Possibility of useful mechanical energy from noise: the solitary wave train problem in the granular chain revisited

A momentary velocity perturbation at an edge of a granular chain with the grains barely touching one another and held between fixed walls propagates as a solitary wave whereas a long lived perturbation, even if it is noisy, ends up as a solitary wave train. Here, we extend our earlier work but with a force instead of a velocity perturbation. Such a perturbation can propagate an extended compression front into the system. We find that a snapshot of the distribution of grain compressions in the solitary wave train shows parabolic as opposed to an approximate exponential decay with the leading edge at the front of the traveling pulse and the trailing edge following it. The system’s time evolution depends on three independent parameters-the material properties, duration of perturbation and the characteristic amplitude of the perturbation. Hence, the coefficients used to describe the parabolic decay of the grain compressions in the solitary wave train depend on these three parameters. When a random finite duration force perturbation is applied we find that the randomness is smoothed out by the system, which in turn suggests that long granular chains (or equivalent systems, such as circuits) can be potentially useful in converting random noisy signals to organized solitary wave trains and hence to potentially usable energy.     

Wave propagation in elasto-plastic granular systems

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.     

Simulation of solitary waves in a monodisperse granular chain using COMSOL multiphysics: localized plastic deformation as a dissipation mechanism

Solitary wave propagation in a monodisperse granular chain was simulated using the finite element method. The model was built to address a discrepancy between numerical and experimental results from Lazaridi and Nesterenko (J Appl Mech Tech Phys 26(3):405–408 1985). In their work, solitary waves were generated in a chain of particles through impact of a piston, and results were quantified by comparing the chains’ reactions to a rigid wall. Their numerical calculations resulted in a solitary wave with a force amplitude of 83 N, while it was measured experimentally to be 71 N. In the present work, the configuration of the granular chain and piston was duplicated from Lazaridi and Nesterenko (J Appl Mech Tech Phys 26(3):405–408, 1985). Qualitatively similar solitary waves were produced, and von Mises stress values indicated that localized plastic deformation is possible, even at low piston impact velocities. These results show that localized plastic deformation was a likely source of dissipation in experiments performed by Lazaridi and Nesterenko.     

Development of plastic nonlinear waves in one-dimensional ductile granular chains under impact loading

A modified split Hopkinson pressure bar (SHPB) was used to load one-dimensional granular chains of metallic spheres under impact loading rates. These homogeneous chains, comprised of brass spherical beads ranging from a single sphere to a chain of sixteen, are of interest because of their unique wave propagation characteristics. In the elastic range, for loads around 10 s of N, nonlinear elastic solitary waves have been observed to form. In this work, loading magnitudes spanning from 9 kN to 40 kN – considerably higher than most previous works on these systems which have been conducted in the elastic regime – cause the granular chains to severely deform plastically. The aim of this study is to identify whether a nonlinear solitary-type wave will be generated under such high load levels, and if so, under what conditions (e.g., chain length, load level, etc.) it will do so. The propagating pulse was found to assume a distinctive shape after travelling through five beads, similar to the elastic case where solitary waves are realized with a traveling wavelength of five bead diameters. The wave speed of the plastic pulses observed here was seen to depend on maximum force, indicating that indeed it is a nonlinear wave in nature and is comparable to the elastic solitary wave. Locally, the plastic dissipation at every contact point through the chains was studied by measuring the residual plastic contact area. It was found that after the formation of the plastic nonlinear solitary wave had occurred there is also decreasing plastic deformation along the chain length except at the end beads in contact with the SHPB, which rebound into the SHPB bar causing larger plastic dissipation locally. To our knowledge this research is the first effort to investigate in detail the development and evolution of solitary-like waves in the plastic regime and will form the basis of future work in this area.     

Two interactional solitary waves propagating in two-dimensional hexagonal packing granular system

We investigate the wave propagation in the two-dimensional (2D) hexagonal packing of spheres in a rectangular region using numerical simulations and theoretical analyses. The impact excitations are resolved into two distinct solitary waves with the same wave amplitude and front velocity. A universal relation between the wave front velocity and the force amplitude is obtained. One-dimensional chain models of the 2D packing are generated. The transfer of energy and momentum leads to the phenomenon of an energy equipartition in the 2D homogeneous hexagonal packed system. Moreover, the solitary wave train containing solitary waves with a decreasing amplitude is also noted. We apply conservation of energy and momentum to the collision process between spheres to predict the amplitude of the solitary wave train. Our analytical results are in good agreement with the numerical simulations.     

Amplitude-dependent attenuation of compressive waves in curved granular crystals constrained by elastic guides

We study the wave propagation in a curved chain of spherical particles constrained by elastic guides under the axial impact of a falling mass. We characterize the force transmission properties of the chain by varying the striker’s mass and the chain’s curvature. Experimental tests demonstrate amplitude-dependent attenuation of compressive waves propagating through the curved chain. In particular, we observe that the curved systems present an improved transmission of small dynamic disturbances relative to that of strong excitations, resulting from the close interplay between the granular particles and the softer elastic medium. We also find that the transmission of the compressive waves through the chains is dependent on the initial curvature imposed to the system. Numerical simulations, based on an approach that combines discrete element and finite element methods, corroborate the experimental results. The findings suggest that hybrid structures composed of granular particles and linear elastic media can be employed as new passive acoustic filtering materials that selectively transmit or mitigate excitations in a desired range of pressure amplitudes.     

Shock dynamics in granular chains: numerical simulations and comparison with experimental tests

The aim of this paper is to simulate the nonlinear wave propagation in granular chains of beads using a recently introduced multiple impact model and to compare numerical results to experimental ones. Different kinds of granular chains are investigated: monodisperse chains, tapered chains and stepped chains. Particular attention is paid to the dispersion effect, and the wave propagation in tapered chains, the interaction of two solitary waves in monodisperse chains, and the formation of solitary wave trains in stepped chains. We show that the main features of the wave propagation observed experimentally in these granular chains are very well reproduced. This proves that the considered multiple impact model and numerical scheme are able to encapsulate the main physical effects that occur in such multibody systems.     

Solitary wave trains in granular chains: experiments,theory and simulations

The features of solitary waves observed in horizontal monodisperse chain of barely touching beads not only depend on geometrical and material properties of the beads but also on the initial perturbation provided at the edge of the chain. An impact of a large striker on a monodisperse chain, and similarly a sharp decrease of bead radius in a stepped chain, generates a solitary wave train containing many single solitary waves ordered by decreasing amplitudes. We find, by simple analytical arguments, that the unloading of compression force at the chain edge has a nearly exponential decrease. The characteristic time is mainly a function involving the grains’ masses and the striker mass. Numerical calculations and experiments corroborate these findings.     

Rectification effect on solitary waves in the symmetric Y-shaped granular chain

The rectification effect on the propagation of solitary waves in the symmetric Y-shaped granular chain is numerically investigated. A heterojunction with mass mismatch occurs at the position of Y-junction by adjusting the branch angle. And the heavy-light heterojunction is more favorable for the solitary wave passing. Based on the characteristics of wave propagation velocity and gap’s opening, we argue that both nonlinearity and collision effects dominate the rectification process. The rectification efficiency can be improved by adjusting the branch angle and the direction of incident solitary wave. The results have particularly practical significance for the potential design of acoustic diode devices.     

Nonlinear wave scattering at the flexible interface of a granular dimer chain

Uncompressed granular dimer chains composed of repetitive pairs of heavy-light spherical, linearly elastic beads exhibit interesting intrinsic responses. The dynamics of these highly discontinuous nonlinear media is governed by the mass ratio scaling the mass disparity of each heavy-light pair of beads. In particular, it has been theoretically and experimentally shown that they support countable infinities of anti-resonances at a discrete set of mass ratios leading to solitary pulses propagating through the dimers with no attenuation or distortion. Conversely, they support countable infinities of resonances at a different discrete set of mass ratios, leading to substantial and rapid attenuation of propagating pulses due to energy scattering from low-to-high frequencies and wavenumbers by means of radiating traveling waves. In this work we computationally study nonlinear scattering of impeding pulses at the interface of an impulsively excited dimer chain with a dispersive elastic boundary, namely, a finite linear string resting on an elastic foundation. We develop a computational algorithm which, through iteration and interpolation at successive time steps, accurately computes (and ensures convergence of) the highly discontinuous contact forces and displacements at the flexible interface of the granular medium. This enables accurate computation of wave transmission, reflection, localization or multi-scale nonlinear scattering at the flexible interface for varying mass ratios of the dimer and the interface parameters. We show that, depending on the mass ratio of the dimer and the stiffness of the elastic foundation, the nonlinear scattering at the flexible interface may lead to significant reduction of the maximum contact force at the interface, and, thus, drastically affect the transmitted and reflected energy at the flexible boundary. In fact, an inverse relation between the stiffness of the elastic foundation and the residual energy transferred from the dimer chain to the flexible boundary is found. Moreover, for sufficiently small mass ratios of the dimer chain transient breathers are realized close to the interface in the form of localized “fast” oscillations of light granules of the dimer that entrap shock energy and then release in a slow time scale back to the chain and the flexible boundary. This work paves the way for studying highly discontinuous and nonlinear scattering phenomena at interfaces of granular media with flexible continua.     

Highly nonlinear solitary waves in chains of hollow spherical particles

We study the dynamic response of one- dimensional granular chains composed of uniform hollow spheres excited by an impulse, and we observe the formation and propagation of highly nonlinear solitary waves. We find that the dynamics of these solitary waves are different from the solitary waves forming in chains composed of uniform solid spheres, because of the changes in the contact interaction between particles. We study the quasi-static contact interaction between two hollows spheres using finite element (FE) simulations, and approximate their response as a power-law type function in the range of forces of interest for this work. The experimental data obtained by testing a chain of particles shows good agreement with theoretical predictions obtained using a long wavelength approximation, and with numerical simulations based on discrete particle and FE models. We also investigate the effect of hollow spheres’ wall thickness on the dynamic response of the chains.     

Evolution of ultrasonic velocity and dynamic elastic moduli with shear strain in granular layers

Ultrasonic wave transmission has been used to investigate processes that influence frictional strength, strain localization, fabric development, porosity evolution, and friction constitutive properties in granular materials under a wide range of conditions. We present results from a novel technique using ultrasonic wave propagation to observe the evolution of elastic properties during shear in laboratory experiments conducted at stresses applicable to tectonic faults in Earth’s crust. Elastic properties were measured continuously during loading, compaction, and subsequent shear using piezoelectric transducers fixed within shear forcing blocks in the double-direct-shear configuration. We report high-fidelity measurements of elastic wave properties for normal stresses up to 20 MPa and shear strains up to 500 % in layers of granular quartz, smectite clay, and a quartz-clay mixture. Layers were 0.1–1 cm thick and had nominal contact area of $5 \mathrm{cm} \!\times \! 5 \mathrm{cm}$ . We investigate relationships among frictional strength, granular layer thickness, and ultrasonic wave velocity and amplitude as a function of shear strain and normal stress. For layers of granular quartz, P-wave velocity and amplitude decrease by 20–70 % after a shear strain of 0.5. We find that P-wave velocity increases upon application of shear load for layers of pure clay and for the quartz-clay mixture. The P-wave amplitude of pure clay and quart-clay mixtures first decreases by $\sim $ 50 and 30 %, respectively, and then increases with additional shear strain. Changes in P-wave speed and wave amplitude result from changes in grain contact stiffness, crack density and disruption of granular force chains. Our data indicate that sample dilation and shear localization influence acoustic velocity and amplitude during granular shear.     

Plane wave propagation in 2D and 3D monodisperse periodic granular media

Plane wave propagation in periodic ordered granular media comprising of elastic spherical particles is investigated. The spheres are under zero precompression and are assumed to interact via the Hertzian contact potential. Various two- and three-dimensional granular structures such as hexagonal packing (2D and 3D), face-centered cubic and body-centered cubic packings are considered in the present study, with the plane impact either normal or oblique to the granular system. For the normal impact case, 1D chains equivalent to the 2D and 3D structures are obtained. A universal relation between the wavefront speed and the force amplitude is derived, valid for all the granular structures studied. In the angular impact case, the shear component of the amplitude of the particle velocity is found to initially decay exponentially and further in a series of linear regimes. By employing simpler models, semi-analytical predictions are obtained for the decay of shearing effect.     

Laser-induced surface acoustic waves for evaluation of elastic stiffness of plasma sprayed materials

The elastic properties of plasma sprayed deposits have been evaluated using a laser-excited surface acoustic wave (SAW) technique and an inversion processing analysis. The SAWs including Lamb and Rayleigh waves were generated in plasma sprayed NiCoCrAlY and ZrO₂, respectively, and their group velocity dispersions were used to determine the elastic properties (i.e.Young's modulus, Poison's ratio and density) of the deposits. Estimated elastic moduli from the velocity dispersions of A₀-mode Lamb waves are in the range of 40–140 GPa for the deposits, which are much lower than the values 220–240 GPa of the comparable dense materials. The dramatic reductions in modulus and density of ZrO₂ deposit have been attributed to the presence of high porosity and particularly microcracks. Moreover, this study has emphasized on exploiting the applicability of each kind of the SAWs for the elastic property evaluation of different sprayed materials. Both Lamb and Rayleigh wave dispersions are useful for the estimation of APS and VPS-deposited NiCrAlY, but S₀-Lamb and Rayleigh waves are exceptional for that of sprayed ZrO₂, because of its characterization of high acoustic attenuation and inconsequent displacement across the weak bonded interface of ZrO₂ and substrate.     

A problem related to the stability of force chains

When a load is applied to a granular material, the stress is not uniformly distributed but concentrates into quasi-linear particle assemblies known as force chains. By the time they can be observed, force chains are apparently quasistatic in the sense that they persist over time scales much longer than elastic timescales (such as a contact time) over which an unstable chain breaks apart. But many force chains attempt to form, but are unstable and break apart after a few contact times. Stability requires that each particle in the chain be pressed against its neighbors by the forces in the chain. Furthermore, each particle must also be in quasistatic equilibrium in the sense that all forces on it must roughly balance. This paper examines a simple linear string of particles in an attempt to estimate how much force is required to hold the chain together and how large a force imbalance can be tolerated. There are two modes of instability, a simple case in which an overloaded contact pushes its particles apart breaking the chain as they separate, and a more complicated mode requiring the interaction of elastic waves traveling along the chain. Overall, dissipation acts as a stabilizing factor, first by reducing the initial energy of the overloaded contact as it unloads, and subsequently, by reducing the energy of the elastic waves.     

Noninvasive method for estimation of complex elastic modulus of arterial vessels

Pulse wave velocity (PWV) is widely used for estimating the stiffness of an artery. It is well-known that a stiffened artery can be associated with various diseases and with aging. Usually, PWV is measured using the "foot-to-foot" method in which the travel time of the wave is measured over a distance. The "foot" of the pressure wave is not clear due to reflected waves and blood noise. Also, PWV is an average indicator of artery stiffness between the two measuring points and, therefore, does not identify local stiffness variations. We propose producing a bending wave in the arterial wall using low-frequency, localized ultrasound radiation force and measuring the wave velocity along the arterial wall. The wave velocity can be measured accurately over a few millimeters. A mathematical model for wave propagation along the artery is developed with which the Young's modulus of the artery can be determined from measured wave velocities. Experiments were conducted on a pig carotid artery cast in a tissue-mimicking gelatin. The wave velocity was measured by the phase change at a known distance for a given frequency. The measured wave velocity is about 3 m/s at 100 Hz and 6.5 m/s at 500 Hz. The real part of complex elastic modulus of the artery is estimated to be 300 kPa.     

南海内孤立波作用下顶张力立管极值响应研究

内孤立波能够引起强剪切流和巨大的冲击荷载,对海洋平台和海洋立管等海上结构物会构成危险,而关于内孤立波对海洋立管作用的研究很少。本文根据描述内孤立波的KdV-mKdV方程,结合改进的Morison公式,采用有限单元法在时域内对内孤立波作用下顶张力立管的极值响应进行了数值模拟,并就内波振幅、立管内流、顶张力、弹性模量和壁厚的变化对极值响应的影响进行了探讨。结果表明,内孤立波会引起立管较大的极值响应,在立管的分析计算中应该加以考虑。     

Using MR elastography to image the 3D force chain structure of a quasi-static granular assembly

We have developed a magnetic resonance elastography (MRE) technique to experimentally investigate the force chain structure within a densely packed 3D granular assembly. MRE is an MRI technique whereby small periodic displacements within an elastic material are measured. We verified our MRE technique using a gel phantom and then extended the method to image the force carrying chain structure within a 3D granular assembly of particles under an initial pre-stressed condition, on top of which is superimposed a small-amplitude vibration. We find that significant coherent displacements form along force chains, where spin phase accumulates preferentially, allowing visualization. This work represents the first time that the internal force chain structure of a dry assembly of granular solids has been fully acquired in three dimensions.     

Modulation of nonlinear waves in a viscous fluid contained in a tapered elastic tube

In the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and the blood as a Newtonian fluid, we have studied the amplitude modulation of nonlinear waves in such a fluid-filled elastic tube, by use of the reductive perturbation method. The governing evolution equation is obtained as the dissipative nonlinear Schrödinger equation with variable coefficients. It is shown that this type of equations admit solitary wave solutions with variable wave amplitude and speed. It is observed that, the wave speed increases with distance for tubes of descending radius while it decreases for tubes of ascending radius. The dissipative effects cause a decay in wave amplitude and wave speed.     

位于弹性半空间上的理想流体层动力反应--平面SV波入射

根据弹性固体与理想流体动力学方程,导出了固-液介质交界面上波的透射与反射系数的理论公式,分析了平面SV波从弹性半空间入射到与理想流体层的交界面时,弹性半空间的弹性模量和密度,平面SV波的入射角和频率,以及流体的体积模量和密度对理想流体层动压力的影响。