deformation does not at all. As indicated by numerical simulations, the discrepancy may arise from the fact that the model does not properly allow the grains to relax from the affine motion approximation. Here we find that the interaction nature at the grain contact could also play a crucial role for the relevant prediction by the model; indeed, such discrepancy can be significantly reduced if the frictional resistance between grains is removed. Another main experimental finding is the influence of the inherent anisotropy of granular packs, realized by different protocols of the sample preparation. Our results reveal that compressional waves are more sensitive to the stress-induced anisotropy, whereas the shear waves are more sensitive to the fabric anisotropy.

Modeling of acoustic penetration into sandy sediments: physical and geometrical aspects

Two different approaches to the problem of acoustic penetration into sandy marine sediments are considered: application of the Buckingham constitutive model for sediment with a plane surface and boundary element analysis of a rough surface of sediment represented as a homogeneous fluid. By a careful modeling of the constitutive behavior for plane seafloors, it is possible to partly reproduce some features of known experimental dependencies for acoustical pressure. However, accounting for roughness appears to be more important. Accordingly, the authors present a detailed numerical analysis of penetration into rough sediments using the boundary element method. The simulation results support conclusions reached by other investigators and demonstrate how local surface irregularities violate the evanescence condition that holds for a plane interface at subcritical incidence, thus considerably increasing penetration. The results apply to the frequency range 0.5-50 kHz and grazing angles larger than approximately 6 degrees -8 degrees at 10-50 kHz. For lower frequencies, when diffraction becomes important, the lowest possible grazing angle strongly depends on the range covered by the incident beam and is, in general, considerably larger. The authors provide several characteristic examples with frequencies 5 and 15 kHz and grazing angles 15 degrees -30 degrees illustrating the impact of roughness on penetration.

Nonlinear effects for coda-type elastic waves in stressed granular media

Experimental results and their interpretations are presented on the nonlinear acoustic effects of multiple scattered elastic waves in unconsolidated granular media. Short wave packets with a central frequency higher than the so-called cutoff frequency of the medium are emitted at one side of the statically stressed slab of glass beads, and received at the other side after multiple-scattering and nonlinear elastic effects. Typical signals are strongly distorted compared to their initially radiated shape both due to nonlinearity and scattering. It is shown that acoustic waves with a deformation amplitude much lower than the mean static deformation of the contacts in the medium can modify the elastic properties of the medium. This addresses the problem of acoustic wave action on granular matter during and after acoustic excitation, which is necessary to understand in the nondestructive testing of the elastic properties of granular media by acoustic methods. Coda signal analysis is shown to be a powerful time-resolved tool in monitoring slight modifications in the elastic response of an unconsolidated granular structure.

Probing dense granular materials by space-time dependent perturbations

The manner in which signals propagate through dense granular systems in both space and time is not well understood. In order to probe this process, we carry out discrete element simulations of the system response to excitations where we control the driving frequency and wavelength independently. Fourier analysis shows that properties of the signal depend strongly on the space-time scales of the perturbation. The features of the response provide a test bed for models that predict statistical and continuum space-time properties. We illustrate this connection between microscale physics and macroscale behavior by comparing the system response to a simple elastic model with damping.

Incremental stress-strain relation from granular elasticity: comparison to experiments

Granular media are reversible and elastic if the stress increments are small enough. An elastic stress-strain relation, employed previously to determine static stress distributions, in this paper is compared to experiments by Kuwano and Jardine [Geotechnique 52, 727 (2002)] on incremental stress-strain relations, and shown to yield satisfactory agreement. In addition, the yield condition is given a firmer footing.

Sound propagation in a constrained lattice of beads: high-frequency behavior and dispersion relation

We report on acoustic wave propagation in a regular array of nominally identical beads under isotropic static stress. The weak polydispersity of the beads makes the contact lattice random. Time-frequency analysis of the acoustic signal is performed and allows measurement of the full lattice dispersion relation. Comparison with the theoretical prediction for a perfect triangular lattice gives an indication of the level of randomness in the contact lattice. The results extend, in a consistent way, a previous study restricted to long wavelength propagation [B. Gilles and C. Coste, Phys. Rev. Lett. 90, 174302 (2003)]: The contact lattice is ordered by increasing the stress, and the smaller the wavelength, the higher the stress required to get regular lattice behavior. Measurements involving ballistic propagation of the coherent wave, whatever its frequency, evidence reversible lattice behavior under compression and/or decompression. Nevertheless, correlations of short wavelength incoherent waves are a sensitive probe of disorder, and allow us to exhibit a small irreversible evolution of the lattice.

Internal states of model isotropic granular packings. II. Compression and pressure cycles

This is the second paper of a series of three investigating, by numerical means, the geometric and mechanical properties of spherical bead packings under isotropic stresses. We study the effects of varying the applied pressure P (from 1 or 10 kPa up to 100 MPa in the case of glass beads) on several types of configurations assembled by different procedures, as reported in the preceding paper [I. Agnolin and J.-N. Roux, Phys. Rev. E 76, 061302 (2007)]. As functions of P , we monitor changes in solid fraction Phi, coordination number z, proportion of rattlers (grains carrying no force) x_(0) , the distribution of normal forces, the level of friction mobilization, and the distribution of near neighbor distances. Assuming that the contact law does not involve material plasticity or damage, Phi is found to vary very nearly reversibly with P in an isotropic compression cycle, but all other quantities, due to the frictional hysteresis of contact forces, change irreversibly. In particular, initial low P states with high coordination numbers lose many contacts in a compression cycle and end up with values of z and x_(0) close to those of the most poorly coordinated initial configurations. Proportional load variations which do not entail notable configuration changes can therefore nevertheless significantly affect contact networks of granular packings in quasistatic conditions.

Internal states of model isotropic granular packings. III. Elastic properties

In this third and final paper of a series, elastic properties of numerically simulated isotropic packings of spherical beads assembled by different procedures, as described in the first companion paper, and then subjected to a varying confining pressure, as reported in the second companion paper, are investigated. In addition to the pressure, which determines the stiffness of contacts because of Hertz's law, elastic moduli are chiefly sensitive to the coordination number z , which should not be regarded as a function of the packing density. Comparisons of numerical and experimental results for glass beads in the 10 kPa-10 MPa pressure range reveal similar differences between dry samples prepared in a dense state by vibrations and lubricated packings, so that the greater stiffness of the latter, in spite of their lower density, can be attributed to a larger coordination number. Effective medium type approaches, or Voigt and Reuss bounds, provide good estimates of bulk modulus B , which can be accurately bracketed, but badly fail for shear modulus G , especially in low z configurations under low pressure. This is due to the different response of tenuous, fragile networks to changes in load direction, as compared to load intensity. In poorly coordinated packings, the shear modulus, normalized by the average contact stiffness, tends to vary proportionally to the degree of force indeterminacy per unit volume, even though this quantity does not vanish in the rigid limit. The elastic range extends to small strain intervals and compares well with experimental observations on sands. The origins of nonelastic response are discussed. We conclude that elastic moduli provide access to mechanically important information about coordination numbers, which escape direct measurement techniques, and indicate further perspectives.

Some aspects of electrical conduction in granular systems of various dimensions

We report on measurements of the electrical conductivity in both a 2D triangular lattice of metallic beads and in a chain of beads. The voltage/current characteristics are qualitatively similar in both experiments. At low applied current, the voltage is found to increase logarithmically in good agreement with a model of widely distributed resistances in series. At high enough current, the voltage saturates due to the local welding of microcontacts between beads. The frequency dependence of the saturation voltage gives an estimate of the size of these welded microcontacts. The DC value of the saturation voltage ( approximately 0.4 V per contact) gives an indirect measure of the number of welded contact carrying the current within the 2D lattice. Also, a new measurement technique provides a map of the current paths within the 2D lattice of beads. For an isotropic compression of the 2D granular medium, the current paths are localized in few discrete linear paths. This quasi-one-dimensional nature of the electrical conductivity thus explains the similarity between the characteristics in the 1D and 2D systems.