Frequency filtering in disordered granular chains
TL;DR: In this article, disorder-induced frequency filtering is studied for one-dimensional systems composed of random, pre-stressed masses interacting through both linear and nonlinear (Hertzian) repulsive forces.
Abstract: The study of disorder-induced frequency filtering is presented for one-dimensional systems composed of random, pre-stressed masses interacting through both linear and nonlinear (Hertzian) repulsive forces. An ensemble of such systems is driven at a specified frequency, and the spectral content of the propagated disturbance is examined as a function of distance from the source. It is shown that the transmitted signal contains only low-frequency components, and the attenuation is dependent on the magnitude of disorder, the input frequency, and the contact model. It is found that increased disorder leads to a narrower bandwidth of transmitted frequencies at a given distance from the source and that lower input frequencies exhibit less sensitivity to the arrangement of the masses. Comparison of the nonlinear and linear contact models reveals qualitatively similar filtering behavior; however, it is observed that the nonlinear chain produces transmission spectrums with a greater density at the lowest frequencies. In addition, it is shown that random masses sampled from normal, uniform, and binary distributions produce quantitatively indistinguishable filtering behavior, suggesting that knowledge of only the distribution’s first two moments is sufficient to characterize the bulk signal transmission behavior. Finally, we examine the wave number evolution of random chains constrained to move between fixed end-particles and present a transfer matrix theory in wave number space, and an argument for the observed filtering based on the spatial localization of the higher-frequency normal modes.
Citations
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TL;DR: In this paper, experimental data on a model soil in a cubical cell are compared with both discrete element (DEM) simulations and continuum analyses and the results show that the generally satisfactory agreement between experimental observations and DEM simulations can be seen as a validation and support the use of DEM to investigate the influence of grain interaction on wave propagation.
Abstract: In this study experimental data on a model soil in a cubical cell are compared with both discrete element (DEM) simulations and continuum analyses. The experiments and simulations used point source transmitters and receivers to evaluate the shear and compression wave velocities of the samples, from which some of the elastic moduli can be deduced. Complex responses to perturbations generated by the bender/extender piezoceramic elements in the experiments were compared to those found by the controlled movement of the particles in the DEM simulations. The generally satisfactory agreement between experimental observations and DEM simulations can be seen as a validation and support the use of DEM to investigate the influence of grain interaction on wave propagation. Frequency domain analyses that considered filtering of the higher frequency components of the inserted signal, the ratio of the input and received signals in the frequency domain and sample resonance provided useful insight into the system response. Frequency domain analysis and analytical continuum solutions for cube vibration show that the testing configuration excited some, but not all, of the system’s resonant frequencies. The particle scale data available from DEM enabled analysis of the energy dissipation during propagation of the wave. Frequency domain analysis at the particle scale revealed that the higher frequency content reduces with increasing distance from the point of excitation.
43 citations
Cites background from "Frequency filtering in disordered g..."
...This confirms that frequency filtering occurs as a function of the wave travelling through a granular system as noted by [49,50] and others....
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TL;DR: This work studies the spreading of initially localized excitations in one-dimensional disordered granular crystals to investigate localization phenomena in strongly nonlinear systems, which it is demonstrated to differ fundamentally from localization in linear and weakly non linear systems.
Abstract: We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to differ fundamentally from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder-an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements)-and for two types of initial conditions (displacement excitations and velocity excitations). We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics depend strongly on the type of initial condition. In particular, for displacement excitations, the long-time asymptotic behavior of the second moment m(2) of the energy has oscillations that depend on the type of disorder, with a complex trend that differs markedly from a power law and which is particularly evident for an Anderson-like disorder. By contrast, for velocity excitations, we find that a standard scaling m(2)∼t(γ) (for some constant γ) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, m(2) and the inverse participation ratio P(-1) satisfy scaling relations m(2)∼t(γ) and P(-1)∼t(-η), and the dynamics is superdiffusive for all of the cases that we consider. Additionally, when precompression is strong, the inverse participation ratio decreases slowly (with η<0.1) for all three types of disorder, and the dynamics leads to a partial localization around the core and the leading edge of a propagating wave packet. For an Anderson-like disorder, displacement perturbations lead to localization of energy primarily in the core, and velocity perturbations cause the energy to be divided between the core and the leading edge. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar (roughly γ≈1.7 and η≈1) for all three types of disorder and for both types of initial conditions.
40 citations
TL;DR: In this paper, a discrete element method (DEM) model of a face-centred cubic packing of uniform spheres was created to simulate bender element tests to investigate this test from a fundamental perspective.
Abstract: While bender element testing is now well-established as a laboratory technique to determine soil stiffness, a robust technique to interpret the data remains elusive. A discrete element method (DEM) model of a face-centred cubic packing of uniform spheres was created to simulate bender element tests to investigate this test from a fundamental perspective. During the DEM simulations transmitter and receiver signals were recorded, analogous to the data available in laboratory tests, and these macro-scale data were supplemented with particle scale measurements (forces, stresses and displacements). A range of approaches previously applied in experimental and numerical studies were used to analyse the resulting data in both the time and frequency domains. The shortcomings in these approaches are clear from the differences in the resultant shear stiffness values and the frequency-dependent nature of the values. The particle-scale data enabled visualization of the passage of the wave through the sample, and it was found not to be possible to precisely link the arrival of the shear wave at the receiver and any of the previously proposed characteristic points along the signal recorded at the receiver. The most reliable determination of the shear wave velocity was obtained by applying a two-dimensional fast Fourier transform (2D FFT) to the data describing the velocity of the particles lying between the transmitter and receiver elements. Use of the DEM model and this 2D FFT approach facilitated the sensitivity of the system response to small variations in the interparticle force–displacement law (the contact model) to be established.
38 citations
01 Jan 1972
32 citations
TL;DR: In this article, discrete element method (DEM) simulations of planar compression wave propagation were performed to generate the data for the study, and the assembly stiffness and mass matrices were extracted from the DEM model and these data were used in an eigenmode analysis that provided significant insight into the observed overall dynamic response.
Abstract: Laboratory geophysics tests including bender elements and acoustic emission measure the speed of propagation of stress or sound waves in granular materials to derive elastic stiffness parameters. This contribution builds on earlier studies to assess whether the received signal characteristics can provide additional information about either the material’s behaviour or the nature of the material itself. Specifically it considers the maximum frequency that the material can transmit; it also assesses whether there is a simple link between the spectrum of the received signal and the natural frequencies of the sample. Discrete element method (DEM) simulations of planar compression wave propagation were performed to generate the data for the study. Restricting consideration to uniform (monodisperse) spheres, the material fabric was varied by considering face-centred cubic lattice packings as well as random configurations with different packing densities. Supplemental analyses, in addition to the DEM simulations, were used to develop a more comprehensive understanding of the system dynamics. The assembly stiffness and mass matrices were extracted from the DEM model and these data were used in an eigenmode analysis that provided significant insight into the observed overall dynamic response. The close agreement of the wave velocities estimated using eigenmode analysis with the DEM results confirms that DEM wave propagation simulations can reliably be used to extract material stiffness data. The data show that increasing either stress or density allows higher frequencies to propagate through the media, but the low-pass wavelength is a function of packing density rather than stress level. Prior research which had hypothesised that there is a simple link between the spectrum of the received signal and the natural sample frequencies was not substantiated.
24 citations
Cites background from "Frequency filtering in disordered g..."
...The FCC sample consisted of 3200 particles (4 × 4 × 200 layers) and so is equivalent to that considered by Mouraille et al. [19] and Mouraille and Luding [12]; it was created by considering the lattice geometry of the packing....
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...Lawney and Luding [15] examined a 1-...
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...Santamarina and Aloufi [10] and Santamarina et al. [11] related the maximum transmitted frequency ( flow−pass) and the associated wavelength (λlow−pass) to particle size, while Mouraille and Luding [12] related λlow−pass to the layer spacing....
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...Santamarina et al. [11] and Santamarina and Aloufi [10] assumed the particle diameter to be an internal scale ( α) of granular materials where λlow−pass = 2α, while Mouraille and Luding [12] took α to be the layer distance for a FCC sample, i.e. α = √2R....
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...D chain model, Lawney and Luding [15] showed that the low-frequency eigenmodes are not affected by small random variations in particle mass. Somfai et al. [30] considered a 2-...
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References
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TL;DR: In this paper, the effects of plasticity in wave propagation in elasto-plastic granular systems were investigated and scaling laws for peak force at each contact point along the chain, velocity of the leading wave, local contact and total dissipation were developed.
Abstract: Due to the nonlinear nature of the inter-particle contact, granular chains made of elastic spheres are known to transmit solitary waves under impulse loading. However, the localized contact between spherical granules leads to stress concentration, resulting in plastic behavior even for small forces. In this work, we investigate the effects of plasticity in wave propagation in elasto-plastic granular systems. In the first part of this work, a force–displacement law between contacting elastic-perfectly plastic spheres is developed using a nonlinear finite element analysis. In the second part, this force–displacement law is used to simulate wave propagation in one-dimensional granular chains. In elasto-plastic chains, energy dissipation leads to the formation and merging of wave trains, which have characteristics very different from those of elastic chains. Scaling laws for peak force at each contact point along the chain, velocity of the leading wave, local contact and total dissipation are developed.
42 citations
"Frequency filtering in disordered g..." refers background in this paper
...The interest in non-linear, realistic wavepropagation phenomena is documented by the most recent publications on contact dissipation mechanisms in one-dimensional [31,36,38] and higher dimensional idealized granular systems [7,24,31]....
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...plastic deformations) [36,38] as well as the deeper understanding of idealized granular systems in higher dimensions [7,24] should help to better understand real system wave propagation in anisotropic, disordered granular matter, where wave propagation is an efficient means to measure the moduli and thus the microstructure of the material, see [19–21] and references therein....
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TL;DR: In this paper, the effects of disorder on the vibration behavior of arrays of micromechanical resonators are investigated, and the consequences for the characteristics of filters comprised of such arrays are described.
42 citations
"Frequency filtering in disordered g..." refers background in this paper
...[18] numerically examine the spectra of disordered micromechanical oscillators, focusing on frequency filtering within the passband of ordered arrays....
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TL;DR: In this paper, it was shown that for vanishingly small loading and large impact condition, it may be possible to generate solitons in a chain of grains that are characterized by Hertzian contacts.
Abstract: We confirm that for vanishingly small loading and large impact condition, it may be possible to generate solitons in a chain of grains that are characterized by Hertzian contacts. For uniform or progressive loading conditions throughout the chain, one generates soft-solitons which are weakly dispersive in space and time. Under conditions of weak impact, one generates acoustic pulses through the chain. We describe the displacements , velocities and accelerations suffered by the individual grains when subjected to solitons, soft-solitons and acoustic pulses and describe the effects of restitution on the propagating pulse.
42 citations
TL;DR: In this paper, the results of a series of discrete element method (DEM) simulations of bender element tests on a simple, idealised granular material were presented. And the authors used a fast Fourier transform to decompose the signal using a fast-Fourier transform.
Abstract: The small-strain (elastic) shear stiffness of soil is an important parameter in geotechnics. It is required as an input parameter to predict deformations and to carry out site response analysis to predict levels of shaking during earthquakes. Bender element testing is often used in experimental soil mechanics to determine the shear (S-) wave velocity in a given soil and hence the shear stiffness. In a bender element test a small perturbation is input at a point source and the propagation of the perturbation through the system is measured at a single measurement point. The mechanics and dynamics of the system response are non-trivial, complicating interpretation of the measured signal. This paper presents the results of a series of discrete element method (DEM) simulations of bender element tests on a simple, idealised granular material. DEM simulations provide the opportunity to study the mechanics of this testing approach in detail. The DEM model is shown to be capable of capturing features of the system response that had previously been identified in continuum-type analyses of the system. The propagation of the wave through the sample can be monitored at the particle-scale in the DEM simulation. In particular, the particle velocity data indicated the migration of a central S-wave accompanied by P-waves moving along the sides of the sample. The elastic stiffness of the system was compared with the stiffness calculated using different approaches to interpreting bender element test data. An approach based upon direct decomposition of the signal using a fast-Fourier transform yielded the most accurate results.
40 citations
"Frequency filtering in disordered g..." refers background in this paper
...The interest in non-linear, realistic wavepropagation phenomena is documented by the most recent publications on contact dissipation mechanisms in one-dimensional [31,36,38] and higher dimensional idealized granular systems [7,24,31]....
[...]
...plastic deformations) [36,38] as well as the deeper understanding of idealized granular systems in higher dimensions [7,24] should help to better understand real system wave propagation in anisotropic, disordered granular matter, where wave propagation is an efficient means to measure the moduli and thus the microstructure of the material, see [19–21] and references therein....
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TL;DR: The influence on the self-demodulation process of the transition from the propagative to the evanescent regime in pump wave propagation that takes place when the pump frequency increases is studied.
Abstract: The self-demodulation process in a nonlinear granular chain of identical beads is studied analytically and numerically. In such a medium, in accordance with the dispersion relation, longitudinal waves that have a frequency higher than the so-called cutoff frequency of the chain are evanescent. Here, we study the influence on the self-demodulation process of the transition from the propagative to the evanescent regime in pump wave propagation that takes place when the pump frequency increases. An analytical solution in discrete coordinates is derived for the case of two primary frequencies mixing into a single difference frequency. This solution is then numerically integrated in order to analyze the demodulation of the acoustic wave packet (i.e., of the harmonic acoustic wave modulated in a pulse mode). Temporal demodulated profiles can be strongly sensitive to the regime (propagative or evanescent) of primary wave transport. This model allows us to detect the cutoff frequency of longitudinal elastic waves in the chain, without receiving the primary waves, but receiving the low frequency nonlinearly radiated signal. The roles of frequency dependent attenuation, velocity dispersion, and observation distance are analyzed.
39 citations
"Frequency filtering in disordered g..." refers background in this paper
...[48] observe the propagation of these low-frequency signals in nonlinear chains, terming itself demodulation....
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