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.
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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: It is shown that a recently proposed effective description of simple decorated chains can be extended to predict pulse properties in chains decorated with small granules of randomly chosen radii, and that the binary-collision approximation can again be used to provide analytic results for this system.
Abstract: We study pulse propagation in one-dimensional chains of spherical granules decorated with small randomly sized granules placed between bigger monodisperse ones. Such "designer chains" are of interest in efforts to control the behavior of the pulse so as to optimize its propagation or attenuation, depending on the desired application. We show that a recently proposed effective description of simple decorated chains can be extended to predict pulse properties in chains decorated with small granules of randomly chosen radii. Furthermore, we also show that the binary-collision approximation can again be used to provide analytic results for this system.
31 citations
"Frequency filtering in disordered g..." refers background in this paper
...One may imagine applications exploiting disorder effects, as could be pursued in the spirit of [5,10,13,14,32,40,47,49] where such systems were engineered to produce a desired output....
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...[14] decorate monodisperse chains with randomly sized small masses and investigate the propagation time and decay of the pulse velocity as a function of system penetration....
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..., deliberate insertion of different sized masses) [10,13,14,27] tapering [5,32,47,49] and controlled variation of the particle material [3,15]....
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TL;DR: This simple model for the propagation of small-amplitude vibrations in a granular material is generalized to include relaxation behavior in the force network---a behavior which is also seen in real granular materials.
Abstract: This paper presents a simple model for the propagation of small-amplitude vibrations in a granular material. In this model, the grains are taken to be spherical balls that interact via linear springs. The positional disorder in the real system is ignored and the particles are placed on the vertices of a square lattice. The only disorder in the system comes from a random distribution of the spring constants. Despite its apparent simplicity, this model is able to reproduce the complex frequency response seen in measurements of sound propagation in a granular system. In order to understand this behavior, the role of the resonance modes of the system is investigated. Finally, this simple model is generalized to include relaxation behavior in the force network---a behavior which is also seen in real granular materials. This model gives quantitative agreement with experimental observations of relaxation.
28 citations
"Frequency filtering in disordered g..." refers background in this paper
...This “Anderson localization” has been confirmed in disordered mechanical systems of vibrating masses [23,41]....
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...This so-called Anderson localization has been observed in many physical contexts including mechanical systems of vibrating masses [23,41]....
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TL;DR: The kth normal mode of vibration (i.e., the atomic displacements) of a disordered one-dimensional lattice (masses and force constants completely arbitrary) with nearest neighbor interaction has precisely k − 1 nodes as discussed by the authors.
Abstract: The kth normal mode of vibration (i.e., the atomic displacements) of a disordered one‐dimensional lattice (masses and force constants completely arbitrary) with nearest‐neighbor interaction has precisely k − 1 nodes. A fortiori, the same is true for ordered one‐dimensional lattices with any number of atoms per unit cell. This theorem exhibits a close relationship between eigenfunctions in monatomic ordered lattices (to which its application has been known for many years) and disordered lattices; a relationship which appears surprising in view of recent demonstrations of the gross differences exhibited in the distribution of eigenvalues. It is thus suggested that some basic concepts of ordered lattice dynamics—propagation vector, phonon momentum, etc.—may retain some simple validity for disordered solids as well. Some numerical examples are given.
27 citations
27 Feb 2009
TL;DR: In this paper, sound wave propagation through different types of dry granular systems is studied with three-dimensional discrete element simulations, theory and experiments, the influence of several micro-scale properties: friction, dissipation, particle rotation, and contact disorder, on the macro-scale sound-wave propagation characteristics are investigated.
Abstract: In this study sound wave propagation through different types of dry confined
granular systems is studied. With three-dimensional discrete element simulations, theory and experiments, the influence of several micro-scale properties: friction, dissipation, particle rotation, and contact disorder, on the macro-scale sound wave propagation characteristics are investigated. Experiments, analyzed with the “Spectral Ratio Technique”, make it possible
to extract frequency-dependent propagation velocities and attenuation. An improved set-up for future investigations is proposed in order to better understand dispersion and propagation of sound in granular materials.
The full dispersion relation of a Face-Centered-Cubic lattice is derived from a
theoretical analysis that involves translations, tangential elasticity, and rotations. The additional displacement and rotation modes and the energy conversion between them is studied using discrete element simulations. Simulations and theory are in perfect quantitative agreement for the regular lattices examined. As a first small step away from order, systems with weak geometrical disorder (system structure) but strong contact disorder, i.e. with an inhomogeneous contact force distribution, are studied next. They reveal nicely the dispersive nature of granular materials and show strong frequency filtering. Low frequencies propagate, whereas high frequencies vanish exponentially. A more detailed study of how energy is transfered between different wavenumber bands shows linearly increasing transfer rates for increasing wavenumbers. A first theoretical approach using a linear Master Equation leads to a quantitative prediction of the energy evolution per band for short times.
A bigger second step in complexity is made by investigating the sound propagation in a realistic tablet made of a sintered frictional and cohesive polydisperse powder and prepared in different ways. These simulations nicely display history dependence and the effect of different material parameters.
As a conclusion, simulations were found to be a valuable tool to complement theoretical and experimental approaches towards the understanding of complex phenomena, such as sound propagation in (dry) granular materials. However, many open issues, in particular concerning the modeling, still remain.
26 citations
"Frequency filtering in disordered g..." refers background in this paper
...Mouraille and Luding [34,35] numerically studied the high-frequency filtering present in three-dimensional packings perturbed from their perfect crystalline geometry by a small random variation in the particle sizes....
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...In comparing the high-frequency-filtering properties of the random one-dimensional systems to the threedimensional packings of Mouraille and Luding [35], we note the importance of the contact geometry in their observations....
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...Frequency mixing due to disorder was also noted by Mouraille [34]....
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...B. P. Lawney · S. Luding (B) Multiscale Mechanics (MSM), MESA+, Faculty of Engineering Technology (CTW), PO Box 217, 7500 AE Enschede, The Netherlands E-mail: s.luding@utwente.nl chains, often motivated by energy modification and shock-protection applications....
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TL;DR: In this article, an independent collision model is proposed for non-linear impulse propagation in granular chains. But the model is not suitable for the case of infinite chains. Andersen et al. showed that independent collision models support solitons in uniform granular chain, and exhibits an exponential dispersion of momentum and energy in tapered chains.
Abstract: An independent-collision model is advanced as a conceptually simple and computationally convenient framework for understanding non-linear impulse propagation in granular chains. In particular, conservation principles are used to show that the independent-collision model supports solitons in uniform granular chains, and exhibits an exponential dispersion of momentum and energy in tapered chains. These results are in qualitative agreement with recent detailed simulations of Hertzian spheres by Sen et al. (Physica A 299 (2001) 551). Near-quantitative agreement is achieved by renormalizing the width of an independent impulse by the solitary wave width. The model is easily extended, as we demonstrate by incorporating the effects of restitution.
25 citations
"Frequency filtering in disordered g..." refers background in this paper
...One may imagine applications exploiting disorder effects, as could be pursued in the spirit of [5,10,13,14,32,40,47,49] where such systems were engineered to produce a desired output....
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..., deliberate insertion of different sized masses) [10,13,14,27] tapering [5,32,47,49] and controlled variation of the particle material [3,15]....
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